information Article
HTCRL: A Range-Free Location Algorithm Based on Homothetic Triangle Cyclic Refinement in Wireless Sensor Networks Dan Zhang 1,2, *, Zhiyi Fang 1 , Hongyu Sun 1 and Jie Cao 1 1 2
*
College of Computer Science and Technology, Jilin University, Changchun 130012, China;
[email protected] (Z.F.);
[email protected] (H.S.);
[email protected] (J.C.) Computer Science and Information Technology Department, Daqing Normal University, Daqing 163712, China Correspondence:
[email protected]; Tel.: +86-138-4592-3219
Academic Editors: Nafaa Jabeur and Nabil Sahli Received: 3 January 2017; Accepted: 23 March 2017; Published: 27 March 2017
Abstract: Wireless sensor networks (WSN) have become a significant technology in recent years. They can be widely used in many applications. WSNs consist of a large number of sensor nodes and each of them is energy-constrained and low-power dissipation. Most of the sensor nodes are tiny sensors with small memories and do not acquire their own locations. This means determining the locations of the unknown sensor nodes is one of the key issues in WSN. In this paper, an improved APIT algorithm HTCRL (Homothetic Triangle Cyclic Refinement Location) is proposed, which is based on the principle of the homothetic triangle. It adopts perpendicular median surface cutting to narrow down target area in order to decrease the average localization error rate. It reduces the probability of misjudgment by adding the conditions of judgment. It can get a relatively high accuracy compared with the typical APIT algorithm without any additional hardware equipment or increasing the communication overhead. Keywords: localization; APIT localization algorithm; sensor nodes; localization error rate; homothetic triangle
1. Introduction With the increasing development of wireless communication technologies, electronic technologies, embedded technologies, computer technologies and sensor technologies, the research and application of WSN (wireless sensor networks) is extensively popular and is a widespread concern. In recent years, wireless sensor have been widely used in the military field, intelligent agriculture, environmental monitoring, intelligent home system, and intelligent transportation system, search, rescue [1,2], etc. WSN consists of a large number of micro sensor nodes and each of them is energy-constrained and low-power dissipation. WSN is a kind of multi-hop and self-organizing network system. The sensors in wireless sensor network are distributed randomly in a designated area. In general, most of the sensor nodes are tiny sensors with small memories and do not acquire their own locations. This means determining the physical locations of the unknown sensor nodes is one of the key issues in WSN. The sensor node location information, however, is of great importance in many WSN applications. The location information could be used in geographical routing; reporting the location of an event; network management; target tracking; etc. [3,4]. In small-scale WSN, we could place the sensor nodes regularly and get the location of each node in advance. However, in practical applications, large numbers of sensor nodes always distributed randomly in large areas. Therefore, most sensor nodes’ locations cannot be known in advance. Since
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manual configuration of sensor nodes is not feasible for large-scale networks with mobile nodes, a possible way to localize sensor nodes is to use the global positioning system (GPS). GPS offers three-dimensional localization based upon direct line of sight with at least four satellites [5]. However, using GPS to location for all the unknown nodes is not feasible. The reasons are as follows. (i) It is unavailable in indoor environments; (ii) the GPS is an energy-consumption module which reduces the lifetime of the sensor networks; (iii) GPS is also a relative large and expensive module which would not reduce the cost of the large scale sensor networks. Therefore, how to locate the location of the unknown node in sensor networks by a feature of the network itself is a significant topic which is of great concern to researchers and manufacturers. Specifically, there are two kinds of nodes in sensor networks. The first is anchor nodes which could determine their own location with hardware equipment such as GPS, and the second is unknown nodes which could not locate themselves with hardware equipment. The purpose of localization algorithms is to locate the unknown nodes through, the features and knowledge of the sensor networks without hardware assistance. Recently, a body of methods and algorithms has been proposed to find solutions for locating the unknown nodes in WSNs, including range-based algorithms and range-free algorithms. Range-based algorithms need node-to-node distances or angles for estimating the locations of unknown nodes. Range-based algorithms use the time of arrival (TOA) [6], Time Difference of Arrival (TDOA) [7], Angle of Arrival (AOA) [8], Radio Interferometry Positioning System (RIPS) [9] and Received Signal Strength Indication (RSSI) [10]. These algorithms have lower localization error rates than the range-free algorithms but require additional hardware to obtain the information on distances or angles. Meanwhile, range-free algorithms depend on connectivity information network knowledge, such as the DV-hop algorithm [11,12], Approximate Point-in-Triangulation test algorithm (APIT) [13], amorphous algorithm and centroid algorithm [14–16]. Therefore, the range-free localization algorithm is widely used in practical applications especially in large scale deployment environment since the lower cost and additional hardware-free features. APIT is a range–free algorithm which is simple and easy to implement. It is low in communication cost and energy consumption. It also has a low location error rate. However, the performance of the APIT algorithm is largely affected by the density of anchor nodes and is restricted for the In-to-Out Error and Out-to-In Error, so there exists space to improve this algorithm. In this paper, an improved APIT algorithm HTCRL (Homothetic Triangle Cyclic Refinement Location) is proposed, which is based on the principle of the homothetic triangle. It involves narrowing the triangle in proportion which contains the unknown node to improve the accuracy of node location. The improved APIT algorithm could get a relative low localization error rate compared with the typical APIT algorithm without any additional hardware or increasing the communication overhead. The rest of this paper are organized as follows: Section 2 presents some related works about APIT localization algorithm. Section 3 proposes the HTCRL algorithm, and in Section 4, experience and analysis is done to verify the performance of the HTCRL algorithm. 2. Related Work In this section, we introduce the APIT algorithm. It is a range-free location algorithm of unknown nodes. The APIT (approximate point in test) [17] algorithm is a range-free WSN localization algorithm. The main principle of APIT algorithm is utilizing the centroid of the overlap sections of the triangles formed by the connected anchors to estimate the location of the unknown nodes. Specifically, the principle is show in Figure 1. In Figure 1, assuming there are n anchors connected to the unknown nodes (the black circle in Figure 1). The unknown node selects 3 nodes from all the connected anchors. Suppose the number of the connected anchors are n. It has Cn3 different triangles. Then, test whether the unknown node is in the triangles that they select. The process of the test is repeated until all of the ternary combination are exhausted or the desired localization error rate is reached. Then, it calculates
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nodes nodes(the (theblack blackcircle circlein inFigure Figure1). 1).The Theunknown unknownnode nodeselects selects33nodes nodesfrom fromall allthe theconnected connectedanchors. anchors. Suppose Suppose the the number number of of the the connected connected anchors anchors are are nn . . ItIt has has CCn3n3 different different triangles. triangles. Then, Then, test test Information 2017, 8, 40 3 of 15 whether whether the the unknown unknown node node isis in in the the triangles triangles that that they they select. select. The The process process of of the the test test isis repeated repeated until until all all of of the the ternary ternary combination combination are are exhausted exhausted or or the the desired desired localization localization error error rate rate isis reached. reached. Then, it calculates the centroid coordinate of the coverage overlap area of all triangles as Then, it calculates the centroid coordinate of the coverage overlap area of all triangles as the the the centroid coordinate of the coverage overlap area of all triangles as the coordinate of the unknown coordinate of the unknown node. As shown in Figure 1, the centroid of the overlap polygon is the coordinate of the unknown node. As shown in Figure 1, the centroid of the overlap polygon is the node. As shown in Figure 1, the centroid of the overlap polygon is the estimated coordinates of the estimated coordinates of the unknown nodes. From Figure 1, the black spot is the unknown node estimatednodes. coordinates the unknown nodes. 1, the black spot is the unknown node unknown From of Figure 1, the black spot From is the Figure unknown node coordinate [18,19]. The main coordinate [18,19]. The main phase of APIT is shown in Figure 2. coordinate [18,19]. The main phase of APIT is shown in Figure 2. phase of APIT is shown in Figure 2.
Figure Figure 1.1.Principles Principles of the APIT algorithm. Figure1. Principlesof ofthe theAPIT APITalgorithm. algorithm.
Figure Figure2.2.The Themain mainphase phaseof ofAPIT. APIT.
Step 1:1:Unknown Unknown nodes gathering the information on the anchor nodes. Step1: Unknownnodes nodesgathering gatheringthe theinformation informationon onthe theanchor anchornodes. nodes. Step In Step 1, all anchor nodes in the area broadcast their location information. The broadcasting InStep Step1,1,all allanchor anchornodes nodesininthe thearea areabroadcast broadcasttheir theirlocation locationinformation. information.The Thebroadcasting broadcasting In message includes its position information and ID. The nodes (includes other anchor nodes and all message includes its position information and ID. The nodes (includes other anchor nodes andall all message includes its position information and ID. The nodes (includes other anchor nodes and unknown nodes) collects the messages and exchange the message (including the value of RSSI) with unknown nodes) collects the messages and exchange the message (including the value of RSSI) with unknown nodes) collects the messages and exchange the message (including the value of RSSI) with their neighbor unknown nodes [20]. The format of the message isisshown shown in Figure 3.3. theirneighbor neighbor unknownnodes nodes[20]. [20].The Theformat formatof ofthe themessage messageis shownin inFigure Figure3. their Information 2017, 8, unknown 40 4 of 16
Figure Figure3.3.Format Formatofofthe themessage messagebroadcasted broadcastedby byanchor anchornodes. nodes.
The unknown nodes test whether they are inside the triangle composed of the three connected The unknown nodes test whether they are inside the triangle composed of the three connected anchors. If the unknown node is inside the triangles the algorithm will calculate the centroid anchors. If the unknown node is inside the triangles the algorithm will calculate the centroid coordinator of all the triangles’ overlap area [21]. coordinator of all the triangles’ overlap area [21]. Step 2: PIT (Point-In-Triangulation Test) test stage. Step 2: PIT (Point-In-Triangulation Test) test stage. The key of the APIT is to judge whether the unknown nodes inside the triangle. In generally, the Perfect Point-In-Triangulation Test method is used to solve this problem in APIT. The principle of the PIT is shown in Figure 4. For any ΔABC and point N on the two dimensional plane, if such a direction exists the unknown node moves along; when the unknown node moves close to one of the anchor nodes A, B and C, it will be far away from the other two nodes, and we can deduce that the unknown node is inside the triangle. The principle is shown in Figure 4a. We also can deduce that if
Figure 3. Format of the message broadcasted by anchor nodes. Figure 3. Format of the message broadcasted by anchor nodes.
The unknown nodes test whether they are inside the triangle composed of the three connected The unknown nodes test whether they are inside the triangle composed of the three connected anchors. If the unknown node is inside the triangles the algorithm will calculate the centroid Information 8, 40 unknown node is inside the triangles the algorithm will calculate the centroid 4 of 15 anchors.2017, If the coordinator of all the triangles’ overlap area [21]. coordinator of all the triangles’ overlap area [21]. Step 2: PIT (Point-In-Triangulation Test) test stage. Step 2: PITthe (Point-In-Triangulation Test) the testunknown stage. The APITisisto tojudge judgewhether whether nodes inside triangle. In generally, The key of the APIT the unknown nodes inside thethe triangle. In generally, the The key of the APIT is to judge whether the unknown nodes inside the triangle. InThe generally, the the Perfect Point-In-Triangulation Test method is used to solve this problem in APIT. principle Perfect Point-In-Triangulation Test method is used to solve this problem in APIT. The principle of Perfect Point-In-Triangulation Testany method usedpoint to solve in APIT. The principle aof of PIT shownininFigure Figure4.4.For For ∆ABCisand and N on onthis the problem thethe PIT is isshown any ΔABC point N the two dimensional plane, if such such a the PIT is shown in Figure 4. For any ΔABC and point N on the two dimensional plane, if such a direction direction exists exists the the unknown unknown node node moves moves along; along; when when the the unknown unknown node node moves moves close close to to one one of of the the direction exists the unknown node moves along; when the unknown node moves close to one of the anchor anchor nodes nodes A, A, BB and and C, C, itit will will be be far far away away from from the the other other two two nodes, nodes, and and we we can can deduce deduce that that the the anchor nodes A, B and C, it will be far away from the other two nodes, and we can deduce that the unknown unknown node node is is inside inside the the triangle. triangle. The The principle principle is is shown shown in in Figure Figure 4a. 4a. We We also also can can deduce deduce that that ifif unknown node is inside thethe triangle. The principle is shown in Figure 4a. We also can deduceaway that if the the unknown unknown node node is is out out of of the triangle triangle when when the the unknown unknown node node moves moves along along aa direction direction far far away the unknown node is out of the triangle when the unknown node moves along a direction far away from from the thethree threeanchor anchornodes nodesA, A,BBand andCCsimultaneously. simultaneously.The Theprinciple principleisisshown shownininFigure Figure4b. 4b. from the three anchor nodes A, B and C simultaneously. The principle is shown in Figure 4b.
(a) (b) (a) (b) Figure 4. APIT localization test principle schematic drawing. (a) principles of PIT (unknown (unknown node in in Figure Figure4.4.APIT APITlocalization localizationtest testprinciple principleschematic schematicdrawing. drawing.(a) (a)principles principlesofofPIT PIT (unknownnode node in the triangle); (b) principles of PIT (unknown node out of the triangle). the node out ofof the triangle). thetriangle); triangle);(b) (b)principles principlesofofPIT PIT(unknown (unknown node out the triangle).
However, the nodes are deployed randomly in WSN and most of them are static, so the nodes However, However,the thenodes nodesare aredeployed deployedrandomly randomlyininWSN WSNand andmost mostofofthem themare arestatic, static,sosothe thenodes nodes cannot move. One solution is with the help of the neighbor nodes. In Figure 5 it is the principle of cannot move. One solution is with the help of the neighbor nodes. In Figure 5 it is the principle cannot move. One solution is with the help of the neighbor nodes. In Figure 5 it is the principleofof APIT test. The SS (Signal Strength) received by the unknown node is inversely proportional to the APIT APITtest. test.The TheSSSS(Signal (SignalStrength) Strength)received receivedby bythe theunknown unknownnode nodeisisinversely inverselyproportional proportionaltotothe the distance from the anchor node. In Figure 5a, assuming that the node M moves to the node 1, 2, 3 and distance the anchor node. InIn Figure 5a,5a, assuming that thethe node MM moves to the node 1, 2,1,32,and 4 distancefrom from the anchor node. Figure assuming that node moves to the node 3 and 4 then tests their SS. The unknown node M exchanges the message with its four neighboring nodes then teststests their SS. The unknown nodenode M exchanges the message with with its four nodesnodes 1, 2, 4 then their SS. The unknown M exchanges the message its neighboring four neighboring 1, 2, 3 and 4. For example, the unknown node M will compare the SS with node 1 and find that the 3 1, and For 4. example, the unknown node M willM compare the SS the withSSnode and find that thethat SS of 2, 34.and For example, the unknown node will compare with1node 1 and find the SS of node M received from node A is lower than node 1. But the SS of node M received from nodes node received from node A node is lower than node 1. But the1.SS of the node nodes B and C SS ofMnode M received from A is lower than node But SSM ofreceived node M from received from nodes B and C is higher than node A. Thus, when node M moves to node 1, it will near node A but is far isBhigher node than A. Thus, nodewhen M moves 1, itto will near but isnode far away and Cthan is higher nodewhen A. Thus, nodetoMnode moves node 1, node it willAnear A butfrom is far away from nodes B and C. This suggests the conclusion that node M is inside ΔABC. In the same way, nodes and C. ThisBsuggests the conclusion that node M is inside ∆ABC. In the ΔABC. same way, it can judge awayBfrom nodes and C. This suggests the conclusion that node M is inside In the same way, it can judge nodes 2, 3 and 4. The conclusion is the same. In Figure 5b, when node M moves to node nodes 2, 3 and 4. The conclusion is the same. In Figure 5b, when node M moves to node 4, it is far it can judge nodes 2, 3 and 4. The conclusion is the same. In Figure 5b, when node M moves to node 4, it is far away from nodes A, B and C; therefore M thought it was outside ΔABC. away from nodesfrom A, B nodes and C;A, therefore Mtherefore thought it outside ∆ABC. 4, it is far away B and C; Mwas thought it was outside ΔABC.
(a) (a)
(b) (b)
Figure 5. 5. APIT localization localization test principle principle schematic.(a) (a) M is in the triangle; triangle; (b)M M is out of of the triangle. triangle. Figure Figure 5.APIT APIT localizationtest test principleschematic. schematic. (a)MMisisininthe the triangle;(b) (b) Misisout out ofthe the triangle.
Step 3: Acquire triangles’ overlapping area. From Step 2, the unknown node encompassed by several triangles then uses the grid scan algorithm to acquire the overlapping area. When the grid is inside any triangle the grids’ count (initialized by 0) in the triangle will be plus 1 in the corresponding area. Step4: Grid scanning. If the unknown node is in the triangle composed of three anchor nodes, then the grid count (initialized by 0) in the triangle will decrease by 1. The principle is shown in Figure 6.
From Step the unknown node encompassed by several triangles then uses the grid scan Step4: Grid 2, scanning. algorithm to acquire the overlapping area. When the grid is inside any triangle count If the unknown node is in the triangle composed of three anchor nodes, thenthe thegrids’ grid count (initialized by by 0) 0) in in the the triangle triangle will will decrease be plus 1 by in the corresponding area. in Figure 6. (initialized 1. The principle is shown Step4: Grid scanning. If the2017, unknown node is in the triangle composed of three anchor nodes, then the grid count Information 8, 40 5 of 15 (initialized by 0) in the triangle will decrease by 1. The principle is shown in Figure 6.
Figure 6. Estimate the location of the unknown node.
From Step 3, we can find the max count of several grids, and then calculate the centroid of the the of node. Figure6. 6.Estimate Estimate thelocation location ofthe theunknown unknown node. polygon composed by the Figure grids. The formulas are as shown in Formulas (1) and (2).
x of+several x 2 + ⋅grids, ⋅ ⋅ + and x n then From Step 3, we can find the max count calculate the centroid of the From Step 3, we can find the max grids, and then calculate the centroid of (1) the x count = 1 of several polygon composed by the grids. The formulas are as shown in Formulas (1) and (2). n polygon composed by the grids. The formulas are as shown in Formulas (1) and (2).
x +x +⋅⋅⋅+x (1) x x == yx111 ++ xy2 22++· ·⋅· ⋅+⋅ x+n y nn (1) (2) y = nn n y1 + y2 + · · · + y n = yhorizontal (2) + y 2 n+ coordinate ⋅ ⋅ ⋅ + y n of the unknown node, y presents In Formulas (1) and (2), x presentsythe 1 (2) y = the vertical coordinate of the unknown node, and (xi,nyi) presents the coordinates of the grids with In Formulas (1) and (2), x presents the horizontal coordinate of the unknown node, y presents the
the max count. vertical coordinate ofand the(2), unknown node, (xi , yi ) presents theof coordinates of the grids with the In Formulas x presents theand horizontal coordinate node, y presents From the four(1) steps above, the coordinates of the unknown nodethe are unknown known. But it cannot meet maxvertical count. coordinate of the unknown node, and (xi, yi) presents the coordinates of the grids with the the extra low localization error rate requirements in some applications. Theoretically, APIT algorithm From the four steps above, the coordinates of the unknown node are known. But it cannot the max could be count. improved by calculate the fine-grained triangles overlap section to decrease the localization meet the extra lowsteps localization error rate requirements in some applications. Theoretically, APIT four above, the the unknown node are known. But it cannot meet errorFrom rate. the This paper depends oncoordinates this theory,ofproposes an improved APIT algorithm, HTCRL algorithm could be improved by calculate the fine-grained triangles overlap section to decrease the the extra low rate requirements in APIT some applications. algorithm, to localization optimize theerror localization error rate of algorithm. Theoretically, APIT algorithm localization error rate. This paper depends on this theory, proposes an improved APIT algorithm, could be improved by calculate the fine-grained triangles overlap section to decrease the localization HTCRL algorithm, to optimize error rate of APIT algorithm. error rate. This paper dependsthe onlocalization this theory, proposes an improved APIT algorithm, HTCRL 3. HTCRL Algorithm algorithm, to optimize the localization error rate of APIT algorithm. 3. HTCRL Algorithm 3.1. Mathematical Model 3. HTCRL Algorithm 3.1. Mathematical Model In order further decrease the localization error rate, in this paper the homothetic triangle is used to the triangle The triangleerror narrows gradually until it satisfiestriangle accuracyis In order further decrease area. the localization rate,down in this paper the homothetic 3.1.narrow Mathematical Modelcoverage requirements. used to narrow the triangle coverage area. The triangle narrows down gradually until it satisfies In order further decrease the localization error rate, in this paper the homothetic triangle is used As anrequirements. improvement of the APIT localization algorithm, the HTCRL algorithm adopts the accuracy to narrow the triangle coverage area. The triangle narrows gradually until it satisfies accuracy “Perpendicular Median Surface to narrow downdown targeted area in order to decrease As an improvement of theCutting” APIT localization algorithm, the HTCRL algorithm adopts the the requirements. localization error rate. It reduces the probability of misjudgment by adding the conditions of “Perpendicular Median Surface Cutting” to narrow down targeted area in order to decrease the As an improvement of the APIT localization algorithm, the HTCRL algorithm adopts the judgment. localization error rate. It reduces the probability of misjudgment by adding the conditions of judgment. “Perpendicular Median Surface Cutting” to narrow down targeted area in order to decrease the localization rate. It reduces the probability of misjudgment the conditions of Definition Homothetic triangle. Suppose that are ififaaadding pair angles Definition 1. 1.error Homothetic triangle. Suppose thattwo twotriangles triangles arehomothetic homotheticby pairofofcorresponding corresponding angles judgment. is isequal equaland andthe thecorresponding corresponding edges edges are are in in proportion. proportion. Information 2017, 8, 40
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Definition 1. Homothetic Suppose that triangles homothetic if a pair of corresponding angles Atwo ==∠ A’, ∠ B ==are ∠ B’, ∠C ==∠C’, ∠C’, InFigure Figure ∆ABCtriangle. In 7,7,ififΔABC ∽ ∆A’B’C’, ΔA’B’C’,then then∠∠A ∠A’, ∠B ∠B’, ∠C is equal and the corresponding edges are in proportion. AB AC BC = = A AB ' B ' = AAC ' C '= BC B ' C. ' . 0 0 C0 0 0 A0 B∠A In Figure 7, if ΔABC ∽ ΔA’B’C’, then =A∠A’, ∠BB= C ∠B’, ∠C = ∠C’,
Figure 7. The principle of homothetic triangle. Figure 7. The principle of homothetic triangle.
Definition 2. The center of the triangle. The angular bisector of the three angles integrates into one point. The point is the center of a triangle’s inscribed circle, which is called the center of a triangle. (The distance between the point and the three sides of the triangle is equal.) An example is shown in Figure 8. As is shown in Figure 8, O is the center of ΔABC, and OD, OE
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Figure 7. The principle of homothetic triangle.
Definition angular bisector of the three angles integrates into into one point. The Definition 2. The Thecenter centerofofthe thetriangle. triangle.The The angular bisector of the three angles integrates one point. pointpoint is theiscenter of a triangle’s inscribed circle, circle, which which is called center a triangle. (The distance between The the center of a triangle’s inscribed is the called the of center of a triangle. (The distance the pointthe andpoint the three sides of the triangle equal.)is equal.) between and the three sides of the is triangle An example is shown in Figure 8. As is shown in Figure 8, O is the center of ΔABC, ∆ABC, and OD, OE andOF OFare arethe theheights heightsofofthe thethree threesides. sides.Then Thenαα 1==αα2,,β β 1=β 2, ,γ 1 ==γγ 2, OD = OE = OF. and β γ , OD = OE = OF. 1 2 1 2 1 2
Figure 8. Diagram Diagram of of triangle’s center.
The improved APIT algorithm adopts the homothetic triangle to narrow down the targeted area. The improved APIT algorithm adopts the homothetic triangle to narrow down the targeted area. According to the theory of the homothetic triangle, the area of the triangle that encompassed the According to the theory of the homothetic triangle, the area of the triangle that encompassed the unknown node will be narrowed down gradually. The principle of the algorithm is shown in Figure 9. unknown node Information 2017, will 8, 40 be narrowed down gradually. The principle of the algorithm is shown in Figure 7 of 16 9.
Figure 9. The main principle of HTCRL localization algorithm. Figure 9. The main principle of HTCRL localization algorithm.
shownininFigure Figure 8, 8, the thethe homothetic triangles is theisorigin coordinates of two- of AsAs shown the center centerofofallall homothetic triangles the origin coordinates ' 0 ' ' ' ' 0 0 0 0 two-dimension which denoted . (oix,oiy,oiy)oi is ) is the estimate coordinator the unknown dimension which is is denoted bybyOOi (i x(oix,oiy,oiy)oi. )(x the estimate coordinator ofof the unknown node N. In the beginning, through PIT test we will get several triangles which encompassed the node N. In the beginning, through PIT test we will get several triangles which encompassed the '
'
'
unknown nodes N. In this stage the purpose is to narrow down the triangle area. ΔAi Bi Ci is the ith '
'
'
(i = 1, 2, 3 … n) that encompasses unknown node N. ΔAi Bi Ci can be narrowed down to Δ Aij' Bij' C ij' , j = (1, 2, 3 …), the variable j represents the times that the triangle narrow. For example, in Figure 8,
ΔAi' Bi'Ci' can be narrowed down to ΔAi'1 Bi'1Ci'1 . ΔAi'1 Bi'1Ci'1 can be narrowed down to ΔAi' 2 Bi'2 Ci' 2 .
In order further In decrease order further the localization decrease error the localization rate, in thiserror paper rate, theinhomothetic this paper triangle the homothetic is used triangle is used to narrow the triangle to narrow coverage the triangle area. The coverage triangle area. narrows The triangle down gradually narrows down until it gradually satisfies accuracy until it satisfies accuracy requirements. requirements. As an improvement As an ofimprovement the APIT localization of the APIT algorithm, localization the algorithm, HTCRL algorithm the HTCRL adopts algorithm the adopts the “Perpendicular “Perpendicular Median Surface Median Cutting” Surface to narrow Cutting” down to targeted narrow down area in targeted order to area decrease in order the to decrease the Information 2017, 8, 40 7 of 15 localization error localization rate. It reduces error rate. the It probability reduces the of misjudgment probability ofby misjudgment adding the by conditions adding the of conditions of judgment. judgment. unknown nodes N. In this stage the purpose is to narrow down the triangle area. ∆Ai 0 Bi 0 Ci 0 is 0 0 0 the ith (i =1.1,Homothetic 2,Definition 3 . . . n)triangle. that encompasses node can becorresponding narrowed down to Definition 1. Homothetic Supposetriangle. thatunknown two Suppose triangles that are N. homothetic two∆A triangles if iaare pair homothetic of if a pair angles of corresponding angles i Bi C 0 0 0 , jcorresponding =is(1, 2, 3and . . . the ),edges the variable j represents times that the triangle narrow. For example, is∆A equal equal corresponding are in proportion. edges are inthe proportion. ij Band ij Cijthe in Figure 8, ∆Ai 0 Bi 0 Ci 0 can be narrowed down to ∆Ai1 0 Bi1 0 Ci1 0 . ∆Ai1 0 Bi1 0 Ci1 0 can be narrowed down to 0 0 0 0 0 0 0 0 0 0 0 ∆Ai2 ∆A BIn ∆A Bi2 0 C of the triangle moves along the i2 Ci2 .7, i C i ∽ ∆A i1 Cthen i1 ∽∠A i2 .= The InBFigure if iΔABC Figure ΔA’B’C’, 7,i1if BΔABC ΔA’B’C’, =i2∠A’, ∠B then ∠B’, ∠A vertex =∠C ∠A’, = ∠C’, ∠B = ∠B’, ∠C = ∠C’, angular bisector simultaneously to narrow down the area. The coordinators of the three vertexes in ∆Ai 0 Bi 0 Ci 0 are A0 ( xi1 0 , yi1 0 ), B0 ( xi2 0 , yi2 0 ) and C 0 ( xi3 0 , yi3 0 ). ∆Ai 0 Bi 0 Ci 0 is narrowed down in proportion that the proportion could be supposed k (i = 1, 2, 3 . . . ), If the triangle has narrowed down j times, the coordinators of ∆Aij 0 Bij 0 Cij 0 are Aij 0 ( xij1 0 , yij1 0 ), Bij 0 ( xij2 0 , yij2 0 ), Cij 0 ( xij3 0 , yij3 0 ), j = 1, 2, 3 . . .. In APIT the nodes Ai ’, Bi ’ and Ci ’are the anchor nodes and their coordinators are known in advance. The coordinators of Ai ’, Bi ’ and Ci ’ are Ai 0 ( xi1 0 , yi1 0 ), Bi 0 ( xi2 0 , yi2 0 ) and Ci 0 ( xi3 0 , yi3 0 ), therefore the coordinator of node Oi ’ can be calculated through Formulas (3) and (4). The coordinator of node Oi 0 is ( xi 0 , yi 0 ). B 0 C 0 · xi1 0 + Ai 0 Ci 0 · xi2 0 + Ai 0 Bi 0 · xi3 0 xi 0 = i i (3) Bi 0 Ci 0 + Ai 0 Ci 0 + Ai 0 Bi 0 yi 0 =
Bi 0 Ci 0 · yi1 0 + Ai 0 Ci 0 · yi2 0 + Ai 0 Bi 0 · yi3 0 Bi 0 Ci 0 + Ai 0 Ci 0 + Ai 0 Bi 0
(4)
In Formulas (3) and (4), Bi 0 Ci 0 =
q
Ai 0 Ci 0 =
q
( xij3 0 − xij2 0 )2 + (yij3 0 − yij1 0 )2
(6)
Ai 0 Bi 0 =
q
( xij2 0 − xij1 0 )2 + (yij2 0 − yij1 0 )2
(7)
( xij3 0 − xij2 0 )2 + (yij3 0 − yij2 0 )2
(5)
In order to calculate the centroid coordinates of the narrowed triangles, for simplicity of calculation it will translate the coordinate axis and the coordinate of Oi 0 will be Oi (0, 0). After translating the coordinate axis the coordinates of Ai 0 , Bi 0 and Ci 0 will be Ai , Bi and Ci , and separately they are Ai ( xi1 0 − xi 0 , yi1 0 − yi 0 ), Bi ( xi2 0 − xi 0 , yi2 0 − yi 0 ), Ci ( xi3 0 − xi 0 , yi3 0 − yi 0 ) A. After narrowing down the translations and translating the coordinate axis, the coordinates are Aij 0 ( xij1 0 , yij1 0 ), Bij 0 ( xij2 0 , yij2 0 ), Cij 0 ( xij3 0 , yij3 0 ). The value of them are shown in Formula (8). Now three vertex of the translation are available. It can be used in node location algorithm. K is the proportion by which the triangles reduced. (
xij1 0 = k( xi1 0 − xi 0 ) , yij1 0 = k(yi1 0 − yi 0 )
(
xij2 0 = k ( xi2 0 − xi 0 ) , yij2 0 = k(yi2 0 − yi 0 )
(
xij3 0 = k( xi3 0 − xi 0 ) yij3 0 = k (yi3 0 − yi 0 )
(8)
3.2. The Main Algorithm Flow Descriptions of HTCRL The improved range free location algorithm based on homothetic triangle cyclic refinement proposed in this paper can decrease the localization error rate efficiently. In APIT algorithm, the size of the overlapping triangle area which embraced by the anchor nodes has a great impact on algorithm performance. Localization accuracy will be greatly improved as the area is reduced gradually. For this, in this paper we proposed a localization algorithm to assist localization, the localization accuracy will be improved by this method. The main process of HTCRL (Homothetic Triangle Cyclic Refinement Location) algorithm is shown in Figure 10.
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Figure 10. 10. The Process of of HTCRL HTCRL Localization Localization Algorithm. Algorithm. Figure The Main Main Process
Step 1: Initialize the network configuration. Step 1: Initialize the network configuration. Step 2: Each anchor node in the area broadcasts the messages including their own location Step 2: Each anchor node in the area broadcasts the messages including their own location information and ID. The nodes (includes other anchor nodes and all unknown nodes collect the information and ID. The nodes (includes other anchor nodes and all unknown nodes collect the messages (ID, position, RSS) and exchange the message with their neighbor unknown nodes. messages (ID, position, RSS) and exchange the message with their neighbor unknown nodes. Step 3: The unknown nodes judge whether they have received the message from more than three Step 3: The unknown nodes judge whether they have received the message from more than three anchor nodes? If the number is more than three, it will go to Step 4. If the number is less than three, anchor nodes? If the number is more than three, it will go to Step 4. If the number is less than three, it will go to Step 7. it will go to Step 7. Step 4: PIT (Point-In-Triangulation Test) test stage. It will judge whether they are inside the Step 4: PIT (Point-In-Triangulation Test) test stage. It will judge whether they are inside the triangle composed by the three neighbor anchors. If the unknown node is inside the triangles, HTCRL triangle composed by the three neighbor anchors. If the unknown node is inside the triangles, HTCRL algorithm begins to narrow down these triangles until the localization error rate is satisfied for the algorithm begins to narrow down these triangles until the localization error rate is satisfied for applications. the applications. Step 5: Acquire triangles overlapping area. Step 5: Acquire triangles overlapping area. Step 6: Use the homothetic triangle cyclic refinement algorithm to calculate the centroid of the Step 6: Use the homothetic triangle cyclic refinement algorithm to calculate the centroid of the overlapping area in Step 5, which is the estimated coordinates of the unknown nodes. overlapping area in Step 5, which is the estimated coordinates of the unknown nodes. Step 7: end. Step 7: end. 4. Simulation and Result Result 4. Simulation and 4.1. The The Parameters Parameters of of Network Network in in Experiments Experiments 4.1. In our our experiments, experiments, several several parameters parameters are are studied studied that that affect the localization localization error rate directly In affect the error rate directly in the HTCRL algorithm. The parameters of network include anchors heard, node density, anchor in the HTCRL algorithm. The parameters of network include anchors heard, node density, anchor percentage, anchor anchor to to node node range range ratio. ratio. In In this this paper, paper, suppose suppose the the radio radiomodel modelisisperfectly perfectlycircular. circular. percentage, Anchors Heard (AH): This parameter is defined as average number of anchors that a node heard Anchors Heard (AH): This parameter is defined as average number of anchors that a node heard and used in experiment. In the network, the more anchors heard the accuracy is better. However, the and used in experiment. In the network, the more anchors heard the accuracy is better. However, more anchors heard the cost is higher. In our experiments, the number of anchors nodes in the network is from 3 to 21.
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the more anchors heard the cost is higher. In our experiments, the number of anchors nodes in the network is from 3 to 21. Node Density (ND): This parameter is defined as average number of nodes in a hop communication range. The nodes include unknown nodes and anchor nodes. Anchor Percentage (AP): This parameter is defined as the percentage of anchors in all sensor nodes. The values can be calculated in Formula (9). In the following, it uses as anchor percentage. That is the number of anchors divided by the number of anchors and the unknown nodes. VAP =
Nanchors × 100% Nanchors + Nunknowns
(9)
Anchor to Node Range Ratio (ANR): This parameter is defined as the average radio range of the anchor divides by the average radio range of the unknown node. From our past experiments, the value most suitable value is 3. 4.2. The Performance of the Localization Algorithm Assume the actual coordinate of the unknown nodes is ( xi , yi ), The estimated coordinate of the unknown nodes is ( xi 0 , yi 0 ). n is the number of the unknown nodes; R is the radio range of the unknown nodes. In the experiment, the localization error is tested through eaverage , which is shown in Formula (10). q n
∑
eaverage =
i =1
( x i − x i 0 )2 + ( y i − y i 0 )2 nR
× 100%
(10)
In order to verify the performance of HTCRL localization algorithm, the algorithm is simulated through MATLAB R2012a. In this section, several parameters which affect the performance of HTCRL localization algorithm are analyzed in our simulations. It is assumed that the experiments are done in a two dimensional 1000 m × 1000 m area. The sensor nodes are distributed randomly and uniformly in the area respectively. Random distribution is when unknown nodes and anchors are distributed randomly. Uniform distribution is when the area is divided into several grids and the unknown nodes and anchors are distributed randomly in the grids. It can ensure the sensors are placed at equal distances. The number of anchors can be adjusted according to the communication connectivity of the network. The experiments are repeated 300 times and the average value is used as the final result. There are 1000 sensor nodes spread out in the area. 1.
Localization Error Varying with Anchor Heard(AH) and Communication Radius (R)
In this experiment, we analyzed the parameters of anchor heard and communication radius (R) that influent localization error rate. In Figure 11a, the parameters are as flows: AH = 3~21, ANR = 3, ND = 8, R = 150 m, K = 0.0050. The sensor nodes are distributed in random distribution. Figure 11a shows that the overall trend of localization error rate in APIT and HTCRL localization algorithms decreases as the number of anchors heard increases. Instead, HTCRL localization algorithms performs better than APIT. The localization error rate of APIT is from 38.2% to 23.1%, while the localization error rate of HTCRL is from 35.3% to 22.5%.
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Figure 11. Localization RateVarying Varying with Anchor Heard. AHANR = 3~21, ANR ND = 8, Figure 11. LocalizationError Error Rate with Anchor Heard. (a) AH(a) = 3~21, = 3, ND = 8,=R3, = 150, R = 150, Random, K = (b) 0.0050; AH = 3~21, ANR NDUniform, = 8, R =K 150, Uniform, = 0.0050; Random, K = 0.0050; AH =(b) 3~21, ANR = 3, ND = 8, = R 3, = 150, = 0.0050; (c) AH K = 3~21, ANR = 3, NDANR = 8, R ==450, Random, (d) AH = 3~21, = 3, ND 8, R ==450, Uniform, (c) AH = 3~21, 3, ND = 8, K R ==0.0050; 450, Random, K =ANR 0.0050; (d)= AH 3~21, ANRK== 0.0050. 3, ND = 8, R = 450, Uniform, K = 0.0050. In Figure 11b, the parameters are as follows: AH = 3~21, ANR = 3, ND = 8, R = 150 m, K = 0.0050. The sensor nodes are distributed in uniform distribution. Figure 11b shows that the overall trend of localization error rate in APIT and HTCRL localization algorithms decreases as the number of anchors heard increases. Instead, HTCRL localization algorithms perform better than APIT. The
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In Figure 11b, the parameters are as follows: AH = 3~21, ANR = 3, ND = 8, R = 150 m, K = 0.0050. The sensor nodes are distributed in uniform distribution. Figure 11b shows that the overall trend of localization error rate in APIT and HTCRL localization algorithms decreases as the number of anchors heard increases. Instead, HTCRL localization algorithms perform better than APIT. The localization error rate of APIT is from 32.2% to 19.0%, while the localization error rate of HTCRL is from 30.1% to 18.9%. From Figure 11a,b, we note that HTCRL localization algorithm is more accurate than the APIT localization algorithm when they have the same parameters. In addition, comparing random distribution and uniform distribution, the performance in uniform distribution is better than in random distribution. In Figure 11c, the parameters are as flows: AH = 3~21, ANR = 3, ND = 8, R = 450 m, K = 0.0050. The sensor nodes are distributed in random distribution. Figure 11c shows that the overall trend of localization error rate in APIT and HTCRL localization algorithms decreases as the number of anchors heard increases. Instead, HTCRL localization algorithms performs better than APIT. The localization error rate of APIT is from 60.1% to 31.2%, while the localization error rate of HTCRL is from 52.1% to 28.9%. In Figure 11d, the parameters are as follows: AH = 3~21, ANR = 3, ND = 8, R = 450 m, K = 0.0050. The sensor nodes are distributed in random distribution. Figure 11d shows that the overall trend of localization error rate in APIT and HTCRL localization algorithms decreases as the number of anchors heard increases. Instead, HTCRL localization algorithms performs better than APIT. The localization error rate of APIT is from 48.2% to 30.1%. However, the localization error rate of HTCRL is from 45.1% to 26.9%. From Figure 11c,d, we note that HTCRL localization algorithm is more accurate than APIT localization algorithm when they have the same parameters. In addition, by comparing random distribution and uniform distribution, the performance in uniform distribution is better than in random distribution. In the above experiments, the communication radius (R) is 150 m and 450 m separately. It can be seen from Figure 11 that the localization error rate is higher when R is 450 m. Therefore, communication radius (R) is not greater when the localization error rate is lower in the two algorithms. In the experiments, the localization error rate decreases when the number of anchors heard is 3 to 21 and the communication radius is constant. When the value of AH increases gradually, it will increase the cost of the network. However, from the results of the experiments, when AH is from 3 to 9, the localization accuracy has been meet the requirements of outdoor localization. Therefore, it does not need to increase the value of AH gradually for little improvement in localization accuracy. Thus, the cost of the network will not increase a lot when it meet the requirements of outdoor localization. 2.
Localization Error Varying Node Density (ND) and Anchor Percentage (AP)
This experiment analyses the parameters of node density (ND) and anchor percentage (AP) that influence localization error rate. In the experiment, the sensor nodes distributed in random distribution. In Figure 12a, the parameters are as follows: AH = 18, ANR = 3, AP = 20%, R = 150 m, K = 0.0050. The sensor nodes are distributed in random distribution. Figure 12a shows that the overall trend of localization error rate in APIT and HTCRL localization algorithms decreases as the node density increases. When the node density is from 3 to 12, the localization error rate decreases dramatically. When the node density is from 12 to 21, the localization error rate changes little. The localization error rate of APIT is about 21%. The localization error rate of HTCRL is about 20%. The localization error rate of APIT is from 52.3% to 21.1%. The localization error rate of APIT is from 48.2% to 20.0%. In general, HTCRL performances better than APIT.
localization error rate in APIT and HTCRL localization algorithms decreases as the node density increases. When the node density is from 3 to 12, the localization error rate decreases dramatically. When the node density is from 12 to 21, the localization error rate changes little. The localization error rate of APIT is about 21%. The localization error rate of HTCRL is about 20%. The localization error rate of APIT from 52.3% to 21.1%. The localization error rate of APIT is from 48.2% to 20.0%. Information 2017, is 8, 40 12 of In 15 general, HTCRL performances better than APIT. 60 55
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(b) Figure 20%, Figure 12. 12. Localization Localization Error Error Rate Rate Varying VaryingNode NodeDensity. Density.(a) (a)ND ND==3~21, 3~21,AH AH== 18, 18, ANR ANR== 3, 3, AP AP = = 20%, R = 150 m Random, K = 0.0050; (b) ND = 3~21, AH = 18, ANR = 3, AP = 30%, R = 150 m Random, R = 150 m Random, K = 0.0050; (b) ND = 3~21, AH = 18, ANR = 3, AP = 30%, R = 150 m Random, K K == 0.0050. 0.0050.
In 0.0050. In Figure Figure 12b, 12b,the theparameters parametersare areasasfollows: follows:AH AH==18, 18,ANR ANR==3,3,AP AP== 30%, 30%, RR== 150 150 m, m, K == 0.0050. The distributed in in random randomdistribution. distribution.Figure Figure12b 12bshows shows that overall trend The sensor nodes are distributed that thethe overall trend of of localization error rate APITand andHTCRL HTCRLlocalization localizationalgorithms algorithmsdecreases decreases as as the the node density localization error rate ininAPIT density increases. When the the node node density density is is from 3 to 9, the localization error rate decreases dramatically. increases. When When the node density is from 9 to 21, the localization error rate changes little. The localization error rate of APIT is from 45.1% to 19.2%. The localization error rate of APIT is from 40.5% to 19.0%. In general, HTCRL performances better than APIT. In the simulation experiments, the sensor nodes are distributed in two kinds of forms. One is random distribution, the other is uniform distribution. It can be seen from the experimental results that the localization error rate is smaller under the uniform distribution than random distribution. In this paper, assume the radius is 450 m and 150 m. The radius is relatively large. When the density of anchors heard and nodes increases continuously, the number of adjacent anchor nodes around the unknown node is equivalent increased. Due to the node location is fixed relatively under the uniform distribution, and the distance between anchor node and unknown node is within a certain range. The distance of the deployed nodes is fixed relatively, so most unknown nodes can communicate with each other. Therefore, the localization error rate is higher. As for the nodes deployed under the random distribution, the location of the nodes and the number of the anchor nodes around the unknown nodes are not fixed, so the localization error rate of the unknown nodes is relatively lower. 3.
Localization Error Rate Varying K
distribution, and the distance between anchor node and unknown node is within a certain range. The distance of the deployed nodes is fixed relatively, so most unknown nodes can communicate with each other. Therefore, the localization error rate is higher. As for the nodes deployed under the random distribution, the location of the nodes and the number of the anchor nodes around the unknown nodes are not fixed, so the localization error rate of the unknown nodes is relatively lower. Information 2017, 8, 40
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Localization Error Rate Varying K
In that influence localization error rate. K isKthe In this this experiment, experiment,ititanalyses analysesthe theparameters parametersofofK K that influence localization error rate. is proportion of triangles reduced. In In thethe experiments, the the proportion of triangles reduced. experiments, thesensor sensornodes nodesdistributed distributedin in random random and and uniform respectively. uniform distribution distribution respectively. In Figure 13a, the = 3, APAP = 20%, R =R450 m, Km,= 0.0020~0.01. The In Figure 13a, theparameters parametersare areasasfollows: follows:AH AH = 3, = 20%, = 450 K = 0.0020~0.01. sensor nodes are distributed in random and and uniform distribution. The sensor nodes are distributed in random uniform distribution. 30 Random Uniform
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(b) Figure 13. Localization Error Error Rate Rate Varying 0.0020~0.01; Figure 13. Localization Varying with with K. K. (a) (a) AH AH == 3, 3, AP AP == 20%, 20%, R R= = 450 450 m, m, K K= = 0.0020~0.01; (b) AH 3, AP AP == 230%, 230%, R R ==450 450m, m,KK==0.0020~0.01. 0.0020~0.01. (b) AH = = 3,
Figure 13a shows that the overall trend of localization error rate in HTCRL localization algorithms decreases as K increases. When K is from 0.002 to 0.04, the localization error rate decreases dramatically. But when K is from 0.004 to 0.01, the localization error rate changes little. The localization error rate of HTCRL is about 15%. The performance is remarkable. The localization error rate of HTCRL is from 29.06% to 16.07% under the random distribution. Under the uniform distribution, the localization error rate of HTCRL is from 26.16% to 15.15% under random distribution. In general, localization error rate of HTCRL decreases when K increases gradually. In Figure 13b, the parameters are as follows: AH = 3, AP = 30%, R = 450 m, K = 0.0020~0.01. The sensor nodes are distributed in random and uniform distribution. Figure 13b shows that the overall trend of localization error rate in HTCRL localization algorithms decreases as K increases. When K is from 0.002 to 0.04, the localization error rate decreases dramatically. But when K is from 0.004 to 0.01, the localization error rate changes little. The localization error rate of HTCRL is about 14%. The performance is remarkable. The localization error rate of HTCRL is from 24.01% to 14.7% under the random distribution. Under the uniform distribution, the localization error
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rate of HTCRL is from 23.05% to 14.12% under the random distribution. In general, localization error rate of HTCRL decreases when K increases gradually. From Figure 13, it can be seen that the localization error rate decreases significantly when K is greater than 0.004 under the two distribution. Moreover, it will decrease gradually with the increasing of R. 5. Conclusions In this paper, we analyzed some related works about APIT localization algorithms. In view of their shortcomings, the paper proposes an improved APIT algorithm HTCRL, which is based on the principle of homothetic triangle. In order to compare the performance of these two algorithms, several experiments are done. We have considered different parameters, including random distributed network, uniform distributed network, anchors heard, node density, anchor percentage, anchor to node range ratio, and communication radius. In all these parameters, the performance of the HTCRL localization algorithm is better than the APIT algorithm without introducing additional hardware or communication overhead. Acknowledgments: This work is supported by the Science and Technology Planning Project of Daqing under grant zd-2016-054. Author Contributions: Dan Zhang and Zhiyi Fang conceived and designed the HTCRL localization algorithm and experiments; Hongyu Sun performed the experiments; Jie Cao analyzed the data; all authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.
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