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doi:10.1016/j.sig

�CorrespondUniversity of O

E-mail addr

nottingham.ac.u

oulu.fi (A. Rab

Ian.Oppermann1Present add

ICT Center, M2Present addr

Nottingham, N3Currently w

Signal Processing 86 (2006) 2153–2171

www.elsevier.com/locate/sigpro

UWB location and tracking for wireless embedded networks

Kegen Yua,1, Jean-philippe Montilleta,2, Alberto Rabbachina,Paul Cheonga,3, Ian Oppermanna,b,�

aCentre for Wireless Communications, University of Oulu, FinlandbNokia Networks, Espoo, Finland

Received 2 January 2005; accepted 4 July 2005

Available online 9 March 2006

Abstract

In this paper, we investigate the performance of different position estimation methods which make use of time-of-arrival

(TOA) of ultra wideband (UWB) signals for low cost/low complexity UWB systems. We first propose a simple and robust

two-stage, non-coherent TOA estimation approach. We then explore positioning algorithms utilizing both non-iterative

and iterative techniques. A review of positioning in distributed networks is also performed and a positioning algorithm is

proposed for node location in multi-hop distributed networks. Furthermore, we consider smoothing techniques to improve

accuracy when tracking moving objects and we propose the use of sinc functions to smooth the estimate of the mobile

position in order to achieve both good accuracy and low complexity. The system modelled and investigated corresponds to

an actual test environment in a ski field where skiers are tracked.

r 2006 Published by Elsevier B.V.

Keywords: Position location/tracking; Iterative/non-iterative estimation; Distributed networks; TOA/TDOA; Sinc smoothing; UWB

1. Introduction

There are many positioning techniques usingradio signals. Signal strength, angle of arrival(AOA), time measurements (time-of-arrival (TOA),

e front matter r 2006 Published by Elsevier B.V.

pro.2005.07.042

ing author. Centre for Wireless Communications,

ulu, Finland. Tel.: +358 407 076 344.

esses: [email protected] (K. Yu), isxjpm@

k (J.-P. Montillet), Alberto.Rabbachin@ee.

bachin), [email protected] (P. Cheong),

@nokia.com, [email protected] (I. Oppermann).

ress: Wireless Technologies Laboratory, CSIRO

arsfield NSW 2122, Australia.

ess: University of Nottingham, University Park,

G7 2RD, UK.

orking in Singapore.

round trip time (RTT), and time difference of arrival(TDOA)) can all be exploited for positioning.

The most straightforward way to estimate posi-tion is to directly solve a set of simultaneousequations [1] based on the TOA/TDOA measure-ments. Therefore, exact solutions can be obtainedfor 2D positioning with three fixed nodes usingthree TOA measurements (with known transmittime) or with four fixed nodes using four TDOAmeasurements. For 3D positioning, five fixed nodesare needed to obtain exact solutions using TDOAmeasurements. Here we assumed one fixed node isemployed to clear the problem of multiple solutions.For an over-determined system (with redundantfixed nodes), several different approaches have beenproposed such as spherical interpolation (SI) [2–5],

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ARTICLE IN PRESSK. Yu et al. / Signal Processing 86 (2006) 2153–21712154

the two-stage maximum likelihood (ML) method[6], and the linear-correction least-square approach[7]. Also several iterative approaches have beeninvestigated for positioning. Taylor series expansioncan be used to iteratively produce a linearized least-square solution [8,9]. A different iterative methodfor positioning comes from nonlinear optimizationtheory. The gradient-based algorithms can beemployed for position estimation [10,11]. One isthe quasi-Newton algorithm [12] which has beenused in the ultra wideband (UWB) precision assetslocation system [13]. The other is the Gauss–Newton type Levenberg–Marquardt method [14].

Some researchers started investigation and appli-cation of UWB signals more than three decades ago[15]. Early UWB applications include positionlocation in Radar systems [16]. Existing andpotential UWB applications result from the manydesirable properties of UWB signals such as the finedelay resolution which enables high-resolutionpositioning and tracking. Very recently UWBtechnology has been successfully employed in theUWB precision assets location systems [13].

The goal of the paper is to investigate/developpractical and feasible algorithms using UWBtechnology for positioning and tracking ASIC(application-specific integrated circuit) UWB de-vices. The main contributions of the paper are asfollows:

�
propose and evaluate a two-stage TOA estima-tion scheme for locating low complexity UWBdevices; � propose and evaluate a positioning algorithm for
localization in distributed sensor networks;
� employ sinc function for smoothing mobile
tracks.

The remainder of the paper is organized asfollows. Section 2 briefly describes the positioningsystem and Section 3 studies TOA estimation.Section 4 presents various positioning algorithmswhile Section 5 discusses about moving devicetracking and smoothing.

2. Brief system description

In the system, there are a limited set of fixednodes which are accurately synchronized by sharingthe same local clock such as through cable connec-tions. The fixed nodes are positioned at knowncoordinates in the area to be monitored. In the

monitored area, there are a certain number of UWBdevices carried by mobile users.

The data transmission is packet based using aTDMA (time division multiple access) scheme.Since each user transmits data in different andpre-assigned time slots, multiple access interferenceis greatly reduced. Due to the drift in the clock ofthe mobile nodes as well as the fixed nodes,synchronization between the fixed nodes and themobile nodes is performed once every second. Thisis achieved by broadcasting beacons from the fixednodes to the mobile nodes. The TOA of the beaconis used as the reference clock for the mobile nodes totransmit data according to the pre-assigned timeslots.

At any given time, the TOA measurements from aspecific group of fixed nodes are collected forposition estimation. Since the mobile node ismoving, the group of fixed nodes will changeover time. In general, the fixed nodes with thestrongest received signal powers are selected toprovide the TOA estimates and their positioncoordinates are employed for the mobile nodeposition estimation.

More information about the system structure canbe found in [17] where a comprehensive systemoverview is provided for both physical layer andMAC (medium access control) layer. Differently,this paper focuses on the techniques and perfor-mance of TOA estimation, positioning algorithms,and tracking/smoothing.

3. Time-of-arrival estimation

To achieve accurate position estimation, we mustfirst acquire accurate TOA measurements. Thereexist numerous TOA estimation algorithms in theliterature. A comprehensive literature review oncode acquisition and delay estimation for direct-sequence spread spectrum signals can be found in[18,19].

The extremely short (usually sub-nanosecond),very low-duty cycle UWB pulses with very lowpower spectral density, poses a challenge forsynchronization in UWB systems. One method inthe literature for UWB timing recovery employs theML criteria [20–22]. A second method appliescorrelators in the traditional way, but makes useof techniques to obtain rapid timing acquisition[23–25]. However, the decrease of the acquisitiontime is not substantial. For low cost and lowcomplexity applications, energy collection-based

ARTICLE IN PRESS

Receivedsignal BPF LNA ( )2 bank of

integratordecision

Fig. 1. Block diagram of TOA estimator.

K. Yu et al. / Signal Processing 86 (2006) 2153–2171 2155

timing acquisition [26] is a promising approachespecially for indoor communications where densemultipath exists. However, the accuracy may not besatisfactory for some applications.

In this work, we employ a two-stage approach forfast timing acquisition to obtain the time-of-arrivalof the desired signal. The scenario considered isbased only on a LOS (line-of-sight) propagation.Since TDMA scheme is employed, no spreading isconsidered and non-coherent detection is employedfor the sake of low complexity. Therefore, a symbolmay consist of either one pulse or a sequence ofpulses. As mentioned in Section 2, synchronizationis performed once every second between the fixedand mobile nodes. For low complexity devices, theclock accuracy/stability may be around 10 parts permillion (ppm). Then, in each second, the uncertaintyof TOA could be around Tu ¼ 10ms in addition tothe propagation delay.

As shown in Fig. 1, after bandpass filtering andlow noise amplification, the received signal issquared4 and then passed through a bank ofintegrators.

3.1. First stage processing

At the first stage, the uncertain region is dividedinto K sectors and the length of each sectorequals T int ¼ Tu=K . Each integrator integratesthe squared signal of one sector5 as shown inFig. 2. In the case of one pulse per symbol, T int maybe chosen to be up to the multipath spread of thechannel. In the case of multiple pulses per symbol,T int may be chosen to be the length of the pulsesequence plus a value less than the multipathspread. In practice, the multipath spread can onlybe estimated/predicted.

Based on the energy measurements, a decision ismade according to a chosen criterion. In thehypothesis testing (decision making), three basiccriteria may be considered. The first is the thresholdcrossing (TC) criterion. With TC, the search is

4The squared operation is necessary if the received pulses can

be both positive and negative. In the event that all the received

signal pulses have the same sign (either positive or negative) or

energy of pulses of one sign is much greater than that of the other

sign, the original signal may be directly integrated.5The integrators may be reused multiple times when K is

greater than the number of integrators. Another option is that

only one single integrator is used to perform integration over the

whole uncertain region provided that no saturation happens.

Then, the output is read at iT int where iX0.

performed serially and is stopped once a measure-ment value crosses the threshold. The correspond-ing sector is then chosen and its startingtime provides the coarse TOA information. Ifnecessary, a verification process may be pursued.In the event that no measurement crosses thethreshold, new measurements are taken andthe search resumes. The TC algorithm requires thesetting of a threshold.

The other approach is the maximum selection(MAX) criterion. With this criterion, measurementsat all sectors are first compared. Then, the maximalmeasurement is produced and the relevantsector is selected. In the event that no appropriatethresholds can be readily obtained, the MAXcriterion could be desirable. Another criterionis the hybrid of MAX and TC. In this hybridcriterion, the maximal measurement is first ob-tained. Then, the maximum is examined against thethreshold. If the threshold is crossed, the relatedintegration sector is selected. If the maximum doesnot cross the threshold, the search resumes. With aprobability dependent on the SNR, the coarse TOAestimate will satisfy

t0 � 12T intot0ot0 þ 1

2T int, (1)

where t0 and t0 are the actual and the coarse TOAestimate, respectively.

3.2. Second stage processing

At the second stage, fine search is constrained tothis reduced uncertain region given by Eq. (1). Thesame bank of integrators can be employed to obtainfine TOA estimate. The difference between the starttime points of two adjacent integrators now can beas small as the clock period. Fig. 3 shows twoexamples of the time sequence and outputs ofthe integrators at the second stage, which representthe best (left) and the worst (right) case TOAestimation.

For simplicity the integration goes over aninterval of t2 � t0 þ Tw for each integratorwhere t0 is the actual TOA and Tw is the pulsewidth. In practice, the integration interval atthe second stage can be chosen based onthe predicted/estimated channel parameters. If

ARTICLE IN PRESS

Tu

MAXselection/

TCselection

TOA estimation(MAX selection)

Integrator 1

Integrator 2

Integrator 1

Integrator 3

Integrator N

Multipathchannel

Fig. 2. Illustration of TOA estimation.

Fig. 3. Time sequence and outputs of the integrators. t0 is the actual TOA. The ith integrator switches on at ti ¼ t0 (left) and at

ti ¼ t0 þ Tw=2 (right). Tw is the pulse width.

6For indoor UWB communications, rich multipath exists and

the multipath spread can be much larger.

K. Yu et al. / Signal Processing 86 (2006) 2153–21712156

appropriate thresholds can be found, the TCcriterion or the hybrid criterion can be employed.Otherwise, the MAX criterion should be considered.The process may continue over a sequence ofsymbols to produce multiple TOA estimates whichcan be further processed to obtain more accurateestimates.

To examine the accuracy of the proposedtwo-stage TOA estimation approach, we consideran outdoor environment with dimensions400m� 100m� 100m. A four path channel model[27] is employed to approximate an outdoor snow

covered environment which provided motivation forthe study. The first path signal has constantamplitude and the other three paths have Nakagamifading amplitudes [28] with a fading figure m ¼ 1:5.The fading amplitudes can be either positive ornegative with the same probability [29]. The delay ofthe second path is assumed to be 2 ns and the fourthpath delay is 12 ns.6 Each bit consists of only onepulse [13] and pulse width is equal to 0.5 ns.

ARTICLE IN PRESSK. Yu et al. / Signal Processing 86 (2006) 2153–2171 2157

It is assumed that the first stage search issuccessful and the TOA is constrained to the regiongiven by Eq. (1). The power ratio (Rician factor) ofthe direct path to the fading paths is equal to one(i.e. 0 dB) and the SNR equals 8 dB and a total of50,000 symbols are examined to produce 50,000TOA estimates. Fig. 4 shows the amplitude dis-tribution of the simulated TOA estimation errorswhile Fig. 5 shows the root mean squared error ofthe TOA estimation with respect to SNR for twodifferent Rician factors. The MAX criterion isemployed.

4

5

6

7

ror

(ns)

RicianK=1RicianK=1(syn)RicianK=10RicianK=10(syn)

4. Positioning algorithms

In this section, we study various positionestimation algorithms of either non-iterative oriterative. Details of several typical algorithmsare presented. Also a positioning algorithm isproposed for node location in distributed sensornetworks.

0 5 10 15 20 25 300

1

2

3

SNR (dB)

RM

S E

r

Fig. 5. RMS of TOA estimation errors. ‘syn’ denotes results

when time instants of the integrators are as shown in the left part

of Fig. 3 and other results are obtained when time instants of the

integrators are as shown in the right part of Fig. 3.

4.1. Non-iterative Methods

There exist various non-iterative position estima-tion algorithms. They include the direct method[1,30], the SI method [4], and other least-squaresrelated techniques [6,7]. The non-iterative algo-rithms are simple and easy to implement comparedto the iterative algorithms to be discussed in Section4.2. For quick reference, the direct method usingTDOA for 3D position estimation can be derived asfollows.

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

TOA Error (ns)

Am

plitu

de D

istr

ibut

ion

(100

%)

Fig. 4. Amplitude distribution of TOA estimation errors when

In the Cartesian system, the range (distance)between fixed node i and the mobile node is given byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xiÞ

2þ ðy� yiÞ

2þ ðz� ziÞ

2

q¼ cðti � t0Þ,

i ¼ 1; 2; 3; 4, ð2Þ

where N is the number of the fixed nodes/basestations, (x; y; z) are the unknown position coordi-nates of the mobile node, and (xi; yi; zi) are theknown coordinates of fixed node i. c is the speed oflight, ti is the signal TOA at fixed node i to beestimated, and t0 is the unknown transmit time atthe mobile node. In the development of the

0 5 10 150

0.05

0.1

0.15

0.2

TOA Error (ns)

Am

plitu

de D

istr

ibut

ion

(100

%)

time instants of the integrators are as shown in Fig. 3.


expressions, we ignore the difference between theactual and the measured TOAs for simplicity.Squaring both sides of (2) gives

ðx� xiÞ2þ ðy� yiÞ

2þ ðz� ziÞ

2¼ c2ðti � t0Þ

2,

i ¼ 1; 2; 3; 4. ð3Þ

Subtracting (3) for i ¼ 1 from (3) for i ¼ 2; 3; 4produces

ct0 ¼c

2ðt1 þ tiÞ þ

1

2cðt1 � tiÞ

�ðbi1 � 2xi1x� 2yi1y� 2zi1zÞ; i ¼ 2; 3; 4,

ð4Þ

where

xi1 ¼ xi � x1; yi1 ¼ yi � y1; zi1 ¼ zi � z1,

bi1 ¼ x2i þ y2

i þ z2i � ðx21 þ y2

1 þ z21Þ.

Define the TDOA between fixed nodes i and j as

Dtij ¼ ti � tj .

Eliminating t0 in (4) yields

a1xþ b1yþ c1z ¼ g1, (5)

where

a1 ¼ Dt12x31 � Dt13x21; b1 ¼ Dt12y31 � Dt13y21,

c1 ¼ Dt12z31 � Dt13z21,

g1 ¼12ðc2Dt12Dt13Dt32 þ Dt12b31 � Dt13b21Þ

and

a2xþ b2yþ c2z ¼ g2, (6)

where

a2 ¼ Dt12x41 � Dt14x21; b2 ¼ Dt12y41 � Dt14y21,

c2 ¼ Dt12z41 � Dt14z21,

g2 ¼12ðc2Dt12Dt14Dt42 þ Dt12b41 � Dt14b21Þ.

Combining (5) and (6) yields

x ¼ Azþ B, (7)

where

A ¼b1c2 � b2c1

a1b2 � a2b1; B ¼

b2g1 � b1g2

a1b2 � a2b1

and

y ¼ CzþD, (8)

where

C ¼a2c1 � a1c2

a1b2 � a2b1; D ¼

a1g2 � a2g1

a1b2 � a2b1.

Then, substitution of (7) and (8) into (4) for i ¼ 2produces

cðt1 � t0Þ ¼ Ezþ F , (9)

where

E ¼1

cDt12ðx21Aþ y21C þ z21Þ,

F ¼cDt12

2þ

1

2cDt12ð2ðx21Bþ y21DÞ � b21Þ.

Substituting (7), (8) and (9) back into (2) for i ¼ 1followed by squaring yields

Gz2 þHzþ I ¼ 0, (10)

where

G ¼ A2 þ C2 þ 1� E2,

H ¼ 2½AðB� x1Þ þ CðD� y1Þ � z1 � EF �,

I ¼ ðB� x1Þ2þ ðD� y1Þ

2þ z21 � F2.

The two solutions to (10) are

z ¼ �H

2G�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH

2G

� �2

�I

G

s. (11)

The two estimated z values (if both are reasonable)are then substituted into (7) and (8) to producethe estimate of the coordinates x and y, respe-ctively. However, there is only one desirablesolution. We remove the one with either no physicalmeaning or which is beyond the monitored area. Ifboth solutions are reasonable and they are veryclose, we may choose the average as the positionestimate. Otherwise, an ambiguity occurs. In thiscase, one more fixed node may be required toclear this ambiguity. The cases where noacceptable results are produced include two com-plex solutions, or when both solutions are beyondthe monitored area.

When the transmit time t0 is available, only threefixed nodes are required to determine the positionvariables and one extra fixed node might be neededto clear the problem of two solutions in some cases.It is trivial to derive the solution of the positioncoordinates in this case. For quick reference, wegive the formulae as follows.

Define

f 1i ¼12½c2ððt1 � t0Þ

2� ðti � t0Þ

2Þ þ bi1�; i ¼ 2; 3.


Also define

A1 ¼x21z31 � x31z21

x31y21 � x21y31

; B1 ¼x31f 12 � x21f 13

x31y21 � x21y31

,

C1 ¼y31z21 � y21z31

x31y21 � x21y31

; D1 ¼y21f 13 � y31f 12

x31y21 � x21y31

.

Then we have

z ¼F 1

E1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF1

E1

� �2

�G1

E1

s, (12)

where

E1 ¼ A21 þ C2

1 þ 1,

F 1 ¼ A1ðy1 � B1Þ þ C1ðx1 �D1Þ þ z1,

G1 ¼ ðx1 �D1Þ2þ ðy1 � B1Þ

2þ z21 � c2ðt1 � t0Þ

2,

and

x ¼ C1zþD1; y ¼ A1zþ B1. (13)

Note that different formulae may be derived forthe direct position calculation, however, the pre-sented method may be desirable since it does notinvolve any matrix operation.

Let us now study the SI method. First map thespatial origin to one of the fixed nodes, say fixednode 1, as shown in Fig. 6. Define

Ri ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

i þ y2i þ z2i

q; pi ¼ ½xi; yi; zi�

T,

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p; di;j ¼ di � dj.

Then, from the Pythagorean theorem, we have

ðRþ di;1Þ2¼ R2

i � 2pTi pþ R2; i ¼ 2; 3; . . . ;N.

(14)

Fig. 6. Illustration for spherical interpolation approach.

In the presence of measurement errors, (14)becomes

�i ¼ R2i � d2

i;1 � 2Rdi;1 � 2pTi p, (15)

where �i is the equation error. Eq. (15) can bewritten in compact form as

e ¼ d� 2Rd� 2Ap, (16)

where e is the equation error vector and

½A�i;1 ¼ xiþ1; ½A�i;2 ¼ yiþ1,

½A�i;3 ¼ ziþ1; i ¼ 1; 2; . . . ;N � 1,

d ¼ ½R22 � d2

2;1;R23 � d2

3;1; . . . ;R2N � d2

N;1�T,

d ¼ ½d2;1; d3;1; . . . ; dN ;1�T.

The standard LS solution for p, given R, is

p ¼ 12ðATAÞ�1AT

ðd� 2RdÞ. (17)

The key idea of the SI approach is to substitute(17) into (16) and minimize the equation error againbut with respect to R. The source location estimateis then obtained as

p ¼1

2ðATWAÞ�1ATW I�

ddTBVB

dTBVBd

� �, (18)

where W and V are weighting matrices and

B ¼ I� AðATWAÞ�1ATW.

Let us examine the performance of the non-iterative methods. Performance evaluation is per-formed in terms of the RMSE of the coordinateestimation and the failure rate. The failure rateincludes the cases where there is no solution or thesolution is unreasonable. With the iterative meth-ods, the failure rate includes situations where thealgorithm does not converge to a solution, themaximum number of function evaluations/itera-tions is exceeded, or the results are beyond the areaexamined. With the non-iterative methods, ‘failures’include cases when: the solutions are beyond themonitored area; both solutions are complex-valued;the two solutions are reasonable but not close toeach other; or inversion of singular matrices isinvolved.

In the simulation results, the monitored areaexamined has dimensions of 90ðlÞ � 90ðwÞ� 10ðhÞm.The positions of the fixed nodes and the mobilenode of interest are randomly generated. At eachtest point, 1000 runs are conducted with newrandom positions of the fixed nodes and the mobilenode at each run. The performance is then averaged.

ARTICLE IN PRESS

0 5 10 15 20 25 300

1

2

3

4

5

6

7

Average SNR (dB)

RM

S E

rror

(m

)

DM4(TDOA)DM3(inacc t0)DM3SI5SI6SI8

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Average SNR (dB)

Failu

re R

ate

DM3(inacc t0)DM3DM4(TDOA)SI5SI6SI8

Fig. 7. RMSE and failure rate of non-iterative position estimation.


The TOA estimation errors are produced using thesynchronization technique described in Section 3.Throughout the rest of the paper, the resultscorresponding to the time instants of the integratorsas shown in the first part of Fig. 3 are employed.

Fig. 7 shows the accuracy and the failure rate ofthe direct method and the SI algorithm [4]. TheTOA-based direct approach (results denoted by‘DM3’) is rather sensitive to the accuracy oftransmit time. ‘DM3(inacc t0)’ represent resultswith transmit time error of 4 ns while DM4(TDOA)denotes results for the TDOA-based directmethod. When transmit time error is in the orderof tens of ns, the TOA-based method does not workwell at all. Therefore, only when nearly error-freetransmit time information (compared to timereference at fixed nodes) is available, (12) to (13)are desirable.

Also shown are the results for the SI methodwith 5, 6 and 8 fixed nodes (denoted by ‘SI5’, ‘SI6’,and ‘SI8’). The technique does not work wellin the case of four fixed nodes as indicated in[4], so the corresponding results are not pre-sented. At relatively high SNR, the SI methodperforms well when at least five fixed nodes areemployed.

4.2. Iterative methods

In the iterative methods, the estimation isperformed iteratively and the iteration will not stopuntil some pre-defined criterion is satisfied. In this

section we investigate Taylor series method andoptimization techniques for position estimation.

4.2.1. Taylor series method

In Taylor series method, a set of nonlinearequations is linearized by expanding it in a Taylorseries around a point (initially an estimate of theactual position) and keeping only terms belowsecond order. The set of linearized equations issolved to produce a new approximate position andthe process continues until a pre-specified criterionis satisfied.

When TDOA measurements are employed, thismethod may be described as follows.

Subtracting (2) for i ¼ 1 from (2) for i ¼

2; 3; . . . ;N producesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xiÞ

2þ ðy� yiÞ

2þ ðz� ziÞ

2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x1Þ

2þ ðy� y1Þ

2þ ðz� z1Þ

2

q¼ cðti � t1Þ; i ¼ 2; 3; . . . ;N. ð19Þ

Define

f iðx; y; zÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xiþ1Þ

2þ ðy� yiþ1Þ

2þ ðz� ziþ1Þ

2q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� x1Þ

2þ ðy� y1Þ

2þ ðz� z1Þ

2

q,

i ¼ 1; 2; . . . ;N � 1,

and let ti be the TOA estimate at fixed node i. Then,

f iðx; y; zÞ ¼ d iþ1;1 þ �iþ1;1; i ¼ 1; 2; . . . ;N � 1,


where d i;1 ¼ cðti � t1Þ and �i;1 is the correspondingrange difference estimation error with covariance R.If xv, yv, and zv are guesses of the actual mobileposition, then,

x ¼ xv þ dx; y ¼ yv þ dy; z ¼ zv þ dz,

where dx, dy, and dz are the position errors to bedetermined. Expanding f i in Taylor series andretaining the first two terms produce

f i;v þ ai;1dx þ ai;2dy þ ai;3dz � d iþ1;1 þ �iþ1;1,

i ¼ 1; 2; . . . ;N � 1, ð20Þ

where

f i;v ¼ f iðxv; yv; zvÞ,

ai;1 ¼qf i

qx

��xv;yv ;zv

¼x1 � xv

d1

�xiþ1 � xv

diþ1

,

d i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxv � xiÞ

2þ ðyv � yiÞ

2þ ðzv � ziÞ

2q

,

ai 2 ¼qf i

qy

��xv;yv;zv

¼y1 � yv

d1

�yiþ1 � yv

diþ1

,

ai;3 ¼qf i

qz

��xv;yv ;zv

¼z1 � zv

d1

�ziþ1 � zv

diþ1

. ð21Þ

Eq. (20) can be rewritten as

Ad ¼ Dþ e, (22)

where

A ¼

a1;1 a1;2 a1;3

a2;1 a2;2 a2;3

..

. ... ..

.

aN�1;1 aN�1;2 aN�1;3

26666664

37777775; d ¼

dx

dy

dz

2664

3775,

D ¼

d2;1 � f 1;v

d3;1 � f 2;v

..

.

dN;1 � f N�1;v

266666664

377777775; e ¼

�2;1

�3;1

..

.

�N ;1

26666664

37777775.

The weighted least-square estimator for (22) pro-duces

d ¼ ½ATR�1A��1ATR�1D. (23)

Given an initial position guess (xv, yv, zv) andcompute d with (23). Then update the positionestimate according to

xv ¼ xv þ dx; yv ¼ yv þ dv; zv ¼ zv þ dz.

Continually refine the position estimate until d issufficiently small.

4.3. Optimization-based methods

After defining the objective function, severaloptimization-based position estimation methodsare studied.

4.3.1. Objective function

An objective function is normally required foroptimization algorithms. Since the aim of position-ing is to obtain an accurate position estimate, it isnatural to define the objective function as the sumof the squared range errors:

F ðx; y; z; t0Þ ¼1

2

XN

i¼1

f 2i ðx; y; z; t0Þ, (24)

where

f iðx; y; z; t0Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xiÞ

2þ ðy� yiÞ

2þ ðz� ziÞ

2

q� cðti � t0Þ.

The optimization purpose is to minimize thisobjective function to produce the optimal positionestimate. For notational simplicity, we define

p ¼ ½x; y; z; t0�T; fðpÞ ¼ ½f 1ðpÞ f 2ðpÞ . . . f N ðpÞ�

T.

Then (24) becomes

F ðpÞ ¼ 12kfðpÞk

2.

There exist many optimization methods to minimizethe pre-defined objective function to achieve theoptimal performance (i.e. the optimal positionestimate in our case). We choose two specificmethods, the Gauss–Newton type methods andthe quasi-Newton methods, for investigation.

4.3.2. Gauss– Newton type methods

Expanding the objective function in the Taylorseries at the current point pk and taking the firstthree terms, we have

F ðpk þ skÞ � F ðpkÞ þ gTk sk þ12sTkGðpkÞsk, (25)

where sk is the directional vector (or incrementvector) to be determined, gk is a vector of the firstpartial derivatives (also called gradient) of theobjective function at pk, and GðpkÞ is the Hessianof the objective function. Minimization of the right-hand side of (25) yields

GðpkÞsk ¼ �gk. (26)


The minimization in which sk is defined by Eq. (26)is termed Newton’s method [10,11]. To avoid thecalculation of the second order information in theHessian, a simplified expression can be approachedfrom (26), resulting in

JTk Jksk ¼ �JT

k fðpkÞ, (27)

where Jk is the Jacobian matrix of fðpÞ at pk. Thisis termed the Gauss–Newton method. When Jk isfull rank, which is the usual case of an over-determined system, we have the linear least-squaressolution

sk ¼ �ðJTk JkÞ

�1JTk fðpkÞ. (28)

The Gauss–Newton method may not perform wellwhen the second order information in the Hessian isnon-trivial. A method to overcome this is theLevenberg–Marquardt method. The Levenberg–Marquardt search direction is defined as thesolution of the equations

ðJTk Jk þ lkIÞsk ¼ �JT

k fðpkÞ,

where lk is a non-negative scalar which controlsboth the magnitude and direction of sk.

To start the iteration, the initial position co-ordinates and the initial transmit time are required.The initial estimated values of the position coordi-nates may be chosen to be the mean positionof the fixed nodes or the area being monitored.The initial estimated transmit time may be chosen tobe some time point earlier than the earliestreceive time. This will depend on the dimen-sion of the monitored area. Certainly, if a more

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Average SNR (dB)

RM

S E

rror

(m

)

DFP4DFP5DFP6DFP8

Fig. 8. RMSE of DFP (left figure) and TS (right figu

accurate initial position and transmit time estimateis available, the performance could be improved.This may be achieved by the non-iterative algo-rithms discussed in Section 4.1 or by positionprediction discussed in Section 5.

4.3.3. Quasi-Newton methods

This method is similar to Newton’s method. TheHessian matrix GðpkÞ in (26) is now approximatedby a symmetric positive definite matrix Bk, which isupdated during each iteration. At the kth iteration,set

sk ¼ �Bkgk.

Using line search along sk produces

pkþ1 ¼ pk þ ask,

where a is the step size. Then Bk is updated to yieldBkþ1. There exist different ways of updating Bk. Onewell known updating formula is the DFP (Davi-don–Fletcher–Powell) formula [12], updating Bk

according to

Bkþ1 ¼ Bk þhkh

Tk

hTkqk

�Bkqkq

TkBk

qTkBkqk

,

where

hk ¼ pkþ1 � pk; qk ¼ gkþ1 � gk.

The initial matrix B1 can be any positive definitematrix. It is usually set to be the identity matrix inthe absence of any better estimate [11]. Anotherimportant formula is the BFGS formula [11]. In this

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

Average SNR (dB)

RM

S E

rror

(m

)

TS4TS5TS6TS8

re) algorithms using 4, 5, 6, and 8 fixed nodes.

ARTICLE IN PRESS

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

Average SNR (dB)

Failu

re R

ate

DFP4DFP5DFP6DFP8

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Average SNR (dB)

Failu

re R

ate

TS4TS5TS6TS8

Fig. 9. Failure rate of position estimation using DFP and TS algorithms.

Table 1

Normalized execution time of four different position estimation

algorithms

No. of FNs 4 5

DFP 61 60

TS 3.1 4.5

SI 1.0

DM 1.0

TS: Taylor series method. SI: spherical interpolation method.

DM: direct calculation method.


formulae, Bk is updated based on

Bkþ1 ¼ Bk þ 1þhTkBkhk

qTkhk

� �qkq

Tk

qTkhk

�qkh

TkBk þ Bkhkq

Tk

qTkhk

.

Clearly the BFGS formula requires significantlygreater computational effort. All the algorithms canbe found in the Matlab optimization toolbox.

4.4. Performance evaluation

In this section we examine the performance of theiterative algorithms through simulation. The simu-lation setup is the same as in Section 4.1.

Figs. 8 and 9 show the RMSE of the positionestimation and the failure rate, respectively, for theDFP and TS algorithms. Four different numbers offixed nodes (i.e. 4, 5, 6 and 8) are examined. Clearlymore fixed nodes achieve better performance.

The Gauss–Newton method did not perform wellat all and the BFGS approach produces very similarperformance to the DFP approach. Since the BFGSapproach is more complicated, the DFP approach ispreferable among the quasi-Newton methods. It isalso shown that the Levenberg–Marquardt algo-rithm produced very similar results to the DFPformulae. Therefore, only results from the DFPapproach and Taylor series method are presented.

Accuracy and failure rate are crucial measures inevaluating a positioning technique. Complexity of aposition estimation algorithm is also an importantissue in the design of a positioning system especially

when low cost and low complexity are involved.Table 1 shows the normalized execution time for thefour different algorithms, i.e. DFP algorithm, Taylorseries method, SI method, and direct method. Thenumber of iterations of TS is set to be 20. TheMatlab optimization toolbox is employed to runDFP algorithm and the maximum number ofiterations is set to be 20. The iterative algorithmsare more complex than the non-iterative ones and theDFP algorithm requires much more computationaltime than any of the other three algorithms.

4.5. Positioning in distributed architectures

In this section, we consider localization in ad hocand distributed networks. An ad hoc network is acollection of wireless nodes that self-configurethemselves to form a network without the aid ofany infrastructure. This sort of networks arecharacterized by large size, need for distributedcoordination and ubiquitous connectivity, powerconstraints and the ability to be ad hoc deployable.

ARTICLE IN PRESS

B2

B3

S3

B1

S1

S2

beacon

sensor

B21

d 2B3

B32

21

13

3B1

d 3B

1

d13

d1B

2 d12

�

�

�

�

�

Fig. 10. Illustration of the proposed location scheme.


In the following subsections, an overview of relatedlocalization approaches is first presented. Then apositioning algorithm is proposed and finally somesimulation results are provided.

4.5.1. Overview

In the recent years, there have been numerousalgorithms (either centralized or distributed) tolocalize sensor nodes in wireless sensor networks.In [31,32], a centralized scheme is proposed whichcollects the entire topology in a server to minimizethe errors using convex optimization. In [33–35],instead of directly solving the set of constraints of thewhole wireless network, multi-dimensional scaling(MDS) is exploited. This technique uses localconnectivity or distance measure to generate relativemaps that represent the relative position of nodes.The main problem of the mentioned algorithms is theneed to have some powerful node or server toperform the large computation. In [36], distributedalgorithms are divided in two sub-families: range-based and range-free algorithms. In range-freelocalization [37,38], beacon nodes broadcast theirpositions to their neighbors that keep an account ofall received beacons. Then, nodes calculate theirpositions based on the received beacon locations, thehop-count from the corresponding beacon and theaverage distance per hop. In [39], the distance perhop is averaged by taking into account the localdensity of nodes. In range-based algorithms, thedistance between two neighboring sensors is firstestimated, for example, by using TOA measure-ments. In [40] and more recently in [41], a distributedmechanism is proposed for GPS-free positioning inmobile ad hoc networks. A slightly modified versionof the GPS-free algorithm is proposed in [42]. In [43],the distance between a sensor and a beacon is directlycalculated using basic triangle rules and simplegeometry. Collinearity is exploited in [44] and factorgraphs are employed in [45].

4.5.2. Proposed algorithm

It is assumed that each node (sensor and beacon)has the ability of measuring both TOA and AOA

d1B1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd13 cos y13 þ d3B1

qd1B2¼ d1B2

,

d1B3¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd12 cos y21 þ d2B3

q

with its 1-hop neighbors. In the first phase, thebeacons broadcast their coordinates. Then, allsensors establish the shortest path with the beacons.As a result, a sensor should have the coordinates ofat least three beacons and the path to reach them.After the shortest path is developed, all sensors andbeacons belonging to the path calculate the TOAand AOA with the neighboring nodes in the path.Then, using the TOA and AOA measurements, asensor is able to estimate the Euclidean distance toeach of at least three beacons.

Fig. 10 gives an illustration of the algorithm. Inthe figure, sensor S1 established a path to each ofthe three beacons (B1, B2 and B3). Then, it stores therelevant information in its data base in the form

S1! B1 : F1 ¼ fðS1;S3;B1Þ; ðy13; y3B1Þ; ðd13; d3B1

Þg,

S1 ! B2 : F2 ¼ fðS1;B2Þ; ðyB21Þ; ðd1B2Þg,

S1! B3 : F3 ¼ fðS1;S2;B3Þ; ðy21; yB32Þ; ðd12; d2B3Þg.

The distance from S1 to the three beacons can bedetermined by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos y3B1

Þ2þ ðd13 sin y13 þ d3B1

sin y3B1Þ2,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos yB32Þ

2þ ðd12 sin y21 þ d2B3

sin yB32Þ2.


After obtaining the distance to each of the beacons,the position coordinates of the node can beestimated using the algorithms studied in Section4. When considering the optimization-based ap-proaches, the cost function is defined as

�ðx; yÞ ¼XNB

k¼1

dk �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxk � xÞ2 þ ðyk � yÞ2

q� �2, (29)

where dk is the estimated distance between thedesired sensor and the kth beacon, and NB is thenumber of beacons available in the network. ðx; yÞand ðxk; ykÞ are the unknown coordinates of thesensor of interest and the known coordinates of thekth beacon, respectively. Although we focused on2D sensor location, it is straightforward to extendthe algorithm to 3D positioning.

4.5.3. Simulation results

The monitored area has dimensions of100ðwÞ � 100ðlÞm. Hundred nodes are randomlypositioned in the area together with a number (5, 15,and 20) of beacon nodes depending on the simula-tion scenarios. The transmission radius of any nodein this network is equal to 30m and it is keptconstant. The position estimation algorithms con-sidered are the DFP algorithm and the directmethod (DM). The performance evaluation isperformed by assuming that the TOA and AOAmeasurement errors are white Gaussian randomvariables of mean zero and variance s2TOA and s2AOA,respectively. At each test point ðsÞ, 30 simulationsare conducted with new random topology of the

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

σangle (degrees)

RM

SE

(m

)

σ =0

σ =0.5ns

σ =2ns

Fig. 11. RMSE of direct method (left) and DFP algorithm (right) wit

different variances of the TOA error (sTOA ¼ f0; 0:5; 2gns).

network and the performance is then averaged. Inthe case of the DFP algorithm, 50 iterations areused to update the estimated coordinates.

Fig. 11 shows the root mean square error of thecoordinates estimations using either the DM orDFP algorithm. In general, the RMS errors of bothalgorithms are under 2m and the DFP algorithmachieves higher accuracy. We also notice the quiteflat curve (except for the case of DM with sTOA ¼ 0)as the AOA measurements error increases. It meansthat the algorithms, to localize the nodes, are notsensitive to the AOA measurement error.

Fig. 12 shows the RMS error with respect to sTOA

at a given sAOA. In this case, the slope of the curvesbecomes sharper. This phenomena tells the fact thatthe algorithms are more sensitive to TOA measure-ment error, comparatively.

5. Tracking moving objects

When considering mobile devices with velocity upto 100 km/h, tracking should be included in thepositioning algorithms. The system should be ableto update the position estimation at a reasonablerate to follow the moving devices. At each timeinstant, a number of TOA measurements arecollected from a specific set of fixed nodes. Someset members can be different as the device moves.Usually the fixed nodes closest to the moving deviceare employed to provide the time measurementssince in general shorter distance means higher signalpower so better performance can be obtained.

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

σangle (degrees)

RM

SE

(m

)

σ =0

σ =0.5ns

σ =2ns

h respect to AOA errors. Five beacons in the network and three

ARTICLE IN PRESS

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

σToA (ns)

RM

SE

(m

)

σ

σ

σ

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

σToA (ns)

RM

SE

(m

)

σ

σ

σ

Fig. 12. RMSE of direct method (left) and DFP algorithm (right) with respect to TOA errors. Five beacons in the network and three

different variances of the AOA error (sAOA ¼ f0; 5; 10g degrees).

signalat FN2

receivedsignalat FNN

Predicted position

signalat FN1

receivedPosition

Estimation

Velocity

PositionSmoothing

TrajectoryDisplay

Position

received

.Estimation

..

Estimation

TOAEstimation

TOA

TOA Estimated Smoothed

At control centerAt fixed nodes

Fig. 13. Block diagram of position location and tracking.


Tracking performance can be improved bysmoothing the individual position results.Kalman filtering has been widely used in moderncontrol systems, tracking and navigation of all sortsof vehicles [46]. Some references about using Kal-man filtering to smooth/filter position/velocityestimate of moving objects can be found in[47–52]. Another filtering approach used forsmoothing position estimates is the linear least-squares approach [53]. The related formula of thetwo smoothing methods is provided in AppendicesA and B, respectively.

When the track is rather irregular/nonlinearand/or the velocity of the moving nodes is time-varying, the LS smoothing approach may not performwell. Kalman filtering requires that the variances ofthe system noise and observation noise are known in apriori. However, in practice, the position estimationnoise variances are usually unknown and implementa-tion of Kalman filtering at low-complexity nodes may

be not feasible due to its relatively high computationalrequirement. For those considerations, we propose toexploit the sinc function to smooth the tracks andimprove the accuracy of the position estimates. Sincfunction has been exploited to interpolate pilotsymbol aided channel estimates [54]. Fig. 13 showsthe block diagram of the proposed location andtracking system.

The principle of sinc smoothing is rather simple.The smoothing function is given by

hðjÞ ¼sinð2pjf MTÞ

2pjf MT; �1ojo1, (30)

where f M is the maximum frequency of the track(function of time) and T is the position updatingperiod. In practice, truncation is used and the lengthof the window is equal to K so that

�K � 1

2pjp

K � 1

2.

ARTICLE IN PRESS

Table 2

Averaged RMSEs of four algorithms at SNR of 16 dB

No. of FNs 4 5 6 8

DFP 4.36 (m) 1.65 (m) 0.63 (m) 0.15 (m)

TS 5.43 2.1 0.88 0.15

SI 1.55 0.33 0.09

DM 7.54

FNs: fixed nodes. TS: Taylor series method. SI: spherical

interpolation method. DM: direct method.

Table 3

Averaged failure rates of four algorithms at SNR of 16 dB

No. of FNs 4 5 6 8

DFP 10.8 (%) 6.6 (%) 5.2 (%) 4.2 (%)

TS 14.3 3.1 1.5 0.9

SI 3.4 1.3 0.9

DM 46.6

Table 4

Averaged RMSEs before and after smoothing

Before smoothing LS KF Sinc

4.13 (m) 2.40 2.30 2.19

2.70 (m) 1.58 1.30 1.33

1.40 (m) 1.02 0.99 0.84

KF: Kalman filtering.


When K is not large such as about 10–20, somewindowing techniques such as Hanning window orHamming window may be required to smooth thetruncation. Let f ~xig be the sequence of the estimatedcoordinate (any of the three coordinates) and fxig bethe corresponding results after smoothing. Then,

xi ¼Xp

j¼�p

hðjÞ ~xiþj ; i4p.

It has been shown in [54] that sinc smoothing isoptimal in the event that the power spectrum of thesignal is bandlimited and has a flat spectrum up tof M. Usually f M is unknown; however, low fre-quency components would be dominant so that itcan be chosen empirically.

6. Simulation results

In this section, we examine the performance ofthe proposed position location and tracking system(as shown in Fig. 13). We use one of the realisticfield structures, a snow covered slope of dimensionsabout 400m� 100m� 100m. The fixed nodes willbe deployed along both sides of the slope andmounted on poles of varying height. Fig. 14 showsthe imagined track for examination. The skiermoves from A to B (120m) at a speed of 8m persecond (m/s). The skier moves from B to C (160m)at a speed of 10m/s and finally from C to D (120m)at a speed of 8m/s.

First we examine the performance of the differentposition estimation algorithms under the morerealistic circumstance. Two hundred different com-binations of fixed node positions are tested and thenthe results are averaged. Tables 2 and 3 compare the

0100

200300

400

2030

4050

6070

800

20

40

60

80

100

120

AB

C

D

Fig. 14. Track for examination.

averaged results of the four algorithms at SNR of16 dB.

For the parameters examined, the SI methodprovides the best tradeoff between performance andcomplexity when there are at least five fixed nodes.To achieve sub-meter accuracy, at least six fixednodes are needed with SNR up to 16 dB.

Let us consider position smoothing by makinguse of the three smoothing techniques discussed inSection 7. Table 4 shows the averaged RMSEsbefore and after smoothing. The estimated tracks(before smoothing) are produced by using the SIalgorithm with five fixed nodes under three sets offixed node configurations. Since the one-Kalman-filter scheme and the three-Kalman-filter schemeproduce the same results, only results from one ofthem are listed. Clearly, the sinc smoothing evenachieves the best results on average.

Fig. 15 shows the original, estimated andsmoothed tracks using the sinc smoothing. The

ARTICLE IN PRESS

0100

200300

400

2030

4050

6070

800

20

40

60

80

100

120

Fig. 15. Original (-), estimated (- -) and sinc smoothed (o) tracks.


smoothing window length equals 11 and the fiveestimated values at both the beginning and the endof the tracks are not processed/smoothed. Theinitial position estimates are produced using the SImethod under one specific combination of the fixednodes and the RMSE before smoothing is 2.70m.The effectiveness of smoothing is clearly demon-strated.

7. Conclusions

In this paper we investigated several positionestimation approaches employing UWB technologyfor outdoor recreational activities. Focus wasplaced on two iterative algorithms (i.e. DFP andTaylor series) and two non-iterative methods(i.e. spherical interpolation and direct method).Performance comparisons of the four methods wereperformed under different scenarios. Using theproposed TOA estimation technique and certainnumber of fixed nodes, accurate position estimatescan be obtained even under a realistic fieldstructure. A sinc smoothing technique was em-ployed to achieve the goal of both low complexityand high accuracy. A new multi-hop locationtechnique was also proposed for distributed sensornetworks.

Acknowledgment

This work is partly funded by the EU 6thframework project PULSERS.

Appendix A. Kalman filtering for position smoothing

Define state vector at time tk

xk ¼ ½xk yk zk vx;k vy;k vz;k�T,

where vx;k, vy;k and vz;k are the velocity componentsalong the x, y, and z coordinates, respectively. Alsodefine the observation vector

pk ¼ ½xk yk zk�T.

Then the system dynamic model (process equation)is

xk ¼ Uk�1xk�1 þ Bk�1wk�1, (31)

where Uk�1 (state transition matrix) and Bk�1 are,respectively, given by

Uk�1 ¼

1 0 0 Dt 0 0

0 1 0 0 Dt 0

0 0 1 0 0 Dt

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

2666666666664

3777777777775,

Bk�1 ¼

0:5Dt2 0 0

0 0:5Dt2 0

0 0 0:5Dt2

Dt 0 0

0 Dt 0

0 0 Dt

2666666666664

3777777777775,

Dt : time increment ð32Þ

and wk�1 is the acceleration noise assumed to bewhite Gaussian random vector of mean zero andcovariance Qk�1. The measurement model (equa-tion) is

pk ¼ Hkxk þ nk, (33)

where Hk is the observation matrix given by

Hk ¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

264

375,

and nk is the observation noise vector alsoassumed to be Gaussian with mean zero andcovariance Rk.

Implementation of Kalman filtering can besummarized as follows. The initial estimate x0 andits error covariance C0 are first given. Let ð�Þ andðþÞ represent the a priori (before update) and the aposteriori (after update) value, respectively. Thestate estimate extrapolation is given by

xkð�Þ ¼ Uk�1xk�1ðþÞ, (34)


and the error covariance extrapolation is performedby

Ckð�Þ ¼ Uk�1Ck�1ðþÞUTk�1 þ Bk�1Qk�1B

Tk�1. (35)

Then the Kalman gain matrix can be computedaccording to

Kk ¼ Ckð�ÞHTk ½HkCkð�ÞH

Tk þ Rk�

�1. (36)

The state estimate is updated by

xkðþÞ ¼ xkð�Þ þ Kk½pk �Hkxkð�Þ�, (37)

and the error covariance is updated by

CkðþÞ ¼ ½I� KkHk�Ckð�Þ. (38)

Therefore, Kalman filter is updated recursively from(34) to (38).

In the case of an individual filter used for eachposition coordinate, the process equation and theobservation equation for the x coordinate become

xk

vx;k

" #|fflfflffl{zfflfflffl}

xk

¼1 Dt

0 1

� �|fflfflfflfflffl{zfflfflfflfflffl}

Uk�1

xk�1

vx;k�1

" #þ

0:5Dt2

Dt

" #|fflfflfflfflfflffl{zfflfflfflfflfflffl}

Bk�1

wk, (39)

pk ¼ ½ 1 0 �|fflfflffl{zfflfflffl}Hk

xk

vx;k

" #þ nk. (40)

In this case the system noise, the observation noise,and the observation are all scalar.

Appendix B. Least-squares approach for position

smoothing

Assume that the target is in a linear motion with aconstant velocity along each coordinate. Thisassumption would be reasonable when a shortdistance is considered, although the whole trackmay not be linear and the velocity can be varyingover a long track. Then the actual target position attime tk is given by

pk ¼ p0 þ vtk; kX1, (41)

where p0 is the position at time t0 which isnormalized to zero. The estimated target positionat time tk is

pk ¼ pk þ ek, (42)

where ek is the estimation error vector. Let us makeuse of a sequence of K position estimates,pk; 1pkpK , for position smoothing. The LS

estimator is found by minimizing

XK

k¼1

kpk � ðp0 þ vtkÞk2.

This minimization may decompose into

minpx;0; vx

XK

k¼1

½px;k � ðpx;0 þ vxtkÞ�2

þ minpy;0;vy

XK

k¼1

½py;k � ðpy;0 þ vytkÞ�2

þ minpz;0;vz

XK

k¼1

½pz;k � ðpz;0 þ vztkÞ�2. ð43Þ

Let us pay attention to the first term in (43), whichcan be written as

minpx;0;vx

ðpx �MyxÞTðpx �MyxÞ, (44)

where

px ¼ ½px;1 px;2 . . . px;K �T; yx ¼ ½px;0 vx�

T,

M ¼1 1 . . . 1

t1 t2 . . . tK

" #T.

The minimization in (44) yields [55]

yx ¼ ðMTMÞ�1MTpx. (45)

After some mathematical manipulations, (45) be-comes

yx ¼1

KPK

k¼1 t2k � ðPK

k¼1 tkÞ2

�

PKk¼1

t2kPKk¼1

px;k �PKk¼1

tk

PKk¼1

tkpx;k

KPKk¼1

tkpx;k �PKk¼1

tk

PKk¼1

px;k

266664

377775. ð46Þ

That is

px;0 ¼

PKk¼1 t2k

PKk¼1 px;k �

PKk¼1 tk

PKk¼1 tkpx;k

KPK

k¼1 t2k � ðPK

k¼1 tkÞ2

,

(47)

vx ¼KPK

k¼1 tkpx;k �PK

k¼1 tk

PKk¼1 px;k

KPK

k¼1 t2k � ðPK

k¼1 tkÞ2

. (48)

Then we have the smoothed x-coordinate positionestimate at time tk as

px;k ¼ px;0 þ vxtk. (49)


In the same way we can find py;0 and pz;0 in the formof (47), vy and vz in the form of (48), and py;k andpz;k in the form of (49).

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