research article a moving source localization...

Research ArticleA Moving Source Localization Method for Distributed PassiveSensor Using TDOA and FDOA Measurements

Zhixin Liu Yongjun Zhao Dexiu Hu and Chengcheng Liu

Zhengzhou Institute of Information Science and Technology Zhengzhou Henan 86-450002 China

Correspondence should be addressed to Dexiu Hu paper hdx126com

Received 30 December 2015 Revised 7 March 2016 Accepted 27 March 2016

Academic Editor Ding-Bing Lin

Copyright copy 2016 Zhixin Liu 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

The conventional moving source localization methods are based on centralized sensors This paper presents a moving sourcelocalization method for distributed passive sensors using TDOA and FDOA measurements The novel method firstly uses thesteepest descent algorithm to obtain a proper initial value of source position and velocity Then the coarse location estimationis obtained by maximum likelihood estimation (MLE) Finally more accurate location estimation is achieved by subtractingtheoretical bias which is approximated by the actual bias using the estimated source location and noisy data measurement Boththeoretical analysis and simulations show that the theoretical bias always meets the actual bias when the noise level is small and theproposedmethod can reduce the bias effectively while keeping the same root mean square error (RMSE) with the originalMLE andTaylor-series method Meanwhile it is less sensitive to the initial guess and attains the CRLB under Gaussian TDOA and FDOAnoise at a moderate noise level before the thresholding effect occurs

1 Introduction

Passive source location has been the focus of considerableresearch efforts for many years It is widely used in manyareas including radar sonar [1 2] microphone arrays [3 4]sensor network [5] and wireless communication [6] Fora stationary emitter the time-difference-of-arrival (TDOA)of a received signal [7] at a number of separated receiverscan be used to obtain the source location estimate EachTDOA defines a hyperbola in which the emitter must lieThe intersection of the hyperbolae gives the source locationestimate [8 9] If there is a relative motion between theemitter and the receivers frequency-difference-of-arrival(FDOA) measurements should be combined with TDOAs toestimate the source position and velocity accurately In thispaper we enforce locating the moving source using TDOAand FDOAmeasurements

Localization of a moving source using TDOA and FDOAis not a trivial task due to the nonlinear nature of theestimation problem [9ndash12] Foy put forward a Taylor-serieslinearization method which requires a proper initial guessclose to the true solution in 1976 [10] However a good initialguess is not easy to obtain in practice In order to avoid it

Ho et al proposed a novel two-step weighted least-squares(TSWLS) [9 12] approach to obtain the source positionand velocity Sun et al applied the total least-squares (TLS)technique [13] to solve this problem

Considering that the noise components of coefficients arelinearly dependent Yu et al used the constrained total least-squares (CTLS) [14 15] in moving source localization

Up to now most passive localization methods usingTDOA and FDOA are based on centralized structure whichis shown in Figure 1(a) For the centralized structure onesensor is chosen as the reference sensor and the others cantransmit their original signals to the reference sensor [16]On the one hand in Electronic Warfare (EW) the receiverswhich intercept signals from the emitter and measure theirphysical parameters play a central role in the passive locationsystem [17] Because the emitter is noncooperating we canonly obtain theTDOAandFDOA ForTDOA-basedmethodcommon practice requires precise time-synchronizationbetween sensors Likewise for FDOA-based method precisefrequency-locking is required Obviously for joint TDOA-and FDOA-basedmethods both types of synchronization aretypically required [18] If a TDOA- and FDOA-based system

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 8625039 12 pageshttpdxdoiorg10115520168625039

2 International Journal of Antennas and Propagation

SensorTarget

(a) Centralized localization

SensorTarget

(b) Distributed localization

Figure 1 Sensor pairing

[18] fails to achieve precise synchronization between devicesfor separation distance operation it is impossible to obtaincorrect measurements from signals sent by receivers andsuch a failure to obtain the accurate values directly affectslocation estimation error [19] For example in order to getmore accuracy of source position and velocity the TSWLSTLS and CTLS also require precise time-synchronizationand frequency-locking among all receivers The price to payis accuracy required for TDOA and FDOA measurementsstrongly depending on synchronization [17] Thus all thereceivers would be precisely synchronized in time which isdifficult to realize and increase the system complexity

On the other hand it is well known that the centralizedmethod is power consuming since raw measurement data isinvolved in the transmissions and there is high computationalcost at a single sensor for centralized localization [20]In addition this kind of method requires a high band-width which is generally limited in wireless sensor networks(WSNs) and alsomay cause a large processing delay [21]Thiswould have significant impacts on the size weight and power(SWaP) requirement of that sensor [22]

Therefore in practice to reduce the requirements for net-work bandwidth synchronization and power consumption[23] the distributed localization is highly desirable As seenin Figure 1(b) several sensor pairs are formed without acommon reference sensor The whole sensors do not need totransmit their original signals to the single reference sensorbut only to their reference sensor of each group Furthermoreas the speed of the target is far less than the signal propagationspeed and the change of target position in synchronizationerror is slightly small that we can ignore it we only needto achieve precise synchronization between two sensors in agroup rather than all sensors and have rough synchronizationbetween each group Therefore the distributed localizationnot only reduces the difficulty of the synchronization betweenall sensors but also decreases the load of data computationand transmission So it is necessary to do some deep

research into distributed localization problem and improvethe estimation accuracy Nowadays the distributed structureis a novel problem in passive localization and there are stillsome problems for distributed passive localization such asthe optimal sensor pairing strategy localization algorithmand optimal baseline between sensors In this paper weaim at the location algorithm problem For conventionalcentralized structure source location methods thanks tothe only reference sensor in the conventional centralizedstructure source location methods they can be transformedfrom a nonlinear problem to a pseudolinear problem [24]For instance Ho and Xu used the position and velocity ofreference sensor to transform the TDOAand FDOAequationto a set of linear equations and then applied linear weightedleast-squares (LS) to obtain the source position [9] As thereare many different reference sensors in distributed structureit is difficult to use the conventional centralized structuresource location methods for estimation Therefore it isessential to put forward a distributed localization algorithm

Meyer et al put forward the distributed sensor self-localization and target tracking method in 2012 [25] Dis-tributed source localization is also a nontrivial problembecause the TDOA and FDOA measurements are nonlin-early related to the source location parameters Due to thisproblem the mean square error (MSE) is composed of twoparts the variance and the bias square When the noiselevel is low and the observation period is short the biasis not significant compared with the variance of the targetposition estimation [26 27] Thus the Cramer-Rao lowerbound (CRLB) which is developed for an unbiased estimatoris often used as a reference for evaluating the performance of alocation estimator However with the increase of observationperiod the location variancewill decrease and the bias cannotbe ignored It had a serious influence on estimation perfor-mance In particular with the development of ultrawideband(UWB) technology the measurements of the target can bemeasured repeatedly in a short time and then the precision

International Journal of Antennas and Propagation 3

of the target location can be improved by using the methodof data fusion [28] but this method can only effectivelyreduce the variance of the estimation of the target and thedeviation does not decrease as the number of measurementsincreases [26 29] In application of target tracking thelocation deviation has a great influence on performance oftarget tracking [27] Therefore how to remove the deviationfrom the estimation of target position and velocity is afocus of research Rui and Ho proposed a bias compensationalgorithm [26] based on time of arrival (TOA) TDOA orangle of arrival (AOA) and verified that the position bias hasgreat influence on location accuracy In order to reduce thisinfluence Hao et al put forward a bias reduction method forpassive source localization using TDOA and gain ratios ofarrival (GROA) [30] However the methods proposed by theabove research can only be applied to the stationary targetFor moving emitters a new bias reduction algorithm usingboth TDOA and FDOA is proposed by Ho [29] It can reducethe estimation error by adding new constraints to the originalposition equationThe performance of estimation is the sameas maximum likelihood estimation (MLE) However thesemethods still have deviation and all need a proper initialguess

Motivated by the above shortcomings of algorithms basedon distributed structure a new bias compensation methodbased on MLE for distributed source localization usingTDOA and FDOA is proposed in this paper We studythe bias of the MLE [31] for source localization becausethe MLE is asymptotically efficient and often serves as abenchmark for performance evaluation The bias of the MLEof a general estimation problem has been investigated in themathematical and statistical literature [32ndash34] The proposedmethod firstly uses the classic steepest descent (SD) method[35 36] which has the features of being more rapid toconvergence and less sensitive to the initial values to getan appropriate initial guess and coarse location estimationis obtained by MLE [26 27] Then more accurate locationestimation is achieved by subtracting theoretical bias whichis approximated by the actual bias using the estimated sourcelocation and noisy data measurement The proposed methodattains the CRLB at a moderate noise level when the TDOAand FDOA noise are Gaussian

The remainder of the paper is organized as follows Thenovel method is introduced in Section 2 Section 3 analysesthe CRLB of distributed passive sensors localization underGaussian measurement noise Section 4 presents simulationto support the theoretical development of the proposedmethod Finally a brief conclusion is given in Section 5

2 The Proposed Algorithm

21 Distributed LocalizationModel Themodel of distributedlocalization is different from centralized model We considera three-dimensional (3D) scenario where an array of 119872

moving sensors is used to determine the position u =

[119909 119910 119911]119879 and velocity u = [ ]

119879 of a moving source usingTDOAs and FDOAsThe sensor position s

119894= [119909119894 119910119894 119911119894]119879 and

velocity s119894

= [119894 119894 119894]119879 119894 = 1 2 119872 are assumed to be

known and 119872 must be even The location problem requiresat least three pairs of receivers (ie 119872 = 6) to producethree TDOAs and three FDOAs This paper focuses on theoverdetermined scenario where the number of receivers islarger than 6We will use the notation (lowast)

119900 to denote the truevalue of the noisy quantity (lowast)

The Euclidean norm between the source and receiver 119894 is

119903119900

119894=

1003817100381710038171003817u119900

minus s119900119894

1003817100381710038171003817 = radic(u119900 minus s119900119894)119879

(u119900 minus s119900119894) (1)

For 119872 sensors there are a total number of 1198722 sensor pairsand TDOAFDOAmeasurements Let

Σ = 2119894 2119894 minus 1 | 1 le 119894 le119872

2 (2)

which denotes the set of all sensor pairs For simplicity andalso without loss of generality the first sensor of each groupis chosen as the reference sensor If the true TDOA of a signalreceived by the receiver pair (2119894 2119894minus1) is 120591

119900

21198942119894minus1and the signal

propagation speed is 119888 the set of equations that relates theTDOAs and the source position is

119903119900

21198942119894minus1= 119888120591119900

21198942119894minus1= 119903119900

2119894minus 119903119900

2119894minus1 (3)

where 119903119900

21198942119894minus1is range difference and 119894 = 1 2 1198722

Note that (3) is nonlinear with respect to u and the 1198722

curves in (3) give the source position estimate The TDOAequations only allow the estimation of the source positionbut not velocity In addition the TDOA equations alone maynot be sufficient to provide enough accuracy to the positionestimate

Due to the moving source the FDOAmeasurements canbe used to improve the accuracy of position estimate andat the same time identify the source velocity The FDOAshave been converted to the range rate difference throughmultiplying by signal propagation speed and dividing by thecenter frequency Let 119903

119900

119894be the true range rate between the

source and receiver 119894 By taking the time derivative of (1) it isequal to

119903119900

119894=

(u119900 minus s119900119894)119879

(u119900 minus s119900119894)

119903119900

119894

(4)

The FDOA between receiver pair 2119894 and 2119894 minus 1 is the timederivative of (3)

119903119900

21198942119894minus1= 119903119900

2119894minus 119903119900

2119894minus1(5)

for 119894 = 1 2 1198722 where 119903119900

21198942119894minus1is range rate difference

derived from the FDOAs [9] Equations (3) and (5) are a set ofnonlinear equations with the source position u and velocityu and solving them from TDOA and FDOA is not an easytask

In practice we cannot obtain the true values of TDOAand FDOA So we let r = [119903

21 11990343

119903119872119872minus1

]119879 and r =

[ 11990321

11990343

119903119872119872minus1

]119879 represent the noisy range differences

and range rate differenceWe will assume that the TDOA andFDOA measurements can be described by the additive noisemodel as

r = r119900 + Δr

r = r119900 + Δr(6)


where

Δr = [Δ11990321

Δ11990343

Δ119903119872119872minus1

]119879

Δr = [Δ 11990321

Δ 11990343

Δ 119903119872119872minus1

]119879

(7)

Equation (6) can be written as the function

m = F (120579119900) + n (8)

where m = [r119879 r119879]119879 is the 119872 times 1 measurement vector with1198722 for range measurement and 1198722 for range differencemeasurement F(120579

119900) = [r119900119879 r119900119879]119879 represents the function

relationship of the noiseless measurement vector in terms ofthe unknown position u119900 and velocity u119900 120579119900 = [u119900119879 u119900119879]119879 isthe location and velocity of the source and n = [Δr119879 Δr119879]119879is the vectors of TDOA and FDOA noiseThey are zero meanGaussian noise and have covariancematrixQWe assume theobservation interval is long enough so that the measurementnoise vectors at different time instants are uncorrelated

22 Initial Solution Using the Steepest Descent Method Esti-mating the position and the velocity of amoving emitter usingthe TDOAFDOAmeasurements has been a challenging taskbecause of the high nonlinearity in the TDOAFDOA signalmodels Under some circumstances where the source andreceiver geometry is not good some localization algorithms[10 26] may fail to attain a more accurate estimation withouta proper initial guess In this paper the appropriate initialvalue is obtained by SD method [35 36] which features fastconvergence speed low computational complexity and lowrequirement of initial value Although the SD method alsorequires initial guess the convergence rate increases as thedifference between initial value and true value grows It isefficient to fill the gap of not having prior knowledge of thetarget Therefore the SD method is chosen as the preferablealternative executing refinement due to its simplicity andvalidity

Equation (6) can be expressed as

120593 = r119900 minus r + Δr

120573 = r119900 minus r + Δr(9)

where 120593 = [1205931 1205932 120593

1198722]119879 and 120573 = [120573

1 1205732 120573

1198722]119879


Φ = F (120579119900) minus m + n (10)

where Φ = [120593119879120573119879]119879 From (10) an objective function is

defined as

120588 (120579) = Φ2

=

1198722

sum

119894=1

(120593119894 (120579))2

+

1198722

sum

119894=1

(120573119894 (120579))2

(11)

sdot represents the Euclidean norm and the solution of (6) isa set of unknown variables which is formulated as

120579 = argmin120579

120588 (120579) (12)

In general the gradient direction G at a certain point is thefastest way to increase the function 120588(120579) on the contrary thedirection of negative gradient minusG is the fastest way to reducethe function value So we can use this principle to find 120579 thatmakes (11) minimal Then we set an arbitrary initial valuedenoted by 120579

0= [u1198790 u1198790]119879 and the gradient direction of 120588(120579)

at the initial values is defined as

G0

= [G11987901

G11987902

]119879

=120597120588

120597120579

100381610038161003816100381610038161003816100381610038161205790

= [120597120588

120597u1198790

120597120588

120597u1198790

]

119879

(13)

We choose the appropriate step size 120582 120583 to find the newestimation that makes (11) minimal along the direction of thenegative gradient minusG So the new estimation is obtained by

1205791

= [u1198791 u1198791]119879

= [u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

]119879

(14)

where

G01

= 2 sdot

1198722

sum

119894=1

120593119894sdot

120597120593119894

120597u+ 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

G02

= 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

120597120593119894

120597u=

12059711990321198942119894minus1

120597u= x1198792119894

minus x1198792119894minus1

120597120573119894

120597u=

120597 11990321198942119894minus1

120597u= k1198792119894

minus k1198792119894minus1

120597120573119894

120597u=

120597120593119894

120597u

(119894 = 1 2 119872

2)

(15)

Appendix A gives details about the evaluation of the deriva-tives for x

119894and k119894

(119894 = 1 2 119872)In order to make (11) minimal within the shortest steps in

new estimation 1205791which is defined as

120588 (1205791) = 120588 (u119879

1 u1198791)

asymp min 120588 (u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

)

(16)


we should find an optimal step size 120582 and 120583 Therefore theTaylor-series expansion of 120588(120579

1) at 1205790up to second order is

120588 (1205791) =

1198722

sum

119894=1

(120593119894(1205791))2

+

1198722

sum

119894=1

(120573119894(1205791))2

asymp

1198722

sum

119894=1

(120593119894)2

+

1198722

sum

119894=1

(120573119894)2

minus 2120583

1198722

sum

119894=1

120573119894(G11987902

120597120573119894

120597u)

minus 2120582

1198722

sum

119894=1

120593119894(G11987901

120597120593119894

120597u) minus 2120582

1198722

sum

119894=1

120573119894(G11987901

120597120573119894

120597u)

+ 1205822

1198722

sum

119894=1

(G11987901

120597120593119894

120597u)

2

+ 1205822

1198722

sum

119894=1

(G11987901

120597120573119894

120597u)

2

+ 1205832

1198722

sum

119894=1

(G11987902

120597120573119894

120597u)

210038161003816100381610038161003816100381610038161003816100381610038161205790

(17)

Representing the gradient of 120588(1205791) with respect to 120582 and 120583

satisfies the equation

120597120588 (1205791)

120597120582= 0

120597120588 (1205791)

120597120583= 0

(18)

Thus the optimal step size is

120582

=

sum1198722

119894=1120593119894(G11987901

(120597120593119894120597u)) + sum

1198722

119894=1120573119894(G11987901

(120597120573119894120597u))

sum1198722

119894=1(G11987901

(120597120593119894120597u))2

+ sum1198722

119894=1(G11987901

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

120583 =

sum1198722

119894=1120573119894(G11987902

(120597120573119894120597u))

sum1198722

119894=1(G11987902

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

(19)

We have obtained the optimal step size which makes theconvergence rate along the direction of negative gradient ofobjective function the fastest Furthermore a proper initialposition and velocity are attained by (14) using (19) until (11)is sufficiently small

23 Bias Compensation Using MLE According to (8) thelogarithm probability density function of the noise data n is

ln119891 (m 120579) = 119896 minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (20)

where 119896 = minus12 ln((2120587)119872

|Q|) is constant and the MLEsolution is

= argmax (119868) (21)

where 119868 is the maximum likelihood cost function equal to

119868 ≜ minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (22)

Due to the nonlinear form of F(120579) iterative or exhaustivesearch method is required to find the ML solution usingthe initial value of (14) Representing the gradient of 119868 withrespect to 120579 as P() satisfies the equation

P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

= 0 (23)

The expectation of the difference between and 120579119900 gives thebias119864[minus120579

119900]We use (23) to obtain the bias without explicitly

solving It assumes that the noise standard deviation relativeto the true values of the TDOA and FDOA measurements issmall Thus the noise terms higher than second order can beignored and the Taylor-series expansion of P() at 120579119900 up tosecond order is

P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

asymp H1015840 + H10158401015840 ( minus 120579119900) + g (120579

119900) = 0 (24)

where

H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

H101584010158401015840119897

=120597

120597120579119897

(1205972119868

120597120579120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900119897 = 1 2 6

g (120579119900) =

1

2

[[[[[[[[[

[

tr (H1015840101584010158401

times [ minus 120579119900] [ minus 120579

119900]119879

)

tr (H1015840101584010158402


119900]119879

)

tr (H1015840101584010158406


119900]119879

)

]]]]]]]]]

]

(25)

We can obtain the theoretical bias from (24)

119864 [ minus 120579119900] = 119864 [minus (H10158401015840)

minus1

H1015840]

+ 119864 [minus (H10158401015840)minus1

g (120579119900)]

(26)

Equation (25) is specifically expressed as

H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= minus (A minus B)

A =120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

B =

119872

sum

119895=1

119872

sum

119894=1

119902119894119895119899119894

1205972119865119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(27)

where 119902119894119895is the element ofQminus1


Next we give a detailed derivation and obtain an algebraicsolution of (26) According to (27) the first term of (26) canbe approximated as

119864 [minus (H10158401015840)minus1

H1015840] = 119864 [(A minus B)minus1 C]

asymp 119864 [Aminus1C] + 119864 [Aminus1BAminus1C]

(28)

Because A does not contain noise 119864[Aminus1C] = 0 and (27) canbe simplified as

119864 [minus (H10158401015840)minus1

H1015840] asymp 119864 [Aminus1BAminus1C] (29)

Substituting the definition of ABC from (27) and 119864[119899119894n] =

Qe119894 the first component of bias is

119864 [minus (H10158401015840)minus1

H1015840]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579)Qminus1 sdot 119864 [119899

119894n]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(30)

where e119894is 119872 times 1 zero vector except that its 119894th element is

unity P119894is

P119894=

119872

sum

119895=1

119902119894119895

1205972F119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(31)

The second bias component 119864[minus(H10158401015840)minus1g(120579119900)] is quite tedious

to evaluate and we will approximate it [26] When the noiselevel is small from (27) we haveH10158401015840 asymp minusA Thus the secondbias component is

119864 [minus (H10158401015840)minus1

g (120579119900)] asymp Α

minus1z (32)

where

z ≜ 119864 [g (120579119900)] asymp

1

2

[[[[[[[

[

tr (119864 [H1015840101584010158401

] times CRLB (120579119900))

tr (119864 [H1015840101584010158402

] times CRLB (120579119900))

tr (119864 [H1015840101584010158406

] times CRLB (120579119900))

]]]]]]]

]

(33)

tr(lowast) represents the trace operation and CRLB(120579119900) is the

CRLB of 120579119900 when its bias is neglected The approximation isvalid for small measurement noise and the fact that MLE isasymptotically efficient In addition

119864 [H101584010158401015840119897

] =

119872

sum

119894=1

[h119879119894e119897P119894+ P119894e119897h119879119894

+ h119894e119879119897P119879119894] (34)

where

h119894=

119872

sum

119895=1

119902119894119895

120597F119895 (120579)

120597120579

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (35)

Utilizing CRLB(120579119900) for a given positioning measurement

type z can be evaluated and the expectation of the bias isequal to

b = 119864 [ minus 120579119900]

= Aminus1(119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894+ z)

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(36)

Equation (36) is the generic form of the bias and thecomplexity of computing bias is 119874(119872

2) When F(120579) takes on

different measurement types the first and second derivativeswill be different yielding different amount of bias in theMLE solution Appendices A and B give details about theevaluation of the derivatives for F(120579)

Equation (36) can accurately predict the bias of the MLEhence the current source position and velocity after biascompensation are given by

= minus b (37)

where can approximately be treated as an unbiased estima-tor of 120579119900 with covariance matrix CRLB(120579

119900) The evolution of

the bias in (36) requires the true value of target position andvelocity which is not known in practice We will replace thetrue value with the ML estimate

In Section 4 we will present the computer simulation tocorroborate our theoretical development and to compare therelative localization accuracy for different methods

3 The CRLB of Distributed Passive SensorsLocalization under Gaussian Distribution

The CRLB is the lowest possible variance that an unbiasedlinear estimator can achieve The measurement vector 120579 isGaussian distributed Hence according to (20) the CRLB isequal to the inverse of the Fisher matrix [31] defined as

J = 119864 [(120597 ln119891 (m 120579)

120597120579)

119879

(120597 ln119891 (m 120579)

120597120579)]

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (38)

where m = [11990321

119903119872119872minus1

11990321

119903119872119872minus1

]119879 is the vector

of range and range rate difference from TDOA and FDOAmeasurements and119891(m 120579) is the probability density functionof m that is parameterized by the vector 120579 The partialderivative of ln119891(m 120579) with respect to 120579 is

120597 ln119891 (m 120579)

120597120579=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C (39)

Thus the CRLB for the underlying problem reduces to

CRLB (120579119900) = Jminus1 = [(

120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900]

minus1

= [A|120579=120579119900]minus1

(40)


The details about the evaluation of the CRLB from (40) areshown in Appendix A

The solution derivation is obtained by evaluating thebias and variance of the location estimate Let be thecoarse location estimation that is obtained by MLE and let be a vector that contains the source position and velocityestimation after bias compensationThus the two estimationresults can be expressed as

= 120579119900

+ Δ

= minus b

(41)

where Δ is the source location bias of We can get the finalsource location bias

Δ = Δ minus b (42)

The expectation of (42) gives the bias

119864 [Δ] = 119864 [Δ minus b] = 119864 [( minus 120579119900) minus 119864 ( minus 120579

119900)]

= 0

(43)

where can approximately be treated as an unbiased estima-tor Under the assumption explained before (24) the high-order noise terms in (24) tend to zero asymptotically asthe measurement noise decreases so that n is zero meanasymptotically at the true value of Therefore from theMLEtheory [31] is asymptotically unbiased under the zeromeanGaussian noise but not unbiased estimator Multiplying (42)by its transpose and taking expectation yields the covarianceof is

cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879

] are the MSEM of and Due to (43) and (44) theMSEMof is equal to the covarianceof and because of (bb119879)

119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)

From (45) we can obtain that the covariance of the proposedmethod is smaller than that of MLE Therefore the biascompensation method is effective to reduce the bias of MLEand the performance of estimation after bias compensation isbetter than that of the original MLE

4 Simulation Results

This section uses numerical simulations to demonstrate theproposed method and to compare its performance withother location estimators The simulation scenario contains

Table 1 Nominal position (in meters) and velocities (in meterssecond) of receivers

Groups Receivernumber 119894

119909119894

119910119894

119911119894

119894

119894

119894

Set 1 1 minus150 minus600 200 10 20 minus302 50 minus750 200 20 30 0

Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20

Set 4 7 minus600 50 400 15 10 minus158 minus750 minus100 500 minus20 minus15 10

Far-field sourceNear-field sourceReceivers

z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000

Figure 2 Localization geometry for simulation

8 receivers (4 groups) and the position and velocity ofreceivers are given in Table 1 as shown in Figure 2 TDOAand FDOA estimates were generated by adding to the truevalues zero mean Gaussian noise The covariance matrices ofTDOA and FDOA were 120590

2

119903R and 01120590

2

119903R where R was set

to 1 in the diagonal elements and 05 otherwise 1205902

119903is the

TDOA noise power multiplied by the square of the signalpropagation speed 3 times 10

8ms and the TDOA and FDOAnoises were uncorrelated The number of ensemble runs was119870 = 10000The estimation bias and accuracy are investigatedfor source as TDOAand FDOA estimates errors increaseTheestimation accuracy in terms of the root mean square error(RMSE) is defined as

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)


Actual biasTheory bias

minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)

0 10 20minus10minus30minus40 minus20minus50

minus40 minus30 minus20 minus10minus50 10 200

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)

Figure 3 Comparison between theoretical and actual bias ofestimation of source position and velocity by MLE for near-fieldsource

The estimation bias in terms of norm of estimation bias isdefined as

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)

where u119900 and u119900 express the true position and velocity of thesource and u

119896and u

119896denote the estimated source position

and velocity at ensemble 119896 and 119870 = 10000 is the numberof ensemble runs In particular all the true values in thebias formula are replaced by the estimated and noisy meas-urement values

41 For Near-Field Source This section concerns near-fieldsource localization The true position and velocity of thesource are u119900 = [500 minus500 600]

119879 and u119900 = [minus30 minus15 20]119879

Figure 3 shows the comparison between theoretical bias andactual bias of estimation of target position and velocity byMLE As shown in the figure the theoretical bias (solid line)matches very well the simulation when the SNR is smallerthan 10 dB With the increase of noise level the theoreticalbias value gradually deviates from the actual bias especiallythe target velocity bias This phenomenon in the proposedmethod is a consequence of ignoring the high-order termsof (24) in deriving the solution which is not valid when thenoise is large Therefore in order to obtain a more accurateestimation of target position we should do the Taylor-seriesexpansion of (24) at 120579119900 up to high order

Figure 4 shows the accuracy of position and velocityestimate of the proposed method in terms of RMSE as thenoise level increases and compares it with the distributed

10

10

minus30 minus299998minus300002

10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)


minus40 minus30 minus20 minus10minus50 0

The proposed methodTaylor-series method

MLECRLB

minus30minus30001 minus29999

10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)

Figure 4 Comparison of root mean square error (RMSE) of theproposedmethod with the original MLE Taylor-series method andthe CRLB for near-field source The accuracy is shown in log scaleas the noise power increase

TDOAFDOA localization algorithmMLE [26] Taylor-seriesmethod [10] and CRLB The initial guess is obtained by SDmethod The estimation precision of the proposed method isalways higher than that of the other localization algorithmsand all the algorithms can attain the CRLB at low tomoderatenoise levelTheTaylor-seriesmethod deviates from the CRLBand gives an inaccurate solution at a noise power that isabout 10 dB lower than the proposed technique Furthermorethe new method achieves about 35 dB reduction in sourceposition RMSE and 23 dB reduction in source velocityRMSE with respect to the one that does not account forthe bias compensation when 120590

2

119903ge 1m2 In the drawing

of partial enlargement the position and velocity estimationbiases of the proposed estimators are all relatively smallwhich indicates that the proposed method exhibits the bestperformance In addition due to the nonlinear nature of thislocalization problem the estimation accuracy of the threemethods grows as the noise level increases

In Figure 5 the estimation results clearly demonstrate thatthe bias of the proposed method is nonetheless smaller thanthe MLE for low noise level The position and velocity biasesof the proposed method are at least 40 dB and 35 dB lowerthan the MLE at low noise level It is efficient to reduce theimpact of the MLE bias on estimation With the increase ofthe noise power the original maximum likelihood estimation(MLE) is affected by the threshold effect which leads to thedecrease of the algorithm performance

42 For Far-Field Source This section concerns far-fieldsource localization The true position and velocity of thesource are u119900 = [2000 minus2500 3000]

119879 and u119900 = [minus30 minus15

20]119879 Figure 6 shows the comparison between theoretical and

actual bias of estimation of target position and velocity by


The proposed methodMLE

minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)

Figure 5 Comparison of the estimation bias of the proposedmethod with the original MLE against the Gaussian TDOA andFDOA noise for near-field source The result is shown in log scaleas the noise power increase




minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)

Figure 6 Comparison between theoretical and actual bias ofestimation of source position and velocity by MLE for far-fieldsource

MLE The trend of the result is the same as Figure 3 and theestimation results clearly indicate that theoretical bias of thefar field is significantly close to the actual bias especially atlow noise level

Figure 7 shows the result for a far-field source about theaccuracy of position and velocity estimate of the proposedmethod and other comparison algorithms in terms of RMSE

10

10minus40 minus30 minus20 minus10minus50 0



MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)


minus299999minus300001

10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)

Figure 7 Comparison of root mean square error (RMSE) of theproposedmethod with the original MLE Taylor-series method andthe CRLB for far-field source The accuracy is shown in log scale asthe noise power increase

as the noise level increases For far-field source the followingtwo conditions are satisfied

119903119900

1asymp 119903119900

2asymp sdot sdot sdot asymp 119903

119900

119872

119903119900

1

119903119900

1

asymp sdot sdot sdot asymp119903119900

119872

119903119900

119872

asymp 0

(48)

The first condition simply indicates that the ranges of thesource to different receivers are approximately the same sincethe source is very far from the receivers Due to the largerange the range rate relative to the range is close to zeroThus the thresholding effect of the far-field source occursearlier than that of near-field source and the location accuracyis generally worse for a far-field source than a near-fieldsource As expected from the theory the proposed solutionmeets theCRLBbefore the thresholding effect occurs at aboutminus10 dB Comparing with the MLE the improvement of theproposed method is about 23 dB in the position and 13 dBin the velocity RMSEs and the estimation precision of theproposed method is always higher than that of the othercomparison algorithmsWhatever the far-field and near-fieldsources with the noise level decrease the improvement ofbias compensation is gradually reducedThis influence in theproposed method is caused by ignoring the high-order termsof (24) in deriving the solution

Figure 8 shows the result for a far-field source about thebias analysis of the proposed method and the trend of theresult is the same as Figure 5 Although the position andvelocity bias of far-field source is about 5 dB smaller than thatof the near-field source because of the two conditions shownin (48) the proposed method is still effective to reduce thebias of the MLE



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)


minus40 minus30 minus20 minus10minus50 10010 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)

Figure 8 Comparison of the estimation bias of the proposedmethod with the original MLE against the Gaussian TDOA andFDOA noise for far-field source The result is shown in log scale asthe noise power increase

5 Conclusion

Due to the disadvantages of centralized localization weproposed a bias compensation algorithm based on MLEfor distributed passive localization using TDOA and FDOAmeasurements which not only reduces the difficulty of thesynchronization between the sensors but also decreases theload of data computation and transmission The proposedmethod tends to reduce the position and velocity bias of amoving source by using the steepest descent method and biascompensation of MLE Computer simulation results showthat the proposed method was able to achieve higher perfor-mance than the existingmethodsTheproposed solution doesnot require initial guess and attains the CRLB for Gaussiannoise before the thresholding effect occurs Meanwhile smallcomputation overhead and small memory demands makeSD method and bias compensation method suitable fordistributed localization schemes of wireless sensor networksIn addition the position of each sensor should not beperfectly known (equip every sensor with a GPS receiver thatis both energy and cost prohibitive) According to [37] wewill take into account the position uncertainty of sensors forfurther study

Appendix

A The First Derivatives of the F(120579) for SourcePosition and Velocity

The first derivatives of F(120579) for source position and velocityare shown in this part Let

r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)

So 120597F(120579)120597120579119879 in (27) can be expressed as

120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)

According to (3) and (5) we let

x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)

Thus the partial derivatives of r(120579) and r(120579) with respect to uand u yield (A4) shown as

120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)

B The Second Derivatives of the F(120579) forSource Position and Velocity

The second derivatives of F(120579) for source position andvelocity are shown in this part

When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579

119879 in (27) can beexpressed as

1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)

where I is the identity matrix Therefore the second-orderpartial derivatives of r(120579) with respect to u and u yield (B3)shown as

1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895

minus X2119895minus1

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r2119895minus1198722119895minus1minus119872 (120579)

120597u120597u1198791205972r2119895minus1198722119895minus1minus119872 (120579)

120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)

The second-order partial derivatives of r(120579) with respect to uand u yield (B6) shown as


120597u120597u119879= Ψ2119895minus119872

minusΨ2119895minus1minus119872


120597u120597u119879= X2119895minus119872

minus X2119895minus1minus119872


120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests

The authors declare that they have no competing interests

References

[1] G C Carter ldquoTime delay estimation for passive sonar signalprocessingrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 29 no 3 pp 463ndash470 1981

[2] E Weinstein ldquoOptimal source localization and tracking frompassive array measurementsrdquo IEEE Transactions on AcousticsSpeech and Signal Processing vol 30 no 1 pp 69ndash76 1982

[3] T Gustafsson B D Rao and M Trivedi ldquoSource localizationin reverberant environments modeling and statistical analysisrdquoIEEE Transactions on Speech and Audio Processing vol 11 no 6pp 791ndash803 2003

[4] B Mungamuru and P Aarabi ldquoEnhanced sound localizationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 34 no 3 pp 1526ndash1540 2004

[5] J C Chen K Yao and R E Hudson ldquoSource localization andbeamformingrdquo IEEE Signal Processing Magazine vol 19 no 2pp 30ndash39 2002

[6] T S Rappaport J H Reed and B D Woerner ldquoPosition loca-tion using wireless communications on highways of the futurerdquoIEEE Communications Magazine vol 34 no 10 pp 33ndash411996

[7] A Yeredor and E Angel ldquoJoint TDOA and FDOA estimation aconditional bound and its use for optimally weighted localiza-tionrdquo IEEE Transactions on Signal Processing vol 59 no 4 pp1612ndash1623 2011

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

[9] K C Ho andW Xu ldquoAn accurate algebraic solution for movingsource location using TDOA and FDOA measurementsrdquo IEEETransactions on Signal Processing vol 52 no 9 pp 2453ndash24632004

[10] W H Foy ldquoPosition-location solutions by taylor-series estima-tionrdquo IEEE Transactions on Aerospace and Electronic Systemsvol AES-12 no 2 pp 187ndash194 1976

[11] R O Schmidt ldquoAn algorithm for two-receiver TDOAFDOAemitter locationrdquo TechnicalMemoTM-1229 ESL IncorporatedSunnyvale Calif USA 1980

[12] K C Ho X Lu and L Kovavisaruch ldquoSource localization usingTDOA and FDOA measurements in the presence of receiverlocation errors analysis and solutionrdquo IEEE Transactions onSignal Processing vol 55 no 2 pp 684ndash696 2007

[13] X Y Sun J D Li P YHuang and J Y Pang ldquoTotal least-squaressolution of active target localization using TDOA and FDOAmeasurements in WSNrdquo in Proceedings of the IEEE 22ndInternational Conference on Advanced Information Networkingand Applications (AINAW rsquo08) pp 995ndash999 Okinawa JapanMarch 2008

[14] K Yang J P An X Y Bu and G C Sun ldquoConstrained totalleast-squares location algorithm using time-difference-of-arrival measurementsrdquo IEEE Transactions on Vehicular Technol-ogy vol 59 no 3 pp 1558ndash1562 2010

[15] H Yu G Huang and J Gao ldquoConstrained total least-squareslocalisation algorithm using time difference of arrival and fre-quency difference of arrival measurements with sensor locationuncertaintiesrdquo IET Radar Sonar and Navigation vol 6 no 9pp 891ndash899 2012

[16] W Meng W Xiao and L Xie ldquoOptimal sensor pairing forTDOA based source localization in sensor networksrdquo in Pro-ceedings of the 8th International Conference on InformationCommunications and Signal Processing (ICICS rsquo11) pp 1ndash5 IEEESingapore December 2011

[17] H Seute J Grandin C Enderli A Khenchaf and J CexusldquoWhy synchronization is a key issue in modern electronic sup-port measuresrdquo in Proceedings of the 16th International RadarSymposium (IRS rsquo15) pp 794ndash799 IEEE Dresden GermanyJune 2015

[18] A Yeredor ldquoOn passive TDOA and FDOA localization usingtwo sensors with no time or frequency synchronizationrdquo inProceedings f the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo13) pp 4066ndash4070 IEEEVancouver Canada May 2013

[19] J-Y Yoon J-W KimW-Y Lee andD-S Eom ldquoA TDoA-basedlocalization using precise time-synchronizationrdquo in Proceedingsof the 14th International Conference on Advanced Communi-cation Technology (ICACT rsquo12) pp 1266ndash1271 PyeongChangRepublic of Korea February 2012


[20] T Li A Ekpenyong and Y-F Huang ldquoSource localization andtracking using distributed asynchronous sensorsrdquo IEEE Trans-actions on Signal Processing vol 54 no 10 pp 3991ndash4003 2006

[21] W Meng L Xie and W Xiao ldquoTDOA sensor pairing in multi-hop sensor networksrdquo in Proceedings of the 11th ACMIEEEConference on Information Processing in Sensing Networks (IPSNrsquo12) pp 91ndash92 ACM Beijing China April 2012

[22] M Pourhomayoun and M L Fowler ldquoDistributed computa-tion for direct position determination emitter locationrdquo IEEETransactions on Aerospace amp Electronic Systems vol 50 no 4pp 2878ndash2889 2014

[23] W Meng L Xie and W Xiao ldquoDecentralized TDOA sensorpairing in multihop wireless sensor networksrdquo IEEE SignalProcessing Letters vol 20 no 2 pp 181ndash184 2013

[24] H-W Wei R Peng Q Wan Z-X Chen and S-F YeldquoMultidimensional scaling analysis for passive moving targetlocalizationwithTDOAandFDOAmeasurementsrdquo IEEETransactions on Signal Processing vol 58 no 3 part 2 pp 1677ndash16882010

[25] FMeyer E Riegler O Hlinka and F Hlawatsch ldquoSimultaneousdistributed sensor self-localization and target tracking usingbelief propagation and likelihood consensusrdquo in Proceedings ofthe 46th Asilomar Conference on Signals Systems andComputers(ASILOMAR rsquo12) pp 1212ndash1216 IEEE Pacific Grove CalifUSA November 2012

[26] L Rui and K C Ho ldquoBias analysis of source localization usingthe maximum likelihood estimatorrdquo in Proceedings of the IEEEInternational Conference on Acoustics Speech and Signal Pro-cessing (ICASSP rsquo12) pp 2605ndash2608 Kyoto Japan March 2012

[27] L Rui and K C Ho ldquoBias compensation for target trackingfrom range based Maximum Likelihood position estimatesrdquoin Proceedings of the IEEE 7th Sensor Array and MultichannelSignal Processing Workshop (SAM rsquo12) pp 193ndash196 IEEEHoboken NJ USA June 2012

[28] R J Barton andD Rao ldquoPerformance capabilities of long-rangeUWB-IR TDOA localization systemsrdquo Eurasip Journal onAdvances in Signal Processing vol 2008 Article ID 236791 1page 2008

[29] K CHo ldquoBias reduction for an explicit solution of source local-ization using TDOArdquo IEEE Transactions on Signal Processingvol 60 no 5 pp 2101ndash2114 2012

[30] B Hao Z Li P Qi and L Guan ldquoEffective bias reductionmeth-ods for passive source localization using TDOA and GROArdquoScience China Information Sciences vol 56 no 7 pp 1ndash12 2013

[31] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory Prentice-Hall Englewood Cliffs NJ USA 1993

[32] M S Bartlett ldquoApproximate confidence intervals II More thanone unknown parameterrdquo Biometrika vol 40 pp 306ndash317 1953

[33] J B S Haldane ldquoThe estimation of two parameters from asamplerdquo Sankhya vol 12 no 4 pp 313ndash320 1953

[34] D R Cox and E J Snell ldquoA general definition of residualsrdquoJournal of the Royal Statistical Society Series B Methodologicalvol 30 no 2 pp 248ndash275 1968

[35] T-L Zhang ldquoSolving non-linear equation based on steepestdescent methodrdquo in Proceedings of the 4th International Con-ference on Information and Computing (ICIC rsquo11) pp 216ndash218IEEE April 2011

[36] Q Q Shi H Huo T Fang and R D Li ldquoUsing steepest descentmethod to refine the node positions in wireless sensor net-worksrdquo in Proceedings of the IET Conference onWireless Mobile

and Sensor Networks (CCWMSN rsquo07) pp 1055ndash1058 ShanghaiChina December 2007

[37] N Wu W Yuan H Wang and J Kuang ldquoTOA-based passivelocalization of multiple targets with inaccurate receivers basedon belief propagation on factor graphrdquoDigital Signal Processingvol 49 pp 14ndash23 2016

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


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



SensorTarget

(a) Centralized localization

SensorTarget

(b) Distributed localization

Figure 1 Sensor pairing

[18] fails to achieve precise synchronization between devicesfor separation distance operation it is impossible to obtaincorrect measurements from signals sent by receivers andsuch a failure to obtain the accurate values directly affectslocation estimation error [19] For example in order to getmore accuracy of source position and velocity the TSWLSTLS and CTLS also require precise time-synchronizationand frequency-locking among all receivers The price to payis accuracy required for TDOA and FDOA measurementsstrongly depending on synchronization [17] Thus all thereceivers would be precisely synchronized in time which isdifficult to realize and increase the system complexity

On the other hand it is well known that the centralizedmethod is power consuming since raw measurement data isinvolved in the transmissions and there is high computationalcost at a single sensor for centralized localization [20]In addition this kind of method requires a high band-width which is generally limited in wireless sensor networks(WSNs) and alsomay cause a large processing delay [21]Thiswould have significant impacts on the size weight and power(SWaP) requirement of that sensor [22]

Therefore in practice to reduce the requirements for net-work bandwidth synchronization and power consumption[23] the distributed localization is highly desirable As seenin Figure 1(b) several sensor pairs are formed without acommon reference sensor The whole sensors do not need totransmit their original signals to the single reference sensorbut only to their reference sensor of each group Furthermoreas the speed of the target is far less than the signal propagationspeed and the change of target position in synchronizationerror is slightly small that we can ignore it we only needto achieve precise synchronization between two sensors in agroup rather than all sensors and have rough synchronizationbetween each group Therefore the distributed localizationnot only reduces the difficulty of the synchronization betweenall sensors but also decreases the load of data computationand transmission So it is necessary to do some deep

research into distributed localization problem and improvethe estimation accuracy Nowadays the distributed structureis a novel problem in passive localization and there are stillsome problems for distributed passive localization such asthe optimal sensor pairing strategy localization algorithmand optimal baseline between sensors In this paper weaim at the location algorithm problem For conventionalcentralized structure source location methods thanks tothe only reference sensor in the conventional centralizedstructure source location methods they can be transformedfrom a nonlinear problem to a pseudolinear problem [24]For instance Ho and Xu used the position and velocity ofreference sensor to transform the TDOAand FDOAequationto a set of linear equations and then applied linear weightedleast-squares (LS) to obtain the source position [9] As thereare many different reference sensors in distributed structureit is difficult to use the conventional centralized structuresource location methods for estimation Therefore it isessential to put forward a distributed localization algorithm

Meyer et al put forward the distributed sensor self-localization and target tracking method in 2012 [25] Dis-tributed source localization is also a nontrivial problembecause the TDOA and FDOA measurements are nonlin-early related to the source location parameters Due to thisproblem the mean square error (MSE) is composed of twoparts the variance and the bias square When the noiselevel is low and the observation period is short the biasis not significant compared with the variance of the targetposition estimation [26 27] Thus the Cramer-Rao lowerbound (CRLB) which is developed for an unbiased estimatoris often used as a reference for evaluating the performance of alocation estimator However with the increase of observationperiod the location variancewill decrease and the bias cannotbe ignored It had a serious influence on estimation perfor-mance In particular with the development of ultrawideband(UWB) technology the measurements of the target can bemeasured repeatedly in a short time and then the precision








[119909 119910 119911]119879 and velocity u = [ ]


119894= [119909119894 119910119894 119911119894]119879 and

velocity s119894

= [119894 119894 119894]119879 119894 = 1 2 119872 are assumed to be




119903119900

119894=

1003817100381710038171003817u119900

minus s119900119894

1003817100381710038171003817 = radic(u119900 minus s119900119894)119879

(u119900 minus s119900119894) (1)


Σ = 2119894 2119894 minus 1 | 1 le 119894 le119872

2 (2)


119900



119903119900

21198942119894minus1= 119888120591119900

21198942119894minus1= 119903119900

2119894minus 119903119900

2119894minus1 (3)

where 119903119900





119900



119903119900

119894=

(u119900 minus s119900119894)119879

(u119900 minus s119900119894)

119903119900

119894

(4)


119903119900

21198942119894minus1= 119903119900

2119894minus 119903119900

2119894minus1(5)

for 119894 = 1 2 1198722 where 119903119900




21 11990343

119903119872119872minus1

]119879 and r =

[ 11990321

11990343

119903119872119872minus1



r = r119900 + Δr

r = r119900 + Δr(6)


where

Δr = [Δ11990321

Δ11990343

Δ119903119872119872minus1

]119879

Δr = [Δ 11990321

Δ 11990343

Δ 119903119872119872minus1

]119879

(7)


m = F (120579119900) + n (8)






120593 = r119900 minus r + Δr

120573 = r119900 minus r + Δr(9)

where 120593 = [1205931 1205932 120593

1198722]119879 and 120573 = [120573

1 1205732 120573

1198722]119879


Φ = F (120579119900) minus m + n (10)


defined as

120588 (120579) = Φ2

=

1198722

sum

119894=1

(120593119894 (120579))2

+

1198722

sum

119894=1

(120573119894 (120579))2

(11)


120579 = argmin120579

120588 (120579) (12)




G0

= [G11987901

G11987902

]119879

=120597120588

120597120579

100381610038161003816100381610038161003816100381610038161205790

= [120597120588

120597u1198790

120597120588

120597u1198790

]

119879

(13)


1205791

= [u1198791 u1198791]119879

= [u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

]119879

(14)

where

G01

= 2 sdot

1198722

sum

119894=1

120593119894sdot

120597120593119894

120597u+ 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

G02

= 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

120597120593119894

120597u=

12059711990321198942119894minus1

120597u= x1198792119894


120597120573119894

120597u=

120597 11990321198942119894minus1

120597u= k1198792119894


120597120573119894

120597u=

120597120593119894

120597u

(119894 = 1 2 119872

2)

(15)


119894and k119894



120588 (1205791) = 120588 (u119879

1 u1198791)

asymp min 120588 (u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

)

(16)




120588 (1205791) =

1198722

sum

119894=1

(120593119894(1205791))2

+

1198722

sum

119894=1

(120573119894(1205791))2

asymp

1198722

sum

119894=1

(120593119894)2

+

1198722

sum

119894=1

(120573119894)2

minus 2120583

1198722

sum

119894=1

120573119894(G11987902

120597120573119894

120597u)

minus 2120582

1198722

sum

119894=1

120593119894(G11987901

120597120593119894

120597u) minus 2120582

1198722

sum

119894=1

120573119894(G11987901

120597120573119894

120597u)

+ 1205822

1198722

sum

119894=1

(G11987901

120597120593119894

120597u)

2

+ 1205822

1198722

sum

119894=1

(G11987901

120597120573119894

120597u)

2

+ 1205832

1198722

sum

119894=1

(G11987902

120597120573119894

120597u)

210038161003816100381610038161003816100381610038161003816100381610038161205790

(17)



120597120588 (1205791)

120597120582= 0

120597120588 (1205791)

120597120583= 0

(18)


120582

=

sum1198722

119894=1120593119894(G11987901

(120597120593119894120597u)) + sum

1198722

119894=1120573119894(G11987901

(120597120573119894120597u))

sum1198722

119894=1(G11987901

(120597120593119894120597u))2

+ sum1198722

119894=1(G11987901

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

120583 =

sum1198722

119894=1120573119894(G11987902

(120597120573119894120597u))

sum1198722

119894=1(G11987902

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

(19)



ln119891 (m 120579) = 119896 minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (20)

where 119896 = minus12 ln((2120587)119872


= argmax (119868) (21)


119868 ≜ minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (22)


P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

= 0 (23)




P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

asymp H1015840 + H10158401015840 ( minus 120579119900) + g (120579

119900) = 0 (24)

where

H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

H101584010158401015840119897

=120597

120597120579119897

(1205972119868

120597120579120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900119897 = 1 2 6

g (120579119900) =

1

2

[[[[[[[[[

[

tr (H1015840101584010158401


119900]119879

)

tr (H1015840101584010158402


119900]119879

)

tr (H1015840101584010158406


119900]119879

)

]]]]]]]]]

]

(25)


119864 [ minus 120579119900] = 119864 [minus (H10158401015840)

minus1

H1015840]

+ 119864 [minus (H10158401015840)minus1

g (120579119900)]

(26)


H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= minus (A minus B)

A =120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

B =

119872

sum

119895=1

119872

sum

119894=1

119902119894119895119899119894

1205972119865119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(27)




119864 [minus (H10158401015840)minus1

H1015840] = 119864 [(A minus B)minus1 C]


(28)


119864 [minus (H10158401015840)minus1




119864 [minus (H10158401015840)minus1

H1015840]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579)Qminus1 sdot 119864 [119899

119894n]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(30)


unity P119894is

P119894=

119872

sum

119895=1

119902119894119895

1205972F119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(31)



119864 [minus (H10158401015840)minus1

g (120579119900)] asymp Α

minus1z (32)

where

z ≜ 119864 [g (120579119900)] asymp

1

2

[[[[[[[

[

tr (119864 [H1015840101584010158401

] times CRLB (120579119900))

tr (119864 [H1015840101584010158402

] times CRLB (120579119900))

tr (119864 [H1015840101584010158406

] times CRLB (120579119900))

]]]]]]]

]

(33)



119864 [H101584010158401015840119897

] =

119872

sum

119894=1

[h119879119894e119897P119894+ P119894e119897h119879119894

+ h119894e119879119897P119879119894] (34)

where

h119894=

119872

sum

119895=1

119902119894119895

120597F119895 (120579)

120597120579

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (35)



b = 119864 [ minus 120579119900]

= Aminus1(119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894+ z)

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(36)





= minus b (37)







J = 119864 [(120597 ln119891 (m 120579)

120597120579)

119879

(120597 ln119891 (m 120579)

120597120579)]

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (38)

where m = [11990321

119903119872119872minus1

11990321

119903119872119872minus1



120597 ln119891 (m 120579)

120597120579=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C (39)


CRLB (120579119900) = Jminus1 = [(

120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900]

minus1

= [A|120579=120579119900]minus1

(40)




= 120579119900

+ Δ

= minus b

(41)





119900)]

= 0

(43)


cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879


119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)






119909119894

119910119894

119911119894

119894

119894

119894


Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20



z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000



2

119903R and 01120590

2



119903is the



RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















[119909 119910 119911]119879 and velocity u = [ ]


119894= [119909119894 119910119894 119911119894]119879 and

velocity s119894

= [119894 119894 119894]119879 119894 = 1 2 119872 are assumed to be




119903119900

119894=

1003817100381710038171003817u119900

minus s119900119894

1003817100381710038171003817 = radic(u119900 minus s119900119894)119879

(u119900 minus s119900119894) (1)


Σ = 2119894 2119894 minus 1 | 1 le 119894 le119872

2 (2)


119900



119903119900

21198942119894minus1= 119888120591119900

21198942119894minus1= 119903119900

2119894minus 119903119900

2119894minus1 (3)

where 119903119900





119900



119903119900

119894=

(u119900 minus s119900119894)119879

(u119900 minus s119900119894)

119903119900

119894

(4)


119903119900

21198942119894minus1= 119903119900

2119894minus 119903119900

2119894minus1(5)

for 119894 = 1 2 1198722 where 119903119900




21 11990343

119903119872119872minus1

]119879 and r =

[ 11990321

11990343

119903119872119872minus1



r = r119900 + Δr

r = r119900 + Δr(6)


where

Δr = [Δ11990321

Δ11990343

Δ119903119872119872minus1

]119879

Δr = [Δ 11990321

Δ 11990343

Δ 119903119872119872minus1

]119879

(7)


m = F (120579119900) + n (8)






120593 = r119900 minus r + Δr

120573 = r119900 minus r + Δr(9)

where 120593 = [1205931 1205932 120593

1198722]119879 and 120573 = [120573

1 1205732 120573

1198722]119879


Φ = F (120579119900) minus m + n (10)


defined as

120588 (120579) = Φ2

=

1198722

sum

119894=1

(120593119894 (120579))2

+

1198722

sum

119894=1

(120573119894 (120579))2

(11)


120579 = argmin120579

120588 (120579) (12)




G0

= [G11987901

G11987902

]119879

=120597120588

120597120579

100381610038161003816100381610038161003816100381610038161205790

= [120597120588

120597u1198790

120597120588

120597u1198790

]

119879

(13)


1205791

= [u1198791 u1198791]119879

= [u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

]119879

(14)

where

G01

= 2 sdot

1198722

sum

119894=1

120593119894sdot

120597120593119894

120597u+ 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

G02

= 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

120597120593119894

120597u=

12059711990321198942119894minus1

120597u= x1198792119894


120597120573119894

120597u=

120597 11990321198942119894minus1

120597u= k1198792119894


120597120573119894

120597u=

120597120593119894

120597u

(119894 = 1 2 119872

2)

(15)


119894and k119894



120588 (1205791) = 120588 (u119879

1 u1198791)

asymp min 120588 (u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

)

(16)




120588 (1205791) =

1198722

sum

119894=1

(120593119894(1205791))2

+

1198722

sum

119894=1

(120573119894(1205791))2

asymp

1198722

sum

119894=1

(120593119894)2

+

1198722

sum

119894=1

(120573119894)2

minus 2120583

1198722

sum

119894=1

120573119894(G11987902

120597120573119894

120597u)

minus 2120582

1198722

sum

119894=1

120593119894(G11987901

120597120593119894

120597u) minus 2120582

1198722

sum

119894=1

120573119894(G11987901

120597120573119894

120597u)

+ 1205822

1198722

sum

119894=1

(G11987901

120597120593119894

120597u)

2

+ 1205822

1198722

sum

119894=1

(G11987901

120597120573119894

120597u)

2

+ 1205832

1198722

sum

119894=1

(G11987902

120597120573119894

120597u)

210038161003816100381610038161003816100381610038161003816100381610038161205790

(17)



120597120588 (1205791)

120597120582= 0

120597120588 (1205791)

120597120583= 0

(18)


120582

=

sum1198722

119894=1120593119894(G11987901

(120597120593119894120597u)) + sum

1198722

119894=1120573119894(G11987901

(120597120573119894120597u))

sum1198722

119894=1(G11987901

(120597120593119894120597u))2

+ sum1198722

119894=1(G11987901

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

120583 =

sum1198722

119894=1120573119894(G11987902

(120597120573119894120597u))

sum1198722

119894=1(G11987902

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

(19)



ln119891 (m 120579) = 119896 minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (20)

where 119896 = minus12 ln((2120587)119872


= argmax (119868) (21)


119868 ≜ minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (22)


P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

= 0 (23)




P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

asymp H1015840 + H10158401015840 ( minus 120579119900) + g (120579

119900) = 0 (24)

where

H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

H101584010158401015840119897

=120597

120597120579119897

(1205972119868

120597120579120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900119897 = 1 2 6

g (120579119900) =

1

2

[[[[[[[[[

[

tr (H1015840101584010158401


119900]119879

)

tr (H1015840101584010158402


119900]119879

)

tr (H1015840101584010158406


119900]119879

)

]]]]]]]]]

]

(25)


119864 [ minus 120579119900] = 119864 [minus (H10158401015840)

minus1

H1015840]

+ 119864 [minus (H10158401015840)minus1

g (120579119900)]

(26)


H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= minus (A minus B)

A =120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

B =

119872

sum

119895=1

119872

sum

119894=1

119902119894119895119899119894

1205972119865119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(27)




119864 [minus (H10158401015840)minus1

H1015840] = 119864 [(A minus B)minus1 C]


(28)


119864 [minus (H10158401015840)minus1




119864 [minus (H10158401015840)minus1

H1015840]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579)Qminus1 sdot 119864 [119899

119894n]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(30)


unity P119894is

P119894=

119872

sum

119895=1

119902119894119895

1205972F119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(31)



119864 [minus (H10158401015840)minus1

g (120579119900)] asymp Α

minus1z (32)

where

z ≜ 119864 [g (120579119900)] asymp

1

2

[[[[[[[

[

tr (119864 [H1015840101584010158401

] times CRLB (120579119900))

tr (119864 [H1015840101584010158402

] times CRLB (120579119900))

tr (119864 [H1015840101584010158406

] times CRLB (120579119900))

]]]]]]]

]

(33)



119864 [H101584010158401015840119897

] =

119872

sum

119894=1

[h119879119894e119897P119894+ P119894e119897h119879119894

+ h119894e119879119897P119879119894] (34)

where

h119894=

119872

sum

119895=1

119902119894119895

120597F119895 (120579)

120597120579

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (35)



b = 119864 [ minus 120579119900]

= Aminus1(119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894+ z)

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(36)





= minus b (37)







J = 119864 [(120597 ln119891 (m 120579)

120597120579)

119879

(120597 ln119891 (m 120579)

120597120579)]

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (38)

where m = [11990321

119903119872119872minus1

11990321

119903119872119872minus1



120597 ln119891 (m 120579)

120597120579=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C (39)


CRLB (120579119900) = Jminus1 = [(

120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900]

minus1

= [A|120579=120579119900]minus1

(40)




= 120579119900

+ Δ

= minus b

(41)





119900)]

= 0

(43)


cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879


119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)






119909119894

119910119894

119911119894

119894

119894

119894


Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20



z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000



2

119903R and 01120590

2



119903is the



RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

Δr = [Δ11990321

Δ11990343

Δ119903119872119872minus1

]119879

Δr = [Δ 11990321

Δ 11990343

Δ 119903119872119872minus1

]119879

(7)


m = F (120579119900) + n (8)






120593 = r119900 minus r + Δr

120573 = r119900 minus r + Δr(9)

where 120593 = [1205931 1205932 120593

1198722]119879 and 120573 = [120573

1 1205732 120573

1198722]119879


Φ = F (120579119900) minus m + n (10)


defined as

120588 (120579) = Φ2

=

1198722

sum

119894=1

(120593119894 (120579))2

+

1198722

sum

119894=1

(120573119894 (120579))2

(11)


120579 = argmin120579

120588 (120579) (12)




G0

= [G11987901

G11987902

]119879

=120597120588

120597120579

100381610038161003816100381610038161003816100381610038161205790

= [120597120588

120597u1198790

120597120588

120597u1198790

]

119879

(13)


1205791

= [u1198791 u1198791]119879

= [u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

]119879

(14)

where

G01

= 2 sdot

1198722

sum

119894=1

120593119894sdot

120597120593119894

120597u+ 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

G02

= 2 sdot

1198722

sum

119894=1

120573119894sdot

120597120573119894

120597u

120597120593119894

120597u=

12059711990321198942119894minus1

120597u= x1198792119894


120597120573119894

120597u=

120597 11990321198942119894minus1

120597u= k1198792119894


120597120573119894

120597u=

120597120593119894

120597u

(119894 = 1 2 119872

2)

(15)


119894and k119894



120588 (1205791) = 120588 (u119879

1 u1198791)

asymp min 120588 (u1198790

minus 120582G11987901

u1198790

minus 120583G11987902

)

(16)




120588 (1205791) =

1198722

sum

119894=1

(120593119894(1205791))2

+

1198722

sum

119894=1

(120573119894(1205791))2

asymp

1198722

sum

119894=1

(120593119894)2

+

1198722

sum

119894=1

(120573119894)2

minus 2120583

1198722

sum

119894=1

120573119894(G11987902

120597120573119894

120597u)

minus 2120582

1198722

sum

119894=1

120593119894(G11987901

120597120593119894

120597u) minus 2120582

1198722

sum

119894=1

120573119894(G11987901

120597120573119894

120597u)

+ 1205822

1198722

sum

119894=1

(G11987901

120597120593119894

120597u)

2

+ 1205822

1198722

sum

119894=1

(G11987901

120597120573119894

120597u)

2

+ 1205832

1198722

sum

119894=1

(G11987902

120597120573119894

120597u)

210038161003816100381610038161003816100381610038161003816100381610038161205790

(17)



120597120588 (1205791)

120597120582= 0

120597120588 (1205791)

120597120583= 0

(18)


120582

=

sum1198722

119894=1120593119894(G11987901

(120597120593119894120597u)) + sum

1198722

119894=1120573119894(G11987901

(120597120573119894120597u))

sum1198722

119894=1(G11987901

(120597120593119894120597u))2

+ sum1198722

119894=1(G11987901

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

120583 =

sum1198722

119894=1120573119894(G11987902

(120597120573119894120597u))

sum1198722

119894=1(G11987902

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

(19)



ln119891 (m 120579) = 119896 minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (20)

where 119896 = minus12 ln((2120587)119872


= argmax (119868) (21)


119868 ≜ minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (22)


P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

= 0 (23)




P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

asymp H1015840 + H10158401015840 ( minus 120579119900) + g (120579

119900) = 0 (24)

where

H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

H101584010158401015840119897

=120597

120597120579119897

(1205972119868

120597120579120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900119897 = 1 2 6

g (120579119900) =

1

2

[[[[[[[[[

[

tr (H1015840101584010158401


119900]119879

)

tr (H1015840101584010158402


119900]119879

)

tr (H1015840101584010158406


119900]119879

)

]]]]]]]]]

]

(25)


119864 [ minus 120579119900] = 119864 [minus (H10158401015840)

minus1

H1015840]

+ 119864 [minus (H10158401015840)minus1

g (120579119900)]

(26)


H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= minus (A minus B)

A =120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

B =

119872

sum

119895=1

119872

sum

119894=1

119902119894119895119899119894

1205972119865119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(27)




119864 [minus (H10158401015840)minus1

H1015840] = 119864 [(A minus B)minus1 C]


(28)


119864 [minus (H10158401015840)minus1




119864 [minus (H10158401015840)minus1

H1015840]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579)Qminus1 sdot 119864 [119899

119894n]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(30)


unity P119894is

P119894=

119872

sum

119895=1

119902119894119895

1205972F119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(31)



119864 [minus (H10158401015840)minus1

g (120579119900)] asymp Α

minus1z (32)

where

z ≜ 119864 [g (120579119900)] asymp

1

2

[[[[[[[

[

tr (119864 [H1015840101584010158401

] times CRLB (120579119900))

tr (119864 [H1015840101584010158402

] times CRLB (120579119900))

tr (119864 [H1015840101584010158406

] times CRLB (120579119900))

]]]]]]]

]

(33)



119864 [H101584010158401015840119897

] =

119872

sum

119894=1

[h119879119894e119897P119894+ P119894e119897h119879119894

+ h119894e119879119897P119879119894] (34)

where

h119894=

119872

sum

119895=1

119902119894119895

120597F119895 (120579)

120597120579

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (35)



b = 119864 [ minus 120579119900]

= Aminus1(119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894+ z)

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(36)





= minus b (37)







J = 119864 [(120597 ln119891 (m 120579)

120597120579)

119879

(120597 ln119891 (m 120579)

120597120579)]

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (38)

where m = [11990321

119903119872119872minus1

11990321

119903119872119872minus1



120597 ln119891 (m 120579)

120597120579=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C (39)


CRLB (120579119900) = Jminus1 = [(

120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900]

minus1

= [A|120579=120579119900]minus1

(40)




= 120579119900

+ Δ

= minus b

(41)





119900)]

= 0

(43)


cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879


119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)






119909119894

119910119894

119911119894

119894

119894

119894


Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20



z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000



2

119903R and 01120590

2



119903is the



RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120588 (1205791) =

1198722

sum

119894=1

(120593119894(1205791))2

+

1198722

sum

119894=1

(120573119894(1205791))2

asymp

1198722

sum

119894=1

(120593119894)2

+

1198722

sum

119894=1

(120573119894)2

minus 2120583

1198722

sum

119894=1

120573119894(G11987902

120597120573119894

120597u)

minus 2120582

1198722

sum

119894=1

120593119894(G11987901

120597120593119894

120597u) minus 2120582

1198722

sum

119894=1

120573119894(G11987901

120597120573119894

120597u)

+ 1205822

1198722

sum

119894=1

(G11987901

120597120593119894

120597u)

2

+ 1205822

1198722

sum

119894=1

(G11987901

120597120573119894

120597u)

2

+ 1205832

1198722

sum

119894=1

(G11987902

120597120573119894

120597u)

210038161003816100381610038161003816100381610038161003816100381610038161205790

(17)



120597120588 (1205791)

120597120582= 0

120597120588 (1205791)

120597120583= 0

(18)


120582

=

sum1198722

119894=1120593119894(G11987901

(120597120593119894120597u)) + sum

1198722

119894=1120573119894(G11987901

(120597120573119894120597u))

sum1198722

119894=1(G11987901

(120597120593119894120597u))2

+ sum1198722

119894=1(G11987901

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

120583 =

sum1198722

119894=1120573119894(G11987902

(120597120573119894120597u))

sum1198722

119894=1(G11987902

(120597120573119894120597u))2

10038161003816100381610038161003816100381610038161003816100381610038161003816120579119900

(19)



ln119891 (m 120579) = 119896 minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (20)

where 119896 = minus12 ln((2120587)119872


= argmax (119868) (21)


119868 ≜ minus1

2[m minus F (120579)]

119879Qminus1 [m minus F (120579)] (22)


P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

= 0 (23)




P () =120597119868

120597120579

10038161003816100381610038161003816100381610038161003816

asymp H1015840 + H10158401015840 ( minus 120579119900) + g (120579

119900) = 0 (24)

where

H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

H101584010158401015840119897

=120597

120597120579119897

(1205972119868

120597120579120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900119897 = 1 2 6

g (120579119900) =

1

2

[[[[[[[[[

[

tr (H1015840101584010158401


119900]119879

)

tr (H1015840101584010158402


119900]119879

)

tr (H1015840101584010158406


119900]119879

)

]]]]]]]]]

]

(25)


119864 [ minus 120579119900] = 119864 [minus (H10158401015840)

minus1

H1015840]

+ 119864 [minus (H10158401015840)minus1

g (120579119900)]

(26)


H1015840 = 120597119868

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579119900=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C

H10158401015840 = 1205972119868

120597120579120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= minus (A minus B)

A =120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

100381610038161003816100381610038161003816100381610038161003816120579=120579119900

B =

119872

sum

119895=1

119872

sum

119894=1

119902119894119895119899119894

1205972119865119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(27)




119864 [minus (H10158401015840)minus1

H1015840] = 119864 [(A minus B)minus1 C]


(28)


119864 [minus (H10158401015840)minus1




119864 [minus (H10158401015840)minus1

H1015840]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579)Qminus1 sdot 119864 [119899

119894n]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(30)


unity P119894is

P119894=

119872

sum

119895=1

119902119894119895

1205972F119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(31)



119864 [minus (H10158401015840)minus1

g (120579119900)] asymp Α

minus1z (32)

where

z ≜ 119864 [g (120579119900)] asymp

1

2

[[[[[[[

[

tr (119864 [H1015840101584010158401

] times CRLB (120579119900))

tr (119864 [H1015840101584010158402

] times CRLB (120579119900))

tr (119864 [H1015840101584010158406

] times CRLB (120579119900))

]]]]]]]

]

(33)



119864 [H101584010158401015840119897

] =

119872

sum

119894=1

[h119879119894e119897P119894+ P119894e119897h119879119894

+ h119894e119879119897P119879119894] (34)

where

h119894=

119872

sum

119895=1

119902119894119895

120597F119895 (120579)

120597120579

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (35)



b = 119864 [ minus 120579119900]

= Aminus1(119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894+ z)

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(36)





= minus b (37)







J = 119864 [(120597 ln119891 (m 120579)

120597120579)

119879

(120597 ln119891 (m 120579)

120597120579)]

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (38)

where m = [11990321

119903119872119872minus1

11990321

119903119872119872minus1



120597 ln119891 (m 120579)

120597120579=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C (39)


CRLB (120579119900) = Jminus1 = [(

120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900]

minus1

= [A|120579=120579119900]minus1

(40)




= 120579119900

+ Δ

= minus b

(41)





119900)]

= 0

(43)


cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879


119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)






119909119894

119910119894

119911119894

119894

119894

119894


Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20



z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000



2

119903R and 01120590

2



119903is the



RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864 [minus (H10158401015840)minus1

H1015840] = 119864 [(A minus B)minus1 C]


(28)


119864 [minus (H10158401015840)minus1




119864 [minus (H10158401015840)minus1

H1015840]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579)Qminus1 sdot 119864 [119899

119894n]

asymp Aminus1119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(30)


unity P119894is

P119894=

119872

sum

119895=1

119902119894119895

1205972F119895 (120579)

120597120579120597120579119879

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(31)



119864 [minus (H10158401015840)minus1

g (120579119900)] asymp Α

minus1z (32)

where

z ≜ 119864 [g (120579119900)] asymp

1

2

[[[[[[[

[

tr (119864 [H1015840101584010158401

] times CRLB (120579119900))

tr (119864 [H1015840101584010158402

] times CRLB (120579119900))

tr (119864 [H1015840101584010158406

] times CRLB (120579119900))

]]]]]]]

]

(33)



119864 [H101584010158401015840119897

] =

119872

sum

119894=1

[h119879119894e119897P119894+ P119894e119897h119879119894

+ h119894e119879119897P119879119894] (34)

where

h119894=

119872

sum

119895=1

119902119894119895

120597F119895 (120579)

120597120579

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (35)



b = 119864 [ minus 120579119900]

= Aminus1(119872

sum

119894=1

P119894Aminus1 (120597

119879F (120579)

120597120579) e119894+ z)

1003816100381610038161003816100381610038161003816100381610038161003816120579=120579119900

(36)





= minus b (37)







J = 119864 [(120597 ln119891 (m 120579)

120597120579)

119879

(120597 ln119891 (m 120579)

120597120579)]

100381610038161003816100381610038161003816100381610038161003816120579=120579119900 (38)

where m = [11990321

119903119872119872minus1

11990321

119903119872119872minus1



120597 ln119891 (m 120579)

120597120579=

120597119879F (120579)

120597120579Qminus1n

100381610038161003816100381610038161003816100381610038161003816120579=120579119900= C (39)


CRLB (120579119900) = Jminus1 = [(

120597119879F (120579)

120597120579Qminus1 120597F (120579)

120597120579119879

)

100381610038161003816100381610038161003816100381610038161003816120579=120579119900]

minus1

= [A|120579=120579119900]minus1

(40)




= 120579119900

+ Δ

= minus b

(41)





119900)]

= 0

(43)


cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879


119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)






119909119894

119910119894

119911119894

119894

119894

119894


Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20



z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000



2

119903R and 01120590

2



119903is the



RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












= 120579119900

+ Δ

= minus b

(41)





119900)]

= 0

(43)


cov () = 119864 [( minus 119864 []) ( minus 119864 [])119879

]

= 119864 [ΔΔ119879

] = 119864 [(Δ minus b) (Δ minus b)119879

]

= 119864 [ΔΔ119879

] minus bb119879

(44)

where 119864[ΔΔ119879

] and 119864[ΔΔ119879


119894119894ge 0 (119894 = 1 2 6)

cov ()119894119894

le (119864 [ΔΔ119879

])119894119894

(45)






119909119894

119910119894

119911119894

119894

119894

119894


Set 2 3 500 minus200 500 minus10 0 104 600 100 600 10 20 15

Set 3 5 100 600 800 minus10 20 206 minus100 400 700 30 0 20



z(m

)

y (m)x (m)

0

500

1000

1500

2000

2500

3000

1000

0

minus1000

minus2000

minus3000

2000

1000

0

minus1000



2

119903R and 01120590

2



119903is the



RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

RMSE (u) =radic

sum119870

119896=1

1003817100381710038171003817u119896 minus u11990010038171003817100381710038172

119870

(46)



minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

bias (u) =1

119870

1003817100381710038171003817100381710038171003817100381710038171003817

119870

sum

119896=1

(u119896

minus u119900)1003817100381710038171003817100381710038171003817100381710038171003817

(47)


119896and u




119879 and u119900 = [minus30 minus15 20]119879



10

10


10minus072

10minus069

10minus2

100

102

Posit

ion

RMSE

(m)

10minus2

100

102

Velo

city

RM

SE (m

s)




MLECRLB


10minus1165

10minus116

10 log(c 120590r ) (dB)

10 log(c21205902r ) (dB)



2





119879 and u119900 = [minus30 minus15





minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus100

minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)





minus50

0

50

Posit

ion

bias

(dB)

minus100

minus50

0

50

Velo

city

bia

s (dB

)

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)




10




MLECRLB

100

102

Posit

ion

RMSE

(m)

10minus2

100

Velo

city

RM

SE (m

s)



10minus073

10minus072

minus30

10minus1178

10minus1177

10minus1176

10 log(c21205902r ) (dB)

10 log(c21205902r ) (dB)



119903119900

1asymp 119903119900


119900

119872

119903119900

1

119903119900

1


119872

119903119900

119872

asymp 0

(48)





minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus100

minus50

0

Posit

ion

bias

(dB)

minus100

minus150

minus50

0

Velo

city

bia

s (dB

)



10 log(c21205902r ) (dB)


5 Conclusion


Appendix



r (120579) = [11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

r (120579) = [ 11990321 (120579) 119903

43 (120579) 119903119872119872minus1 (120579)]

119879

F (120579) = [r119879 (120579) r119879 (120579)]119879

(A1)


120597F (120579)

120597120579119879

=

[[[[

[

120597r (120579)

120597u119879120597r (120579)

120597u119879

120597r (120579)

120597u119879120597r (120579)

120597u119879

]]]]

]

(A2)


x119894= 119903119894 (120579)minus1

(u minus s119894)119879

k119894= 119903119894 (120579)minus1

(u minus s119894)119879

minus 119903119894 (120579)minus2

119903119894 (120579) (u minus s

119894)119879

(A3)


120597r (120579)

120597u119879=

120597r (120579)

120597u119879=

[[[[

[

x2

minus x1

x119872

minus x119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879=

[[[[

[

k2

minus k1

k119872

minus k119872minus1

]]]]

](1198722)times3

120597r (120579)

120597u119879= 0(1198722)times3

(A4)



When 1 le 119895 le 1198722 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

1205972r21198952119895minus1 (120579)

120597u120597u1198791205972r21198952119895minus1 (120579)

120597u120597u119879

]]]]]

]6times6

(B1)


X119895

= (I minus x119879119895x119895) 119903119895 (120579)minus1

(B2)


1205972r21198952119895minus1 (120579)

120597u120597u119879= X2119895


1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879=

1205972r21198952119895minus1 (120579)

120597u120597u119879

= 03times3

(B3)


When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










When 1198722 + 1 le 119895 le 119872 1205972F119895(120579)120597120579120597120579


1205972F119895 (120579)

120597120579120597120579119879

=

[[[[[

[



120597u120597u119879



120597u120597u119879

]]]]]

]6times6

(B4)

Let

Y119895

= 119903119895 (120579) 119903

119895 (120579)minus2

(3x119879119895x119895

minus I)

w119895

= 119903119895 (120579)minus1

(u minus s119895)119879

W119895

= 119903119895 (120579)minus1 x119879119895w119895

Ψ119895

= Y119895

minus W119895

minus W119879119895

(B5)



120597u120597u119879= Ψ2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= X2119895minus119872



120597u120597u119879= 03times3

(B6)

Competing Interests


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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