research article robust tdoa localization algorithm for...

Research ArticleRobust TDOA Localization Algorithm for AsynchronousWireless Sensor Networks

Hui Xiong Zhiyuan Chen Wei An and Beiya Yang

School of Electronic Science and Engineering National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Hui Xiong 1907985967qqcom

Received 18 May 2014 Revised 29 July 2014 Accepted 14 August 2014

Academic Editor Nianbo Liu

Copyright copy 2015 Hui Xiong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a tag localization algorithm based on the time-difference-of-arrival (TDOA) of mobile tag signal forasynchronous wireless sensor network (WSN) withN anchors (nodes with known locations) and a large number of mobile tags Toobtain time synchronization all anchors broadcast signals periodically relative clock offsets and skews of anchor pairs are estimatedby the least-square (LS) method using the times-of-arrival (TOAs) of broadcast signals at anchors When a tag transmits signal theTOA of tag signal at each anchor is stamped and errors in original TDOAs of tag signal due to relative clock offsets and skews ofanchor pairs are eliminated Based onGaussian noisemodelmaximum likelihood estimation (MLE) for the tag position is obtainedPerformance issues are addressed by evaluating the Cramer-Rao lower bound of synchronization and localization algorithms Sincethe tag can be located via a single transmission least power consumption of tag is required and large number of tags can be servedin WSNThe proposed algorithm is simple and effective with performance close to that of synchronous TDOA algorithm

1 Introduction

With the advances of wireless communications and micro-electronics technologies wireless sensor networks (WSNs)have received more attention over the recent decade due totheir potential application to a wide variety of diverse areas[1] such as environmental monitoring space explorationmilitary applications target tracking and healthcare

One of the most important issues for WSN is that thelocation of each sensor node must be determined Howeverit is expensive and impractical to equip all sensors in thenetwork with a global positioning system (GPS) receiver[2] To obtain accurate location estimates of sensor nodesrange-based localization algorithms are more favorable thanrange-free ones [3ndash5] In general the range-based algorithmsalways follow two steps [6 7] they first measure somemetrics bearing location information the so-called rangingor bearing and second estimate the positions based on thosemetrics the so-called location information fusion There aremainly four metrics time-of-arrival (TOA) or time-of-flight(TOF) [8] time-difference-of-arrival (TDOA) [5 6] angle-of-arrival (AOA) [9] and received signal strength (RSS)[10] The ranging methods using RSS can be implemented

by energy detectors but they can only achieve a coarseresolution Antenna arrays are required for AOA-basedmethods which encumbers their popularity On the otherhand high accuracy and potentially low cost implementationmake TOA or TDOAbased on ultrawideband system (UWB)and broadband spread spectrum system a promising rangingmethod [11 12]

Since TOA and TDOA measurements are time-basedclock synchronization is essential for high accuracy local-ization and low cost implementation Many protocols havebeen proposed to resolve the synchronization problem ofWSN such as the reference broadcast synchronization (RBS)protocol [13] the timing-sync protocol for sensor networks(TPSN) [14] and flooding time synchronization protocol(FTSP) [15] On the other hand clock synchronization canalso be handled by estimating the relative clock offsetsand skews among sensor nodes using maximum likelihoodestimator (MLE) [3ndash5 7] the theoretical performance limitsare evaluated by Cramer-Rao lower bound (CRLB)

Due to the close relationship between synchronizationand localization problems joint estimation of the relativeclock skew the relative clock offset and the positions of

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 598747 10 pageshttpdxdoiorg1011552015598747

2 International Journal of Distributed Sensor Networks

source nodes are proposed in [4 5] for network with asyn-chronous anchors The broadcasting property is exploitedto jointly estimate clock parameters and source positionusing asymmetrical time-stamping and a passive listeningprotocol in [5 16] Although these algorithms achieve goodperformance complex protocols such as two-way rangingprotocol are necessary Moreover frequent transmissions ofsource node may cause more energy dissipation and limit thenumber of tags in the service area of WSN

In this paper we propose a robust TDOA localizationalgorithm for asynchronous wireless sensor networks InWSN sensor nodes are divided into two categories anchorand tag Tags are mobile sensor nodes anchor nodes areused as reference points to localize tag nodes and assumedto have known positions and good power supplies Thereare many limitations for tag nodes such as size costand energy dissipation which is concerned intensively forpractical implementation In order to synchronize anchorclocks all anchors broadcast signal periodically TOAs ofbroadcasting signal at other anchors are stamped accordingto their own local clocks respectively A localization serveron the backbone network collects all these timestamps toestimate the relative clock offsets and skews of anchor pairsWhen anchors receive a broadcasting signal from a tagthe TOAs of tag signal at anchors are also stamped anddelivered to the localization server The error in originalTDOA of tag signal due to relative clock offsets and skewsamong anchors can be easily canceled by a compensationoperation Based on a Gaussian measurement noise modelmaximum likelihood estimation (MLE) for tag position isobtained Performance issues are addressed by evaluating theCramer-Rao lower bound (CRLB) of synchronization andlocalization algorithm Only anchor-anchor synchronizationis required in this paper and tag node is localized via a signaltransmission so power consumption of tag can beminimizedand the number of tags to be served can be maximized

This paper is organized as follows The sensor networkmodel for localization problem considered in the paper isgiven in Section 2 Estimation of relative clock offsets andskews of anchor pairs and performance analysis are presentedin Section 3 The localization of tags is studied in Section 4and the statistical performance analysis of the localizationalgorithm is also presented in this section Section 5 presentssome numerical results and Section 6 concludes the paper

2 Sensor Network Model for Localization

Wireless sensor network localization is the process by whichsensor nodes determine their location In simple terms local-ization is a mechanism for discovering spatial relationshipsbetween objectsThe various approaches taken in literature tosolve this localization problem differ in the assumptions theymake about their respective network and sensor capabilitiesA detailed but not exhaustive list of assumptions madeinclude assumptions about network infrastructure devicehardware signal propagation models timing and energyrequirements operational environment time synchroniza-tion communication costs error requirements and nodemobility

MT or tag

Anchor Anchor

AnchorAnchor

Localization server

TOA l20(t)

TOA l10(t) TOA li0(t)

TOA l30(t)

Figure 1 Radio WSN model for tag localization (anchor nodesare represented by red rectangles and tags are represented by greencircles)

In this paper sensor nodes are divided into two cate-gories anchor node and tag node Anchor nodes are usedas reference points to locate tag nodes they are alwaysconnected to network infrastructure and well powered Toreduce network complexity and costs the quantity of anchoris limited and positions of anchor nodes are planned inadvance to guarantee the coverage of service area The tagsare sensor nodes to be located and must be designed forlow cost and low power consumption and operation for along lifetime To further reduce the complexity and powerconsumption of tag the processing capability is restricted forpractical implementation For example a WSN based patientmanagement consists of limited anchors which serve for largequantity of tags carried by patients

The radio sensor network model considered in the paperis depicted in Figure 1 all anchors receive broadcastingsignals from other anchors or tags stamp TOAs accordingto their own clocks and route the TOA information to thelocalization server where synchronization and localizationalgorithms are implemented The only work of tag is totransmit signals with certain period which is determined byworking condition and requirement measuring TOA andextra processing function are not needed in tag node For tagswith low mobility a large value of period can be chosen toprolong the working time

In Figure 1 all sensor nodes broadcast signals periodicallywith respect to their own local clocks The message ofbroadcasting signal does not include a timestamp generatedby the sender nor is it important exactly when it is sentAs a matter of fact the broadcast of tag does not even needto be a dedicated packet for localization Almost any extantbroadcast can be used to tag localization for instance ARPpackets or the broadcast control traffic in wireless networks(eg RTSCTS exchanges or route discovery packets)

Consider a radio WSN with 119873 anchors at known fixedpositions x

119894for 119894 = 1 119873 and a tag at an unknown position

x0(since tags are located using TOAs arriving at anchors

independently the proximity of tags cannot be taken intoconsideration for simplicity only one tag is considered in thepaper) The position vectors x

119894 119894 = 0 1 119873 are all 2D or

International Journal of Distributed Sensor Networks 3

Anchor j Anchor q Anchor p Anchor i

ljq(k)lqq(k)

liq(k)

ljp(k + 1)

lip(k + 1)

lpp(k + 1)

Figure 2 Anchor blinks and local TOA measurements

3DThroughout the paper the subscripts 119894 = 1 119873 refer toanchor 119894 and subscript 0 refers to the tag Each anchor has aquartz clock that operates at the same nominal frequency 119891

0

with unknown clock skew 120576119894 The frequency 119891

119894of anchor 119894 is

[6]

119891119894= 1198910(1 + 120576

119894) (1)

Considering random clock drift due to frequency fluc-tuation the frequency 119891

119894may change slightly over times in

practice In this paper the following assumptions are madefor wireless sensor network localization problem

(1) The coverage range of sensor node is within 300mwhich is a typical value for short range radio transmit-ter working in ISM (industry science and medicine)band such as IEEE 80211 bng and IEEE 802154a

(2) All sensor nodes are equipped with low cost quartzoscillators the typical clock skew is in the range of[10minus6

10minus4

](3) Since the anchors are connected towired network and

well powered the power consumption of anchor isnot concerned in this paper However the power con-sumption of tag is critical and there is no limitation onsignal transmitting period andmoving velocity of tagLocalization server where synchronization and local-ization algorithms are implemented is also assumedto have enough processing capability (if the numberof tag exceeds a threshold more localization serversrunning on desktop computers can be adopted tobalance the computation load there is no need torevise the design of anchor and tag)

3 Estimation of Relative ClockOffsets and Skews

31 TOA and TDOA for Anchor Blink Suppose anchorsbroadcast signals by turns with a period of 119879 with respect totheir own local clocksThe broadcasting packet or pulse of theanchor is called ldquoanchor blinkrdquo Two consecutive blinks andTOA measurements are plotted in Figure 2

Assuming anchor 119902 broadcasts 119896th blink the TOA atanchor 119895 is

119897119895119902 (119896) =

119895119902 (119896) + 119899119895 (119896) (2)

where 119895119902

(119896) is the true value of TOA and 119899119895(119896) is measure-

ment noise that can be modeled as additive white Gaussiannoise (AWGN) with mean of zero and variance of 1205902

119899[6 17]

In this paper we focus on the line-of-sight (LOS) propagationbetween sensor nodes Define 120593

119894119895(119896) and 119910

119894119895(119896) to be the

true and measured relative clock offset between anchor 119894 andanchor 119895 when anchor 119902 transmits blink 119896 Consider

120593119894119895 (119896) =

119894119902 (119896) minus 119895119902 (119896) minus (119889

119894119902minus 119889119895119902) (3)

119910119894119895 (119896) = 119897

119894119902 (119896) minus 119897119895119902 (119896) minus (119889

119894119902minus 119889119895119902) (4)

where 119889119895119901

= (1119888)x119895minus x119901 is propagation delay between

anchor 119895 and anchor 119901 and 119888 is the velocity of lightSubstituting (2) and (3) into (4) we have

119910119894119895 (119896) = 120593

119894119895 (119896) + 119899119894 (119896) minus 119899

119895 (119896) (5)

The true value (119896 + 1 119896) of the interval of blink 119896 canbe obtained from the TOAs of two consecutive blinks at eachanchor

(119896 + 1 119896) =

119894119901 (119896 + 1) minus 119889

119894119901minus [119894119902 (119896) minus 119889

119894119902]

1 + 120576119894

=

119895119901 (119896 + 1) minus 119889

119895119901minus [119895119902 (119896) minus 119889

119895119902]

1 + 120576119895

(6)

Since the clock skew of a typical low cost quartz oscillator isin the range of [10minus6 10minus4] the first-order approximation of(6) is

119894119901 (119896 + 1) minus

119894119902 (119896) minus [119889119894119901

minus 119889119894119902]

119895119901 (119896 + 1) minus

119895119902 (119896) minus [119889119895119901

minus 119889119895119902]

asymp 1 + 120576119894minus 120576119895 (7)

Define 119890119894119895

= 120576119894minus 120576119895to be the relative clock skew between

anchor 119894 and anchor 119895 Substituting (3) into (7) we have

120593119894119895 (119896 + 1) = 120593

119894119895 (119896) + 119890119894119895[119895119901 (119896 + 1) minus

119895119902 (119896) minus (119889119894119901

minus 119889119894119902)]

(8)

Notice that the coverage range of sensor node is within300m (119889

119894119901minus 119889119894119902) is smaller than 1 120583s and (119889

119894119901minus 119889119894119902)119890119894119895is

much smaller than 10minus10 and may be ignored If the noise

of TOA measurement is also much smaller than 1120583s (TOAmeasurement standard deviation is inversely proportionalto the effective bandwidth of signal [18] for ultrawidebandsystem (UWB) complying with IEEE 802154a standards themeasurement noise may be only several nanoseconds [12])we have

120593119894119895 (119896 + 1) asymp 120593

119894119895 (119896) + 119890119894119895[119897119895119901 (119896 + 1) minus 119897

119895119902 (119896)] (9)

Comparing (8) and (9) it can be seen that the true valueof blink interval can be replaced by local TOAmeasurementDefine119879

119895(119896+1 119896) = 119897

119895119901(119896+1)minus119897

119895119902(119896) to be the time interval of

two consecutive TOAs at anchor 119895with respect to local clockThe relative clock offset becomes

120593119894119895 (119896 + 1) = 120593

119894119895 (119896) + 119890119894119895119879119895 (119896 + 1 119896) (10)


32 Relative Clock Offset and Skew Estimation Supposeanchor 119894 transmits blink 119896 + 119894 minus 1 and we can obtain the119873 lowast (119873 minus 1) noisy measurement of relative clock offset from119873 blinks

1199101198941 (119896 + 119898) = 120593

1198941 (119896 + 119898) + 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

119894 = 2 119873 119898 = 0 119873 minus 1

(11)

From (10) we have

1205931198941 (119896 + 119898) = 120593

1198941 (119896 + 119873 minus 1) minus 11989011989411198791 (119896 + 119873 minus 1 119896 + 119898)

(12)

Since clock frequency changes slightly over times the relativeclock skews of anchors are assumed to be constant during119873 blinks if value of blink interval is appropriately selectedSubstituting (12) into (11) we can get 119873 minus 1 measurementequations for 119894 = 2 119873

1199101198941 (119896 + 119898) = 120593


+ 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

(13)

Stacking 119873 minus 1 measurement equation into matrix form

y (119896 + 119898) = h (119896 + 119898) s (119896) + C0n (119896 + 119898) (14)

where

y (119896 + 119898) = [11991021 (119896 + 119898) 1199101198731 (119896 + 119898)]

119879

s (119896) = [s11987921

(119896) s11987931

(119896) sdot sdot sdot s1198791198731

(119896)]119879

s1198941 (119896) = [120593

1198941 (119896 + 119873 minus 1) 1198901198941]119879

h (119896 + 119898) = 119868119873minus1

otimes [1 minus119879 (119896 + 119873 minus 1 119896 + 119898)]

C0=

[[[[

[

minus1 1

minus1 1

dminus1 1

]]]]

](119873minus1)times119873

n (119896 + 119898) = [1198991 (119896 + 119898) 1198992 (119896 + 119898) sdot sdot sdot 119899

119873 (119896 + 119898)]119879

119899119894 (119896 + 119898) sim 119873(0 120590

2

119899)

(15)

It should be addressed that I119873represents 119873 times 119873 matrix

of identity and the operator ldquootimesrdquo denotes Kronecker productrespectively

After each anchor blinks once we can get 119873 lowast (119873 minus 1)

measurement equations for consecutive119873 blinks combiningall these equations into matrix form we have

Y (119896) = H (119896) s (119896) + C sdot u (119896) (16)

where

Y (119896) = [y119879 (119896) y119879 (119896 + 1) y119879 (119896 + 119873 minus 1)]119879

H (119896) = [h119879 (119896) h119879 (119896 + 1) sdot sdot sdot h119879 (119896 + 119873 minus 1)]119879

u (119896) = [n119879 (119896) n119879 (119896 + 1) sdot sdot sdot n119879 (119896 + 119873 minus 1)]119879

C = I119873

otimes C0

(17)

Since the noise of TOA measurement is independent

Q119906= 1205902

119899sdot I1198732 (18)

The LS estimation of s(119896) is [19]

s (119896) = [H119879 (119896)Qminus1119888H (119896)]

minus1

H119879 (119896)Qminus1119888Y (119896) (19)

whereQ119888= C119879Q

119906C The covariance of s(119896) is

Q119904(119896)

= [H119879 (119896)Qminus1119888H (119896)]

minus1

(20)

For the Gaussian linear model the CRLB of element 119899 of s(119896)is diagonal element 119899 ofQ

119904(119896)[19]

The procedure of the relative clock offset and skewestimation is described as follows

(1) The localization server collects the TOAs of 119873 con-tinuous anchor blinks (each anchor blinks once)

(2) Evaluate the value of blink interval using119879119895(119896+1 119896) =

119897119895119901

(119896+1)minus119897119895119902

(119896) and formmeasurementmatrixH(119896)(3) Calculate measurement values of anchor relative

clock offset using (4) and combine these values intomeasurement vector Y(119896)

(4) Calculate covariance matrix of measurement noise ofrelative clock offsets usingQ

119888= C119879Q

119906C

(5) Get the LS estimation and CRLB of relative clockoffset and skew based on (19) and (20) respectively

4 Localization of Tag

41 Calibration of TDOA Measurement Since the tag trans-mits signal based on its own clock some kind ofmedia accessmethod may be adopted to avoid collision of node transmis-sions The TOAs of tag signals at anchors are depicted inFigure 3

Define120595119894119895(119896119879+120591

0) as the original TDOA of the tag signal

between anchor 119894 and anchor 119895

120595119894119895(119896119879 + 120591

0) = 1198971198940

(119896119879 + 1205910) minus 1198971198950

(119896119879 + 1205910) (21)

120595119894119895(119896119879 + 120591

0) = 120593119894119895(119896119879 + 120591

0) + 1198891198940

minus 1198891198950

+ 1198991198940

(119896119879 + 1205910) minus 1198991198950

(119896119879 + 1205910)

(22)

where 1205910is the elapsed time from the last anchor blink

From (22) we can see that the original TDOA is corruptedby relative clock offset which should be removed to get an


ljj(k minus 1)

lji(k)

lj0(kT + 1205910)120595ij(kT + 1205910)

dij

Tag

lij(k minus 1)

lii(k)

li0(kT + 1205910)

Anchor j Anchor i

Figure 3 TOAs of tag signals at anchors

accurate TDOA The estimated relative clock offset at time119896119879 + 120591

0is

1205931198941

(119896119879 + 1205910) = 1205931198941 (119896) + 120591

01198901198941 (119896) (23)

Comparing (23) and (9) the elapsed time from the last anchorblink can be replaced by local TOA measurements of anchor1 we have

1205910asymp 11989710

(119896119879 + 1205910) minus 1198971119894 (119896) (24)

The calibrated TDOA 1199111198941(119896119879 + 120591

0) is obtained by sub-

tracting the estimated relative clock offset 1205931198941(119896119879 + 120591

0) from

1205951198941(119896119879 + 120591

0)

1199111198941

(119896119879 + 1205910) = 1205951198941

(119896119879 + 1205910) minus [120593

1198941 (119896) + 12059101198901198941 (119896)] (25)

Combine the 119873 minus 1 calibrated TDOA into matrix form

z = 119891 (x0) + w (26)

where z = [11991121(119896119879 + 120591

0) 119911

1198731(119896119879 + 120591

0)]119879 and

119891 (x0) = [119889

20minus 11988910

11988930

minus 11988910

1198891198730

minus 11988910

]119879 (27)

w = Ds119890 (119896) + C

0u0(119896119879 + 120591

0) (28)

where

s119890 (119896) = s (119896) minus s (119896)

u0(119896119879 + 120591

0)

= [11989910

(119896119879 + 1205910) 11989920

(119896119879 + 1205910) 119899

1198730(119896119879 + 120591

0)]119879

u0(119896119879 + 120591

0) sim 119873 (0 120590

2

119899I119873)

D =

[[[[

[

1 1205910

0 sdot sdot sdot sdot sdot sdot 0

0 0 1 1205910

0 sdot sdot sdot

sdot sdot sdot sdot sdot sdot sdot sdot sdot 0

0 sdot sdot sdot 0 1 1205910

]]]]

](119873minus1)times(2119873minus2)

(29)

Since s119890(119896) and u

0(119896119879 + 120591

0) are independent the covari-

ance matrix of w is

Q119908

= 119864 (ww119879) = CQ119906C119879 + DQ

119904(119896)D119879 (30)

The likelihood function is

119901 (z x0) =

1

(2120587)(119873minus1)2 det (Q

119908)12

sdot exp [minus1

2(z minus 119891 (x

0))119879Qminus1119908

(z minus 119891 (x0))]

(31)

Notice that the clock skews of anchors are assumed to beconstant in (19) and (25) Considering random clock drift dueto frequency fluctuation blink interval 119879 must be selectedcarefully to guarantee the accuracy of localizationThe choiceof blink interval will be further discussed in Section 5

42 Maximum Likelihood Estimation TheMLE of x0is [19]

x0= arg min [(z minus 119891(x

0))119879Qminus1119908

(z minus 119891 (x0))] (32)

Under the Gaussian noise assumption the MLE has aleast-squares interpretation But a closed-form solution of(32) does not exist in general due to the nonlinear function119891(x0) Numerical minimization is thus needed a successive

linearization procedure [6 17 19 20] is summarized asfollows

(1) Let the estimation at the 119898th iteration be x0(119898) and

x0= x0(119898) + Δ(119898) Linearizing 119891(x

0) around x

0(119898) yields

119891 (x0) asymp 119891 (x

0 (119898)) + 119866 (x0 (119898)) Δ (119898) (33)

where 119866(x0(119898)) is the Jacobian matrix

G (x0) =

120597119891 (x0)

120597x0

(34)

Substituting (33) into (32) and solving the linearized mini-mization problem for Δ(119898) yield

Δ (119898) = [G119879(x0(119898))Qminus1

119908G(x0(119898))]minus1

sdot G119879 (x0 (119898))Qminus1

119908[119911 minus 119891 (x

0 (119898))]

(35)

where

G (x0) =

1

119888[r1198792(x0) minus r1198791(x0) r119879

119873(x0) minus r1198791(x0)]119879

(36)

r119894(x0) defines a unit-norm direction vectors

r119894(x0) =

x0minus x119894

1003817100381710038171003817x0 minus x119894

1003817100381710038171003817

119894 = 1 119873 (37)

(2) The estimation at the (119898 + 1)th iteration is

x0 (119898 + 1) = x

0 (119898) + Δ (119898) (38)

The iteration starts with an initial guess x0(0) and

terminates at convergence When a tag transmits signals


periodically previous estimate can serve as good initialguess of the current value of x

0 However the iteration

may stop at a local minimum and may not converge when[G119879(x

0(119898))Qminus1

119908G(x0(119898))]minus1 is large A two-step process

starting with a coarse grid search and continuing with aniterative procedure can be adopted to search for the globalminimum

The procedure of the tag localization algorithm isdescribed as follows

(1) The localization server collects the tag signalrsquos TOAsat anchors and calculates original 119873 minus 1 TDOA valueusing (21)

(2) Calibrate TDOAs by eliminating relative clock offseterrors using (22)

(3) Calculate covariance matrix Q119908 set 119898 = 1 and give

an initial value x0(1) of tag position randomly

(4) Calculate 119891(x0(119898)) and step vector Δ(119898) using (27)

and (35) respectively

(5) Update tag position with (38) if position change islarger than 01mm set119898 = 119898+1 and go to step (4) tocontinue iteration Otherwise the iteration stops andestimate of tag position is achieved

43 CRLB for Tag Localization Algorithm The localizationperformance of the proposed algorithm is analyzed using theCRLB The CRLB is the inverse of the Fisher informationmatrix ForGaussian noise the Fisher informationmatrix hasthe form [6 19]

119869 (x0) = (

120597119891 (x0)

120597x0

)

119879

Qminus1119908

(120597119891 (x0)

120597x0

) = 119866119879(x0)Qminus1119908

119866 (x0)

(39)

Assuming that 119869minus1(x0) exists and the 119899th diagonal element of

119869minus1

(x0) is denoted by [119869

minus1(x0)]119899119899

for 119899 = 1 2 3 the varianceof any element 119909

0119899of x0is bounded below by [119869

minus1(x0)]119899119899The

CRLB of the position estimation is

1205902

119909=

3

sum

119899=1

[119869minus1

(x0)]119899119899

(40)

Another performancemeasure commonly used in sourcelocalization systems is the geometric dilution of precision(GDOP) [21] The GDOP is the magnification in localizationerror due to the geometric relationship between the anchorsand tags Let Cov(x

0) be the covariancematrix of an unbiased

position estimate of x0 The GDOP is defined and related to

the CRLB as

GDOP =

radictr (Cov (x0))

119888120590119899

=120590119909

119888120590119899

(41)

where tr(sdot) denotes matrix trace

10minus7

10minus8

10minus9

10minus10

10minus710minus810minus910minus10

Cloc

k off

set e

stim

atio

n er

ror120590

e

TOA measurement standard deviation 120590n

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Figure 4 RCRLB and RMSE for relative clock offset estimationversus TOA measurement standard deviation (clock drift notconcerned)

5 Numerical Results

In this section numerical results for both estimations ofanchorsrsquo clock parameters and tagrsquos localization are presentedA two-dimensional localization scene is considered in thispaper four anchors are placed evenly at coordinates of (0 0)(100m 0) (0 100m) and (100m 100m) respectively Theanchors stamp TOAs of signal emitted by other anchors ortags and deliver the TOA information to localization serverwhere estimation of clock parameters and tag location areimplemented

Firstly Monte Carlo simulations were carried out toevaluate the statistical performance of the proposed algo-rithm without taking the clock drift into consideration Thesimulation result unfolds some theoretical features of theproposed algorithm Secondly simulations in the presenceof clock drift are carried out and the appropriate value ofblink interval is analyzed At last the GDOP performancesfor different layouts of anchors are evaluated

51 Performance without Clock Drift

511 Results of Relative Clock Offset and Skew EstimationWithout considering clock drift the performance of synchro-nization was evaluated as a function of TOA measurementThe actual root mean square error (RMSE) and root CRLB(RCRLB) of relative clock offset and skew estimates withdifferent values of blink interval119879 are plotted in Figures 4 and5 The RMSE results were averaged over 50000 independentnoise measurements

From Figure 4 it can be seen that the value of blinkinterval has no effect on the performance of relative clock


10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e

Figure 5 RCRLB and RMSE for clock skews estimation versus TOAmeasurement standard deviation (clock drift not concerned)

offset estimation We can see that the RMSE and CRLB ofrelative clock skew estimation are inversely proportional tothe value of blink interval in Figure 5 It is inferred in (20)that estimate of relative clock skew is a weighted average ofTOAs over time span (119873minus1)119879 so larger value of blink intervalmeans less error of relative clock skew estimate Althoughwe can get more accurate estimate with larger value of blinkinterval theoretically it is not practical because it is implied in(20) that clock skews of anchors are assumed to be constantand clock drifts are not taken into account

The performance of tag localization is given in Figure 6It seems that blink interval has no effect on the performanceof tag localization It has been concluded that the RMSEand RCRLB of relative clock skew estimation are inverselyproportional to the value of blink interval it can be seenin (23) that 120591

01198901198941(119896) is proportional to the value of elapsed

time from the last blink Since the average value of 1205910is 1198792

the standard deviation of residual clock offset in calibratedTDOA 119911

1198941(119896119879 + 120591

0) remains constant whatever the value

of blink interval is Notice that the performance curves inFigure 5 are achieved under the assumptions that clock skewsof anchors are constant and clock drifts are not taken intoaccount It is unreasonable to calibrate the original TDOAwith the timing estimates of an hour agoThe clock driftmustbe taken into consideration for practical implementation

52 Performance with Clock Drift521 Results of Clock Difference Estimation Clock drift isinevitable for all kinds of clock source it must be takeninto account for real implementation In this paper clock

101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Figure 6 RCRLB and RMSE for tag position estimation versusTOA measurement for different blink intervals (clock drift notconcerned)

frequency fluctuation of quartz oscillator is modeled asAWGN [22]

119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)

where 119899119891is AWGN with variance of 1205902

119891 which is unknown

andmay depend on the characteristics of the quartz oscillatorthe ambient temperature the working voltage and so on

The performance of synchronization in the presenceof clock drift was also evaluated as a function of TOAmeasurement Since blink interval 119879 greatly influences theaccuracy of localization the actual root mean square error(RMSE) and root CRLB (RCRLB) of relative clock offset andskew estimates with different values of blink interval 119879 areplotted in Figures 7 and 8 The RMSE results were averagedover 50000 independent noise measurements

There is an obvious gap between the RMSE and corre-sponding RCRLB in Figures 7 and 8 because clock skew istreated as constant in the computation procedure of RCRLBHowever frequency fluctuations are inevitable in practicethe impact of frequency fluctuations is more significant whenTOA measurement standard deviation 120590

119899is close to 120590

119891in

(42) and 120590119891forms low bound of clock skew estimate The

RCRLB curves of relative clock offset in Figure 7 for differentvalues of 119879 converge when noise level increases because 120590

119891

is much smaller than 120590119899

522 Tag Localization Result The RMSE and RCRLB fortag localization with different blink interval 119879 are plotted




10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e

Figure 7 RCRLB and RMSE for relative clock offset estimationversus TOA measurement standard deviation 120590

119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e

Figure 8 RCRLB andRMSE for clock skews estimation versus TOAmeasurement standard deviation 120590

119891= 10minus9

in Figure 9 For simplicity 1205910is set to be 1198792 which is the

average elapsed time from the last anchor blink There is alsoan obvious gap between RMSE and corresponding RCRLBbecause relative clock skew is treated as constant in (19) TheRCRLBs for different values of 119879 converge when noise levelincreases because residual clock drift in calibrated TDOA ismuch smaller than noise level of TOA measurement

RCRLB in [4]RMSE in [7]RCRLB in [7]

101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)



RCRLB T = 01RMSE T = 01RCRLB T = 02RMSE T = 02RCRLB T = 04RMSE T = 04RCRLB T = 08RMSE T = 08

RCRLB T = 16RMSE T = 16

Figure 9 RCRLB andRMSE for tag position estimation versus TOAmeasurement for different anchor blink intervals The simulationresults for algorithms in [4 7] are also given 120590

119891= 10minus9

The performance of the synchronous TDOA algorithmin [7] is presented for comparison in Figure 9 The dottedred lines with markers ldquo998795rdquo and ldquo+rdquo represent RCRLB andRMSE of [7] respectively We can conclude that the RMSEand RCRLB values of our algorithm decrease and approachthose of the synchronous TDOA algorithm if a smaller valueof 119879 is selectedThis is due to the fact that the impact of clockdrift on calibrated TDOA becomes less significant as the timeinterval of anchor blinks is reduced Although more accurateestimation of the positions of tags can be achieved with asmaller blink interval 119879 the number of tags to be served willdecrease due to the large number of anchor blink signals Wecan conclude that a value in the range [01 04] is adequatefor practical implementation

Furthermore the algorithm in this paper is comparedwith the algorithms in [4] where a joint synchronization andlocalization algorithm for asynchronousWSNwas proposedThe RCRLB curve of this algorithm is plotted with bluemarker ldquolozrdquo in Figure 9 The joint algorithm in [4] performsbetter than our algorithm due to two-way message exchangescheme and the averaging of noisy TOAs of multiple signalsfrom the same tag We can also see that the differencebetween pairs of algorithms in localization precision is within3 dB which has little effect in practice for typical standarddeviation in the range of [10

minus9 10minus8

] [12] Since the tagperforms two-way message exchange with each anchor TOAmeasurement capability and complex protocol are neededfor tags and energy dissipation of tag may increase with the


2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)

Figure 10 Constant contour of GDOPs as a function of the taglocation for both the asynchronous algorithm in this paper 119903 (solidline) and synchronous algorithm in [7] (red dotted line)

number of anchors However the algorithm in this paper issimple and effective only periodical broadcasting of anchoris required and the tag can be located via single transmissionwhichmeans least power consumption is required for mobiletag node

Although the algorithm proposed in this paper cannotreach the performance of algorithms in [4 7] there areless assumptions required in this paper Clock skews in[4 7] are treated as constant Furthermore it is assumedthat all anchors are synchronized and their positions areknown in [7] it is also assumed that there are 119871 (119871 ge

3) anchors with known timing and locations in [4] Thelocations of anchors can be achieved easily in practice but itis difficult and impractical to get accurate timing parametersof anchors equipped with low cost quartz oscillators becausequartz clock skew is influenced by working voltage ambienttemperature and initial working condition On the contrarythe algorithm in this paper deals with this problem andestimates the relative clock offsets and skews of anchor pairsThere is no need for external sophisticated and expensiveclock sources such as GPS receivers and atom clocks andsuitable for low cost implementation Compared with thesynchronized TDOAalgorithmof [4 7] when an appropriateblink interval is selected performance loss is within 3 dBwhich is acceptable for many applications

53 Performance of GDOP The GDOP of the proposedasynchronous localization method is plotted in Figure 10 as afunction of the tag position and it is comparedwith that of thesynchronous algorithm in [7] As expected the asynchronousmethod had a worse geometric condition but we notice thatthe tag can be located with GDOP performance (less than2) close to synchronous algorithm when the tag is in therectangle formed by four anchors

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)

Figure 11 Constant contour of GDOPs (anchors locate on (00) (100 0) (50 866) and (50 289)) for both the asynchronousalgorithm in this paper (solid line) and synchronous algorithm in[7] (red dotted line)

The GDOP with different layout of anchors is plotted inFigure 11 Three anchors locate on the vertex of an equilateraltriangle whose edge is 100m the fourth anchor locates on thecenter of gravity The GDOP of the synchronous algorithmin [7] is plotted for comparison We can also conclude thatGDOP performance (less than 2) is close to synchronousalgorithmwhen the tag is in the region formed by the outsidethree anchors

The influence of the number of anchors is also studiedin the simulation For 2D localization at least three anchorsare needed for tag localization and four anchors are neededfor 3D localization The GDOP of triangular layout of threeanchors is plotted in Figure 12 three anchors locate on thevertex of an equilateral triangle whose edge is 100m TheGDOP of four anchors (the fourth anchor locates on thecenter of gravity) is also plotted for comparison It can beseen that the GDOP performance of the three anchors isworse than that of the four anchors Statistically more TOAmeasurements are helpful to reduce the locating error oftag Furthermore redundant TOAmeasurements can also beused to cope with the problem of non-LOS by selecting theTOAmeasurements with stronger receiving signal or shorterpropagation delay

6 Conclusion

In this paper a novel TDOA tag localization algorithm forWSNs is presented To synchronize clocks of anchors eachanchor broadcasts blink signals periodically relative clockoffsets and skews of anchor pairs are estimated by the LSmethod using the TOAs of broadcast signals at anchorsWhen a tag transmits a signal the TDOA error due to the


1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)

Figure 12 Comparison ofGDOPwith three anchors (blue solid lineanchor locates on (0 0) (100 0) and (50 866)) and GDOP withfour anchors (red dotted line anchor locates on (0 0) (100 0) (50866) and (50 289))

relative clock offset of the anchor pair can be eliminated usinga compensation operation Moreover a linearized MLE isadopted to estimate the position of the tag Compared withprevious methods the algorithm proposed in this paper issimple energy-efficient and particularly suitable for low costand fully asynchronous WSNs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I F Akyildiz W Su Y Sankarasubramaniam and E Cayirci ldquoAsurvey on sensor networksrdquo IEEE Communications Magazinevol 40 no 8 pp 102ndash105 2002

[2] H Liu H Darabi P Banerjee and J Liu ldquoSurvey of wirelessindoor positioning techniques and systemsrdquo IEEE Transactionson Systems Man and Cybernetics Part C Applications andReviews vol 37 no 6 pp 1067ndash1080 2007

[3] B Sundararaman U Buy and A D Kshemkalyani ldquoClocksynchronization for wireless sensor networks a surveyrdquoAdHocNetworks vol 3 no 3 pp 281ndash323 2005

[4] J Zheng and Y-C Wu ldquoJoint time synchronization and local-ization of an unknown node in wireless sensor networksrdquo IEEETransactions on Signal Processing vol 58 no 3 pp 1309ndash13202010

[5] Y Wang X Ma and G Leus ldquoRobust time-based localizationfor asynchronous networksrdquo IEEE Transactions on Signal Pro-cessing vol 59 no 9 pp 4397ndash4410 2011

[6] T Li A Ekpenyong and Y-F Huang ldquoSource localizationand tracking using distributed asynchronous sensorsrdquo IEEE

Transactions on Signal Processing vol 54 no 10 pp 3991ndash40032006

[7] R M Vaghefi and R M Buehrer ldquoAsynchronous time-of-arrival-based source localizationrdquo in Proceedings of the 38thIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo13) pp 4086ndash4090 May 2013

[8] K W Cheung H C So W-K Ma and Y T Chan ldquoLeastsquares algorithms for time-of-arrival-based mobile locationrdquoIEEE Transactions on Signal Processing vol 52 no 4 pp 1121ndash1128 2004

[9] D Niculescu and B Nath ldquoAd hoc positioning system (APS)using AOArdquo in Proceedings of the 22nd Annual Joint Conferenceon the IEEE Computer and Communications Societies pp 1734ndash1743 San Francisco Calif USA April 2003

[10] P Bahl and V Padmanabhan ldquoRADAR an in-building RF-based user location and tracking systemrdquo in Proceedings ofthe 19th Annual Joint Conference of the IEEE Computer andCommunications Societies (IEEE INFOCOM rsquo00) vol 2 pp775ndash784 Tel Aviv Israel March 2000

[11] S Gezici Z Tian G B Giannakis et al ldquoLocalization via ultra-wideband radios a look at positioning aspects for future sensornetworksrdquo IEEE Signal Processing Magazine vol 22 no 4 pp70ndash84 2005

[12] IEEE Computer Society IEEE Standard 802154a 2007[13] J Elson L Girod and D Estrin ldquoFine-grained network time

synchronization using reference broadcastsrdquo ACM SIGOPSOperating Systems Review vol 36 no SI pp 147ndash163 2002

[14] S Ganeriwal R Kumar and M B Srivastava ldquoTiming-syncprotocol for sensor networksrdquo in Proceedings of the ACM1st International Conference on Embedded Networked SensorSystems (SenSys rsquo03) pp 138ndash149 Los Angeles Calif USANovember 2003

[15] M Maroti B Kusy G Simon and A Ledeczi ldquoThe floodingtime synchronization protocolrdquo in Proceedings of the 2nd Inter-national Conference on Embedded Networked Sensor Systems(SenSys rsquo04) pp 39ndash49 November 2004

[16] S P Chepuri R T Rajan G Leus and A-J van der VeenldquoJoint clock synchronization and ranging asymmetrical time-stamping and passive listeningrdquo IEEE Signal Processing Lettersvol 20 no 1 pp 51ndash54 2013

[17] Y T Chan and K C Ho ldquoSimple and efficient estimator forhyperbolic locationrdquo IEEE Transactions on Signal Processingvol 42 no 8 pp 1905ndash1915 1994

[18] H V Poor An Introduction to Signal Detection and EstimationSpringer New York NY USA 2nd edition 1994

[19] SMKay Fundamentals of Statistical Signal ProcessingVolume IEstimationTheory Pearson Education Prentice Hall PTR 1993

[20] J O Smith and J S Abel ldquoClosed-form least-squares sourcelocation estimation from range-difference measurementsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 35 no 12 pp 1661ndash1669 1987

[21] M A Spirito ldquoOn the accuracy of cellular mobile station loca-tion estimationrdquo IEEE Transactions on Vehicular Technologyvol 50 no 3 pp 674ndash685 2001

[22] EM Oliveira Jr M L O Souza H K Kuga and R V F LopesldquoClock synchronization via Kalman filterrdquo in Proceedings of the8th BrazilianConference onDynamics Control andApplicationsMay 2009

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



source nodes are proposed in [4 5] for network with asyn-chronous anchors The broadcasting property is exploitedto jointly estimate clock parameters and source positionusing asymmetrical time-stamping and a passive listeningprotocol in [5 16] Although these algorithms achieve goodperformance complex protocols such as two-way rangingprotocol are necessary Moreover frequent transmissions ofsource node may cause more energy dissipation and limit thenumber of tags in the service area of WSN

In this paper we propose a robust TDOA localizationalgorithm for asynchronous wireless sensor networks InWSN sensor nodes are divided into two categories anchorand tag Tags are mobile sensor nodes anchor nodes areused as reference points to localize tag nodes and assumedto have known positions and good power supplies Thereare many limitations for tag nodes such as size costand energy dissipation which is concerned intensively forpractical implementation In order to synchronize anchorclocks all anchors broadcast signal periodically TOAs ofbroadcasting signal at other anchors are stamped accordingto their own local clocks respectively A localization serveron the backbone network collects all these timestamps toestimate the relative clock offsets and skews of anchor pairsWhen anchors receive a broadcasting signal from a tagthe TOAs of tag signal at anchors are also stamped anddelivered to the localization server The error in originalTDOA of tag signal due to relative clock offsets and skewsamong anchors can be easily canceled by a compensationoperation Based on a Gaussian measurement noise modelmaximum likelihood estimation (MLE) for tag position isobtained Performance issues are addressed by evaluating theCramer-Rao lower bound (CRLB) of synchronization andlocalization algorithm Only anchor-anchor synchronizationis required in this paper and tag node is localized via a signaltransmission so power consumption of tag can beminimizedand the number of tags to be served can be maximized

This paper is organized as follows The sensor networkmodel for localization problem considered in the paper isgiven in Section 2 Estimation of relative clock offsets andskews of anchor pairs and performance analysis are presentedin Section 3 The localization of tags is studied in Section 4and the statistical performance analysis of the localizationalgorithm is also presented in this section Section 5 presentssome numerical results and Section 6 concludes the paper

2 Sensor Network Model for Localization

Wireless sensor network localization is the process by whichsensor nodes determine their location In simple terms local-ization is a mechanism for discovering spatial relationshipsbetween objectsThe various approaches taken in literature tosolve this localization problem differ in the assumptions theymake about their respective network and sensor capabilitiesA detailed but not exhaustive list of assumptions madeinclude assumptions about network infrastructure devicehardware signal propagation models timing and energyrequirements operational environment time synchroniza-tion communication costs error requirements and nodemobility

MT or tag

Anchor Anchor

AnchorAnchor

Localization server

TOA l20(t)

TOA l10(t) TOA li0(t)

TOA l30(t)

Figure 1 Radio WSN model for tag localization (anchor nodesare represented by red rectangles and tags are represented by greencircles)

In this paper sensor nodes are divided into two cate-gories anchor node and tag node Anchor nodes are usedas reference points to locate tag nodes they are alwaysconnected to network infrastructure and well powered Toreduce network complexity and costs the quantity of anchoris limited and positions of anchor nodes are planned inadvance to guarantee the coverage of service area The tagsare sensor nodes to be located and must be designed forlow cost and low power consumption and operation for along lifetime To further reduce the complexity and powerconsumption of tag the processing capability is restricted forpractical implementation For example a WSN based patientmanagement consists of limited anchors which serve for largequantity of tags carried by patients

The radio sensor network model considered in the paperis depicted in Figure 1 all anchors receive broadcastingsignals from other anchors or tags stamp TOAs accordingto their own clocks and route the TOA information to thelocalization server where synchronization and localizationalgorithms are implemented The only work of tag is totransmit signals with certain period which is determined byworking condition and requirement measuring TOA andextra processing function are not needed in tag node For tagswith low mobility a large value of period can be chosen toprolong the working time

In Figure 1 all sensor nodes broadcast signals periodicallywith respect to their own local clocks The message ofbroadcasting signal does not include a timestamp generatedby the sender nor is it important exactly when it is sentAs a matter of fact the broadcast of tag does not even needto be a dedicated packet for localization Almost any extantbroadcast can be used to tag localization for instance ARPpackets or the broadcast control traffic in wireless networks(eg RTSCTS exchanges or route discovery packets)

Consider a radio WSN with 119873 anchors at known fixedpositions x

119894for 119894 = 1 119873 and a tag at an unknown position

x0(since tags are located using TOAs arriving at anchors

independently the proximity of tags cannot be taken intoconsideration for simplicity only one tag is considered in thepaper) The position vectors x

119894 119894 = 0 1 119873 are all 2D or



ljq(k)lqq(k)

liq(k)

ljp(k + 1)

lip(k + 1)

lpp(k + 1)



0



[6]

119891119894= 1198910(1 + 120576

119894) (1)






10minus4






119897119895119902 (119896) =

119895119902 (119896) + 119899119895 (119896) (2)

where 119895119902



119899[6 17]


119894119895(119896) and 119910

119894119895(119896) to be the


120593119894119895 (119896) =

119894119902 (119896) minus 119895119902 (119896) minus (119889

119894119902minus 119889119895119902) (3)

119910119894119895 (119896) = 119897

119894119902 (119896) minus 119897119895119902 (119896) minus (119889

119894119902minus 119889119895119902) (4)

where 119889119895119901



119910119894119895 (119896) = 120593

119894119895 (119896) + 119899119894 (119896) minus 119899

119895 (119896) (5)


(119896 + 1 119896) =

119894119901 (119896 + 1) minus 119889

119894119901minus [119894119902 (119896) minus 119889

119894119902]

1 + 120576119894

=

119895119901 (119896 + 1) minus 119889

119895119901minus [119895119902 (119896) minus 119889

119895119902]

1 + 120576119895

(6)


119894119901 (119896 + 1) minus

119894119902 (119896) minus [119889119894119901

minus 119889119894119902]

119895119901 (119896 + 1) minus

119895119902 (119896) minus [119889119895119901

minus 119889119895119902]

asymp 1 + 120576119894minus 120576119895 (7)

Define 119890119894119895



120593119894119895 (119896 + 1) = 120593

119894119895 (119896) + 119890119894119895[119895119901 (119896 + 1) minus

119895119902 (119896) minus (119889119894119901

minus 119889119894119902)]

(8)



119894119901minus 119889119894119902)119890119894119895is



120593119894119895 (119896 + 1) asymp 120593

119894119895 (119896) + 119890119894119895[119897119895119901 (119896 + 1) minus 119897

119895119902 (119896)] (9)


119895(119896+1 119896) = 119897

119895119901(119896+1)minus119897



120593119894119895 (119896 + 1) = 120593

119894119895 (119896) + 119890119894119895119879119895 (119896 + 1 119896) (10)



1199101198941 (119896 + 119898) = 120593

1198941 (119896 + 119898) + 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

119894 = 2 119873 119898 = 0 119873 minus 1

(11)

From (10) we have

1205931198941 (119896 + 119898) = 120593


(12)


1199101198941 (119896 + 119898) = 120593


+ 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

(13)


y (119896 + 119898) = h (119896 + 119898) s (119896) + C0n (119896 + 119898) (14)

where

y (119896 + 119898) = [11991021 (119896 + 119898) 1199101198731 (119896 + 119898)]

119879

s (119896) = [s11987921

(119896) s11987931

(119896) sdot sdot sdot s1198791198731

(119896)]119879

s1198941 (119896) = [120593

1198941 (119896 + 119873 minus 1) 1198901198941]119879

h (119896 + 119898) = 119868119873minus1


C0=

[[[[

[

minus1 1

minus1 1

dminus1 1

]]]]

](119873minus1)times119873

n (119896 + 119898) = [1198991 (119896 + 119898) 1198992 (119896 + 119898) sdot sdot sdot 119899

119873 (119896 + 119898)]119879

119899119894 (119896 + 119898) sim 119873(0 120590

2

119899)

(15)





Y (119896) = H (119896) s (119896) + C sdot u (119896) (16)

where

Y (119896) = [y119879 (119896) y119879 (119896 + 1) y119879 (119896 + 119873 minus 1)]119879



C = I119873

otimes C0

(17)


Q119906= 1205902

119899sdot I1198732 (18)


s (119896) = [H119879 (119896)Qminus1119888H (119896)]

minus1

H119879 (119896)Qminus1119888Y (119896) (19)

whereQ119888= C119879Q


Q119904(119896)

= [H119879 (119896)Qminus1119888H (119896)]

minus1

(20)


119904(119896)[19]




119897119895119901

(119896+1)minus119897119895119902




119888= C119879Q

119906C




Define120595119894119895(119896119879+120591



120595119894119895(119896119879 + 120591

0) = 1198971198940

(119896119879 + 1205910) minus 1198971198950

(119896119879 + 1205910) (21)

120595119894119895(119896119879 + 120591

0) = 120593119894119895(119896119879 + 120591

0) + 1198891198940

minus 1198891198950

+ 1198991198940

(119896119879 + 1205910) minus 1198991198950

(119896119879 + 1205910)

(22)




ljj(k minus 1)

lji(k)

lj0(kT + 1205910)120595ij(kT + 1205910)

dij

Tag

lij(k minus 1)

lii(k)

li0(kT + 1205910)

Anchor j Anchor i



0is

1205931198941

(119896119879 + 1205910) = 1205931198941 (119896) + 120591

01198901198941 (119896) (23)


1205910asymp 11989710

(119896119879 + 1205910) minus 1198971119894 (119896) (24)




0) from

1205951198941(119896119879 + 120591

0)

1199111198941

(119896119879 + 1205910) = 1205951198941

(119896119879 + 1205910) minus [120593

1198941 (119896) + 12059101198901198941 (119896)] (25)


z = 119891 (x0) + w (26)

where z = [11991121(119896119879 + 120591

0) 119911

1198731(119896119879 + 120591

0)]119879 and

119891 (x0) = [119889

20minus 11988910

11988930

minus 11988910

1198891198730

minus 11988910

]119879 (27)

w = Ds119890 (119896) + C

0u0(119896119879 + 120591

0) (28)

where

s119890 (119896) = s (119896) minus s (119896)

u0(119896119879 + 120591

0)

= [11989910

(119896119879 + 1205910) 11989920

(119896119879 + 1205910) 119899

1198730(119896119879 + 120591

0)]119879

u0(119896119879 + 120591

0) sim 119873 (0 120590

2

119899I119873)

D =

[[[[

[

1 1205910


0 0 1 1205910

0 sdot sdot sdot



]]]]


(29)

Since s119890(119896) and u

0(119896119879 + 120591


ance matrix of w is

Q119908

= 119864 (ww119879) = CQ119906C119879 + DQ

119904(119896)D119879 (30)


119901 (z x0) =

1

(2120587)(119873minus1)2 det (Q

119908)12

sdot exp [minus1

2(z minus 119891 (x

0))119879Qminus1119908

(z minus 119891 (x0))]

(31)




0))119879Qminus1119908

(z minus 119891 (x0))] (32)





0) around x

0(119898) yields

119891 (x0) asymp 119891 (x

0 (119898)) + 119866 (x0 (119898)) Δ (119898) (33)


G (x0) =

120597119891 (x0)

120597x0

(34)


Δ (119898) = [G119879(x0(119898))Qminus1

119908G(x0(119898))]minus1

sdot G119879 (x0 (119898))Qminus1

119908[119911 minus 119891 (x

0 (119898))]

(35)

where

G (x0) =

1

119888[r1198792(x0) minus r1198791(x0) r119879

119873(x0) minus r1198791(x0)]119879

(36)


r119894(x0) =

x0minus x119894

1003817100381710038171003817x0 minus x119894

1003817100381710038171003817

119894 = 1 119873 (37)


x0 (119898 + 1) = x

0 (119898) + Δ (119898) (38)







0(119898))Qminus1












119869 (x0) = (

120597119891 (x0)

120597x0

)

119879

Qminus1119908

(120597119891 (x0)

120597x0

) = 119866119879(x0)Qminus1119908

119866 (x0)

(39)


119869minus1


minus1(x0)]119899119899



minus1(x0)]119899119899The


1205902

119909=

3

sum

119899=1

[119869minus1

(x0)]119899119899

(40)




the CRLB as

GDOP =

radictr (Cov (x0))

119888120590119899

=120590119909

119888120590119899

(41)


10minus7

10minus8

10minus9

10minus10


Cloc

k off

set e

stim

atio

n er

ror120590

e


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16


5 Numerical Results







10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e







1198941(119896119879 + 120591




101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16



119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)






119899is close to 120590

119891in



119891






10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ljq(k)lqq(k)

liq(k)

ljp(k + 1)

lip(k + 1)

lpp(k + 1)



0



[6]

119891119894= 1198910(1 + 120576

119894) (1)






10minus4






119897119895119902 (119896) =

119895119902 (119896) + 119899119895 (119896) (2)

where 119895119902



119899[6 17]


119894119895(119896) and 119910

119894119895(119896) to be the


120593119894119895 (119896) =

119894119902 (119896) minus 119895119902 (119896) minus (119889

119894119902minus 119889119895119902) (3)

119910119894119895 (119896) = 119897

119894119902 (119896) minus 119897119895119902 (119896) minus (119889

119894119902minus 119889119895119902) (4)

where 119889119895119901



119910119894119895 (119896) = 120593

119894119895 (119896) + 119899119894 (119896) minus 119899

119895 (119896) (5)


(119896 + 1 119896) =

119894119901 (119896 + 1) minus 119889

119894119901minus [119894119902 (119896) minus 119889

119894119902]

1 + 120576119894

=

119895119901 (119896 + 1) minus 119889

119895119901minus [119895119902 (119896) minus 119889

119895119902]

1 + 120576119895

(6)


119894119901 (119896 + 1) minus

119894119902 (119896) minus [119889119894119901

minus 119889119894119902]

119895119901 (119896 + 1) minus

119895119902 (119896) minus [119889119895119901

minus 119889119895119902]

asymp 1 + 120576119894minus 120576119895 (7)

Define 119890119894119895



120593119894119895 (119896 + 1) = 120593

119894119895 (119896) + 119890119894119895[119895119901 (119896 + 1) minus

119895119902 (119896) minus (119889119894119901

minus 119889119894119902)]

(8)



119894119901minus 119889119894119902)119890119894119895is



120593119894119895 (119896 + 1) asymp 120593

119894119895 (119896) + 119890119894119895[119897119895119901 (119896 + 1) minus 119897

119895119902 (119896)] (9)


119895(119896+1 119896) = 119897

119895119901(119896+1)minus119897



120593119894119895 (119896 + 1) = 120593

119894119895 (119896) + 119890119894119895119879119895 (119896 + 1 119896) (10)



1199101198941 (119896 + 119898) = 120593

1198941 (119896 + 119898) + 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

119894 = 2 119873 119898 = 0 119873 minus 1

(11)

From (10) we have

1205931198941 (119896 + 119898) = 120593


(12)


1199101198941 (119896 + 119898) = 120593


+ 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

(13)


y (119896 + 119898) = h (119896 + 119898) s (119896) + C0n (119896 + 119898) (14)

where

y (119896 + 119898) = [11991021 (119896 + 119898) 1199101198731 (119896 + 119898)]

119879

s (119896) = [s11987921

(119896) s11987931

(119896) sdot sdot sdot s1198791198731

(119896)]119879

s1198941 (119896) = [120593

1198941 (119896 + 119873 minus 1) 1198901198941]119879

h (119896 + 119898) = 119868119873minus1


C0=

[[[[

[

minus1 1

minus1 1

dminus1 1

]]]]

](119873minus1)times119873

n (119896 + 119898) = [1198991 (119896 + 119898) 1198992 (119896 + 119898) sdot sdot sdot 119899

119873 (119896 + 119898)]119879

119899119894 (119896 + 119898) sim 119873(0 120590

2

119899)

(15)





Y (119896) = H (119896) s (119896) + C sdot u (119896) (16)

where

Y (119896) = [y119879 (119896) y119879 (119896 + 1) y119879 (119896 + 119873 minus 1)]119879



C = I119873

otimes C0

(17)


Q119906= 1205902

119899sdot I1198732 (18)


s (119896) = [H119879 (119896)Qminus1119888H (119896)]

minus1

H119879 (119896)Qminus1119888Y (119896) (19)

whereQ119888= C119879Q


Q119904(119896)

= [H119879 (119896)Qminus1119888H (119896)]

minus1

(20)


119904(119896)[19]




119897119895119901

(119896+1)minus119897119895119902




119888= C119879Q

119906C




Define120595119894119895(119896119879+120591



120595119894119895(119896119879 + 120591

0) = 1198971198940

(119896119879 + 1205910) minus 1198971198950

(119896119879 + 1205910) (21)

120595119894119895(119896119879 + 120591

0) = 120593119894119895(119896119879 + 120591

0) + 1198891198940

minus 1198891198950

+ 1198991198940

(119896119879 + 1205910) minus 1198991198950

(119896119879 + 1205910)

(22)




ljj(k minus 1)

lji(k)

lj0(kT + 1205910)120595ij(kT + 1205910)

dij

Tag

lij(k minus 1)

lii(k)

li0(kT + 1205910)

Anchor j Anchor i



0is

1205931198941

(119896119879 + 1205910) = 1205931198941 (119896) + 120591

01198901198941 (119896) (23)


1205910asymp 11989710

(119896119879 + 1205910) minus 1198971119894 (119896) (24)




0) from

1205951198941(119896119879 + 120591

0)

1199111198941

(119896119879 + 1205910) = 1205951198941

(119896119879 + 1205910) minus [120593

1198941 (119896) + 12059101198901198941 (119896)] (25)


z = 119891 (x0) + w (26)

where z = [11991121(119896119879 + 120591

0) 119911

1198731(119896119879 + 120591

0)]119879 and

119891 (x0) = [119889

20minus 11988910

11988930

minus 11988910

1198891198730

minus 11988910

]119879 (27)

w = Ds119890 (119896) + C

0u0(119896119879 + 120591

0) (28)

where

s119890 (119896) = s (119896) minus s (119896)

u0(119896119879 + 120591

0)

= [11989910

(119896119879 + 1205910) 11989920

(119896119879 + 1205910) 119899

1198730(119896119879 + 120591

0)]119879

u0(119896119879 + 120591

0) sim 119873 (0 120590

2

119899I119873)

D =

[[[[

[

1 1205910


0 0 1 1205910

0 sdot sdot sdot



]]]]


(29)

Since s119890(119896) and u

0(119896119879 + 120591


ance matrix of w is

Q119908

= 119864 (ww119879) = CQ119906C119879 + DQ

119904(119896)D119879 (30)


119901 (z x0) =

1

(2120587)(119873minus1)2 det (Q

119908)12

sdot exp [minus1

2(z minus 119891 (x

0))119879Qminus1119908

(z minus 119891 (x0))]

(31)




0))119879Qminus1119908

(z minus 119891 (x0))] (32)





0) around x

0(119898) yields

119891 (x0) asymp 119891 (x

0 (119898)) + 119866 (x0 (119898)) Δ (119898) (33)


G (x0) =

120597119891 (x0)

120597x0

(34)


Δ (119898) = [G119879(x0(119898))Qminus1

119908G(x0(119898))]minus1

sdot G119879 (x0 (119898))Qminus1

119908[119911 minus 119891 (x

0 (119898))]

(35)

where

G (x0) =

1

119888[r1198792(x0) minus r1198791(x0) r119879

119873(x0) minus r1198791(x0)]119879

(36)


r119894(x0) =

x0minus x119894

1003817100381710038171003817x0 minus x119894

1003817100381710038171003817

119894 = 1 119873 (37)


x0 (119898 + 1) = x

0 (119898) + Δ (119898) (38)







0(119898))Qminus1












119869 (x0) = (

120597119891 (x0)

120597x0

)

119879

Qminus1119908

(120597119891 (x0)

120597x0

) = 119866119879(x0)Qminus1119908

119866 (x0)

(39)


119869minus1


minus1(x0)]119899119899



minus1(x0)]119899119899The


1205902

119909=

3

sum

119899=1

[119869minus1

(x0)]119899119899

(40)




the CRLB as

GDOP =

radictr (Cov (x0))

119888120590119899

=120590119909

119888120590119899

(41)


10minus7

10minus8

10minus9

10minus10


Cloc

k off

set e

stim

atio

n er

ror120590

e


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16


5 Numerical Results







10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e







1198941(119896119879 + 120591




101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16



119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)






119899is close to 120590

119891in



119891






10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1199101198941 (119896 + 119898) = 120593

1198941 (119896 + 119898) + 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

119894 = 2 119873 119898 = 0 119873 minus 1

(11)

From (10) we have

1205931198941 (119896 + 119898) = 120593


(12)


1199101198941 (119896 + 119898) = 120593


+ 119899119894 (119896 + 119898) minus 119899

1 (119896 + 119898)

(13)


y (119896 + 119898) = h (119896 + 119898) s (119896) + C0n (119896 + 119898) (14)

where

y (119896 + 119898) = [11991021 (119896 + 119898) 1199101198731 (119896 + 119898)]

119879

s (119896) = [s11987921

(119896) s11987931

(119896) sdot sdot sdot s1198791198731

(119896)]119879

s1198941 (119896) = [120593

1198941 (119896 + 119873 minus 1) 1198901198941]119879

h (119896 + 119898) = 119868119873minus1


C0=

[[[[

[

minus1 1

minus1 1

dminus1 1

]]]]

](119873minus1)times119873

n (119896 + 119898) = [1198991 (119896 + 119898) 1198992 (119896 + 119898) sdot sdot sdot 119899

119873 (119896 + 119898)]119879

119899119894 (119896 + 119898) sim 119873(0 120590

2

119899)

(15)





Y (119896) = H (119896) s (119896) + C sdot u (119896) (16)

where

Y (119896) = [y119879 (119896) y119879 (119896 + 1) y119879 (119896 + 119873 minus 1)]119879



C = I119873

otimes C0

(17)


Q119906= 1205902

119899sdot I1198732 (18)


s (119896) = [H119879 (119896)Qminus1119888H (119896)]

minus1

H119879 (119896)Qminus1119888Y (119896) (19)

whereQ119888= C119879Q


Q119904(119896)

= [H119879 (119896)Qminus1119888H (119896)]

minus1

(20)


119904(119896)[19]




119897119895119901

(119896+1)minus119897119895119902




119888= C119879Q

119906C




Define120595119894119895(119896119879+120591



120595119894119895(119896119879 + 120591

0) = 1198971198940

(119896119879 + 1205910) minus 1198971198950

(119896119879 + 1205910) (21)

120595119894119895(119896119879 + 120591

0) = 120593119894119895(119896119879 + 120591

0) + 1198891198940

minus 1198891198950

+ 1198991198940

(119896119879 + 1205910) minus 1198991198950

(119896119879 + 1205910)

(22)




ljj(k minus 1)

lji(k)

lj0(kT + 1205910)120595ij(kT + 1205910)

dij

Tag

lij(k minus 1)

lii(k)

li0(kT + 1205910)

Anchor j Anchor i



0is

1205931198941

(119896119879 + 1205910) = 1205931198941 (119896) + 120591

01198901198941 (119896) (23)


1205910asymp 11989710

(119896119879 + 1205910) minus 1198971119894 (119896) (24)




0) from

1205951198941(119896119879 + 120591

0)

1199111198941

(119896119879 + 1205910) = 1205951198941

(119896119879 + 1205910) minus [120593

1198941 (119896) + 12059101198901198941 (119896)] (25)


z = 119891 (x0) + w (26)

where z = [11991121(119896119879 + 120591

0) 119911

1198731(119896119879 + 120591

0)]119879 and

119891 (x0) = [119889

20minus 11988910

11988930

minus 11988910

1198891198730

minus 11988910

]119879 (27)

w = Ds119890 (119896) + C

0u0(119896119879 + 120591

0) (28)

where

s119890 (119896) = s (119896) minus s (119896)

u0(119896119879 + 120591

0)

= [11989910

(119896119879 + 1205910) 11989920

(119896119879 + 1205910) 119899

1198730(119896119879 + 120591

0)]119879

u0(119896119879 + 120591

0) sim 119873 (0 120590

2

119899I119873)

D =

[[[[

[

1 1205910


0 0 1 1205910

0 sdot sdot sdot



]]]]


(29)

Since s119890(119896) and u

0(119896119879 + 120591


ance matrix of w is

Q119908

= 119864 (ww119879) = CQ119906C119879 + DQ

119904(119896)D119879 (30)


119901 (z x0) =

1

(2120587)(119873minus1)2 det (Q

119908)12

sdot exp [minus1

2(z minus 119891 (x

0))119879Qminus1119908

(z minus 119891 (x0))]

(31)




0))119879Qminus1119908

(z minus 119891 (x0))] (32)





0) around x

0(119898) yields

119891 (x0) asymp 119891 (x

0 (119898)) + 119866 (x0 (119898)) Δ (119898) (33)


G (x0) =

120597119891 (x0)

120597x0

(34)


Δ (119898) = [G119879(x0(119898))Qminus1

119908G(x0(119898))]minus1

sdot G119879 (x0 (119898))Qminus1

119908[119911 minus 119891 (x

0 (119898))]

(35)

where

G (x0) =

1

119888[r1198792(x0) minus r1198791(x0) r119879

119873(x0) minus r1198791(x0)]119879

(36)


r119894(x0) =

x0minus x119894

1003817100381710038171003817x0 minus x119894

1003817100381710038171003817

119894 = 1 119873 (37)


x0 (119898 + 1) = x

0 (119898) + Δ (119898) (38)







0(119898))Qminus1












119869 (x0) = (

120597119891 (x0)

120597x0

)

119879

Qminus1119908

(120597119891 (x0)

120597x0

) = 119866119879(x0)Qminus1119908

119866 (x0)

(39)


119869minus1


minus1(x0)]119899119899



minus1(x0)]119899119899The


1205902

119909=

3

sum

119899=1

[119869minus1

(x0)]119899119899

(40)




the CRLB as

GDOP =

radictr (Cov (x0))

119888120590119899

=120590119909

119888120590119899

(41)


10minus7

10minus8

10minus9

10minus10


Cloc

k off

set e

stim

atio

n er

ror120590

e


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16


5 Numerical Results







10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e







1198941(119896119879 + 120591




101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16



119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)






119899is close to 120590

119891in



119891






10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ljj(k minus 1)

lji(k)

lj0(kT + 1205910)120595ij(kT + 1205910)

dij

Tag

lij(k minus 1)

lii(k)

li0(kT + 1205910)

Anchor j Anchor i



0is

1205931198941

(119896119879 + 1205910) = 1205931198941 (119896) + 120591

01198901198941 (119896) (23)


1205910asymp 11989710

(119896119879 + 1205910) minus 1198971119894 (119896) (24)




0) from

1205951198941(119896119879 + 120591

0)

1199111198941

(119896119879 + 1205910) = 1205951198941

(119896119879 + 1205910) minus [120593

1198941 (119896) + 12059101198901198941 (119896)] (25)


z = 119891 (x0) + w (26)

where z = [11991121(119896119879 + 120591

0) 119911

1198731(119896119879 + 120591

0)]119879 and

119891 (x0) = [119889

20minus 11988910

11988930

minus 11988910

1198891198730

minus 11988910

]119879 (27)

w = Ds119890 (119896) + C

0u0(119896119879 + 120591

0) (28)

where

s119890 (119896) = s (119896) minus s (119896)

u0(119896119879 + 120591

0)

= [11989910

(119896119879 + 1205910) 11989920

(119896119879 + 1205910) 119899

1198730(119896119879 + 120591

0)]119879

u0(119896119879 + 120591

0) sim 119873 (0 120590

2

119899I119873)

D =

[[[[

[

1 1205910


0 0 1 1205910

0 sdot sdot sdot



]]]]


(29)

Since s119890(119896) and u

0(119896119879 + 120591


ance matrix of w is

Q119908

= 119864 (ww119879) = CQ119906C119879 + DQ

119904(119896)D119879 (30)


119901 (z x0) =

1

(2120587)(119873minus1)2 det (Q

119908)12

sdot exp [minus1

2(z minus 119891 (x

0))119879Qminus1119908

(z minus 119891 (x0))]

(31)




0))119879Qminus1119908

(z minus 119891 (x0))] (32)





0) around x

0(119898) yields

119891 (x0) asymp 119891 (x

0 (119898)) + 119866 (x0 (119898)) Δ (119898) (33)


G (x0) =

120597119891 (x0)

120597x0

(34)


Δ (119898) = [G119879(x0(119898))Qminus1

119908G(x0(119898))]minus1

sdot G119879 (x0 (119898))Qminus1

119908[119911 minus 119891 (x

0 (119898))]

(35)

where

G (x0) =

1

119888[r1198792(x0) minus r1198791(x0) r119879

119873(x0) minus r1198791(x0)]119879

(36)


r119894(x0) =

x0minus x119894

1003817100381710038171003817x0 minus x119894

1003817100381710038171003817

119894 = 1 119873 (37)


x0 (119898 + 1) = x

0 (119898) + Δ (119898) (38)







0(119898))Qminus1












119869 (x0) = (

120597119891 (x0)

120597x0

)

119879

Qminus1119908

(120597119891 (x0)

120597x0

) = 119866119879(x0)Qminus1119908

119866 (x0)

(39)


119869minus1


minus1(x0)]119899119899



minus1(x0)]119899119899The


1205902

119909=

3

sum

119899=1

[119869minus1

(x0)]119899119899

(40)




the CRLB as

GDOP =

radictr (Cov (x0))

119888120590119899

=120590119909

119888120590119899

(41)


10minus7

10minus8

10minus9

10minus10


Cloc

k off

set e

stim

atio

n er

ror120590

e


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16


5 Numerical Results







10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e







1198941(119896119879 + 120591




101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16



119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)






119899is close to 120590

119891in



119891






10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0(119898))Qminus1












119869 (x0) = (

120597119891 (x0)

120597x0

)

119879

Qminus1119908

(120597119891 (x0)

120597x0

) = 119866119879(x0)Qminus1119908

119866 (x0)

(39)


119869minus1


minus1(x0)]119899119899



minus1(x0)]119899119899The


1205902

119909=

3

sum

119899=1

[119869minus1

(x0)]119899119899

(40)




the CRLB as

GDOP =

radictr (Cov (x0))

119888120590119899

=120590119909

119888120590119899

(41)


10minus7

10minus8

10minus9

10minus10


Cloc

k off

set e

stim

atio

n er

ror120590

e


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16


5 Numerical Results







10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e







1198941(119896119879 + 120591




101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16



119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)






119899is close to 120590

119891in



119891






10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus7

10minus6

10minus8

10minus9

10minus10



RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e







1198941(119896119879 + 120591




101

102

100

10minus1


Sour

ce p

ositi

on es

timat

ion

erro

r (m

)


RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16



119897119894 (119905) = 119891

119894 (119905) + 119899119891 (42)






119899is close to 120590

119891in



119891






10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












10minus7

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01


RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k off

set e

stim

atio

n er

ror120590

e


119891= 10minus9



10minus7

10minus6

10minus8

10minus9

10minus10

RMSE T = 01

CRLB T = 01

RMSE T = 02

CRLB T = 02

RMSE T = 04

CRLB T = 04

RMSE T = 08

CRLB T = 08

RMSE T = 16

CRLB T = 16

Cloc

k sk

ew es

timat

ion

erro

r120590e


119891= 10minus9




101

102

100

10minus1

10minus2

Tag

posit

ion

estim

atio

n er

ror (

m)






119891= 10minus9



minus9 10minus8



2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2

2

2

2

3

3

3

3

4

4

4

4

44

4

4

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

66

7

7

7

7

7

8

8

8

8

8

9

9

9

9

10

10

10

10

11

11

11

12

12

12

12

2

22

2

2

2

2

2

3

3

3

3

3

3

3

3

4 4

44

4

4

4

4

4

4

445

5

5

5

5

5

5

6 6

6

6

6

7

77

7

7

7

8

8

8

8

9

minus50 0 50 100 150minus50

0

50

100

150

x-axis (m)

y-a

xis (

m)






minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

2

2

2

3

3

3

3 3

4

4

4

4

4

5

5

5

5

6

6

6

6 6

6

7

7 7

7

8

8 8

9

9 9

10

10

10

11

11

12

12

12

13

13

13

14

1414

14

1515

15

16

16

171890

2

2

2

2

2

3

33

3

3

3

4

4

4

44

4

4

5

5

5

5 5

5

6

66

6

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

99

9

10

10 10

0

11

11 1111

12

1212

1213

13

13

14

11516

100

120

x-axis (m)

y-a

xis (

m)




6 Conclusion



1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 12 2

2

2

2

3

3

3

3

3

3

33 3

4

4

4

4

4

4

4

4

4 4

5

5

5

5

5

5

5

5

5 5

6

6

6

6

6 6

6

6

7

7

7

7 77

7

8

8

8

88

8

8

9

9

9

9

9 99

9

10

10

10

10

1010 10

10

11

11

11

111

11

11

1

12

12

12

12

12

12

12

13

13

13

13

13

13

13

14

14

14

14

14

15

15

15

15

1515

15

15

16 16

16

16

16

16

16

16

17 17

17

17

17

17

17

18

18

18

18

18

18

18

1818

19

19

19

19

19

19

19

1920

20

20

20

20

20

20

2

2

2

2

2

3

3

3

3

3

3

3

3

4

4

44

4

4

45

5

5

5

5 5

5

5

6

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

88

88

8

9

9

9

9

9

9

9

10

10

10

10

10

10

11

11

1

11

11

11

12 12

12

12

1212

12

12

13

13

13

13

13

13

14

1414

14

15

1515

1516

1616

16

7

17

1717

17

17

18

8

18

18

18

1919

19

1920

20

20

2

minus20 0 20 40 60 80 100 120

minus20

0

20

40

60

80

100

120

x-axis (m)

y-a

xis (

m)





References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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