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A New WSN Localization Algorithm Based on Regularization Method

Wang Lei

Department of Measurement and Instrumentation

School of Control Science and Engineering

Shandong University, Jinan, Shandong Province, P. R. China

E-mail: leiwang@sdu.edu.cn

Wang XiaoPeng

Department of Measurement and Instrumentation

School of Control Science and Engineering

Shandong University, Jinan, Shandong Province, P. R. China

E-mail: wangxiaopeng321@yahoo.cn

ABSTRACT

AbstractNode localization is a key problem of wireless sensornetwork (WSN) to complex monitoring and tracking in wide

applications. The distance measurement errors lead the

localization to an ill-posed problem which means that the

localization solution is not unique that is one of the three ill-

posed problem terms. In this paper, based on Tikhonov

regularization method, a new localization algorithm is proposed

for the ill-posed problem in the localization, which includes

location model building, and regularization parameter selecting.

2-dimensioned and 3-dimensioned localization test results show

that the location precision of the proposed algorithm is betterthan that of the Maximum Likelihood Estimation (MLE)

method, and the measurement errors are about 1 meter while

regularization parameter is about 55.

Keywords: wireless sensor network; localization; ill-posed

problem; regularization;

I. INTRODUCTIONThe wireless sensor network (WSN) is composed of a

large number of sensor nodes with wireless communications,computations, and sensing capabilities, which are denselydeployed either inside the monitoring phenomenon or veryclose to it[1]. These tiny WSN nodes which are capable ofcommunicating with each other, acquiring various physicalvalues, performing computations can cooperatively achieve adesired task through specific protocol. This makes it possiblefor the WSN to a variety of monitoring and trackingapplications in military, agriculture, and industry, etc[2].

An important aspect in most of the WSN applications isthe accurate localization of the individual node[3]. WSN nodesare usually placed in different environment to accomplishdifferent tasks, in which the locations of these nodes arerandom and unknown in advance. However, data acquired bythese nodes are only available when connected with locationinformation. Therefore the node localization technology hasaroused widespread attention and become one of the hotspotsof WSN study.

WSN node localization is that WSN node is located by asmall number of known nodes in the networks, which meansthat coordinates are established with the location informationof these konwn nodes and the coordinate of unknown nodescan be obtained using a special locating algorithm[4].

WSN node localization methods can be classified into twocategories: the Range-free methods and the Range-basedmethods[5]. The former can realize node localization usingonly the information of network connectivity without theinformation of point to point distances, while the latter

depends on the information of point to point distances.

Both methods discussed above can be used in differentconditions with their own advantages and disadvantages[6].The Range-free methods mainly include DV-hop algorithm,

Amorphous algorithm and Approximate Point-In-Triangulation Test (APIT) algorithm. These methods haverelatively low hardware requirements, need only certaincommunication expenses and computation complexity torealize localization. However, a plenty of known nodesdensely deployed are necessary for these methods[7]. Besides,these methods are localization methods of low-accuracy.

The Range-based methods realize localization withdistances between unknown and known nodes via localizationalgorithm[8]. The Range-based methods have high locationaccuracy and low computation complexity, but they need to

measure distances, which increase power consumption.Besides, study on the accuracy of metrical distances isindispensable. There are several methods to measure thedistances between nodes: Received Signal Strength Indicator

RSSI , Time of Arrival TOA , Time of Difference ofArrival TDOA and so on. Among these, the RSSI methodis commonly used, in which knowledge of the power of thetransmitted signal, the path loss model and the power of thereceived signal are used to determine the distance between thereceiver and the transmitter[9]. This method is a low-powerand low-cost one whose major drawback is that multi-pathreflections, non line-of-sight conditions, shadowing effectsmay lead to erroneous distance estimates which make thenode localization an ill-posed problem. In this paper, aninnovative algorithm based on Tikhonov regularization is

proposed[10].

II. RSSI LOCALIZATION MODELSuppose the coordinates of unknown and known nodes

are ( , , )x y z and 1 1 1 2 2 2 3 3 3( , , ) ( , , ) ( , , )...( , , )n n ny z x y z x y z x y z . The

distances from known nodes to unkonwn node are

1 2 3, , ... nd d d d . The multilateral localization model can be

denoted by

2 2 2 2

1 1 1 1

2 2 2 2

2 2 2 2

2 2 2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) .n n n n

x x y y z z d

x x y y z z d

x y y z z d

- + + =

+ + = + + =

# (1)

By subtracting one equation from rests, we get

AX B= (2) where

1 1 1

1 1 1

2( ) 2( ) 2( )

2( ) 2( ) 2( )

n n n

n n n n n n

x y y z z

A

x x y y z z

=

# # #

=

zy

x

X

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2 2 2 2 2 2 2 2

1 1 1 1

1 2 2 2 2 2 2 2 2

2 2 2 2

1 2 2 2 2 2 2 2 21 1 1 1

n n n n

n n n n

nn n n n n n n n

d d z y x z y xb

d d z y x z y xB

bd d z y x z y x

+ + +

+ + + = = + + +

##

.

The distances from the known nodes to the unknown nodecan be determined by the RSSI in the localization process[11].

The power of received signal at distancei

d ( 1, 2,3... )i n= is

typically modeled by

0

0

( ) ( ) 10 lg( )ii id

PL d PL d nd

= + , (3)where

id denotes the distance from transmitting node to

receiving one,0

d denotes the reference distance, n is the

path-loss exponent typically being 2~4, ( )iPL d denotes the

received power at distancei

d ,0

( )PL d denotes the received

power at the reference distance0

d , andi

( 1, 2,3... )i n=

denote the noises[12].

Generally, the vector B in (2) is the parameter vectorunder ideal conditions. Due to the noises of RSSI and theerror of the empirical model, the vector we get is only anapproximation of B , which can be defined by B + ,

where calls the error vector. That is, the actual equation

used to localization is denoted by

AX B = + . (4)With the definition B B = + , equation (4) can be expressed

as

AX B

= . (5)The errors discussed above lead the localization problem to anill-posed problem, which means that the equation does notfulfill the following terms for all admissible data:(i) a solutionexists; (ii) the solution is unique; (iii) the solution depends oncontinuously on the data. Many application problems are ill-

posed, the difficulty of which is that the solutions do not

depend on the input data continuously, which may lead to theinstability of these equations. Getting the solution of the ill-posed problem is indispensable in order to identify thelocation information of the unknown node. Usually wereplace the linear system by a nearby system that is lesssensitive to the error and consider the computed solution of

the latter system as an approximation of the solution of theformer system, this replacement is known as regularization.There are several regularization methods to solve ill-posed

problem: Tikhonov regularization method, TSVDregularization method and some kinds of iteration methods. Inthis paper, Tikhonov regularization method is proposed tosolve the ill-posed problem in localization.

III. LOCALIZATION ALGORITHM BASED ONTIKHONOV REGULARIZATION METHODTikhonov regularization method is a general and effective

way to solve various ill-posed equations[13]. The key issue of

this method is selection of reasonable and feasibleregularization matrix and regularization parameter for

different concrete situations, in which selecting is the core

problem. Tikhonov regularization method refers the

vectorX that has the minimization of X as the solution to

(5), where vectorXsatisfies the following

AX B

, (6)

where = . That is, we have to compute the minimization

of the function

2( )F X X= , (7)

under inquality (6). Equation (7) will reach its minimum if

AX B

= . (8)

Therefore, we can refer this as the minimization problem asthe following

min ( ),{ : }F X X AX B

= . (9)

Function (9) can be converted to the minimization problem of(10) by introducing a positive Lagrange multiplier ,

2 2AX B X

+ , (10)

which is equivalent to solve

( )T TI A A X A B

+ = . (11)

Solving the equation, then we get

1( )T TX I A A A B

= + (12)

where is regularization parameter, and X is the

approximation solution.

Equation(12) shows that determining parameter

is thekey issue of working out the vectorX. Generally speaking,the Xgets closer to the solution to (5) when gets smaller.

However, the state of (12) gets worse at the same time.Consequently, selection of a compromise is critical[14].

Tikhonov regularization method tells us that the valueof has the same order of magnitude with . In this paper we

try to determine an estimate value of with the following

method, and try to do further modification on the value of .

The minimization problem of (9) can also be converted tothe minimization problem of (13) by introducing positiveLagrange multipliers , i.e.,

2 2X AX B

+ . (13)

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This minimization problem satisfies the followingqualification

( ) 2 0T TA AX A B X

+ = . (14)

Hence,

( 2 )C X += , (15)

where C+ is the generalized inverse matrix ( )T TA AX A B

.

During the tests, the unknown nodes coordinate ismeasurable. In practical application, we could substitute oneknown nodes coordinate to vectorX in (15), then we get avalue of . We may estimate by

1

= . (16)

The method proposed above to estimate is based on

Morozov discrepancy criterion, which is an effective methodto select regularization parameter[15]. After the determinationof , localization computation can be realized.

IV. TESTSA. 2-dimensioned node localization

A tracking test is carried out in order to investigate theperformance of the proposed algorithm to the mobile nodes as

below: 4 nodes are chosen as known nodes, and the unknownnode is moved along the straight line at uncertain intervals,then several positions in the line are localized. The

performance of the test is shown in Fig.1.

Figure1. The results of t racking simulation.Fig.1 indicates that errors are obvious at a small number of

nodes due to environment impacts and input data errors, butfor most nodes the estimate locations are close to the realones, whose errors are acceptable.

B. 3-dimensioned node localizationFive reference nodes are positioned randomly at the top of

the room. As shown in Fig.2, the asterisks denote knownnodes and the pluses denote unknown nodes.

)LJXUH Distribution map of node in experiment of localizationIn the simulation, regularization parameter is evaluated

by Morozov discrepancy principle. Putting the actualcoordinate of nodes into the equation (16), the value of

ranges from 40 to 70. To prove the validity of the method, welocate every node with the ranging from 10 to 1000. The

errors between the known node ( , , )i i i

X Y Z and the unkonwn

node ( , , )i i iy z is

2 2 2( ) ( ) ( )i i i i i i i

E X x Y y Z z= + + . (17)

The errors accompanying changes with are shown in

Fig.3. The minimum errors are got when is around 55.

Figure3. Localization results with different Selecting regularization parameter for 55 and

comparing the localization errors of Tikhonov regularizationmethod to that of Maximum Likelihood Estimate(MLE)method, the result is shown in Table I.

15 20 25 30 35 400

5

10

15

20

25

30

x(m)

Ym

circle: real trailstar : estimate trail

0

1

2

3

4

0

2

4

6

8

0

0.5

1

1.5

2

2.5

3

X mY m

Zm

4

3

1

(1)

(2)(5)(6)

(3)

(4)(7)

(8)

(9)(10)

(11)(12)

25

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

error(m)

node 1

node2

node 3

node 4

node 5

node 6

node 7

node 8

node 9

node 10

node 11

node 12

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Table I. Compare the localization errors of Tikhonov regularization to that of MLEunknown node 1 2 3 4 5 6 7 8 9 10 11 12 e

Tikhonov 0.42 4.04 1.53 2.32 2.04 1.98 1.65 2.41 2.75 2.26 1.09 2.44 2.04

MLE 9.42 10.29 3.74 27.54 5.80 11.59 1.02 7.31 29.40 3.27 4.68 6.95 10.08

The average location error can be calculated with the

following

2 2 2

1

( ) ( ) ( )n

i i i i i i

i

X x Y y Z z

en

=

+ +

=

. (18)

The simulation shows that the Tikhonov regularizationmethod is more effective than MLE. Choosing the appropriateregularization parameter, the average error is 2.04 metres and

the least error is less than 1 metre. Based on MLE, the errorsof node 4 and node 9 approach 30 metres. Such results cant

be accepted. The error of distance measurement will causeserious localization deviation that is one of the three ill-posed

problem terms. The Tikhonov regularization method cansolve the problems effectively and reduce positioning errorgreatly.

V. CONCLUSIONS AND DISSCUSIONSIn this paper, an innovative localization algorithm based

on Tikhonov regularization method has been presented for theill-posed problem in the localization. With the algorithm

proposed here, the localization error was reduced, comparing

with that of the MLE method.The regularization parameter has been studied in this

paper. The simulation showed that localization accuracy wasgood and the minimum localization errors was got with

55 = .

Although the above-mentioned algorithm has the samecomputation expense as the MLE method, it has the higherlocalization accuracy than that of the MLE method.Furthermore, localization environments and area sizes caneffect the value of and localization accuracy. So, it is

necessary to adjust the value of according to the real

localization environment in practice.

ACKNOWLEDGEMENTS

This work is supported by the Outstanding YoungScientists Incentive Funding of Shandong Province undergrant 2007BS01009.

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