research article a mathematical algorithm of locomotive...

Research ArticleA Mathematical Algorithm of Locomotive Source LocalizationBased on Hyperbolic Technique

Homayun Kabir Jeevan Kanesan Ahmed Wasif Reza and Harikrishnan Ramiah

Faculty of Engineering Department of Electrical Engineering University of Malaya 50603 Kuala Lumpur Malaysia

Correspondence should be addressed to Homayun Kabir hkabirumsiswaumedumy

Received 18 November 2014 Revised 15 March 2015 Accepted 25 March 2015

Academic Editor Longjun Dong

Copyright copy 2015 Homayun Kabir et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recent trend shows that sensors situated on an axis in two-dimensional scenario measuring the time difference of arrival (TDOA)and frequency difference of arrival (FDOA) of the emitting signal from a moving source can estimate the emitting signalrsquos positionand velocity from the intersection point of hyperbola which derives from TDOA and FDOA However estimating the location ofan emitter based on hyperbolic measurements is a highly nonlinear problem with inconsistent data which are created due to themeasurement noise the deviation between assumptionmodel and actual field of the velocity and so forth In addition the coefficientmatrix of TDOA and FDOA equations set is singular in the linear sensor array network (LSAN) In this paper a noniterative andsimpler method is proposed to locate the instantaneous position of the moving source in LSAN by estimating the position andvelocity based on TDOA and FDOA which does not have the convergence problem In addition the method avoids the singularityproblem of LSAN by introducing the nuisance variables The proposed method achieved the theoretical lower bound for near tofar field with same and different velocity and different baseline of sensors in low to moderate noise

1 Introduction

In recent decades localization awareness has aroused greatinterest in wireless sensor network (WSN) applicationssuch as in tracking the mobile homeland security defensecommand and control ship navigation and disaster manage-ment [1ndash6] The instantaneous position of signal emitter isdetermined by using relative position information as bearing[7] distance received signal strength (RSS) [8] receivedsignal strength ratio (RSSR) [9] time of arrival (TOA) [10]and time difference of arrival (TDOA) [11] of sensors whichcapture the signal from the source in WSN In this paper thesource position was estimated through a passive localizationtechnique called TDOA represented in Figure 1 as thistechnique does not require timestamping Furthermore itis easy to implement in the practical localization set-up[12] For the moving source velocity estimation applies thevital rule for instantaneous localization of the source [13]Hence frequency difference of arrival (FDOA)was combinedwith the TDOA for improving the localization accuracy ofmobile source [14 15] Two types of geometrical shapes of

sensors in WSN were used namely arbitrary sensor arraynetwork (ASAN) and linear sensor array network (LSAN)In ASAN geometry the two-step weighted least square (LS)method was derived in [14] to estimate the position andvelocity of moving source based on TDOA and FDOAFurthermore an accurate and closed-form solution wasproposed based on multidimensional scaling (MDS) analysisthat optimized a cost function related to the MDSrsquos scalarproduct matrix in [16] In addition a closed-form of LSestimation was derived for circular array sensor networkin [17] An analytical solution for cuboid network withoutpremeasured wave velocity was proposed in [18] AnotherLS method in [19] and an effective closed-form solution in[20] were derived to locate the mobile source position byconsidering the source and reference sensor motions whichcreate ASAN Most of the source localization algorithmsare for estimating the position of stationary sources Astationary source localization algorithm using LS estimationthat utilized the multiple-input multiple-output radar systemwith widely separated antennas was derived in [6] Also anoniterativemethodwith comparative performance based on

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 384180 9 pageshttpdxdoiorg1011552015384180

2 International Journal of Distributed Sensor Networks

d3

d2

d1

Receiver 3Receiver 2 Receiver 1

TDOA21 = d2 minus d1 TDOA31 = d3 minus d1

Data processor

Figure 1 TDOA based localization technique in LSAN

TDOA for Global Positioning System (GPS) was proposed in[21 22] In this paper we have focused our research based onthe LSAN to locate the position and velocity ofmoving sourceat near to far field

It has been found from the literature that the TDOA andFDOA equations with high nonlinearity form hyperbolaswhich may not be intersected at a single point due tothe inconsistent data such as measurement error and thedeviation between assumption model and actual field of thevelocity in LSAN [14 23] Also the singularity problemarises in the coefficient matrix of hyperbolic equations setin 2D LSAN [23 24] To overcome the challenges in the2D scenario a mathematical solution by approximate MLestimation based on TOA was developed in [25] which wasonly applicable for three linear sensors Additionally the MLestimation in [26] a closed-form of LS estimation in [19 23]and the geometric solution in [23] were proposed based onthe TDOA to predict the position of stationary source inLSAN

Moreover the trigonometric mathematical approach in[5] was derived to estimate the position of a moving sourcein LSAN for only a special case where the range betweenthe source and receivers was large compared to the spacingbetween receivers (far field) To the best of the authorsrsquoknowledge no literature has been found with the linearsensor array in a 2D scenario for estimating the position aswell as velocity of near to far field moving emitter In the 2Dscenario for LSAN each element in the array lies in sameaxis [23] The coordinate of the source position in that axis isalso absent in the nonlinear equation set Hence coefficientmatrix of the LSAN is singular [24] For overcoming theseissues we have proposed a mathematical approach based onTDOA and FDOA to estimate the position and velocity ofa moving source in LSAN In this paper we simulated theproposed method to estimate the position and velocity ofmoving source at near to far field with the same and differentvelocity in LSAN where Gaussian noise is considered

The paper is arranged in the following manner Inthe following section the proposed mathematical model ispresented Next the derivation of Cramer-Rao lower bound(CRLB) is provided After that the results are analyzed andperformance evaluation is explainedThen the limitation andfuturework are discussed Finally the concluding remarks areprovided

2 Proposed Method

The119873 linear stationary sensors of LSAN are considered in a2D space to estimate the locomotive emitter with unknownposition 119901 = [119909 119910]

119879 and the velocity V = [V119909 V119910]119879 utilizing

the TDOA and the FDOA localization techniques wherematrix transpose operation is denoted by 119879 represented inFigure 1 The signals emitted from the source is captured by119873 stationary sensors of LSAN which are situated at 119877

119894=

[119909119894 119910119894]119905 where 119910

119894= 0 and 119894 = 1 2 3 119873 The range

between the 119894th sensor and emitter is

1198890

119894=1003816100381610038161003816119901 minus 119877119894

1003816100381610038161003816 =radic(119901 minus 119877

119894)119879(119901 minus 119877

119894) (1)

The true path difference of sensor pair 119894th and reference thatassumed sensor 1 is

1198890

1198941= 1198881199051198941= 1198890

119894minus 1198890

1 (2)

With regard to (2) 119888 is the velocity of signal propagationand 1199051198941is the time difference of sensor pair 119894th and reference

Equation (2) is rearranged as 11988901198941+1198890

1= 1198890

119894and then both sides

are squared Substituting the 11988901and 1198890

119894from (1) the TDOA

equation set obtained is

1198890

1198941

2

+ 21198890

11198890

1198941

= 119877119879

119894119877119894minus 119877119879

11198771minus 2 (119877

119894minus 1198771)119879119901 where 119894 = 2 3 119873

(3)

Equation (3) is a nonlinear set with unknown 119901 and 11988901

which created 119873 minus 1 hyperbolic curves with focus 119877119894=

[119909119894 119910119894]119905 where 119894 = 1 2 3 119873 Those hyperbolic curves

intersect at a point which gives the estimated position ofthe source In the 2D scenario at least two hyperbolic curvesare required to solve the localization problem by utilizing theTDOA represented in Figure 2

Not only the position but also the velocity estimationis essential for determining the instantaneous position ofmoving emitter Conversely TDOA equations set may notbe adequate to provide the needed localization accuracy ofmoving emitter as TDOA estimates only the position of thesource The FDOA measurement which is obtained fromthe relative velocity between the source [13] and sensors isapplied to improve instantaneous localization accuracy of thesourceThe relation between the path rate and source positionparameters is obtained by the time derivative of (1) as follows

V0119894=V119879 (119901 minus 119877

119894)

1198890

119894

(4)

International Journal of Distributed Sensor Networks 3

0 5 10

0

2

4

6

8

10

Reference sensor

Source

minus10minus10

minus5

minus6

minus4

minus2

minus8

X (m)

Y(m

)

TDOA21TDOAN1

TDOA31

Figure 2 Source position estimation by the hyperbolic technique

The FDOA is calculated by the time derivative of (3) asfollows

2 (V011989411198890

1198941+ V011989411198890

1+ 1198890

1198941V01)

= minus2 (119877119894minus 1198771)119879 V where 119894 = 2 3 119873

(5)

In terms of (5) the path difference rate is denoted by V01198941

which is obtained from FDOAThe unknown position 119901 andvelocity V of the source are estimated by solving the obtainedTDOA and FDOA equations setThe 119899

1198941and 119899V

1198941are additive

white Gaussian noise summed with the true range differenceand the rate of range difference of sensor pair respectively asshown in (6) and (7)

1198891198941= 1198890

1198941+ 1198991198941 (6)

V1198941= V01198941+ 119899V1198941 (7)

The vector of noisy TDOA and FDOA 119863 =

[11988921 11988931 sdot sdot sdot 1198891198731] and 119881 = [V21 V31sdot sdot sdot V1198731] have a

covariance matrix [14] Consider

119876 = 119864[119863119879119881119879]119879

[119863119879119881119879] = 120590

2[

1198761119874

0 1198761

] (8)

Here 1205902 is the variance of zero mean Gaussian noise and 0 isa zero square matrix and

1198761=

[[[[[[[[[

[

1 05 05 sdot sdot sdot 05




]]]]]]]]]

]

which is a (119873 minus 1) times (119873 minus 1) matrix

(9)

Substituting the 1198891198941and V1198941in (3) and (5) the two nonlinear

equation sets obtained are

2 (119877119894minus 1198771)119879119901 + 2119889

11989411198890

1= minus1198892

1198941+ 119877119879

119894119877119894minus 119877119879

11198771

2V11989411198890

1+ 2 (119877

119894minus 1198771)119879 V + 2119889

1198941V01= minus2119889

1198941V1198941

(10)

In the LSAN 119910-axis coordinates of sensors are equal to zeroHence the singularity problem arises in (10) For mitigatingthe singularity problem these equations can be rewritten as

minus2 (119909119894minus 1199091) 119909 minus 2119889

11989411198890

1= 1198892

1198941+ 1199092

1minus 1199092

119894 (11)

minus2V11989411198890

1minus 2 (119909

119894minus 1199091) V119909minus 21198891198941V01= 21198891198941V1198941 (12)

With regard to (12) and (13) 119909 11988901 V119909 and V0

1are unknown

The solutions of the above equations are obtained by ini-tially denoting an auxiliary vector 120579 = [119909 119889

0

1V119909

V01]119879

which contains the 119909-axis unknown position parametersand two nuisance variables 1198890

1and V0

1(nuisance variables

are associated with the variation of the dependent variableas an outcome which is extraneous to the effects of theindependent variables) The reference sensor position isconsidered at the origin and the error vector of the aboveequations (11) and (12) is

120576 = [120576119905 120576119891]119879

= ℎ minus 119892120579 (13)

where 120576119905and 120576119891are the error vector of TDOA and FDOA

equation sets respectively whereas

ℎ =

[[[[[[[[[[[[[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

211988921V21

211988921V31

21198891198731V1198731

]]]]]]]]]]]]]]]]]]]

]

119892 = minus2

[[[[[[[[[[[[[[[[[[[

[

1199092minus 119909111988921

0 0

1199093minus 119909111988931

0 0

119909119873minus 11990911198891198731

0 0

0 V21

1199092minus 119909111988921

0 V31

1199093minus 119909111988931

0 V1198731

119909119873minus 11990911198891198731

]]]]]]]]]]]]]]]]]]]

]

(14)

To minimize the error vector (13) can be written as

⟨120579⟩ = arg min120579

(ℎ minus 119892120579) (15)


Here the weight least square method is used for minimizingthe error vector We get the unknown vector as

⟨120579⟩ = (119892119879119882119892)minus1

119892119879119882ℎ (16)

where 119882 = 119876minus1 From the above equation only 119909-axis

unknown position ⟨119909⟩ and velocity ⟨V119909⟩ of the locomotive

source and two nuisance variables ⟨11988901⟩ and ⟨V0

1⟩ are esti-

mated To determine the 119910-axis position parameter ⟨119909⟩ ⟨V119909⟩

⟨1198890

1⟩ and ⟨V0

1⟩ values are substituted into (1) and then the

equation can be rewritten as

⟨119910⟩ = radic⟨1198890

1⟩2minus ⟨119909⟩2 (17)

Finding the appropriate sign of 119910-axis source position coor-dinate the equation cost error function 119869 = 120576

119879

11198761205761of (11)

has to be minimized Regarding equation cost error function1205761= ℎ1minus 11989211205791 and

ℎ1=

[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

]]]]]]]

]

1198921= minus2

[[[[[[

[

1199092minus 119909111988921

1199093minus 119909111988931

119909119873minus 11990911198891198731

]]]]]]

]

1205791= [119909 119889

0

1]119879

(18)

This solution is identical to the one given in [26] Finally theobtained 119910-axis position parameter value is put into (4) toestimate the 119910-axis velocity parameter as indicated below

⟨V119910⟩ =

⟨1198890

1⟩ ⟨V01⟩ minus ⟨119909⟩ ⟨V119909⟩⟨119910⟩

(19)

3 Cramer-Rao Lower Bound

It is important to know the optimum achievable localizationaccuracy that can be attained with the available measurementset The CRLB provides a lower bound on the covariancethat is asymptotically achievable by any unbiased estimationalgorithm [14] Therefore the CRLB sets a benchmark ofan unbiased estimation which has been compared with theproposed method However the CRLB is equal to the sumof the diagonal elements of covariance matrix estimation Toestimate the covariancematrix of source position and velocityin LSAN based on the TDOA and FDOA we perturbed therandom quantities in 120579 mentioned as auxiliary vector andproceed as before to obtain the following [23]

119864 [Δ120579 Δ120579119879] = (119892

0119879

120593minus11198920)

minus1

(20)

With regard to (20) 120593 = 11986111198761198611 1198611= [119861 119874

1198610119861] 119861 =

2 diag (119889021198890

31198890

4sdot sdot sdot 1198890

119873) 1198610 = 2 diag (V0

2V03V04sdot sdot sdot V0119873)

andO is a zero square matrix Following (1) 1205791= [Δ119909 Δ119910]

119879

and Δ11988901are related to

Δ1198890

1=

Δ120579119879

1

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(21)

According to (4) 1205792= [ΔV119909 ΔV119910]

119879 and ΔV01are related to

ΔV01=

Δ120579119879

2

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(22)

Equation (22) is formulated due to 12059711988901120597119909 = 120597V0

1120597V119909and

1205971198890

1120597119910 = 120597V0

1120597V119910 From (21) and (22) we obtain

Δ120579 = 119870 [Δ1205791 Δ1205792]

=

[[[[[[[[[

[

1 0 0 0

1199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

0 0

0 0 1 0

0 01199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

]]]]]]]]]

]

[[[[[

[

Δ119909

Δ119910

ΔV119909

ΔV119910

]]]]]

]

(23)

Therefore the required CRLB is

CRLB (120579) = 119870minus1 (1198920119879

120593minus11198920)

minus119879

119870minus119879

= (1198701198791198920119879

119861minus1

1119876minus11198920119879

119870)

minus1

(24)

4 Results and Discussion

Simulation results to estimate the position and velocity basedon TDOA and FDOA in LSAN for 2D scenario are presentedin this section The sensors positions are (119909

119894= 119894 and 119910

119894= 0)

and (119909119894= minus(119894 minus 1) and 119910

119894= 0) when 119894 (119894 = 1 2 10)

is even and odd number respectively The near to far fieldsources are situated at A (8m 22m) B (0m 50m) C (minus30m25m) andD (minus50m 250m) [23]The source velocities are1198811(minus2ms 15ms) 1198812 (1ms 2ms) and 1198813 (minus4ms 2ms)In all cases the proposed method and Taylorrsquos series [27 28]are compared against CRLB [23] calculated from (24) andzero mean Gaussian noise is considered as dB = 10 log(1205902)The MSE of the proposed method is calculated via MSE

119901=

sum119872

1119901 minus 119901

02119872 and MSEV = sum

119872

1V minus V02119872 for position

and velocity of the moving source where 119872 = 11198645 is thequantity of random generation to maintain the covarianceof Gaussian noise 1199010 is A or B or C or D and V0 is 1198811 or1198812 or1198813The deviation percentages of the proposed methodand Taylorrsquos series from the CRLB are measured via ((119875MSE119901 minus

119862MSE119901)119862MSE119901)times100 ((119879MSE119901minus119862MSE119901)119862MSE119901)times100 ((119875MSEVminus

119862MSEV)119862MSEV) times 100 and ((119879MSEV minus 119862MSEV)119862MSEV) times 100 forposition and velocity where the symbols119875119862 and119879 representthe proposed CRLB and Taylorrsquos methods respectively


The MSE of position and velocity estimation of theproposed method against Taylorrsquos series method and CRLBis represented in Figure 3 at the noise levels minus100 dB to minus5 dBThe position and velocity estimation through the proposedmethod reaches the CRLB when the noise is below minus30 dBThe proposed method diverges from CRLB with an increasein the noise level starting from minus30 dB The MSEs in bothposition and velocity of the proposed method are 1005 121and 285 times higher than the CRLB at noise level minus80 dBminus25 dB and minus5 dB respectively Below minus35 dB noise theratio of the proposed method varies between 1005 and 105against CRLB It should be noted here that Taylorrsquos seriesmethod is also simulated for comparison with the proposedmethod in the same simulation environment This methoduses truncated Taylorrsquos series expansion (avoiding the higherorder terms) to linearize the TDOA and FDOA nonlinearequations with iterative solution The MSEs of the emitterby Taylorrsquos series depended on the initial guess Also theyonly converges to the local minimum solution Hence itgives good position and velocity accuracy when the initialguess is also approximate with actual position and velocityof the emitter at very low level noise On the other handthe MSEs are large compared to proposed method Now toavoid the initialization limitation of Taylorrsquos series 12 timesof actual values of position and velocity are assumed as theinitial values of source position and velocity In this case aminimum of 4 iterations is needed for each solution whenthe noise is less than minus40 dB The deviation of Taylorrsquos seriesresults from CRLB starts from minus40 dB noise The iterationnumber and the deviation increased with the increment ofthe noise level The MSE of Taylorrsquos series is higher than theproposed method and CRLB which is shown in Figure 3 forposition and velocity estimation of moving source

TheMSE comparison of source position and velocity esti-mation at near field for the proposed method Taylorrsquos seriesand CRLB at noise minus40 dB is represented in Tables 1 and 2respectively where the number of sensors in LSAN is variedAt 10 sensors in LSAN the MSE of Taylorrsquos series is slightlyless than the CRLB and proposed method due to the loweffective noise level Tables 1 and 2 show that the MSE atthe near field source position and velocity reduced with theincrement of a number of sensors In addition the deviationrate of Taylorrsquos series from CRLB is higher than the proposedmethod at 7 to 3 sensors and 8 to 3 sensors for position andvelocity estimation respectively

Figure 4 shows the obtained MSE estimation of positionand velocity at far field using the proposed method Taylorrsquosseries and CRLB at noise ranges from minus100 dB to minus20 dBTheMSEs of the proposed method Taylorrsquos series and CRLBare almost same that is below minus50 dB noise After minus50 dBtheMSE of the proposedmethod and Taylorrsquos series becomeshigher with an increment in noise However the rising slopeof the proposed method (position and velocity estimation ofdistant sources) is less than Taylorrsquos series which is clearlydepicted in Figure 4

The MSEs of far field source position and velocity esti-mation differ through variation of the baseline of LSAN asillustrated in Tables 3 and 4 In this simulation minus50 dB noisesare considered The true baseline of LSAN is decreased by

Table 1 MSE comparison of source position at near field

Number of sensors Proposed method CRLB Taylorrsquos series(m2) (m2) (m2)

10 0009532 00095923 000949969 0017638 00176473 001764738 0028882 00288823 002897517 0062159 00619726 00625216 01219 01208001 012285235 0356023 03557193 035672354 1121039 10944217 11822113 829721 75660109 84193549

Table 2 MSE comparison of source velocity at near field

Number of sensors Proposed method CRLB Taylorrsquos series(m2s2) (m2s2) (m2s2)

10 0001085 0001082 00010729 0001954 0001933 00019758 000318 0003146 00032047 0006872 000681 0006946 0013721 0013588 00138425 0039103 0038908 00390254 0128801 0122769 01332233 1017037 0923471 1076407

Table 3 Comparison of MSE source position for the proposedmethod CRLB and Taylorrsquos series at far field


10 10856 10896 108469 18907 18909 189088 35365 35361 353717 70232 70093 704936 15773 15562 160215 43068 40971 449714 15709 144989 18138

Table 4 Comparison of MSE source velocity for the proposedmethod CRLB and Taylorrsquos series at far field


10 11007 11006 110059 18332 18328 183438 34095 34071 341957 67421 67679 687926 15728 15359 162835 43499 40464 449374 18259 16337 20341

reducing the number of sensors from the network Also thecomparative baseline (ratio of the true baseline and the rangebetween source and sensor network) of LSAN decreases


Proposed methodCRLB

Taylorrsquos series

minus100 minus80 minus60 minus40 minus20

10minus2

10minus4

10minus6

10minus8

102

100

0

MSE

(m2)

(1205902)10 log

(a)


10minus2

10minus4

10minus6

10minus8

102

100

0

Proposed methodCRLB

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)

Figure 3 Comparison of (a) position and (b) velocity MSE of the proposed method with Taylorrsquos series and CRLB for near field

Table 5 The comparative MSE of the proposed method and Taylorrsquos series with respect to CRLB at near to far source position with samevelocity

Position Method ()5 numbers of sensors Noise minus40 dB

Noise minus50 dB Noise minus30 dB 7 numbers of sensors 6 numbers of sensorsPosition Velocity Position Velocity Position Velocity Position Velocity

A (8m 22m)Proposed methodCRLB (MSE)Taylorrsquos series

02146 02475 12784 09871 04251 05101 09123 0958800356 000389 35636 03885 00619 00068 01208 0013611024 10125 41358 35784 12146 11572 17265 1857

B (0m 50m)Proposed methodCRLB (MSE)Taylorrsquos series

10001 09578 51279 68795 10127 19245 15723 2478205645 00589 57557 58953 096703 00999 2152 0222735789 34978 112345 135789 26987 18912 26987 27682

C (minus30m 25m)Proposed methodCRLB (MSE)Taylorrsquos series

21579 24798 81256 95879 21987 10124 32783 3432412233 01452 139561 13906 20938 02368 55342 0612162879 75679 16248 15871 45789 61234 72453 87152

D (minus50m 250m)Proposed methodCRLB (MSE)Taylorrsquos series

51183 75013Large Large

99821 11278 17563 1820140971 40464 84184 61733 185651 13549763 11005 16782 15721 31298 29458

In addition direction control becomes weaker and weakerdue to the reduction of true baseline Here it describes theaccuracy of the position and the velocity for a pair of sensorsand is mostly limited to one direction which is parallelto these two sensors and perpendicular to the LSAN [2930] Hence the large variation of position and velocity isobserved Most interestingly it can be observed from Tables3 and 4 that the MSEs of position and velocity are too largewhen the number of sensors is three due to the too weakdirection control In addition the MSEs of Taylorrsquo series

are significantly larger than the proposed method when thenumber of sensors is less (4 or 5) due to its linearization errors[23 31]The baseline of the network increases with increasingthe number of sensors as a result the linearization errorreduces In addition favorable initial guess is also needed forTaylorrsquos series In practice this is not possible and solutiondivergence may occur

The comparative position and velocity MSE of the pro-posed method and Taylorrsquos series with respect to theoreticalposition and velocity MSE are presented in Table 5 where


Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)

Figure 4 Comparison of (a) position and (b) velocity MSE of the proposed method with Taylorrsquos series and CRLB for far field

different source positions A B C and D and same velocity1198811 for all source positions are considered First of all thedeviation percentages of positions are 215 and 628 (whennoise is minus50 dB and the number of sensors is 5 in LSAN)and the deviation percentages of velocity are 247 and 756for the proposed and Taylorrsquos series respectively at sourceposition C In addition the results of position and velocityobtained for the proposed method are 05 and 06 timesTaylorrsquos series at position C when noise is minus30 dB Secondlysix and seven numbers of sensors in LSAN at noise minus40 dBare also considered to measure the comparative positionand velocity MSE in Table 5 Here the deviation of MSEpercentage of Taylorrsquos series is 2 to 5 times compared to theproposed method Moreover the deviation percentage goeshigher with the reduction of the number of sensors in LSAN

The position and velocity MSErsquos results of the proposedmethod and Taylorrsquos series with respect to the theoreticalMSE are observed at minus40 dB noise and 6 numbers of sensorsin LSAN where combination of different positions andvelocity of sources are considered in Table 6 For all cases theproposed methodrsquos results are found to be 2 to 3 times betterthan Taylorrsquos series In addition the deviation percentage ofposition MSE is marginally increased through the incrementof the source velocity It is to be noted that the deviationpercentage of velocity MSE is higher than the deviationpercentage of position MSE

The most interesting observation from Tables 5 and 6 isthat the MSE at source position C is larger than at position Bdespite the long distance between the source position B andnetwork This happens because (1) the source C is situatedbehind the outside to LSAN where the control of 119909-direction

is extremelyweak and (2) the source B is on the perpendicularline of the LSAN Therefore the source position and velocitywill be undefinedwhen it is on the axis but outside (either leftor right) of the LSAN On the other hand the source positionand velocity will be effectively estimated when its position ison the perpendicular line of the LSAN [29 30]

In conclusion it is apparent that the MSE of positionand velocity of the far field source is higher than the nearfield because of the geometric spreading that is the abilityto estimate the position and the velocity of emitter becomesweaker and weaker as the position moves away from thesensor network [29 30] In our simulation results theproposed method yields better results than Taylorrsquos seriesdue to the initialization problem local minimum solutionlinearization errors and so forth of Taylorrsquos seriesThereforethe proposed method in close proximity with the CRLB fromnear to far field source with same and various velocities anddifferent baseline of network at varying noise levels

5 Conclusion

Thenonlinear localization equations set measurement noiseand singularity problem in LSAN pose the challenges tolocate the position and velocity of the locomotive sourcein the 2D scenario based on TDOA and FDOA measure-ments To overcome these challenges nuisance variables areintroduced in this study These variables have contributedto avoidance of the singularity problem of LSAN in non-linear localization equations set and to improvement ofthe instantaneous source location estimation The proposedmethod is found to be noniterative of low complexity


Table 6 The comparative MSE of the proposed method and Taylorrsquos series with respect to CRLB at near to far source position with differentvelocity

Position Method ()Nose minus40 dB and 6 numbers of sensors

1198811 (2ms minus15ms) 1198812 (1ms 2ms) 1198813 (minus4ms 2ms)Position Velocity Position Velocity Position Velocity


09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257

and attractive and does not have convergence problem andinitialization problems as in Taylorrsquos series The proposedmethod accomplished the CRLB for low to moderate noiselevel in case of moving source which is positioned at near tofar field with same and different velocity under the Gaussiannoise

Conflict of Interests

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

Acknowledgments

This research work is supported by the ERGS fund (ER011-2013A) Ministry of Education (MOE) Malaysia and Uni-versity of Malaya Research Grant (UMRG) scheme (RG286-14AFR)

References

[1] V A Gordienko N V Krasnopistsev V N Nekrasov and VN Toropov ldquoLocalization of sources on a ship hull using com-bined receiver and high-resolution spectral analysisrdquoAcousticalPhysics vol 57 no 2 pp 168ndash179 2011

[2] M Moradi J Rezazadeh and A S Ismail ldquoA reverse localiza-tion scheme for underwater acoustic sensor networksrdquo Sensorsvol 12 no 4 pp 4352ndash4380 2012

[3] S Coraluppi ldquoMultistatic sonar localizationrdquo IEEE Journal ofOceanic Engineering vol 31 no 4 pp 964ndash974 2006

[4] M Zhou Y-B Xu L Ma and S Tian ldquoOn the statisticalerrors of RADAR location sensor networks with built-in Wi-Figaussian linear fingerprintsrdquo Sensors vol 12 no 3 pp 3605ndash3626 2012

[5] E Weinstein ldquoOptimal source localization and tracking frompassive array measurementsrdquo IEEE Transactions on AcousticsSpeech and Signal Processing vol 30 no 1 pp 69ndash76 2012

[6] M Dianat M R Taban J Dianat and V Sedighi ldquoTarget local-ization using least squares estimation for MIMO radars withwidely separated antennasrdquo IEEETransactions onAerospace andElectronic Systems vol 49 no 4 pp 2730ndash2741 2013

[7] Q-L An J-F Chen and Z-H Yin ldquoSource localization foruniform noise maximum likelihood estimation method anditerative algorithm based on AOArdquo in Computer Applicationsfor Communication Networking and Digital Contents vol 350of Communications in Computer and Information Science pp10ndash16 Springer Berlin Germany 2012

[8] A Bel J L Vicario and G Seco-Granados ldquoLocalizationalgorithmwith on-line path loss estimation and node selectionrdquoSensors vol 11 no 7 pp 6905ndash6925 2011

[9] Q Kong X Yang and X Xie ldquoA novel localization algorithmbased on received signal strength ratiordquo in Proceedings ofthe International Conference on Wireless Communications Net-working and Mobile Computing (WiCOM rsquo08) pp 1ndash6 October2008

[10] I Guvenc and C-C Chong ldquoA survey on TOA based wirelesslocalization andNLOSmitigation techniquesrdquo IEEE Communi-cations Surveys amp Tutorials vol 11 no 3 pp 107ndash124 2009

[11] C Jing-Min W He-Wen and Y Jian ldquoWeighted constrainedtotal least-square algorithm for source localization using TDOAmeasurementsrdquo in Future Wireless Networks and InformationSystems vol 143 of Lecture Notes in Electrical Engineering pp739ndash746 Springer Berlin Germany 2012

[12] 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

[13] P C Chestnut ldquoEmitter location accuracy using TDOA anddifferential dopplerrdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 18 no 2 pp 214ndash218 1982

[14] 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

[15] K C Ho X Lu and L Kovavisaruch ldquoSource localization usingTDOA and FDOA measurements in the presence of receiver


location errors analysis and solutionrdquo IEEE Transactions onSignal Processing vol 55 no 2 pp 684ndash696 2007

[16] H-WWei R Peng QWan Z-X Chen and S-F Ye ldquoMultidi-mensional scaling analysis for passive moving target localiza-tion with TDOA and FDOAmeasurementsrdquo IEEE Transactionson Signal Processing vol 58 no 3 pp 1677ndash1688 2010

[17] B Friedlander ldquoA passive localization algorithm and its accu-racy analysisrdquo IEEE Journal of Oceanic Engineering vol 12 no1 pp 234ndash245 1987

[18] LDong X Li Z ZhouGChen and JMa ldquoThree-dimensionalanalytical solution of acoustic emission source location forcuboid monitoring network without pre-measured wave veloc-ityrdquo Transactions of Nonferrous Metals Society of China vol 25no 1 pp 293ndash302 2015

[19] J S Abel Localization using range differences [PhD thesis]Stanford University Stanford Calif USA 1989

[20] X-B Li and L-J Dong ldquoAn efficient closed-form solutionfor acoustic emission source location in three-dimensionalstructuresrdquo AIP Advances vol 4 no 2 Article ID 027110 2014

[21] S Bancroft ldquoAn algebraic solution of the GPS equationsrdquo IEEETransactions on Aerospace and Electronic Systems vol 21 no 1pp 56ndash59 1985

[22] J Abel and J Chaffee ldquoDirect GPS solutionsrdquo in Proceedings ofthe 49th Annual Meeting of the Institute of Navigation pp 1905ndash1915 1993

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

[24] G Strang Linear Algebra and Its Applications Academic PressNew York NY USA 2nd edition 1980

[25] Y-T Chan H Y C Hang and P-C Ching ldquoExact andapproximate maximum likelihood localization algorithmsrdquoIEEE Transactions on Vehicular Technology vol 55 no 1 pp 10ndash16 2006

[26] J S Abel and J O Smith ldquoSource range and depth estimationfrommultipath range difference measurementsrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 8pp 1157ndash1165 1989

[27] W H Foy ldquoPosition-location solutions by Taylorrsquos series esti-mationrdquo IEEETransactions onAerospace and Electronic Systemsvol 12 no 2 pp 187ndash194 1976

[28] K Yu J-P Montillet A Rabbachin P Cheong and I Opper-mann ldquoUWB location and tracking for wireless embeddednetworksrdquo Signal Processing vol 86 no 9 pp 2153ndash2171 2006

[29] M Ge Optimization of transducer array geometry for acousticemissionmicroseismic source location [PhD thesis] The Penn-sylvania State University 1988

[30] M Ge and H R Hardy Jr ldquoA statistical method for evaluationof AEMS source location accuracy and transducer arraygeometryrdquo in Rock Mechanics A Guide for Efficient Utilizationof Natural Resource pp 663ndash670 1989

[31] M Ge ldquoAnalysis of source location algorithms part II iterativemethodsrdquo Journal of Acoustic Emission vol 21 no 1 pp 29ndash512003

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



d3

d2

d1

Receiver 3Receiver 2 Receiver 1

TDOA21 = d2 minus d1 TDOA31 = d3 minus d1

Data processor

Figure 1 TDOA based localization technique in LSAN

TDOA for Global Positioning System (GPS) was proposed in[21 22] In this paper we have focused our research based onthe LSAN to locate the position and velocity ofmoving sourceat near to far field

It has been found from the literature that the TDOA andFDOA equations with high nonlinearity form hyperbolaswhich may not be intersected at a single point due tothe inconsistent data such as measurement error and thedeviation between assumption model and actual field of thevelocity in LSAN [14 23] Also the singularity problemarises in the coefficient matrix of hyperbolic equations setin 2D LSAN [23 24] To overcome the challenges in the2D scenario a mathematical solution by approximate MLestimation based on TOA was developed in [25] which wasonly applicable for three linear sensors Additionally the MLestimation in [26] a closed-form of LS estimation in [19 23]and the geometric solution in [23] were proposed based onthe TDOA to predict the position of stationary source inLSAN

Moreover the trigonometric mathematical approach in[5] was derived to estimate the position of a moving sourcein LSAN for only a special case where the range betweenthe source and receivers was large compared to the spacingbetween receivers (far field) To the best of the authorsrsquoknowledge no literature has been found with the linearsensor array in a 2D scenario for estimating the position aswell as velocity of near to far field moving emitter In the 2Dscenario for LSAN each element in the array lies in sameaxis [23] The coordinate of the source position in that axis isalso absent in the nonlinear equation set Hence coefficientmatrix of the LSAN is singular [24] For overcoming theseissues we have proposed a mathematical approach based onTDOA and FDOA to estimate the position and velocity ofa moving source in LSAN In this paper we simulated theproposed method to estimate the position and velocity ofmoving source at near to far field with the same and differentvelocity in LSAN where Gaussian noise is considered

The paper is arranged in the following manner Inthe following section the proposed mathematical model ispresented Next the derivation of Cramer-Rao lower bound(CRLB) is provided After that the results are analyzed andperformance evaluation is explainedThen the limitation andfuturework are discussed Finally the concluding remarks areprovided

2 Proposed Method

The119873 linear stationary sensors of LSAN are considered in a2D space to estimate the locomotive emitter with unknownposition 119901 = [119909 119910]

119879 and the velocity V = [V119909 V119910]119879 utilizing

the TDOA and the FDOA localization techniques wherematrix transpose operation is denoted by 119879 represented inFigure 1 The signals emitted from the source is captured by119873 stationary sensors of LSAN which are situated at 119877

119894=

[119909119894 119910119894]119905 where 119910

119894= 0 and 119894 = 1 2 3 119873 The range

between the 119894th sensor and emitter is

1198890

119894=1003816100381610038161003816119901 minus 119877119894

1003816100381610038161003816 =radic(119901 minus 119877

119894)119879(119901 minus 119877

119894) (1)

The true path difference of sensor pair 119894th and reference thatassumed sensor 1 is

1198890

1198941= 1198881199051198941= 1198890

119894minus 1198890

1 (2)

With regard to (2) 119888 is the velocity of signal propagationand 1199051198941is the time difference of sensor pair 119894th and reference

Equation (2) is rearranged as 11988901198941+1198890

1= 1198890

119894and then both sides

are squared Substituting the 11988901and 1198890

119894from (1) the TDOA

equation set obtained is

1198890

1198941

2

+ 21198890

11198890

1198941

= 119877119879

119894119877119894minus 119877119879

11198771minus 2 (119877

119894minus 1198771)119879119901 where 119894 = 2 3 119873

(3)

Equation (3) is a nonlinear set with unknown 119901 and 11988901

which created 119873 minus 1 hyperbolic curves with focus 119877119894=

[119909119894 119910119894]119905 where 119894 = 1 2 3 119873 Those hyperbolic curves

intersect at a point which gives the estimated position ofthe source In the 2D scenario at least two hyperbolic curvesare required to solve the localization problem by utilizing theTDOA represented in Figure 2

Not only the position but also the velocity estimationis essential for determining the instantaneous position ofmoving emitter Conversely TDOA equations set may notbe adequate to provide the needed localization accuracy ofmoving emitter as TDOA estimates only the position of thesource The FDOA measurement which is obtained fromthe relative velocity between the source [13] and sensors isapplied to improve instantaneous localization accuracy of thesourceThe relation between the path rate and source positionparameters is obtained by the time derivative of (1) as follows

V0119894=V119879 (119901 minus 119877

119894)

1198890

119894

(4)


0 5 10

0

2

4

6

8

10

Reference sensor

Source

minus10minus10

minus5

minus6

minus4

minus2

minus8

X (m)

Y(m

)

TDOA21TDOAN1

TDOA31



2 (V011989411198890

1198941+ V011989411198890

1+ 1198890

1198941V01)


(5)



1198941and 119899V

1198941are additive


1198891198941= 1198890

1198941+ 1198991198941 (6)

V1198941= V01198941+ 119899V1198941 (7)




119876 = 119864[119863119879119881119879]119879

[119863119879119881119879] = 120590

2[

1198761119874

0 1198761

] (8)


1198761=

[[[[[[[[[

[





]]]]]]]]]

]


(9)



2 (119877119894minus 1198771)119879119901 + 2119889

11989411198890

1= minus1198892

1198941+ 119877119879

119894119877119894minus 119877119879

11198771

2V11989411198890

1+ 2 (119877

119894minus 1198771)119879 V + 2119889

1198941V01= minus2119889

1198941V1198941

(10)



11989411198890

1= 1198892

1198941+ 1199092

1minus 1199092

119894 (11)

minus2V11989411198890

1minus 2 (119909

119894minus 1199091) V119909minus 21198891198941V01= 21198891198941V1198941 (12)


1are unknown


0

1V119909

V01]119879


1and V0



120576 = [120576119905 120576119891]119879

= ℎ minus 119892120579 (13)



ℎ =

[[[[[[[[[[[[[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

211988921V21

211988921V31

21198891198731V1198731

]]]]]]]]]]]]]]]]]]]

]

119892 = minus2

[[[[[[[[[[[[[[[[[[[

[

1199092minus 119909111988921

0 0

1199093minus 119909111988931

0 0

119909119873minus 11990911198891198731

0 0

0 V21

1199092minus 119909111988921

0 V31

1199093minus 119909111988931

0 V1198731

119909119873minus 11990911198891198731

]]]]]]]]]]]]]]]]]]]

]

(14)


⟨120579⟩ = arg min120579

(ℎ minus 119892120579) (15)



⟨120579⟩ = (119892119879119882119892)minus1

119892119879119882ℎ (16)




1⟩ are esti-


⟨1198890

1⟩ and ⟨V0



⟨119910⟩ = radic⟨1198890

1⟩2minus ⟨119909⟩2 (17)


119879

11198761205761of (11)


ℎ1=

[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

]]]]]]]

]

1198921= minus2

[[[[[[

[

1199092minus 119909111988921

1199093minus 119909111988931

119909119873minus 11990911198891198731

]]]]]]

]

1205791= [119909 119889

0

1]119879

(18)


⟨V119910⟩ =

⟨1198890

1⟩ ⟨V01⟩ minus ⟨119909⟩ ⟨V119909⟩⟨119910⟩

(19)



119864 [Δ120579 Δ120579119879] = (119892

0119879

120593minus11198920)

minus1

(20)

With regard to (20) 120593 = 11986111198761198611 1198611= [119861 119874

1198610119861] 119861 =

2 diag (119889021198890

31198890


119873) 1198610 = 2 diag (V0



119879


Δ1198890

1=

Δ120579119879

1

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(21)



ΔV01=

Δ120579119879

2

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(22)


1120597V119909and

1205971198890

1120597119910 = 120597V0


Δ120579 = 119870 [Δ1205791 Δ1205792]

=

[[[[[[[[[

[

1 0 0 0

1199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

0 0

0 0 1 0

0 01199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

]]]]]]]]]

]

[[[[[

[

Δ119909

Δ119910

ΔV119909

ΔV119910

]]]]]

]

(23)


CRLB (120579) = 119870minus1 (1198920119879

120593minus11198920)

minus119879

119870minus119879

= (1198701198791198920119879

119861minus1

1119876minus11198920119879

119870)

minus1

(24)



119894= 119894 and 119910

119894= 0)


119894= 0) when 119894 (119894 = 1 2 10)


119901=

sum119872

1119901 minus 119901


119872












10 0009532 00095923 000949969 0017638 00176473 001764738 0028882 00288823 002897517 0062159 00619726 00625216 01219 01208001 012285235 0356023 03557193 035672354 1121039 10944217 11822113 829721 75660109 84193549



10 0001085 0001082 00010729 0001954 0001933 00019758 000318 0003146 00032047 0006872 000681 0006946 0013721 0013588 00138425 0039103 0038908 00390254 0128801 0122769 01332233 1017037 0923471 1076407



10 10856 10896 108469 18907 18909 189088 35365 35361 353717 70232 70093 704936 15773 15562 160215 43068 40971 449714 15709 144989 18138



10 11007 11006 110059 18332 18328 183438 34095 34071 341957 67421 67679 687926 15728 15359 162835 43499 40464 449374 18259 16337 20341



Proposed methodCRLB

Taylorrsquos series


10minus2

10minus4

10minus6

10minus8

102

100

0

MSE

(m2)

(1205902)10 log

(a)


10minus2

10minus4

10minus6

10minus8

102

100

0

Proposed methodCRLB

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)






02146 02475 12784 09871 04251 05101 09123 0958800356 000389 35636 03885 00619 00068 01208 0013611024 10125 41358 35784 12146 11572 17265 1857


10001 09578 51279 68795 10127 19245 15723 2478205645 00589 57557 58953 096703 00999 2152 0222735789 34978 112345 135789 26987 18912 26987 27682


21579 24798 81256 95879 21987 10124 32783 3432412233 01452 139561 13906 20938 02368 55342 0612162879 75679 16248 15871 45789 61234 72453 87152



99821 11278 17563 1820140971 40464 84184 61733 185651 13549763 11005 16782 15721 31298 29458





Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)







5 Conclusion







09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10

0

2

4

6

8

10

Reference sensor

Source

minus10minus10

minus5

minus6

minus4

minus2

minus8

X (m)

Y(m

)

TDOA21TDOAN1

TDOA31



2 (V011989411198890

1198941+ V011989411198890

1+ 1198890

1198941V01)


(5)



1198941and 119899V

1198941are additive


1198891198941= 1198890

1198941+ 1198991198941 (6)

V1198941= V01198941+ 119899V1198941 (7)




119876 = 119864[119863119879119881119879]119879

[119863119879119881119879] = 120590

2[

1198761119874

0 1198761

] (8)


1198761=

[[[[[[[[[

[





]]]]]]]]]

]


(9)



2 (119877119894minus 1198771)119879119901 + 2119889

11989411198890

1= minus1198892

1198941+ 119877119879

119894119877119894minus 119877119879

11198771

2V11989411198890

1+ 2 (119877

119894minus 1198771)119879 V + 2119889

1198941V01= minus2119889

1198941V1198941

(10)



11989411198890

1= 1198892

1198941+ 1199092

1minus 1199092

119894 (11)

minus2V11989411198890

1minus 2 (119909

119894minus 1199091) V119909minus 21198891198941V01= 21198891198941V1198941 (12)


1are unknown


0

1V119909

V01]119879


1and V0



120576 = [120576119905 120576119891]119879

= ℎ minus 119892120579 (13)



ℎ =

[[[[[[[[[[[[[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

211988921V21

211988921V31

21198891198731V1198731

]]]]]]]]]]]]]]]]]]]

]

119892 = minus2

[[[[[[[[[[[[[[[[[[[

[

1199092minus 119909111988921

0 0

1199093minus 119909111988931

0 0

119909119873minus 11990911198891198731

0 0

0 V21

1199092minus 119909111988921

0 V31

1199093minus 119909111988931

0 V1198731

119909119873minus 11990911198891198731

]]]]]]]]]]]]]]]]]]]

]

(14)


⟨120579⟩ = arg min120579

(ℎ minus 119892120579) (15)



⟨120579⟩ = (119892119879119882119892)minus1

119892119879119882ℎ (16)




1⟩ are esti-


⟨1198890

1⟩ and ⟨V0



⟨119910⟩ = radic⟨1198890

1⟩2minus ⟨119909⟩2 (17)


119879

11198761205761of (11)


ℎ1=

[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

]]]]]]]

]

1198921= minus2

[[[[[[

[

1199092minus 119909111988921

1199093minus 119909111988931

119909119873minus 11990911198891198731

]]]]]]

]

1205791= [119909 119889

0

1]119879

(18)


⟨V119910⟩ =

⟨1198890

1⟩ ⟨V01⟩ minus ⟨119909⟩ ⟨V119909⟩⟨119910⟩

(19)



119864 [Δ120579 Δ120579119879] = (119892

0119879

120593minus11198920)

minus1

(20)

With regard to (20) 120593 = 11986111198761198611 1198611= [119861 119874

1198610119861] 119861 =

2 diag (119889021198890

31198890


119873) 1198610 = 2 diag (V0



119879


Δ1198890

1=

Δ120579119879

1

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(21)



ΔV01=

Δ120579119879

2

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(22)


1120597V119909and

1205971198890

1120597119910 = 120597V0


Δ120579 = 119870 [Δ1205791 Δ1205792]

=

[[[[[[[[[

[

1 0 0 0

1199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

0 0

0 0 1 0

0 01199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

]]]]]]]]]

]

[[[[[

[

Δ119909

Δ119910

ΔV119909

ΔV119910

]]]]]

]

(23)


CRLB (120579) = 119870minus1 (1198920119879

120593minus11198920)

minus119879

119870minus119879

= (1198701198791198920119879

119861minus1

1119876minus11198920119879

119870)

minus1

(24)



119894= 119894 and 119910

119894= 0)


119894= 0) when 119894 (119894 = 1 2 10)


119901=

sum119872

1119901 minus 119901


119872












10 0009532 00095923 000949969 0017638 00176473 001764738 0028882 00288823 002897517 0062159 00619726 00625216 01219 01208001 012285235 0356023 03557193 035672354 1121039 10944217 11822113 829721 75660109 84193549



10 0001085 0001082 00010729 0001954 0001933 00019758 000318 0003146 00032047 0006872 000681 0006946 0013721 0013588 00138425 0039103 0038908 00390254 0128801 0122769 01332233 1017037 0923471 1076407



10 10856 10896 108469 18907 18909 189088 35365 35361 353717 70232 70093 704936 15773 15562 160215 43068 40971 449714 15709 144989 18138



10 11007 11006 110059 18332 18328 183438 34095 34071 341957 67421 67679 687926 15728 15359 162835 43499 40464 449374 18259 16337 20341



Proposed methodCRLB

Taylorrsquos series


10minus2

10minus4

10minus6

10minus8

102

100

0

MSE

(m2)

(1205902)10 log

(a)


10minus2

10minus4

10minus6

10minus8

102

100

0

Proposed methodCRLB

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)






02146 02475 12784 09871 04251 05101 09123 0958800356 000389 35636 03885 00619 00068 01208 0013611024 10125 41358 35784 12146 11572 17265 1857


10001 09578 51279 68795 10127 19245 15723 2478205645 00589 57557 58953 096703 00999 2152 0222735789 34978 112345 135789 26987 18912 26987 27682


21579 24798 81256 95879 21987 10124 32783 3432412233 01452 139561 13906 20938 02368 55342 0612162879 75679 16248 15871 45789 61234 72453 87152



99821 11278 17563 1820140971 40464 84184 61733 185651 13549763 11005 16782 15721 31298 29458





Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)







5 Conclusion







09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











⟨120579⟩ = (119892119879119882119892)minus1

119892119879119882ℎ (16)




1⟩ are esti-


⟨1198890

1⟩ and ⟨V0



⟨119910⟩ = radic⟨1198890

1⟩2minus ⟨119909⟩2 (17)


119879

11198761205761of (11)


ℎ1=

[[[[[[[

[

1198892

21+ 1199092

1minus 1199092

2

1198892

31+ 1199092

1minus 1199092

3

1198892

1198731+ 1199092

1minus 1199092

119873

]]]]]]]

]

1198921= minus2

[[[[[[

[

1199092minus 119909111988921

1199093minus 119909111988931

119909119873minus 11990911198891198731

]]]]]]

]

1205791= [119909 119889

0

1]119879

(18)


⟨V119910⟩ =

⟨1198890

1⟩ ⟨V01⟩ minus ⟨119909⟩ ⟨V119909⟩⟨119910⟩

(19)



119864 [Δ120579 Δ120579119879] = (119892

0119879

120593minus11198920)

minus1

(20)

With regard to (20) 120593 = 11986111198761198611 1198611= [119861 119874

1198610119861] 119861 =

2 diag (119889021198890

31198890


119873) 1198610 = 2 diag (V0



119879


Δ1198890

1=

Δ120579119879

1

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(21)



ΔV01=

Δ120579119879

2

100381610038161003816100381610038161199010minus 1198771

10038161003816100381610038161003816

1198890

1

(22)


1120597V119909and

1205971198890

1120597119910 = 120597V0


Δ120579 = 119870 [Δ1205791 Δ1205792]

=

[[[[[[[[[

[

1 0 0 0

1199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

0 0

0 0 1 0

0 01199090minus 1199091

1198890

1

1199090minus 1199091

1198890

1

]]]]]]]]]

]

[[[[[

[

Δ119909

Δ119910

ΔV119909

ΔV119910

]]]]]

]

(23)


CRLB (120579) = 119870minus1 (1198920119879

120593minus11198920)

minus119879

119870minus119879

= (1198701198791198920119879

119861minus1

1119876minus11198920119879

119870)

minus1

(24)



119894= 119894 and 119910

119894= 0)


119894= 0) when 119894 (119894 = 1 2 10)


119901=

sum119872

1119901 minus 119901


119872












10 0009532 00095923 000949969 0017638 00176473 001764738 0028882 00288823 002897517 0062159 00619726 00625216 01219 01208001 012285235 0356023 03557193 035672354 1121039 10944217 11822113 829721 75660109 84193549



10 0001085 0001082 00010729 0001954 0001933 00019758 000318 0003146 00032047 0006872 000681 0006946 0013721 0013588 00138425 0039103 0038908 00390254 0128801 0122769 01332233 1017037 0923471 1076407



10 10856 10896 108469 18907 18909 189088 35365 35361 353717 70232 70093 704936 15773 15562 160215 43068 40971 449714 15709 144989 18138



10 11007 11006 110059 18332 18328 183438 34095 34071 341957 67421 67679 687926 15728 15359 162835 43499 40464 449374 18259 16337 20341



Proposed methodCRLB

Taylorrsquos series


10minus2

10minus4

10minus6

10minus8

102

100

0

MSE

(m2)

(1205902)10 log

(a)


10minus2

10minus4

10minus6

10minus8

102

100

0

Proposed methodCRLB

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)






02146 02475 12784 09871 04251 05101 09123 0958800356 000389 35636 03885 00619 00068 01208 0013611024 10125 41358 35784 12146 11572 17265 1857


10001 09578 51279 68795 10127 19245 15723 2478205645 00589 57557 58953 096703 00999 2152 0222735789 34978 112345 135789 26987 18912 26987 27682


21579 24798 81256 95879 21987 10124 32783 3432412233 01452 139561 13906 20938 02368 55342 0612162879 75679 16248 15871 45789 61234 72453 87152



99821 11278 17563 1820140971 40464 84184 61733 185651 13549763 11005 16782 15721 31298 29458





Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)







5 Conclusion







09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















10 0009532 00095923 000949969 0017638 00176473 001764738 0028882 00288823 002897517 0062159 00619726 00625216 01219 01208001 012285235 0356023 03557193 035672354 1121039 10944217 11822113 829721 75660109 84193549



10 0001085 0001082 00010729 0001954 0001933 00019758 000318 0003146 00032047 0006872 000681 0006946 0013721 0013588 00138425 0039103 0038908 00390254 0128801 0122769 01332233 1017037 0923471 1076407



10 10856 10896 108469 18907 18909 189088 35365 35361 353717 70232 70093 704936 15773 15562 160215 43068 40971 449714 15709 144989 18138



10 11007 11006 110059 18332 18328 183438 34095 34071 341957 67421 67679 687926 15728 15359 162835 43499 40464 449374 18259 16337 20341



Proposed methodCRLB

Taylorrsquos series


10minus2

10minus4

10minus6

10minus8

102

100

0

MSE

(m2)

(1205902)10 log

(a)


10minus2

10minus4

10minus6

10minus8

102

100

0

Proposed methodCRLB

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)






02146 02475 12784 09871 04251 05101 09123 0958800356 000389 35636 03885 00619 00068 01208 0013611024 10125 41358 35784 12146 11572 17265 1857


10001 09578 51279 68795 10127 19245 15723 2478205645 00589 57557 58953 096703 00999 2152 0222735789 34978 112345 135789 26987 18912 26987 27682


21579 24798 81256 95879 21987 10124 32783 3432412233 01452 139561 13906 20938 02368 55342 0612162879 75679 16248 15871 45789 61234 72453 87152



99821 11278 17563 1820140971 40464 84184 61733 185651 13549763 11005 16782 15721 31298 29458





Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)







5 Conclusion







09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Proposed methodCRLB

Taylorrsquos series


10minus2

10minus4

10minus6

10minus8

102

100

0

MSE

(m2)

(1205902)10 log

(a)


10minus2

10minus4

10minus6

10minus8

102

100

0

Proposed methodCRLB

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)






02146 02475 12784 09871 04251 05101 09123 0958800356 000389 35636 03885 00619 00068 01208 0013611024 10125 41358 35784 12146 11572 17265 1857


10001 09578 51279 68795 10127 19245 15723 2478205645 00589 57557 58953 096703 00999 2152 0222735789 34978 112345 135789 26987 18912 26987 27682


21579 24798 81256 95879 21987 10124 32783 3432412233 01452 139561 13906 20938 02368 55342 0612162879 75679 16248 15871 45789 61234 72453 87152



99821 11278 17563 1820140971 40464 84184 61733 185651 13549763 11005 16782 15721 31298 29458





Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)







5 Conclusion







09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Proposed methodCRLB


10minus2

10minus4

102

100

106

104

MSE

(m2)

Taylorrsquos series

(1205902)10 log

(a)

Proposed methodCRLB


10minus2

10minus4

104

102

100

MSE

(m2s2)

Taylorrsquos series

(1205902)10 log

(b)







5 Conclusion







09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














09123 09588 08826 09009 0932 1102901208 00136 01189 00129 012247 00188917265 1857 1638 14289 1987019 21748


15723 24782 13978 14123 171423 185672152 02227 21222 02191 221976 0271926987 27682 26127 26212 297845 30784


32783 34324 32315 33012 329723 3925655342 06121 54994 06023 55685 0937372453 87152 72021 85278 737258 90157


17563 18201 16928 13859 1928794 17758185651 1354 185604 1353 185669 1365131298 29458 30661 27972 3356872 31257




Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a mathematical algorithm of locomotive...

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