[ieee 2011 seventh international conference on mobile ad-hoc and sensor networks (msn) - beijing,...

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

Ziqi Fan1, Yuanfang Chen1, Lei Wang1∗, Lei Shu2, Takahiro Hara21School of Software, Dalian University of Technology, China

Email: [email protected], yuanfang [email protected], [email protected] of Multimedia Engineering, Osaka University, Japan

Email: [email protected], [email protected]

Abstract—In Wireless Sensor Networks (WSNs) a multitudeof location-dependent applications have been proposed recently,which is very intriguing for researchers to discover and designmore accurate and cost-effective localization algorithms. Inanisotropic networks, the Euclidean distance between a pair ofnodes may not correlate closely with the hop count betweenthem because the corresponding shortest path may have to curvearound intermediate holes, resulting in poor distance estimation.And without the help of a large number of uniformly deployedseed nodes, those schemes fail in anisotropic WSNs. To addressthis issue and improve the accuracy of localization, we proposethe Removing Heavily Curved Path (RHCP) scheme in this paper.RHCP takes advantage of selecting the paths which are notheavily affected by the holes to recalculate the location of eachunknown node. Through simulation, the results reveal that RHCPperforms better than original DV-Hop in anisotropic networkswith different shape of holes. In addition, through iterationsof RHCP, the results get improved for different anchor nodedensities.

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 cost-

effective localization algorithms [1], [2]. Existing approaches

fall into two categories: Range-based approaches and Range-

free 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 di-

minishes 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 spe-

cial functionality of hardware. Approximate point in triangle

∗ Corresponding author: Lei Wang, [email protected].

(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 transfor-

mation 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 hop-

count 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. HopItemswith 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:

HopDistancei =

∑j �=i

√(xi − xj)2 + (yi − yj)2∑

j �=i hij, (1)

where (xi,yi), (xj ,yj) are coordinates of anchor i and anchor j,

hij is the hops between anchor i and anchor j. Then, average

per hop distance is flooded to the entire network. After receiv-

ing 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

2011 Seventh International Conference on Mobile Ad-hoc and Sensor Networks

978-0-7695-4610-0/11 $26.00 © 2011 IEEE

DOI 10.1109/MSN.2011.67

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978-0-7695-4610-0/11 $26.00 © 2011 IEEE

DOI 10.1109/MSN.2011.67

75


978-0-7695-4610-0/11 $26.00 © 2011 IEEE

DOI 10.1109/MSN.2011.67

75

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, anisotrop-

ic networks are more likely to exist in realistic situations

caused by mountains, lakes, buildings, etc. In [11], the au-

thors propose the Rendered Path (REP) protocol, a range-

free 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]: Pa-

per [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 = {s1, s2, ..., sn} the set of nodes including bothanchor nodes and unknown nodes

E the set of communication linksri transmission radius of sensor node i

||si − sj || the Euclidean distance between si and sjSa anchor nodesSu unknown nodes

EEPset of Estimated Euclidean Paths

from each unknown node to each anchor node

EEP [i]set of Estimated Euclidean Paths fromeach anchor node to unknown node si

si.P reRHCP store pre-processed coordinate of node isi.RHCP store RHCP localized coordinate of node i

x,y variables of spline curvexi,yi control points of spline curve

a,b,c,d... coefficients of spline curveM 4 ∗ 4 matrixC column vector storing coefficientsY column vector storing the value of y1 to y4fi() function of piecewise polynomial curveslpi slope of curve at sensor node si

tivariable between 0 and 1and depended on xi or yi

α(s) unit-speed curves arc length

φtangential angle measured counterclockwise

from the x-axis to the unit tangentk curvaturen total number of nodes in WSNs

TABLE IMAIN 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 = {s1, s2, ..., sn} 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 si and sj is ||si − sj || ≤ ri, where ||si − sj ||is the Euclidean distance between si and sj . Our network

includes two types of sensor nodes Sa (anchor nodes) and

Su (unknown nodes). Unknown nodes are randomly deployed

with a density ρSu within an area Ω, and a set of special sensor

nodes Sa with known location, are also randomly deployed

767676

Fig. 1. In DV-Hop, for each unknown node, it uses all of the paths betweenthe unknown node and anchor nodes to locate. But in RHCP, we use the pathsAB, AC and AD to locate, removing the heavily curved paths AE and AF, toachieve better accuracy.

with a density ρSa. 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 repre-

sent 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 HopDistancewith 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 EEPoutput: si.P reRHCP and si.RHCP

1 begin2 Step 1:

3 i=0

4 while i<numberOfUnknownNode do5 Multilaterating EEP [i] to calculate

si.P reRHCP6 i++

7 Calculate the average estimated error of PreRHCP

8 Step 2:

9 i=0

10 while i<numberOfUnknownNode do11 while j<numberOfHopItem in HopTable do12 Construct Space Piecewise Polynomial Curve

13 Calculate curvature based on Space Piecewise

Polynomial Curve

14 j++

15 Select the average curvature satisfied paths

calculating si.RHCP16 i++

17 Calculate the average estimated error of RHCP

18 Step 3:

19 i=0

20 while i<numberOfUnknownNode do21 si.P reRHCP = si.RHCP22 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+ cx2 + dx3 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+ cx2 + dx3 is the one

that is most typically chosen for constructing smooth curves.

777777

Fig. 2. An example of a curve which has more than one inflection point. Black points represent sensor nodes. Red points represents inflection points. Thecurve 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+ cx2 + dx3 = y, (2)

and the constants a, b, c, d can be calculated using anchor

nodes coordinates (the Formula 3):

⎛⎜⎜⎝

1 x1 x21 x311 x2 x22 x321 x3 x23 x331 x4 x24 x34

⎞⎟⎟⎠

⎛⎜⎜⎝

abcd

⎞⎟⎟⎠ =

⎛⎜⎜⎝

y1y2y3y4

⎞⎟⎟⎠ , (3)

where the elements xi and yi are x-axis coordinate and y-axis

coordinate of nodes si, respectively. The Formula 3 can be

denoted as: MC = Y , and then the C =M−1Y .

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, fi(x) = ai + bix + cix2 + dix

3 is the function

representing the curve between sensor nodes si and si+1.

For each cubic polynomial function, there are four coeffi-

cients. 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: ai + bixi + cixi2 +

dixi3 = yi and ai + bixi+1 + cixi+1

2 + dixi3 = yi+1.

Step two: The curve segments should have the same slope

where they join together. This gives the third linear equa-

tion for each segment bi + 2cixi+1 + 3dixi+12 − bi+1 −

2ci + 1xi+1 − 3di + 1xi + 12 = 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

2ci+6dixi+1 = 2ci+1+6di+1xi+1 or 2ci+6dixi+1−2ci+1−6di+1xi+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 slp0 and slpn, getting f ′0(x0) = slp0 and f ′n−1(xn) =slpn, so b0 + c0x0 + 2d0x0

2 = slp0 and bn−1 + cn−1xn +2dn−1xn

2 = slpn 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

si and si+1, we need to construct two functions fxi =axi+bxiti+cxit

2i +dxit

3i and fyi = ayi+byiti+cyit

2i +dyit

3i

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

787878

(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 xand 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 t0, t1 t2 ... ti ti+1 ... which vary between 0 and 1

as we step along the x axis between control points. For each

pair of sensor nodes si and si+1,

ti =x− xi

xi+1 − xi , withdtidx

=1

xi+1 − xi . (4)

Each curve segment is specified by a parametric cubic curve

fyi(ti) = ayi + byiti + cyiti2 + dyiti

3. (5)

At the left side of segment i, ti = 0 and at the right ti = 1,

so step one constraint in the above subsection is fyi(0) =ayi = yi and fyi(1) = ayi + byi + cyi + dyi = yi+1.

For step two constraint, we differentiate once with respect

to x using the chain rule:

Dxfyi =∂fyi∂ti

dtidx

= f ′yi1

xi+1 − xi . (6)

For step three constraint, we differentiate twice

Dx2fyi = (

∂f ′yi∂ti

dtidx

)dtidx

= f ′′yi1

(xi+1 − xi)2. (7)

We force fyi and fy(i+1) to match slopes by

(Dxfyi)(1) = (Dxfy(i+1))(0). (8)

We force fyi and fy(i+1) to match curvatures by

(Dx2fyi)(1) = (Dx

2fy(i+1))(0). (9)

Remember that we provided two extra equations by spec-

ifying slopes at the 2 endpoints slp0 and slpn. So this gives

f ′y0(0) = slp0 and fy(n−1)(1) = slpn.

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)

797979

RHCP

DV-Hop

(a) Circle shape hole

RHCP

DV-Hop

(b) Concave shape hole

RHCP

DV-Hop

(c) Face shape hole

RHCP

DV-Hop

(d) Spiral shape hole

Fig. 5. Anchor Node Density vs. Error in anisotropic sensor networks.

IV. SIMULATIONS

A. Simulation Setup

In order to check the performance of RHCP localization

algorithm, we implemented it into our NetTopo WSN simula-

tor [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∑k=1

√Δxk +Δyk

nR. (12)

Our WSN’s size is 500*500m2. 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 SpiralRHCP 0.414 0.563 0.344 0.973

DV-Hop 0.699 0.788 0.589 1.512

TABLE IIAVERAGE ESTIMATED ERROR OF DIFFERENT ANCHOR NODE DENSITIES

FOR EACH ANISOTROPIC NETWORK.

808080

1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time of repeating

Estim

ate

dE

rror

R()

AnchorNodeDensity(%)=8.3


AnchorNodeDensity(%)=25




(a) Circle shape hole

1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

Time of repeating

Estim

ate

dE

rror

R()







(b) Concave shape hole

1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.3

0.4

0.5

0.6

0.7

0.8

Time of Iterations

Estim

ate

dE

rro

rR(

)







(c) Face shape hole

1 1.5 2 2.5 3 3.5 4 4.5 5

0.8

1

1.2

1.4

1.6

1.8

2

Time of Iterations

Estim

ate

dE

rro

rR(

)







(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: RHCPoutperforms original DV-Hop in different kind of anisotropicnetworks.

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, theaverage localization Error for each anchor node density isgoing smaller. But till the fourth and fifth time, the effect ofoptimizing reaches its limit, and the line goes smooth.

V. CONCLUSION

Given the condition that many previous localization algo-

rithms 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 dif-

ferent anchor node density in different anisotropic networks.

ACKNOWLEDGEMENTS

This work is partially supported by Natural Science Founda-

tion of China under Grant No. 61070181, the Fundamental Re-

search Funds for the Central Universities No. DUT10ZD110,

818181

and Natural Science Foundation of Liaoning Province under

Grant No. 20102021.

Lei Shu’s research in this paper was supported by Grant-in-

Aid for Scientific Research (S) (21220002) of the Ministry of

Education, Culture, Sports, Science and Technology, Japan.

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