research article indoor mobile localization in mixed...

Research ArticleIndoor Mobile Localization in Mixed Environment withRSS Measurements

Zhengguo Cai Lin Shang Dan Gao Kang Zhao and Yingguan Wang

Shanghai Institute of Microsystem and Information Technology Chinese Academy of Sciences Shanghai 200050 China

Correspondence should be addressed to Zhengguo Cai caizieegmailcom

Received 3 June 2014 Revised 24 September 2014 Accepted 29 September 2014

Academic Editor Shigeng Zhang

Copyright copy 2015 Zhengguo Cai 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

Mobile localization is a significant issue for wireless sensor networks (WSNs) However it is a problem for the indoor localizationusing received signal strength (RSS) measurements that the signal is contaminated by the anisotropy fading and interferencedue to walls and furniture Standard schemes such as Kalman filter are inadequate as the random transition of line-of-sight(LOS)non-line-of-sight (NLOS) conditions occurs frequently This paper proposes an indoor mobile localization scheme withRSS measurements in a mixed LOS and NLOS environment First a new efficient composite measurement model is inducedand validated which approximates the complex effects of LOS and NLOS channels Second a greedy anchor sensor selectionstrategy is adopted to break through the constraints of hardware consistency and the multipath interference Third for the Markovtransition between LOS andNLOS conditions an effective unscented Kalman filter (UKF) based interactive multiple model (IMM)is proposed to estimate not only the posterior model probabilities but also a weighted-sum position estimation with the aid oflikelihood function To evaluate the proposed algorithm a complete hardware and software platform for mobile localization hasbeen constructed The numerical study relying on the actual experiments illustrates that the proposed UKF based IMM achievesa substantial gain in precision and robustness compared with other works

1 Introduction

Mobile location estimation has already been a popularresearch topic for decades Different solutions based on angleof arrival (AOA) time difference of arrival (TDOA) time ofarrival (TOA) and received signal strength (RSS) have beenreported in literature [1 2] Applications of these techniquesarise such as in emergency services location-based billingsmart home fleetmanagement and intelligent transportationsystems (ITS) [2] Here we concentrate on mobile terminaltracking based on RSS measurements in wireless sensornetworks (WSNs)

Accurate position estimation would be feasible using anefficient filtering if a direct physical connection between themobile terminal (MT) and the anchor sensor (AS) existsthat is the channel can be considered as line-of-sight (LOS)However in certain environments especially in indoor areasreflection and diffraction occur often between the MTand AS The worsened propagation decay correspondingto non-line-sight (NLOS) leads to a misestimated range

The erroneous position estimations would occur whenusing normal strategies Meanwhile the frequent transitionbetween LOS and NLOS will cause a serious measurementerror for the range estimation because the estimated covari-ance of the measurement noise is not adaptively to match thetrue covariance in the LOS andNLOS cases So the estimatorswhich are robust against a mixed LOSNLOS environmentare required

This burning question has been of considerable interestfor many years [3ndash14] These localization algorithms can begrouped into two categories detection-based approach andestimation-based approachThe detection-based localizationalgorithms rely on either the residual or the prior informationof the NLOS error in the detection hypotheses on thebasis of statistical decision theory [15ndash17] One limitation ofthese algorithms is that it requires at least 3 measurementsLocating a MT in NLOS environments is also consideredin [18 19] to improve accuracy by smoothing the mea-surements or overweighting the LOS ASs Another methodbased on modified probability data association (MPDA) is

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 106475 13 pageshttpdxdoiorg1011552015106475

2 International Journal of Distributed Sensor Networks

proposed in [14]Thismethod constructs different subgroupsof measurements Then intermediate results of the LOSsubgroups are combined to reach the final estimated positionof the moving target However in severe NLOS conditionsit is very likely that no subgroup is detected as LOS at manytime instances As a result no new measurement is availableto update the filtering

While the estimation-based approaches utilize the statis-tical properties of the LOS and NLOS noises and reconstructposterior estimated position [3ndash5 19] they are robust inmany NLOS conditions but are not efficient as the highcomputational cost and incomplete measurement modelsAs for frequent LOSNLOS transition in a mixed environ-ment some good progress with interactive multiple model(IMM) has been made in this category It [20ndash22] hasbeen demonstrated as one of the most effective schemesfor position estimation in a hybrid dynamic system underuncertain environments Yang et al [23] present a locationestimation algorithm using fuzzy-based IMM estimator In[24] IMM is adopted for the MT location estimation as wellHowever the indoor environment is quite different from theurban structure in [24] The system model and estimatorseems to be an oversimplification for an indoor applicationsince it fails to consider the signal attenuated by walls orfurniture

Most existing research works on a localization algo-rithm under the assumption of different probability distri-bution functions of the LOS and NLOS noise Howeversome other significant distinctions exist between LOS andNLOS propagation resulting in various new measurementmodels

In this paper we present an efficient MT positioningalgorithm using UKF based IMM with individual RSS mea-surements in a mixed LOSNLOS indoor environment Inaddition the proposed framework has been evaluated on apractical platform in a real application situation The mainworks are as follows

(i) An effective measurement model is derived todescribe the influence of fading caused by the indoorenvironment and defined as a superposition RIM andGaussian noise Furthermore the model has beenverified in an actual indoor building containing theLOS and NLOS channel conditions

(ii) A greedyAS selection strategy is introduced to choosethe AS with the largest RSS measurement in eachtime instant The proposed strategy is much easier toimplement in the practical application to reduce thecommunication traffic and lower the requirement ofthe hardware

(iii) The unscented Kalman filter (UKF) is employed totreat the estimator The point of sampling conquersthe severe nonlinear of the mapping of RSS torange By integrating the filter into IMM the modelprobabilities are updated according to the likelihoodsof errors A combination of the weighted respectiveestimations is achieved as the output result

a b

c

Anchor sensorMT trace

Figure 1 A corridor situation with LOS and NLOS conditions

(iv) A complete mobile localization platform whichworks on STM32F107VCT and AT86RF231 is estab-lished by applying the algorithm into a real indoorscenarioThe experiments illustrate that the proposedestimator outperforms some other localization meth-ods [14 24 25]

The rest of this paper is organized as follows Section 2briefly states the problem of mobile localization in a time-variant environment Section 3 derives a composite measure-ment model and validates it in a real application scenarioSection 4 presents the proposed UKF based IMM schemewith both theoretical analysis and formula derivation Anumerical study relying on the practical experiment data ispresented in Section 5 to illustrate the merits of the proposedalgorithm and some deep analysis compared with otherworks Finally conclusions are drawn in Section 6

2 Problem Statement

A synchronous network of 119873 anchor sensors (AS) fixed atthe known positions is assumed Their positions are denotedby a set of m-dimensional vectors 119860 = [119860

11198602sdot sdot sdot 119860119899]

respectively 119898 = 2 is considered in this paper although119898 = 3 is also possible for a stereoscopic situation Thus for119894 isin 1 2 119873 the 119894th AS is fixed at 119860

119894= [119909119904

119894 119910119904

119894]

In order to state the problem a corridor situation inour lab building is illustrated in Figure 1 The solid trianglesrepresent the ASs the dashed line represents the trace of aMT All the ASs are located in the rooms along the corridorDue to the doors someASs can be exposed to theMT at somecertain locations

The MT moves along the corridor The unknown posi-tions of the MT at time instant 119905 are denoted by a 2119898-dimensional vector

119883119905= [119909119905 119905 119910119905 119910119905] (1)

International Journal of Distributed Sensor Networks 3

b

Wall

Anchor sensor

Mobile terminal

a

c

Figure 2 Propagating conditions alteration

where (119909119905 119910119905) denotes the coordinates of the MT and (

119905 119910119905)

denotes the velocity of the MT in 119909 and 119910 direction TheMTrsquos state updates over time according to the random forcemodel

119883119905= 119865119883119905minus1

+ 119866120596119905minus1 (2)

where

119865 =

[[[[[

[

1 0 119879 0

0 1 0 0

0 0 1 119879

0 0 0 1

]]]]]

]

119866 =

[[[[[[[[

[

1198792

20

01198792

2119879 0

0 119879

]]]]]]]]

]

(3)

119883119905minus1

represents the state of the MT at time instant 119905 minus 1 Theprocess noise 120596

119905minus1is modeled as zero mean iid Gaussian

noise with covariance matrix 119876When theMTmoves a discovery signal can be measured

by the ASs As the typical case in Figure 2 shows thediscovery signal sent by the MT at 119887 transmits indirectly tothe AS at 119888 The signal path between 119886 and 119888 is regarded asLOS propagation condition while the one between 119887 and 119888 isregarded as NLOS propagation condition due to the wall

Because of the presence of the walls between the MT andthe AS the signal may be reflected diffracted and scatteredThe signal path loss may be increased by the encounteredobjects and sharp corners As a result of the absorptionof the walls the propagation medium is also normallydifferent in diverse directions On account of the complicatedeffects on the RSS measurement the measurement model amathematical mapping of RSS to range is hardly denoted bya single curve

As a consequence the measurement model RSS = 119867(119889)

is illustrated as the superimposed effects on the path lossbetween the MT and the AS In addition the measurementmodel is time variant altering between LOS andNLOS fadingchannel which makes it difficult to estimate the range basedon the RSS values Some other works [24 25] demonstrate themultiple measurement models in a mixed environment andestimate the locations in the time-dependent fading channelsHowever the approximate description of the alternative

measurement model is not complete lack of the irregularityof a radio pattern And the estimator is not robust enoughespecially when the ASs are not deployed densely whichindicates that the nonlinearmapping enlarges the filter errors

From the above an appropriate measurement model isan urgent issue to approximate the mapping of RSS to rangeMore importantly it is essential to propose an adaptivelocalization estimator in order to accommodate the time-variant fading channels

3 A Composite Measurement Model

In order to approximate the time-variant measurementmodel especially the irregular effect on wireless signal fadingcaused by the obstacles the RIM model has been introduced[26] Furthermore as a measurement the power measure-ment 119875

119894(119905) corresponding to the RSS value between the AS

119894

and the MT at the time instant 119905 can be modeled as

119875119894(119905) = 119875

119879minus 1198751198710minus 10120578log

10(119889119894

1198890

)

1+DOI+ ] (119905)

119894 = 1 2 119873

(4)

The parameter DOI (degree of irregularity) is defined asthe maximum path loss percentage variation per unit degreechange in the direction of the radio propagation [26] Asshown in Figure 3 when the DOI is set to 0 there is no rangevariation and the communication range is a perfect sphereHowever when the DOI value increases the communicationrange becomes more and more irregular

For different fading channels of LOS and NLOS thereare various DOI values respectively marked as DOILOSDOINLOS To simplify model (4) a general expression for theLOS and NLOS channels is as follows

119875119894(119905) = 119875

119879minus 1198751198710minus 10120578LOSlog10 (

119889119894

1198890

) + ]LOS (119905)

119894 = 1 2 119873

119875119894(119905) = 119875

119879minus 1198751198710minus 10120578NLOSlog10 (

119889119894

1198890

) + ]NLOS (119905)

119894 = 1 2 119873

(5)

where ]LOS is the RSS measurement noise modeled as awhite Gaussian with 119873(120583LOS 120590

2

LOS) and ]NLOS is the RSSmeasurement noise modeled as a white Gaussian with119873(120583NLOS 120590

2

NLOS) The parameters of 120578 119875119879 1198890 and 119875

0are

channel attenuation coefficient transmitting power of theRF model reference distance between the MT and the ASand the path loss at 119889

0 119889119894is the Euclidean distance between

the MT (119909119905 119910119910) and the 119894th AS (119909

119904

119894 119910119904

119894) which is defined

as

119889119894= radic(119909

119905minus 119909119904

119894)2

+ (119910119905minus 119910119904

119894)2

119894 = 1 2 119873 (6)


minus100 0 100 200minus200

minus100

0

100

Propagation distance (m)

Prop

agat

ion

dist

ance

(m)

minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)

minus100 0 100minus100

minus50

0

50

100


Prop

agat

ion

dist

ance

(m)

DOI Modelpound∘DOI = 0

DOI Modelpound∘DOI = 001 DOI Modelpound∘DOI = 002


minus100

0

100

200

Propagation distance (m)Pr

opag

atio

n di

stan

ce (m

)


Figure 3 RIM model with DOI = 0 0005 001 002

Figure 4 Mobile node and anchor node

To determine the key parameters above we make anexperiment of RSS measurements collection using 8 ASs and1 MT over a set of distances ranging from 1m to 28m withLOS andNLOS conditionsThewhole experiment takesmorethan 6 hours with a sampling period 1 s Figure 4 shows theanchor node and the mobile node The node is composed ofa STM32F107VCT MCU running with an embedded system120583COSII and a ATRF231 RF model working at 24GHz

The experiment dataset contains about 380 sets of mea-surements and each set corresponds to a certain MTrsquos


0 5 10 15 20 25 30minus80

minus70

minus60

minus50

minus40

Distance (m)

RSSI

(dBm

)LOS

Practical samplesBest-fit samples

Distance (m)


0 5 10 15 20 25 30minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS

Figure 5 Mapping of RSSI to range at various locations

0 05 1 15minus80

minus70

minus60

minus50

minus40

RSSI

(dBm

)

LOS



04 06 08 1 12 14 16 18minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS

log10(distance) (m) log10(distance) (m)

Figure 6 Mapping of RSSI to logarithmic range at various locations

location namely the sampling location Each set of datacontains more than 28800 pairs of RSSs and distances Forany pair of RSS measurement and the known distance aconstraint should be satisfied

min120578120583

sum

119894119895

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

RSS119894119895minus

[[[

[

119875119879minus 1198751198710

minus10120578log10(

radic(119909119895minus 119909119904

119894)2

+ (119910119895minus 119910119904

119894)2

1198890

)+ 120583]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119894 = 1 2 119873 119895 = 1 2 119872

(7)119873 is the number of ASs and 119872 is the quantity of samplinglocations RSS119894

119895indicates the mean of RSS measurements to

119894th AS (119909119894 119910119894) at jth sampling location (119909

119895 119910119895) For any pair

of dataset a ldquoLOSrdquo or ldquoNLOSrdquo indicator is given by using alaser rangefinder Therefore we get two categories of datasetone is for LOS and the other is for NLOS For either categorythe best-fit 120578 and 120583 are achieved utilizing LS (least square)estimator as follows

119875119879minus 1198751198710minus 10120578log

10(

119889119894

119895

1198890

) + 120583 (119895) = RSS119894119895

119894 isin 1119873 119895 isin 1119872

(8)

[[[[[[[[[[

[

1 minus10log10(1198891

1

1198890

)

1 minus10log10(

119889119894

119895

1198890

)

]]]]]]]]]]

]

[120583

120578] =

[[

[

RSS11minus 119875119879+ 1198751198710

RSS119894119895minus 119875119879+ 1198751198710

]]

]

(9)

where (9) is the matrix format of (8) For convenience wedefine the left matrix of (9) as119860 and the right matrix of (9) as119861 Then the optimal solution is given below

[120583

120578] = (119860

119879119860)minus1

119860119879119861 (10)

In order to validate the efficiency of the optimal 120583 and120578 for LOS or NLOS we examine it in another experimentwhere 7 ASs and 1 MT are set at different positions Figure 5compares the fitting RSSs according to 120583 and 120578 above with thereal RSS measurements at a series of sampling positions inLOS and NLOS conditions respectively Figure 6 illustratesthe approximation result in the form of logarithmic rangesIt is clear that the fitting curve can approximate the fadingtrend of RSS measurements along with the increasing rangesin both propagation conditions The proposed compositemeasurement model makes a more complete description ofthe complex fading model for LOS and NLOS propagationchannels

119885119894(119905) is the RSS measurement measured by the 119894th AS

at time instant 119905 and 119867(119883119905AS119894) indicates the mathematical

mapping of the RSS measurement and the MTrsquos state 119883119905


Model 1NLOS

Model 2LOSp11

p12

p21

p22

Figure 7 Markov switching system

As a consequence the composite measurement model ofa LOSNLOS mixed fading channel is defined as follows

119885119894(119905) = 119867 (119883

119905AS119894)

=


119889119894

1198890

) + ]LOS (119905)

for LOS


119889119894

1198890

) + ]NLOS (119905)

for NLOS(11)

where 120578LOS = 13063 120578NLOS = 19508 ]LOS = 119873(minus0591367567) and ]NLOS = 119873(57512 447445) The definitions of119875119879 1198751198710 1198890 and 119889

119894are identical to those in (5)

4 UKF Based IMM Localization Estimation

As mentioned above since the signal propagation indoor iscomplicated the fading condition alters between LOS andNLOS cases The transmission channels between the AS andthe MT are considered as a switching mode system In otherwords a LOSNLOS transition occurs when the MT movesinto an environment with the different properties of thepropagationmediumA two-stateMarkov process in Figure 7is employed to describe the switching system

A singlemeasurementmodel corresponding to one prop-agation condition cannot adjust to both LOS and NLOSsituations It is necessary to introduce a mixed and adaptivescheme against this challenge Therefore an UKF basedIMM localization estimator is adopted for the LOSNLOSenvironment

41 General Concept The flowchart of the UKF based IMMlocalization estimation is illustrated in Figure 8

First as the MT moves it broadcasts a discovery signalBased on greedy anchor sensor selection the AS with thelargest RSS measurement 119885(119905) is chosen from the candidateswhich have received the discovery signal It should be notedthat a larger RSS measurement indicates either a LOS prop-agation model or a shorter distance between the transmitterand receiver It is instinctive and easy to implement

Then the state 119883(119905 | 119905) for the 119894th AS is simultaneouslyestimated by two parallel UKFs according to the LOS andNLOS models respectively The mode probabilities of thepresent measurement model can be calculated and updated

by a likelihood function via the respective estimation errorAfterwards the IMM structure combines the independentestimation results with their different mode probabilities Forthe next time instant the prior state transition probabilitiesrely on a constant Markov switching matrix and the previousmode probabilities

42 Greedy Anchor Sensor Selection As the MT broadcaststhe discovery signal in a constant transmitting power eachASreceives this singal and obtains a different RSS measurementdue to the different distance After that each AS enables atimer with an initial 119879

119894 119894 isin 1 2 119873

119879119894=

1198790

119885119894(119905)

(12)

where 119885119894(119905) is the RSS measured by the 119894th AS at time instant

119905 1198790is a constant to adjust each 119879

119894to a practical value for the

hardware clock According to (12) a larger RSS measurementindicates a shorter 119879

119894 Therefore the AS with the largest RSS

times out firstly then it replies to theMT amessage includingits own coordinates and the largest RSS 119885

119894(119905) Once the MT

received any reply it broadcasts a stop-reply signal to all theASs The subsequent ASs abort their timers and return to thestate of monitoring

In this way the largest RSS is collected to the MT In(11) the RSS is related to 119875

119879 However in [24 25] 119875

119879is

assumed to be identical and time-invariant for all the ASsIt is usually unpractical for most of the applications On thisview the proposed anchor sensor selection ensures all RSSmeasurements are based on one 119875

119879which is emitted by the

MT rather than any AS in each time instantFurthermore in a practical system it is hard to distin-

guish the ASs if a subtle difference between the first smallest119879119894and the second smallest119879

119895happens It is injudiciousness to

enlarge 1198790 because it leads to a more significant delay From

this view an improved scheme is presented in Figure 9

43 UKF-IMM Algorithm The proposed algorithm consistsof three major stages interaction filtering and combination

431 Interaction 119894 119895 isin 1 2 1 for the LOS estimator and 2for the NLOS estimator

We suppose that the currentmultiple-mode states dependon the previousmodes and all the transition probabilities areknownThemixing probability frommode 119894 tomode 119895 can bedenoted as

120583119894|119895(119905 minus 1 | 119905 minus 1) =

119901119894119895120583119894(119905 minus 1 | 119905 minus 1)

119888119895

(13)

where 119901119894119895is the Markov transition probability frommode 119894 to

mode j 120583119894(119905 minus 1 | 119905 minus 1) is the probability of mode 119894 at time

instant 119905 minus 1 and 119888119895is a normalization factor for the prior

mode and is expressed as

119888119895= sum

119894

119901119894119895120583119894(119905 minus 1 | 119905 minus 1) (14)


Interaction

IMM structure

Mixingprobability

Anchor sensor selection

Markov switchingprobability

Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)

X01(t minus 1 | t minus 1)


120583i|j(t minus 1 | t minus 1)

120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)

Figure 8 Flowchart of the UKF based IMM localization estimation

Themixed prior state1198830119895(119905minus1 | 119905minus1) and covariance119875

0119895(119905minus1 |

119905minus1) for the jth mode-matched estimator at time instant 119905minus1can be obtained by

1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894

[119883119894(119905 minus 1 | 119905 minus 1) minus 119883

0119894(119905 minus 1 | 119905 minus 1)]

times [119883119894(119905 minus 1 | 119905 minus 1) minus 119883

0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)

where119883119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are the state estimation

and covariance for the 119894th mode-matched estimator at timeinstant 119905minus1 respectively119883

119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are

prepared by the previous mode-matched unscented Kalmanfiltering stage

432 Filtering Based on the prior knowledge that the mea-surement models are quite different between LOS and NLOSconditions two unscented Kalman filters are designed forthese two measurement models

Initializing For either estimator the initial state 119883119894(0) and

119875119894(0) are obtained from the system initialization

Sampling A set of sigma points 119878119895= 120594119895119882119895 is generated

so that the mean and the covariance of the samples are

MT broadcasts a discovery signal

ASs measure RSSs and set

MT receives the anchor info

A subtle difference between MT chooses the AS which is

MT broadcasts a stop-reply signal

Yes

No

Ti

and RSS with the smallest Ti

Ti and Tj happens nearest to FX(t | t minus 1)

s

Figure 9 The scheme of greedy anchor sensor selection

119883119894(119905 minus 1 | 119905 minus 1) and 119875

119894(119905 minus 1 | 119905 minus 1) The samples are not

drawn randomly but according to a specific deterministicalgorithm as follows

1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

+ (radic(119899 + 120581) 119875119894(119905 minus 1 | 119905 minus 1))

119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

minus (radic(119899 + 120581) 119875119894(119905 minus 1 | 119905 minus 1))

119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)

where 119899 is the dimension of the state estimation119883119894(119905 minus 1 | 119905 minus 1) and 120581 is the scaling factor which determines

the approximating precision When the state estimation119883119894(119905 minus 1 | 119905 minus 1) is assumed Gaussian an useful heuristic is to

select 119899 + 120581 = 3 [27]The weights 119882119898

119895 119882119888119895should also meet some constraint

principles Here 120572 determines the ldquosizerdquo of the sigma pointdistribution It is recommended to be a small value to avoidsampling nonlocal effects when the system is nonlinearstrongly 120573 in (17) is a nonnegative weighting term toincorporate knowledge of the higher order components of thedistribution For a Gaussian assumption the optimal 120573 is 2This parameter can also control the deviation in the kurtosiswhich affects the ldquoheavinessrdquo of the tails of the posterior statedistribution [28]

TimeUpdate Instantiate each point in (16) by the state updatefunction and the measurement function to yield the set oftransformed sigma points

120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)

The mean is given by the weighted sum of the transformedpoints And the covariance is the weighted outer product ofthe transformed sigma points

119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)

where 119876 is the covariance of Gaussian process noise asmentioned in (2)

Measurement Update With the chosen RSS measurement119885119894(119905) a measurement update is computed

119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)

where119870 is the Kalman gain and 119877 is the measurement noiseFor a LOS model 119877 = 120590

2

LOS for a NLOS model 119877 = 1205902

LOS]119894(119905) 119875119894(119905 | 119905) and 119883

119894(119905 | 119905) should be substituted to the

process of combination to derive the mode probabilities andthe weighted estimation result

]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)

433 Combination When the estimated states are obtainedby both estimators respectively the model likelihoods andprobabilities are required to be calculated in the combinationmodule

Firstly the model likelihood Λ119894(119905) is measured by a

Gaussian density function of residual error ]119894(119905) in (21)

with zero mean and covariance 119878119894(119905) = 119875

119885119894119885119894in (20) The

updated 120583119894(119905 | 119905) is a normalized weighted sum of the model

likelihoods and the previous prior mode probabilities in (14)

Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)

According to the posterior mode probability 120583119894(119905 | 119905) the

combined estimation can be derived as

119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)

119894 = 1 2 respectively for LOS and NLOS model(24)

Here the combined result is exported in the formof119883(119905 | 119905) =(119909119905 119910119905) For each estimator the estimated 119883

119894(119905 | 119905) and 119875

119894(119905 |

119905) return to the interaction process at the next time instant


Table 1 The set of the experiment parameters

Experiment parameters ValuesMonitoring region 50m times 50mMonitoring time 88 sNumber of ASs119873 14AS deployment error 01mSampling period 119879 1 s120578 13063 for LOS 19508 for NLOS

Measurement noise ] 119873(minus0591 367567) for LOS119873(57512 447445) for NLOS

Process noise 120596 [0012

0

0 0012]

Transmitting power 119875119879 3 dBm

Path loss at 1198890PL0 46 dBm

5 Numerical Study

In this section we use the experiment platform which ismentioned in Section 3 to evaluate the performance of theproposed localization algorithm Firstly we describe ourexperiment environment and parameters Then we definethe performance metrics to compare the proposed algorithmwith other works

51 Experiment Environment We set up an indoor wirelesssensor network with N ASs to monitor a 50m times 50m archfield as shown in Figure 1 The total monitoring time is88 s and the sampling period 119879 is 1 s All the ASs withinthe monitoring region have the same structure Each AShas the ability to obtain the RSS of the MTrsquos signal Forthe LOS propagation channel 120578LOS is set to 13063 and theGaussian noise ]LOS is set to 119873(minus0591 367567) for theNLOS propagation channel 120578NLOS is set to 19508 and theGaussian noise ]NLOS is set to 119873(57512 447445) The MTbroadcasts its discovery signal at a power output of 3 dBmand the 119875119871

0is 46 dBm All the parameters are acquired from

the experiment in Section 3 In both cases the process noise120596 119890119905119886 119875

119879 and 119875119871

0are illustrated in Table 1

In order to describe the actual experiment we set theinitial states as follows A MT starts to move along thecorridor at 119905 = 0 with an initial position and velocity[0226 06 12 0]

119879 Then the MT makes a turn at the cornerof the corridor and continues to move

52 Performance Metrics To evaluate the performance ofthe proposed algorithm and other frameworks we calculatethe root of mean square errors (RMSEs) of localizationestimations at each time instant The RMSE metric [14] isdefined as follows

RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)

(119909119905 119910119905) is the estimated coordinate of MT at 119905 time instant

while (119909119905 119910119905) is the true position at that time instant A time

series of RMSEs of positions and velocities will be given inthe following subsection

0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)

AnchorReal traceIMM-UKF

Figure 10 Tracking results of the proposed algorithm

For the mixed and switching channel situation an auxil-iary but important issue should be observed The posteriormodel probability 119875(Model | 119885(119905)) is also a major pointWe also compare the estimated posterior probability with theactual model probability

53 Results and Analysis

531 Performance of Localization In order to validate theperformance of localization accuracy a comparison with anEKF based LOS model an EKF based on NLOS modelIMM-EKF [24 25] and MPDA [14] is carried out below InFigure 10 the tracking trajectory obtained by the proposedalgorithm is shown Compared with the real trace which isdenoted by the solid line the trajectory generated by the UKFbased IMM algorithm is quite close and follows the movingtrend although a maneuver turn happens at 48 s

It is clear that the whole trajectory can be divided into twoparts one is from the starting position to the sudden turningpoint and the other is from the turning point to the endpoint The trajectory estimated by UKF based IMM schemehas larger errors in the second part than in the first part Itis noticed that the parameters of the measurement model areobtained in a similar scenario to the first part It is appreciablethat the prior knowledge about the fading conditions fits thefirst part of the trace more precisely On the other hand theproposed algorithm is able to offset the errors produced by abiased measurement model in some degree

As shown in Figure 11 the performance of UKF basedIMM is obviously superior to other works During the first20 s the fading channel is mainly LOS condition with a slightpropagation variation Then the channel condition changesto NLOS The proposed algorithm remains a smaller RMSEduring the next 20 s In the rest of the monitoring time thefading channel switchovers several times The EKF based


Table 2 The time-averaging localization RMSEs of the proposed algorithm and other works

RMSE IMM-UKF IMM-EKF EKF based on LOS EKF based on NLOS MPDAV119909[ms] 0013 0038 01656 00334 00868

V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)

IMM-UKFEKF based on LOSEKF based on NLOS

MPDAIMM-EKF

Figure 11 RMSEs of positions estimated by UKF based IMM andother works

on LOS gradually diverges with accumulative errors due toan improper measurement model in some certain periodThe EKF based on NLOS performs better than the EKFbased on LOS because the channel remains in the state ofNLOS much longer Neither of the works relying on a singlemodel or a single measurement reach a satisfactory levelThe MPDA benefits from the multiple measurements butfails to adapt to the time-variant environment especiallywhen there is no subset of available measurements in somepositions It is the key for an estimator not only to updatethe prediction state with the current measurement but alsoto adjust to an actual-matched system model The proposedalgorithm also performs better than IMM-EKF although themain ideas are the same However the UKF based IMMconquers amore serious nonlinear system and the greedy ASselection strategy weakens the influence of the poor-qualitymeasurements

In Figure 12 the comparison of the velocities of 119883

and 119884 directions is summarized Table 2 lists the time-averaging localization RMSEs of the proposed algorithmand other works Considering the maximum velocity in 119883

or 119884 direction during the maneuvering is about 06msthe velocity errors estimated by MPDA and EKF based onLOS are sizable It is obvious that the proposed UKF basedIMM algorithm remains much smaller errors in velocitiesAt 47 s in Figure 12 the RMSEs of our algorithm undergoes

Table 3 Average localization RMSEs for different numbers of ASs

Algorithms [m] Number of ASs14 12 10 8 6

UKF based IMM 085 091 089 105 119IMM-EKF 143 141 172 180 200EKF based on LOS 1008 1074 920 110 1061EKF based on NLOS 218 209 223 248 304MPDA 518 69 840 1127 1594

a estimator transition while some other works encounter thebreaking points

The model transition details are shown in Figure 13The marked line depicts the posterior model probability119875(Model | 119885(119905)) estimated by UKF based IMM and the solidline indicates the real model probability Noticing that 0 is forNLOS channel 1 is for LOS channelThe proposed algorithmseizes most of the inflection points and its estimated modelprobability approximates to the real situation With the119875(Model | 119885(119905)) the efficient method can choose a matchedmeasurement model and combine the filters with a set ofreasonable weights

532 Performance of Robustness For the referencedworks in[22 24 25] localization estimators using IMM and EKF withTOA or RSS measurements in a mixed propagation modelwere presented The employed EKF achieves an acceptableperformance in those cases However according to thefading channels and deployment environment the quotativeestimator encounters a performance degradation especiallywhen the distance between the MT and the AS increasesTable 3 also shows that as the numbers of AS decrease theaverage RMSEs of other works increase obviously or remainas a larger level Some discussion in detail comes below

An UKF recommended by the proposed algorithm per-forms better than an EKF It is proved that the approximationprecision is closely 3rd-order of Taylor expansion at leastwhile an EKF depends on 1st-order of Taylor expansionBesides that UKFrsquos computation complexity of 1198992 is mucheasier to implement in a practical application due to noexplicit calculation of a Jacobians or Hessians like an EKFdoes In Figure 14 each marker point represents an averagelocalization RMSE during the monitoring time for a certainnumber of anchors As the numbers of anchors decrease thedistance between the MT and any specific anchor increasesThen the referenced IMM-EKF decays rapidly whereas theproposed UKF based IMM is hardly affected by the sparsityof anchors


0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)

Figure 12 RMSEs of velocity estimated by UKF based IMM and other works

0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)

Real channel conditionEstimated channel condition

Prob

abili

ty (m

odel|Z(t))

Figure 13 Model probabilities of environment (estimated by UKFbased IMM and the real situation)

Through [29] let 119883119898 be a random variable with mean 119883and covariance119875

119883119883119885 is related to119883119898 through the nonlinear

transformation namely the measurement model (5)

119885 (119883119898) = 119867 (119883

119898AS119894) (26)

The EKF used refers to the Taylor series expansion of thisequation Let 119883119898 = 119883

119898

+ 120575119883119898 where 120575

119883119898 is a zero mean

random variable with covariance 119875119883119883

Expanding119867(sdot) about119883119898

119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752

119883119898 + sdot sdot sdot

= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF

Figure 14 Comparison of the average RMSEs of UKF based IMMand IMM-EKF for different numbers of AS

where the 1st-order term in the multidimensional is

nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909

119904 119910119904) is the coordinate of

the specific AS 119888 is a constant Considering in (27) asthe numbers of anchors decrease the measurement rangebecomes larger which leads to a nonignorable term Inconsequence an EKF estimator fails to approximate thehigher order term in (27)

It is crucial for an UKF that it approximates an arbitrarynonlinear system with the weighted sigma points Thesepoints are deterministically chosen so that certain propertiesmatch those of the prior distribution With this set of points


an UKF guarantees the same performance as the truncated3rd-order filter

6 Conclusion

In this paper we address the problem of robust position-ing of a mobile terminal using RSS measurements in amixed LOSNLOS environment The original measurementmodels have been reformulated as nonlinear ones whichindicates the anisotropy caused by the indoor obstaclesin a NLOS case We construct the measurement modelswhich completely describe the differences between LOS andNLOS conditions for an indoor application and validateour composite measurement model in a real scenario Inparticular the UKF based IMM localization estimator isproposed for mobile location estimation in a practical roughwireless environments An UKF works better than an EKFdue to its superior ability to approximate the nonlinear systemin a higher order With the aid of the likelihood functionto determine the mode probabilities in LOS and NLOSthe proposed UKF based IMM could accurately estimaterange distance between the MT and the AS even withthe channels switching randomly between LOS and NLOSconditions

The real experiment results illustrate that the perfor-mance of our proposed algorithm achieves high accuracyeven in a complex environment where the LOS and NLOSchannel conditions switch frequently with obviously differentfading Furthermore the UKF based IMM scheme manifestsrobustness against the sparse deployment of ASs It makes itmore practical to utilize a localization system widely

Conflict of Interests

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

Acknowledgments

This work is supported in part by the Strategic PriorityResearch Program of the Chinese Academy of Sciences(CAS) under Grant no XDA06020300 and the IoT NationalStandards System Research and Industrial Application andDemonstration based on Information Perception and Iden-tification Technology of Shanghai Science and TechnologyCommission (SSTC) research projects under Grant no12DZ0500100

References

[1] F Gustafsson and F Gunnarsson ldquoMobile positioning usingwireless networks possibilities and fundamental limitationsbased on availablewireless networkmeasurementsrdquo IEEE SignalProcessing Magazine vol 22 no 4 pp 41ndash53 2005

[2] A H Sayed A Tarighat and N Khajehnouri ldquoNetwork-basedwireless location challenges faced in developing techniques foraccurate wireless location informationrdquo IEEE Signal ProcessingMagazine vol 22 no 4 pp 24ndash40 2005

[3] M McGuire K N Plataniotis and A N VenetsanopoulosldquoRobust estimation of mobile terminal positionrdquo ElectronicsLetters vol 36 no 16 pp 1426ndash1428 2000

[4] T Perala and R Piche ldquoRobust extended Kalman filtering inhybrid positioning applicationsrdquo in Proceedings of the 4thWork-shop on Positioning Navigation and Communication (WPNCrsquo07) pp 55ndash63 Hannover Germany March 2007

[5] G-L Sun andW Guo ldquoBootstrapping M-estimators for reduc-ing errors due to non-line-of-sight (NLOS) propagationrdquo IEEECommunications Letters vol 8 no 8 pp 509ndash510 2004

[6] C Ma R Klukas and G Lachapelle ldquoA nonline-of-sight error-mitigation method for TOAmeasurementsrdquo IEEE Transactionson Vehicular Technology vol 56 no 2 pp 641ndash651 2007

[7] H Miao K Yu and M J Juntti ldquoPositioning for NLOSpropagation algorithm derivations and Cramer-Rao boundsrdquoIEEE Transactions on Vehicular Technology vol 56 no 5 pp2568ndash2580 2007

[8] S Bartelmaos K Abed-Meraim and E Grosicki ldquoGeneralselection criteria for mobile location in NLoS situationsrdquo IEEETransactions on Wireless Communications vol 7 no 11 pp4393ndash4403 2008

[9] K G Yu and Y J Guo ldquoStatistical NLOS identification basedon AOA TOA and signal strengthrdquo IEEE Transactions onVehicular Technology vol 58 no 1 pp 274ndash286 2009

[10] L Cong and W Zhuang ldquoNonline-of-sight error mitigation inmobile locationrdquo IEEE Transactions on Wireless Communica-tions vol 4 no 2 pp 560ndash573 2005

[11] U Hammes and A M Zoubir ldquoRobust mobile terminal track-ing in NLOS environments based on data associationrdquo IEEETransactions on Signal Processing vol 58 no 11 pp 5872ndash58822010

[12] F Quitin C Oestges F Horlin and P deDoncker ldquoPolarizationmeasurements and modeling in indoor NLOS environmentsrdquoIEEE Transactions onWireless Communications vol 9 no 1 pp21ndash25 2010

[13] L Yi S G Razul Z Lin and C-M See ldquoRoad-constraintassisted target tracking in mixed LOSNLOS environmentsbased on TDOA measurementsrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems (ISCAS rsquo12)pp 2581ndash2584 Seoul Republic of Korea May 2012

[14] UHammes EWolsztynski andAM Zoubir ldquoRobust trackingand geolocation for wireless networks in NLOS environmentsrdquoIEEE Journal on Selected Topics in Signal Processing vol 3 no 5pp 889ndash901 2009

[15] W Wei X Jin-Yu and Z Zhong-Liang ldquoA new NLOS errormitigation algorithm in location estimationrdquo IEEE Transactionson Vehicular Technology vol 54 no 6 pp 2048ndash2053 2005

[16] S Marano W M Gifford H Wymeersch and M Z WinldquoNLOS identification and mitigation for localization based onUWB experimental datardquo IEEE Journal on Selected Areas inCommunications vol 28 no 7 pp 1026ndash1035 2010

[17] L Yi S G Razul Z Lin and C M See ldquoTarget trackingin mixed LOSNLOS environments based on individual mea-surement estimation and LOS detectionrdquo IEEE Transactions onWireless Communications vol 13 no 1 pp 99ndash111 2014

[18] J M Huerta A Giremus J Vidal and J-Y Tourneret ldquoJointparticle filter and UKF position tracking under strong nlossituationrdquo in Proceedings of the IEEESP 14th Workshop onStatistical Signal Processing (SSP rsquo07) pp 537ndash541 IEEE August2007


[19] J Zhen and S Zhang ldquoAdaptive AR model based robustmobile location estimation approach in NLOS environmentrdquo inProceedings of the 59th IEEE Vehicular Technology Conference(VTC rsquo04) vol 5 pp 2682ndash2685 May 2004

[20] H A P Blom and Y Bar-Shalom ldquoInteracting multiple modelalgorithm for systems with Markovian switching coefficientsrdquoIEEE Transactions on Automatic Control vol 33 no 8 pp 780ndash783 1988

[21] E Mazor A Averbuch Y Bar-Shalom and J Dayan ldquoInteract-ing multiple model methods in target tracking a surveyrdquo IEEETransactions on Aerospace and Electronic Systems vol 34 no 1pp 103ndash123 1998

[22] J-F Liao and B-S Chen ldquoRobust mobile location estimatorwith NLOS mitigation using interacting multiple model algo-rithmrdquo IEEE Transactions on Wireless Communications vol 5no 11 pp 3002ndash3006 2006

[23] C-Y Yang B-S Chen and F-K Liao ldquoMobile locationestimation using fuzzy-based IMM and data fusionrdquo IEEETransactions onMobile Computing vol 9 no 10 pp 1424ndash14362010

[24] B-S Chen C-Y Yang F-K Liao and J-F Liao ldquoMobile loca-tion estimator in a rough wireless environment using extendedKalman-based IMM and data fusionrdquo IEEE Transactions onVehicular Technology vol 58 no 3 pp 1157ndash1169 2009

[25] Y Z Zhang W Y Fu D F Wei J J Jiang and B YangldquoMoving target localization in indoor wireless sensor networksmixed with LOSNLOS situationsrdquo Eurasip Journal on WirelessCommunications and Networking vol 2013 no 1 article 2912013

[26] G Zhou T He S Krishnamurthy and J A Stankovic ldquoModelsand solutions for radio irregularity in wireless sensor networksrdquoACMTransactions on Sensor Networks vol 2 no 2 pp 221ndash2622006

[27] S J Julier and J K Uhlmann ldquoA new extension of the Kalmanfilter to nonlinear systemsrdquo in Proceedings of the InternationalSymposium onAerospaceDefense Sensing Simulation and Con-trols vol 3 p 32 Orlando Fla USA 1997

[28] R van der Merwe A Doucet N de Freitas and E WanldquoThe unscented particle filterrdquo in Proceedings of the NeuralInformation Processing Systems Conference (NIPS rsquo00) pp 584ndash590 2000

[29] S J Julier ldquoThe scaled unscented transformationrdquo in Proceed-ings of the American Control Conference vol 6 pp 4555ndash4559May 2002

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

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



proposed in [14]Thismethod constructs different subgroupsof measurements Then intermediate results of the LOSsubgroups are combined to reach the final estimated positionof the moving target However in severe NLOS conditionsit is very likely that no subgroup is detected as LOS at manytime instances As a result no new measurement is availableto update the filtering

While the estimation-based approaches utilize the statis-tical properties of the LOS and NLOS noises and reconstructposterior estimated position [3ndash5 19] they are robust inmany NLOS conditions but are not efficient as the highcomputational cost and incomplete measurement modelsAs for frequent LOSNLOS transition in a mixed environ-ment some good progress with interactive multiple model(IMM) has been made in this category It [20ndash22] hasbeen demonstrated as one of the most effective schemesfor position estimation in a hybrid dynamic system underuncertain environments Yang et al [23] present a locationestimation algorithm using fuzzy-based IMM estimator In[24] IMM is adopted for the MT location estimation as wellHowever the indoor environment is quite different from theurban structure in [24] The system model and estimatorseems to be an oversimplification for an indoor applicationsince it fails to consider the signal attenuated by walls orfurniture

Most existing research works on a localization algo-rithm under the assumption of different probability distri-bution functions of the LOS and NLOS noise Howeversome other significant distinctions exist between LOS andNLOS propagation resulting in various new measurementmodels

In this paper we present an efficient MT positioningalgorithm using UKF based IMM with individual RSS mea-surements in a mixed LOSNLOS indoor environment Inaddition the proposed framework has been evaluated on apractical platform in a real application situation The mainworks are as follows

(i) An effective measurement model is derived todescribe the influence of fading caused by the indoorenvironment and defined as a superposition RIM andGaussian noise Furthermore the model has beenverified in an actual indoor building containing theLOS and NLOS channel conditions

(ii) A greedyAS selection strategy is introduced to choosethe AS with the largest RSS measurement in eachtime instant The proposed strategy is much easier toimplement in the practical application to reduce thecommunication traffic and lower the requirement ofthe hardware

(iii) The unscented Kalman filter (UKF) is employed totreat the estimator The point of sampling conquersthe severe nonlinear of the mapping of RSS torange By integrating the filter into IMM the modelprobabilities are updated according to the likelihoodsof errors A combination of the weighted respectiveestimations is achieved as the output result

a b

c

Anchor sensorMT trace

Figure 1 A corridor situation with LOS and NLOS conditions

(iv) A complete mobile localization platform whichworks on STM32F107VCT and AT86RF231 is estab-lished by applying the algorithm into a real indoorscenarioThe experiments illustrate that the proposedestimator outperforms some other localization meth-ods [14 24 25]

The rest of this paper is organized as follows Section 2briefly states the problem of mobile localization in a time-variant environment Section 3 derives a composite measure-ment model and validates it in a real application scenarioSection 4 presents the proposed UKF based IMM schemewith both theoretical analysis and formula derivation Anumerical study relying on the practical experiment data ispresented in Section 5 to illustrate the merits of the proposedalgorithm and some deep analysis compared with otherworks Finally conclusions are drawn in Section 6

2 Problem Statement

A synchronous network of 119873 anchor sensors (AS) fixed atthe known positions is assumed Their positions are denotedby a set of m-dimensional vectors 119860 = [119860

11198602sdot sdot sdot 119860119899]

respectively 119898 = 2 is considered in this paper although119898 = 3 is also possible for a stereoscopic situation Thus for119894 isin 1 2 119873 the 119894th AS is fixed at 119860

119894= [119909119904

119894 119910119904

119894]

In order to state the problem a corridor situation inour lab building is illustrated in Figure 1 The solid trianglesrepresent the ASs the dashed line represents the trace of aMT All the ASs are located in the rooms along the corridorDue to the doors someASs can be exposed to theMT at somecertain locations

The MT moves along the corridor The unknown posi-tions of the MT at time instant 119905 are denoted by a 2119898-dimensional vector

119883119905= [119909119905 119905 119910119905 119910119905] (1)


b

Wall

Anchor sensor

Mobile terminal

a

c



119905 119910119905)


119883119905= 119865119883119905minus1

+ 119866120596119905minus1 (2)

where

119865 =

[[[[[

[

1 0 119879 0

0 1 0 0

0 0 1 119879

0 0 0 1

]]]]]

]

119866 =

[[[[[[[[

[

1198792

20

01198792

2119879 0

0 119879

]]]]]]]]

]

(3)

119883119905minus1













119894


119875119894(119905) = 119875

119879minus 1198751198710minus 10120578log

10(119889119894

1198890

)

1+DOI+ ] (119905)

119894 = 1 2 119873

(4)



119875119894(119905) = 119875


119889119894

1198890

) + ]LOS (119905)

119894 = 1 2 119873

119875119894(119905) = 119875


119889119894

1198890

) + ]NLOS (119905)

119894 = 1 2 119873

(5)


2


2


0are




119904

119894 119910119904


as

119889119894= radic(119909

119905minus 119909119904

119894)2

+ (119910119905minus 119910119904

119894)2

119894 = 1 2 119873 (6)


minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)

minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)


minus50

0

50

100


Prop

agat

ion

dist

ance

(m)




minus100

0

100

200


opag

atio

n di

stan

ce (m

)







0 5 10 15 20 25 30minus80

minus70

minus60

minus50

minus40

Distance (m)

RSSI

(dBm

)LOS


Distance (m)


0 5 10 15 20 25 30minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS


0 05 1 15minus80

minus70

minus60

minus50

minus40

RSSI

(dBm

)

LOS



04 06 08 1 12 14 16 18minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS




min120578120583

sum

119894119895

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


[[[

[

119875119879minus 1198751198710

minus10120578log10(

radic(119909119895minus 119909119904

119894)2

+ (119910119895minus 119910119904

119894)2

1198890

)+ 120583]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119894 = 1 2 119873 119895 = 1 2 119872




119895 119910119895) For any pair


119875119879minus 1198751198710minus 10120578log

10(

119889119894

119895

1198890

) + 120583 (119895) = RSS119894119895

119894 isin 1119873 119895 isin 1119872

(8)

[[[[[[[[[[

[

1 minus10log10(1198891

1

1198890

)

1 minus10log10(

119889119894

119895

1198890

)

]]]]]]]]]]

]

[120583

120578] =

[[

[

RSS11minus 119875119879+ 1198751198710

RSS119894119895minus 119875119879+ 1198751198710

]]

]

(9)


[120583

120578] = (119860

119879119860)minus1

119860119879119861 (10)






Model 1NLOS

Model 2LOSp11

p12

p21

p22



119885119894(119905) = 119867 (119883

119905AS119894)

=


119889119894

1198890

) + ]LOS (119905)

for LOS


119889119894

1198890

) + ]NLOS (119905)

for NLOS(11)











119894 119894 isin 1 2 119873

119879119894=

1198790

119885119894(119905)

(12)







119894(119905) Once the MT



119879 However in [24 25] 119875

119879is











120583119894|119895(119905 minus 1 | 119905 minus 1) =

119901119894119895120583119894(119905 minus 1 | 119905 minus 1)

119888119895

(13)





119888119895= sum

119894

119901119894119895120583119894(119905 minus 1 | 119905 minus 1) (14)


Interaction

IMM structure

Mixingprobability



Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)




120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)



0119895(119905minus1 |


1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894


0119894(119905 minus 1 | 119905 minus 1)]


0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)




119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are












Yes

No

Ti



s


119883119894(119905 minus 1 | 119905 minus 1) and 119875



1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










b

Wall

Anchor sensor

Mobile terminal

a

c



119905 119910119905)


119883119905= 119865119883119905minus1

+ 119866120596119905minus1 (2)

where

119865 =

[[[[[

[

1 0 119879 0

0 1 0 0

0 0 1 119879

0 0 0 1

]]]]]

]

119866 =

[[[[[[[[

[

1198792

20

01198792

2119879 0

0 119879

]]]]]]]]

]

(3)

119883119905minus1













119894


119875119894(119905) = 119875

119879minus 1198751198710minus 10120578log

10(119889119894

1198890

)

1+DOI+ ] (119905)

119894 = 1 2 119873

(4)



119875119894(119905) = 119875


119889119894

1198890

) + ]LOS (119905)

119894 = 1 2 119873

119875119894(119905) = 119875


119889119894

1198890

) + ]NLOS (119905)

119894 = 1 2 119873

(5)


2


2


0are




119904

119894 119910119904


as

119889119894= radic(119909

119905minus 119909119904

119894)2

+ (119910119905minus 119910119904

119894)2

119894 = 1 2 119873 (6)


minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)

minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)


minus50

0

50

100


Prop

agat

ion

dist

ance

(m)




minus100

0

100

200


opag

atio

n di

stan

ce (m

)







0 5 10 15 20 25 30minus80

minus70

minus60

minus50

minus40

Distance (m)

RSSI

(dBm

)LOS


Distance (m)


0 5 10 15 20 25 30minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS


0 05 1 15minus80

minus70

minus60

minus50

minus40

RSSI

(dBm

)

LOS



04 06 08 1 12 14 16 18minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS




min120578120583

sum

119894119895

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


[[[

[

119875119879minus 1198751198710

minus10120578log10(

radic(119909119895minus 119909119904

119894)2

+ (119910119895minus 119910119904

119894)2

1198890

)+ 120583]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119894 = 1 2 119873 119895 = 1 2 119872




119895 119910119895) For any pair


119875119879minus 1198751198710minus 10120578log

10(

119889119894

119895

1198890

) + 120583 (119895) = RSS119894119895

119894 isin 1119873 119895 isin 1119872

(8)

[[[[[[[[[[

[

1 minus10log10(1198891

1

1198890

)

1 minus10log10(

119889119894

119895

1198890

)

]]]]]]]]]]

]

[120583

120578] =

[[

[

RSS11minus 119875119879+ 1198751198710

RSS119894119895minus 119875119879+ 1198751198710

]]

]

(9)


[120583

120578] = (119860

119879119860)minus1

119860119879119861 (10)






Model 1NLOS

Model 2LOSp11

p12

p21

p22



119885119894(119905) = 119867 (119883

119905AS119894)

=


119889119894

1198890

) + ]LOS (119905)

for LOS


119889119894

1198890

) + ]NLOS (119905)

for NLOS(11)











119894 119894 isin 1 2 119873

119879119894=

1198790

119885119894(119905)

(12)







119894(119905) Once the MT



119879 However in [24 25] 119875

119879is











120583119894|119895(119905 minus 1 | 119905 minus 1) =

119901119894119895120583119894(119905 minus 1 | 119905 minus 1)

119888119895

(13)





119888119895= sum

119894

119901119894119895120583119894(119905 minus 1 | 119905 minus 1) (14)


Interaction

IMM structure

Mixingprobability



Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)




120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)



0119895(119905minus1 |


1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894


0119894(119905 minus 1 | 119905 minus 1)]


0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)




119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are












Yes

No

Ti



s


119883119894(119905 minus 1 | 119905 minus 1) and 119875



1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)

minus100 0 100 200minus200

minus100

0

100


Prop

agat

ion

dist

ance

(m)


minus50

0

50

100


Prop

agat

ion

dist

ance

(m)




minus100

0

100

200


opag

atio

n di

stan

ce (m

)







0 5 10 15 20 25 30minus80

minus70

minus60

minus50

minus40

Distance (m)

RSSI

(dBm

)LOS


Distance (m)


0 5 10 15 20 25 30minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS


0 05 1 15minus80

minus70

minus60

minus50

minus40

RSSI

(dBm

)

LOS



04 06 08 1 12 14 16 18minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS




min120578120583

sum

119894119895

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


[[[

[

119875119879minus 1198751198710

minus10120578log10(

radic(119909119895minus 119909119904

119894)2

+ (119910119895minus 119910119904

119894)2

1198890

)+ 120583]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119894 = 1 2 119873 119895 = 1 2 119872




119895 119910119895) For any pair


119875119879minus 1198751198710minus 10120578log

10(

119889119894

119895

1198890

) + 120583 (119895) = RSS119894119895

119894 isin 1119873 119895 isin 1119872

(8)

[[[[[[[[[[

[

1 minus10log10(1198891

1

1198890

)

1 minus10log10(

119889119894

119895

1198890

)

]]]]]]]]]]

]

[120583

120578] =

[[

[

RSS11minus 119875119879+ 1198751198710

RSS119894119895minus 119875119879+ 1198751198710

]]

]

(9)


[120583

120578] = (119860

119879119860)minus1

119860119879119861 (10)






Model 1NLOS

Model 2LOSp11

p12

p21

p22



119885119894(119905) = 119867 (119883

119905AS119894)

=


119889119894

1198890

) + ]LOS (119905)

for LOS


119889119894

1198890

) + ]NLOS (119905)

for NLOS(11)











119894 119894 isin 1 2 119873

119879119894=

1198790

119885119894(119905)

(12)







119894(119905) Once the MT



119879 However in [24 25] 119875

119879is











120583119894|119895(119905 minus 1 | 119905 minus 1) =

119901119894119895120583119894(119905 minus 1 | 119905 minus 1)

119888119895

(13)





119888119895= sum

119894

119901119894119895120583119894(119905 minus 1 | 119905 minus 1) (14)


Interaction

IMM structure

Mixingprobability



Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)




120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)



0119895(119905minus1 |


1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894


0119894(119905 minus 1 | 119905 minus 1)]


0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)




119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are












Yes

No

Ti



s


119883119894(119905 minus 1 | 119905 minus 1) and 119875



1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30minus80

minus70

minus60

minus50

minus40

Distance (m)

RSSI

(dBm

)LOS


Distance (m)


0 5 10 15 20 25 30minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS


0 05 1 15minus80

minus70

minus60

minus50

minus40

RSSI

(dBm

)

LOS



04 06 08 1 12 14 16 18minus100

minus80

minus60

minus40

RSSI

(dBm

)

NLOS




min120578120583

sum

119894119895

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816


[[[

[

119875119879minus 1198751198710

minus10120578log10(

radic(119909119895minus 119909119904

119894)2

+ (119910119895minus 119910119904

119894)2

1198890

)+ 120583]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119894 = 1 2 119873 119895 = 1 2 119872




119895 119910119895) For any pair


119875119879minus 1198751198710minus 10120578log

10(

119889119894

119895

1198890

) + 120583 (119895) = RSS119894119895

119894 isin 1119873 119895 isin 1119872

(8)

[[[[[[[[[[

[

1 minus10log10(1198891

1

1198890

)

1 minus10log10(

119889119894

119895

1198890

)

]]]]]]]]]]

]

[120583

120578] =

[[

[

RSS11minus 119875119879+ 1198751198710

RSS119894119895minus 119875119879+ 1198751198710

]]

]

(9)


[120583

120578] = (119860

119879119860)minus1

119860119879119861 (10)






Model 1NLOS

Model 2LOSp11

p12

p21

p22



119885119894(119905) = 119867 (119883

119905AS119894)

=


119889119894

1198890

) + ]LOS (119905)

for LOS


119889119894

1198890

) + ]NLOS (119905)

for NLOS(11)











119894 119894 isin 1 2 119873

119879119894=

1198790

119885119894(119905)

(12)







119894(119905) Once the MT



119879 However in [24 25] 119875

119879is











120583119894|119895(119905 minus 1 | 119905 minus 1) =

119901119894119895120583119894(119905 minus 1 | 119905 minus 1)

119888119895

(13)





119888119895= sum

119894

119901119894119895120583119894(119905 minus 1 | 119905 minus 1) (14)


Interaction

IMM structure

Mixingprobability



Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)




120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)



0119895(119905minus1 |


1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894


0119894(119905 minus 1 | 119905 minus 1)]


0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)




119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are












Yes

No

Ti



s


119883119894(119905 minus 1 | 119905 minus 1) and 119875



1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Model 1NLOS

Model 2LOSp11

p12

p21

p22



119885119894(119905) = 119867 (119883

119905AS119894)

=


119889119894

1198890

) + ]LOS (119905)

for LOS


119889119894

1198890

) + ]NLOS (119905)

for NLOS(11)











119894 119894 isin 1 2 119873

119879119894=

1198790

119885119894(119905)

(12)







119894(119905) Once the MT



119879 However in [24 25] 119875

119879is











120583119894|119895(119905 minus 1 | 119905 minus 1) =

119901119894119895120583119894(119905 minus 1 | 119905 minus 1)

119888119895

(13)





119888119895= sum

119894

119901119894119895120583119894(119905 minus 1 | 119905 minus 1) (14)


Interaction

IMM structure

Mixingprobability



Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)




120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)



0119895(119905minus1 |


1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894


0119894(119905 minus 1 | 119905 minus 1)]


0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)




119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are












Yes

No

Ti



s


119883119894(119905 minus 1 | 119905 minus 1) and 119875



1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Interaction

IMM structure

Mixingprobability



Likelihoodfunction

Combination

LOSUKF estimator

Modeprobability

NLOSUKF estimator

IMM structure

Z(t)

pij

Λij(t)




120583ij(t | t)

X1(t | t)

X2(t | t)

X(t | t)



0119895(119905minus1 |


1198830119895(119905 minus 1 | 119905 minus 1) = sum

119894

119883119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

1198750119895(119905 minus 1 | 119905 minus 1)

= sum

119894


0119894(119905 minus 1 | 119905 minus 1)]


0119894(119905 minus 1 | 119905 minus 1)]

119879

+ 119875119894(119905 minus 1 | 119905 minus 1) 120583

119894|119895(119905 minus 1 | 119905 minus 1)

(15)




119894(119905minus1 | 119905minus1) and119875

119894(119905minus1 | 119905minus1) are












Yes

No

Ti



s


119883119894(119905 minus 1 | 119905 minus 1) and 119875



1205940

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)

120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 1 119899


120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120594119895

119894(119905 minus 1 | 119905 minus 1) = 119883

119894(119905 minus 1 | 119905 minus 1)


119895

119895 = 119899 + 1 2119899

(16)

119882119898

0=

120581

119899 + 120581

119882119898

0=

120581

119899 + 120581

119882119898

119895= 119882119888

119895=

120581

2 (119899 + 120581)

(17)







120594119895

119894(119905 | 119905 minus 1) = 119865120594

119895

119894(119905 minus 1 | 119905 minus 1) 119895 = 0 1 2119899

120595119895

119894(119905 | 119905 minus 1) = 119867 (120594

119895

119894(119905 | 119905 minus 1)) 119895 = 0 1 2119899

(18)


119883119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120594119895

119894(119905 | 119905 minus 1)

119885119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119898

119895120595119895(119905 | 119905 minus 1)

119875119894(119905 | 119905 minus 1) =

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

119879

+ 119876

(19)



119875119885119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

+ 119877

119875119883119894119885119894

=

2119899

sum

119895=0

119882119888

119895[120594119895

119894(119905 | 119905 minus 1) minus 119883

119894(119905 | 119905 minus 1)]

times [120595119895

119894(119905 | 119905 minus 1) minus 119885

119894(119905 | 119905 minus 1)]

119879

119870119894= 119875119883119894119885119894

119875minus1

119885119894119885119894

(20)


2


LOS]119894(119905) 119875119894(119905 | 119905) and 119883



]119894(119905) = 119885

119894(119905) minus 119885

119894(119905 | 119905 minus 1)

119875119894(119905 | 119905) = 119875

119894(119905 | 119905 minus 1) minus 119870

119894119875119885119894119885119894

119870119879

119894

(21)

119883119894(119905 | 119905) = 119883

119894(119905 | 119905 minus 1) + 119870

119894]119894(119905) (22)





119885119894119885119894in (20) The



Λ119894(119905) = 119873 (]

119894(119905) 0 119878

119894(119905))

120583119894(119905 | 119905) =

Λ119894(119905) 119888119894

119888

119888 = sum

119894

Λ119894(119905) 119888119894

(23)



119883 (119905 | 119905) = sum

119894

119883119894(119905 | 119905) 120583

119894(119905 | 119905)



119894(119905 | 119905) and 119875

119894(119905 |







0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














0

0 0012]



5 Numerical Study





119879 and 119875119871





RMSE (119905) = radic(119909119905minus 119909119905)2

+ (119910119905minus 119910119905)2

(25)




0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

40

45

x (m)

y (m

)











V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












V119910[ms] 00148 00199 01066 00417 00409

radic1199092 + 1199102 [m] 084539 14254 100769 21763 51728

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

t (s)

RMSE

(m)


MPDAIMM-EKF














0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20 30 40 50 60 70 80 900

02

04

06

08

t (s)


MPDAIMM-EKF

RMSE

of X

-vel

ocity

(ms

)

0 10 20 30 40 50 60 70 80 900

01

02

03

04

t (s)


MPDAIMM-EKF

RMSE

of Y

-vel

ocity

(ms

)


0 10 20 30 40 50 60 70 80 900

01

02

03

04

05

06

07

08

09

1

t (s)


Prob

abili

ty (m

odel|Z(t))





119885 (119883119898) = 119867 (119883

119898AS119894) (26)


119898

+ 120575119883119898 where 120575




119867(119883119898AS119894) = 119867 (119883

119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla21198671205752


= 119867 (119883119898

AS119894) + nabla119867120575

119883119898 +

1

2nabla2119867119875119883119883

+ sdot sdot sdot

(27)

6789101112131408

1

12

14

16

18

2

Numbers of anchors

RMSE

(m)

IMM-UKFIMM-EKF



nabla119867 = [nabla119867119883

nabla119867119884

]

=

[[[[

[

119888 sdot119909119898minus 119909119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

119888 sdot119910119898minus 119910119904

(119909119898 minus 119909119904)

2+ (119910119898 minus 119910119904)

2

]]]]

]

(28)

Here 119883119898

= (119909119898 119910119898) and (119909






6 Conclusion





Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











6 Conclusion





Acknowledgments


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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