Recurrent Algorithm for TDOA Localization in Sensor ... - Scielo.br

doi: 10.5028/jatm.v9i4.727

Recurrent Algorithm for TDOA Localization in Sensor Networks Igor Olegovych Tovkach1, Serhii Yakovych Zhuk1

ABSTRACT: Using the mathematical apparatus of the extended Kalman Filter, the recurrent algorithm of the passive location in sensor networks — based on the Time Difference of Arrival method in case of correlated errors of measurements — is developed. The initial estimates of Radio Frequency Sources coordinates and the correlation matrix of the vector estimation are determined based on the method of the least squares in case of 3 difference measurement distances. Efficiency analysis of recurrent adaptive algorithm and its comparison with the quadratic correction one are performed by statistical modeling. A comparison of them with the lower limit of the Cramer-Rao was carried out. The implementation of the recurrent adaptive algorithm requires 2.7 times less computational cost than the quadratic correction one. KEYWORDS: Passive location, Time Difference of Arrival method, Extended Kalman Filter, Recurrent adaptive algorithm, Sensor network.

INTRODUCTION The problem of passive position determination of Radio Frequency Sources (RFS) is widely met in the monitoring of the surrounding space, disaster management, in intelligent transport and security systems. Currently, sensor networks are used for its solution (Rullan-Lara et al. 2013; Amar and Leus 2010). One of the main approaches of passive position determination of RFS is based on the application of the Time Difference of Arrival method (TDOA), which uses the time difference of reception of signals received by the various sensors and the network reference sensor. This method has a significant advantage in the ease of implementation, being widely used in practice ITU (2014). The accuracy of determining the RFS coordinates based on the TDOA depends on the errors from measuring time signal reception sensors of the sensor network. The errors from the determination of the difference in signal reception times are correlated, because they contain the error in the reference sensor measurement. In the known methods for determining the RFS coordinates based on the TDOA (Buzuverov 2008), the coordinate calculation is performed after receiving the measurements from all sensors. In this study, based on the mathematical apparatus of the extended Kalman Filter, the algorithm is developed, which — after the formulation of the initial conditions based on the measurement of time for receiving signals from 4 sensors — allows to recurrently specify the location of RFS as the measurement proceeds from the other sensors. The developed algorithm also evaluates the time error from receiving the signal by the reference sensor measurement that allows considering the synthesized algorithm as adaptive.

1.National Technical University of Ukraine – Kyiv Polytechnic Institute – Department of Radio Engineering Devices and Systems – Kyiv – Ukraine. Author for correspondence: Igor Olegovich Tovkach | National Technical University of Ukraine – Kyiv Polytechnic Institute – Department of Radio Engineering Devices and Systems | Politekhnichna St. 12 | 03056 – Kyiv – Ukraine | Email: [email protected] Received: Jun. 21, 2016 | Accepted: Mar. 30, 2017

J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.489-494, Oct.-Dec., 2017

490

Tovkach IO, Zhuk SYa

FORMULATION OF THE PROBLEM The sensors network transducers have the coordinates y iS ), i = 0, n. The position of the RFS is characterized by a point with coordinates (x, y). When determining the coordinates of RFS on the x-y plane, the sensor network should consist of no less than 4 sensors. Figure 1 shows block diagrams of sensor networks on the x-y plane, consisting of 9 (n = 8) and 4 (n = 3) sensors. When using TDOA, the time difference in the reception of signals between the sensors i = 1, n and the reference probes is measured:

(x iS,

Δi 0 = ti − t0 + ni − n0 , i = 1, n ,

(1)

(1)

distances vi0, i = 1,n , are correlated because they contain the reference sensor measurement error v0. The presence of correlated errors makes it difficult to use traditional recurrent target coordinates evaluation algorithms. This difficulty can be avoided through the introduction of v0 into the state vector of the estimated parameters. It is believed that, during the difference measurement of the distances between the RFS and the network sensors, its coordinates do not change. Synthesizing a recurrent algorithm is required, and, after the formation of the initial conditions based on the measurement of signals receiving time from 4 sensors, it allows to specify recurrently the location RFS in process of receipt of measurements from the other sensors of the reference sensor (1)measurement. Δi 0and = ttoi −estimate t0 + nithe − nerror 0 , i = 1, n ,

where: ti is the time of signal reception of the i − m sensor; DEVELOPMENT t0 is the time of signal reception by the reference sensor; ni is of R the 1, n , = c ti + ni − t the − c uncorrelated t0 + n0 − t error = Riof−time R0 +measurements vi − v0 = di0R=i − c signal t0i ++nviby − c t0 + n0 (2) − t = Ri − R0 + vi − v0 = Ri − R0 + vi 0 , i = 1, n , i 0−, t i = The coordinates of the RFS position (xk, yk) should be reception of the i − m sensor (Buzuverov 2008) with dispersion (1)the measurement time of distance difference between Δi 0 = ti − t0 + ni − n0 , i = 1, n , assessed; σ 2n, i = 0, n. Because of the transformation in Eq. 1, the TDOA equations Δithe as(1) its coordinates, =RFS t −and t +the n network − n , i =sensors, 1, n , as well (3) (3) n; vi 00 = vi i −0v0 , ii = 1,0n ; i ,− v0 , i = 1, ,vii0==1,vncan (1) = ti − t0 + ni −for n0 network be represented as: do not change. The error in measurements of the reference (1) Δi 0 =Eq.ti 2−does t0 + not ni −change n0 , i as = 1, n ,in 1 measurement sensor from well d = c t + n − t − c t + n − t = R − R + v − v = Rcycle, , (2)the dynamics of the estimated vi 0Sequation , 2i = 1, ndescribing i − R0so+the Ri −i0R0 = (i x − ixiS )2 + ( y −0yiS )20 − x 2 + yi2 0 (4)i (2)R0i − Rparameters (4) + ( yform − yiS )2 − x 2 + y 2 0 = ( x − xhas i ) the

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(

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(2)

di0 = c (ti + ni − t ) − c (t0 + n0 − t ) = Ri − R0 + vi − v0 = Ri − R0 + vi 0 , i = 1, n , (2) (2) = Ri − R0 + vi − v0 = Ri − R0 + vi 0 , i = 1, n , (5) nd ; = c (t + n − tu)k(3) uk = uk −1 , (5) −=c (ut0k −+1n, 0 − t ) = Ri − R0 + v(5) i − v0 = Ri − R0 + vi 0 , i = 1, n , i i (1) i0 Δi 0 = ti − t0 + ni − n0 ,vii0==1,vni , − v0 , i = 1, where: Ri is the distance between the i − m sensor and the RFS, (3) vi 0 = vi − v0 , i = 1, Tn ; i where: uk = (xk, yk, v0k) is the state vector including position = vi − v0 , i ==1,1,nn;; R0 is the distance(3)between the reference sensor and (6) (ukmeasured ) + vk , difference of , and the measurement (6) (ukof ) +thevkRFS the RFS;rkdi0=ishthe error(3) of the Ri − R0 = ( x − xiS )2distances + ( y − yiSi )=2 1,− n;x 2 + coordinates y 2rk = hv(4) i 0 = vi − v0 , i = 1, n ; on the current time step k. The index k c is the spreading speed of electromagnetic waves; t is the reference sensor v S 2 0k S 2 2 2 (1) Δ tti − t0 + ni − n0 , i = 1, n , (4) in0 = R − R = ( x − x ) + ( y − y ) − x + y c (ti + ni − t − c t + − , (2) = R − R + v − v = R − R + v , i = 1, n 0 i i i characterizes the incomings sequence of measured differences 0 (4)i vi0 is 0 the error i 0of difference i0 0 RFS = ( x − xiS )2)+ moment ( y(−0 yiS )2of − )xS2signal + yi2 emission; (5) 2 S u2k = u2k −1 , 2 S 2 measurement between the distances: SS 22 2 ; (7) h(uk ) = ( xk − xi ) + ( yk − yi ) − xk + yk − v0k h(uk ) = in(distances. xR xk2y+S )y2k2−− vx0k2 ;+ y(7) (4) k i−−xR i 0) =+ ((yxk −−xyi i )) +−( y − i , (5) uThe = u equation describing the measured kth k measurement k −1 uk = uk −1 , (3) distance difference considering Eqs. 2, 4 and 5 can be viewed as (3) vi 0 = vi − v0 , i(5) = 1, n ; = c (ti + ni − t ) − c (t0 + n0 − t ) = Ri − Rrk0 +=vhi (−uvk0)=+Rvik−, R−01 + vi 0 , i = 1, (6) (2) uk = uk −1 , (5) n, i0 −1 T T ⎡ ∂h(uˆk −1 ) ⎤ ∂hT (uˆk −1 ) ⎡ ∂h(uˆk −1 ) ˆ ∂hT (uˆk −1 ) ˆ ˆ h u h u ∂ ∂ ( ) ( ) 2 ⎤ ; (8) 2 k k − 1 − 1 ˆ K k = Pkwhere: Pk −1 random variables + σ v ⎥ having vi are the K k = Pˆk −1 ⋅ rk = h(u⎢k ) + vk , Pˆk −1 σ v ⎥ ; (8) (6) +(6) ⎢ uncorrelated −1 ⋅ uk S 2⎣ (6) ∂uk S 2 ∂2uk 2 ∂uk ∂uk ∂uk rk = h(uk ) + the vk ,meaning∂of ⎦ ⎣ ⎦ (4)the RFS Ri − R0 = ( x −the xi )distance + ( y −measurement yi ) − xS errors +2 y between S 2 2 2 , (6)the nonr = h ( u ) + v 2 2 2 ; (7) ) = 1,a(ndispersion x ; − x ) +σ (=y c −σy(3) − v h(u − vh(u, iwith k k , )i =−0, n.x + ywhere: ) kis a non-linear function described by and thevnetwork i 0 = vi sensors uˆk = uˆk −1 +0 Kk k ⎡⎣rk − kh(uˆki−1 )⎤⎦ ;v k ni (9) k k uˆk 0=k uˆkSk−12 + Kk ⎡rk S−2h(uˆk −12)⎤ ; 2 (9) ⎣− y ) − x ⎦+ y − v ; (7) The difference in distance Ri – R0 is determined by the formula h(uk ) =linear ( xk measurement − xi ) + ( yk expression: 0k i k k (5) S 2 S u2k = uk2 −1 , 2 ( xk − xi ) + ( yk − yi ) − xk + yk − v0k ; (7) ˆ ∂h(u ) ∂h(uSˆk −21 ) ˆ 2 ˆ ˆ −1 T ( ky−k1 −, yiS ) 2 − (10) xk2 + yk2 − v0k ; (7)(7) (4) (4) Ri P −ˆkR=0 =Pˆk −1( x−−KxkiS )2 + (ky−1−∂hyPˆTiSk(−)u12ˆk, −−1 ) ⎡x∂2h+(uˆyk(10) ˆk −1P)k h=(uP2k ⎤−)1=− K( xkk − xi ) + P h u ∂ ) ( − 1 ˆ ˆ ∂ku−1k ⋅ Kk = P Pk −1 + σ v ⎥ ; (8) ∂uk ⎢ −1 ∂uk ∂uk rk = h(uk ) + vk , −1 ∂uk ⎣ (6) ∂hT (⎦uˆk −1 ) ⎡ ∂h(uˆk −1 ) ˆ ∂hT (uˆk −1 ) 2 ⎤ ; (8) ˆ T T of the sensor 4, the u coordinates v is the⎢measurement K k = Pk −where: Pk −error + σ v with ⎥ coordinates ⎤of the reference sensor 1⋅ 1 ˆk u ∂h (uˆk −1 ) ⎡ ∂h(uˆk −1In ) Eq. (5)were relied −1k)−1 , 2 ˆ ∂uhk (= ∂ukk ∂uk ∂uk −1 σ v ⎥ ; (8) +difference ⎣ ⎦ ⎢ to be P k −The 1 T T , y . x 0. errors of measurement between the uˆk = uˆk −1 + Kk ⎡⎣rSk − h(uˆk −1 S)⎤⎦ ; i i ˆ (9) ∂h (uˆk −1 ) ⎡ ∂h(uˆk −1 ) ˆ ∂h (uSˆk −1 ) ∂uk ∂uk 2 ⎤ ; (8) ⎣ˆ ∂uk S ⎦ ⎤ + σ v ⎥ xk −1 − xi xˆk −1 S 2 ∂h(uˆk −21 ) yˆ2k⎡−1 − yi xˆk −1 − xi Kkyˆ k=−1Pk −1 ⋅ x∂ˆku−1 ⎢ ∂u Pkyˆ−k1−1 −∂yui yˆk −1 S−2 .k k ⎡r⎣k; − h(kuˆk −1 )⎤ ; ⎥ ; 1 ; − − ˆ ˆ (9) k ⎢ u u K = + ; −1 = − ⎦− ; (7) h ( u ) = ( x − x ) + ( y − y ) − x + y − v k k − 1 0 k S 2 iManag., 2São k José2idos Campos,kVol.9, SN k∂ k ⎣ ⎦ Technol. 4, pp.489-494, Oct.-Dec., 2017 S 2 kJ. Aerosp. 2 S 2 2 2 S 2 S 2 2 2 S 2 S 2 2 uˆ(kxˆk=−1uˆ−k −x1i + + k( yˆ⎡⎣kr−k1 −− yhir(ku + yˆ ) K )ˆk=−1h)⎤⎦(xuˆ;kk−1)++yˆvkP−kˆ1, =(9)P(∂ˆxˆukk−1−−Kxi ⎢⎣) h+((x(6) ˆu(ˆkyk−ˆ1−k 1−−1) −xPˆi y)i ),+ ( yˆ k −1xˆ−k −1y(10) ˆ ⎥1 + yˆ k −1 ( xˆk −1 − xi ) + ( yˆ k −1 − yi ) xˆk −1 + yˆ k2−1 i ) k −1 xk −⎦ k k −1 k k −1 (9) uˆk ∂=hu(ˆukˆk−−11+) Kk ⎡⎣rk − h(uˆk −1 )⎤⎦ ; ∂uk ˆ ˆ ˆ , (10) P = P − K P ˆ ∂ h u ( ) k k − 1 k k − 1 k −1 ˆ −1 ∂uk T (10) = Pˆk −1 − K k PTk −1 ,

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∂h (uˆ

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∂h (uˆ

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⎤

(ti + niR−i t−)R−0c=(t0 (+Rxin−−0xR−iS0)t2)=+S =(2(yxR−−i −yxiSi R))20 S+− (2vyix−2 +vyv0i y2i)0=2 =R−i2v−ix R−(4)+0v+y0 v, ii0 ,=i1,=(4)n1,;n , Ri − R0 = ( x − xi ) + ( y − yi ) − x + y

(3)

(2)

(4)

(5) uk = uk −1 , S 2 uvk ==uvk −− = ( x − x ) + ( y − yiS )2 −xS x 2y+S y 2d 1 , v , i = 1, n ; Ri − R(5) 0 i (3) − v0 = Ri − R0 +i 0viu0 k, i=i=u1,k n−01,, (2) (5) 1 1 10 x1S

S

S

(4) xS + y S − d 2 1 1 10 S

S

2

S S for TDOA 2 Sx S x +2 y in − Sensor Networks y Algorithm d110 y1S = dx102S yRecurrent 1 d 20 1 ; d10 =− dx102S + yLocalization A 2 1 d 20x11 ;+ yb 2 − S S S S S 2S b = Sx +2y − d 2 ; SA = Sx(5) yS y2S dx20 2 d 30 2 20 d30x22 + dy220− ;dx20 3 ; y3 b2 = 3 +; y3 −

u = uk −A (6) r = h(u ) + v , 1 ,= xS 2 (6) k 2 r = h(uk ) k+Sv2k , k S 2k 2 (4) S S Ri(3) −kR0r=k =( h x (−uxki )) ++v(ky, − yi ) − x 2 + y(6) S 2 x3S xy33SS + dy330 x3S + y3S − d30 x3 y3 d30 − d30 2 2 2 ⎡ 2σ v σ v the model (Eqs. 6 and 7), a recurrent (1) estimation σ v ⎤ ΔUsing i 0 = ti − t0 + ni − n0 , i = 1, n , ⎢(6)2 2 ⎡ 2 2 2 ⎥ 2 S 2 S 2 2 2 , 2 2 r = h ( u ) + v algorithm can be obtained k;the (7) ⎡ 2σ v Σσ=v ⎢ σ vσ v ⎤22σσv v σσv2v ⎥ . σ v ⎤ h(uuk k)S ==2ofu(the x−k1−state xk + based ykk − von , xi S) vector 2+ ( yk − 2 yi ) 2 −k (5) 0 k k ; (7) ⎢ ⎥ h2 (uk )2 = extended ( xk − xi ) Kalman +S (2yk −Filter yi ) S−(Welch x + y − v ⎢ ⎥ 2 k Bishop 20 k ⎢ 2 =2 ⎢ σ2 2 2σ2 2⎥ σ 2 ⎥ . 2006), being ( xk − xi ) + ( yk − yi ) 2 −k xand − x + yh(uk ) = (4) σ v ⎥σ. v v 2σ v v⎥ Σ = ⎢ σ v2 2σ v2⎢ σ vΣ v k + yk − v0 k ; (7) ⎣ ⎦ described by

⎢

σ2

σ2

⎢⎥ 2 2σ 2 ⎢ σ v

σ v2

491

(13)

⎥ 2σ v2 ⎥

S 2 ⎢⎣ v The vcorrelation matrix Ω of the⎦ vector estimation error ω huˆ+ (uvk)) −= v (∂= xhkT(6) −(−uˆx1−iS R ))2 ++(vyk⎤,−−1iy= − , xk2 +(2) yk2 − v0vk⎣⎥⎦; (7) vRk)i ⎡,−∂R i )1, n c (ti + n(5) − t ) − cT(rt0k +=nh0 (−∂uhtkT))(uˆ+ = R h ( 0 i 0 i 0 i 0 i 2 T k k k − 1 − 1 − 1 ; (8) is defined (Amar−and h (=uˆ Pˆ ) ⎡⋅ ∂h(uˆ ) ∂K ∂h (uˆ Pˆk)−1 2 ⎤ ; −(8) 1 + σ v ⎥ (8) Leus 2010) by 1 K k = Pˆk −1 ⋅ k ∂hkT−k(1−uˆ1k⎢−1 ) ⎡∂∂ukh−k(1uˆkPˆ−k1⎢⎣−)1 ∂uk∂hkT−(1u +−1σ) v ∂⎥uk2 ⎤ ˆ ⎛ 0 T −1 0 ⎞ k ⎦ ˆ ; (8) , (12) K k = Pˆk −1 ⋅∂uk P σ + u u ∂ ∂ A A Ω = Η −1 ⎜ k −1 k −1 ⎣ ⎢ k ⎦ v ⎥ T ⎟

∂uk

∂uk

∂uk

⎛

0 T

( )

⎝

( ⎞) ⎛ 0

( ) ⎠ 0

−1

0 ⎞

⎣ ⎦ Ω = A Η −1 TAΩ = ⎜, A Η−1 A (12) S 2 2 ⎟ , ⎤ h(uˆ⎜k −1 ) (9) ˆ2kv+=0(,u ;;⎡ ∂(7) h(6) (uk ) = v(i 0xk =− vxiS )u yˆkik −−=1 y1, )n k;−⎡rˆkx− +h∂(hyuˆTk2k(−−uˆ1kv)−0⎤1k)(3) K + − i k ˆ ∂h (uˆ⎟⎠k −1⎝) + σ 2 ⎥ ; (8)⎠ uˆk = uˆk −1 + Kk ⎡⎣rk −Rˆ hK (=uˆkk −=(1⎣xˆ)P⎤⎦k−−;1x⋅S )2 + (yˆ ⎦(9) S 2 ⎝ (9)Pk −1 ⎢ v ; kj = 1,3 . ∂uk uˆk = uˆk −1 + Kk ⎡⎣rkj − h(uˆ0k −1 )⎤⎦j ; ∂uk 0 − ⎣y (9) j ) ∂u where: H is the⎦matrix ​​determined S 2 S 2

(12)

(14)

by the formula H = B∑B;

S 2 ˆˆ =) ( x ˆS 2 + (y .ˆ 0 − y j ) ; j = 1,3 . kj(y −1ˆ 0 −ˆ y 0j )− ;xjj )= 1,3 ˆ Rˆ=∂j hP=ˆ(uˆk(−−xˆ10)K− x∂j h) (uR+ , 2 −1 (10) P P B = diag {Rˆ1, Rˆ2, Rˆ3}; Rˆj = √(xˆ0 – x S0 )2 + (yˆ0 – y S0 )2; j = 1, 3. k k − 1 k k − 1 ˆ ˆ ˆ S 2 S 2 2 ⎡ 2 2 Pk = Pk −1T− K k ˆ ˆ (9) , (10) P u u K r h uˆk −1 )⎤⎦ ; = + − ( ˆ T ∂ h u ( ) ∂ u k − 1 (4) k k − 1 k k − x −⎡x∂i uh)(uˆ+k −k(1−y)1− Pyˆi ∂)kh, −(uˆkx−1 )+ y(10) xk + yk − vR ˆ R(7) ∂=0hP=ˆ (uˆk(− ⎣ ˆ 2 ⎤ ; (8) (10) 0ˆki ;P The initial correlation −1 )K k∂ ˆ 0 ⎤P0 has the block form k k − 1 k − 1 ⎡ Ω 2×2matrix K k = Pk −1 ⋅ Pk −1 + σ v ⎥ ⎢ k ∂u The ˆ ˆ P has the block form , where Ω2×2 is determined based on Ω, initial correlation matrix = P ∂uk ∂uk ⎢ 0 2 ⎥ ⎣ ∂uk k ⎦ ∂h(uˆ 0 ) 0 ⎤ ⎡ Ω Ω 0 2×⎦2 0 σ ⎡ ⎤ 2 × 2 k − 1 v ˆ ⎣ ˆ th matrixP P hasˆ the block form P = ⎢ Ω2×2 is 2determined based o The initial correlation 2×2 is determined Pˆkvector = Pˆk(5) −ˆat0 Kthe ⎥ , where Ωbased has block ˆform on Ω, correlation matrix , (15) where: uˆu is the estimation of the state = uThe −u1kP k kthestep; k −01 , P0 = ⎢ (10) 2 ⎥ , 0where k −1 initial kk 0 σ v ⎦ ∂ u −1 0 σ S S ⎣ ⎤ ⎡ ⎤ k v ⎣ ⎦ T ⎡ ˆ ˆ ˆ ; (9) uxikcorrelation K r xof h(ustate )vector = uk −1 + matrix ˆk −1estimation ˆ−the xPˆˆ⎤ is−the − yi yˆ yˆ⎤k −1 S ∂h xˆ(uˆk− ⎦ yS of ythe . xˆkth−1 − k ⎣ k k −1 ykˆ−k;1−1 − ; −1⎥ 56’[p-;;;;;;n = ⎢k −1 −1 )xi+ σ vS2 ⎥kk−1 ; (8) S i k −1 − S 2 S 2 2 2 S 2 S 2 2 coefficient 56734r5 ned based on Ω, by uk−ˆat the k step; ; −ˆ Ω1⎥2×2+. isyˆ⎤2determi −gain: xˆk2−1 ;Kxˆk is the yˆ(kxˆ−1 of −−the yxi )filter yˆwhere ∂S⎢u2k x(ˆxkˆ−1 −−xixerror k2 −1 x ⎥ ˆ ˆ y y y y y + − + + − ) ( ) ( ) ⎦ S 2 2 S 2 S 2 2 . i k −1− k −;1⎥− 1⎥k −1 ⎦ and the third column. (uˆ2k k−(−1,;x1ˆ)k −ˆ1 −k −x1 i ) +S (k2y−ˆ1k −(6) ˆk −1 i+2 yˆdeleting y x − xi ⎣) +S (k2−yˆ1k −1 −i yiˆ ) Srkˆk2−1=−xˆkh−1(i+u2kyˆ∂)k −h+ − ) the third row S 2 2 1 v 1 i k 1 − ˆ Pk y=i P)k −1 −⎡Kxˆkk−1 + yˆ k −k1 P(kxˆ−k1−S,1 − xi ) + (10) xˆk −1 + yˆ k −1 ⎦ ⎥⎦ S ( yˆ k −1 − yi ) k −1 − xi ) + ( yk −1 − uˆk −1 ) xˆk −1 yˆk −the yi yˆk −1conditions⎤based ∂h((9) ∂uk xˆk −1 − xi After of the initial (uˆk −1 )⎦⎤ ; 1 − formation ; −1⎥ . on the ; − = ⎢ − S 2 S 2 2 2 S 2 S 2 2 2 time measurements to receive signals from ∂uk ⎢⎣ ( xˆkT−1 −−1xi ) −+1 ( yˆ kT−1 −−1yi ) xˆk −1 + yˆ k −1 ( xˆk −1 − xi ) + ( yˆ k −1 − yi ) xˆk −1 + yˆ k −1 the ⎥⎦4 sensors, the −2A 1 Σ A S 2A Σ 2 b , 2 S (11) ω = 0.5 synthesized algorithm (Eqs. 8 – 10) allows to recurrently specify, −1 y ) − x + y − v ; (7)(11) (0.5 uk ) =AT(Σ xk−T1−A x−i 1) +A−(1TyΣ ω =h(10) 0k k −T b i,−1 k (11) k −1 , S = 0.5 A Σ A S , (11) ω A Σ b step k, ⎤ the location of the RFS as the measurement xˆk −1 − xi xˆk −1 yˆk −1 − yi yˆatk −each 1 ⎥ .the other sensors k = 1, n – 3. ; 1 ; − − − proceeds from −1 S 2 −1− y S ) 2 T xˆ−21 + yˆ 2 Ty ˆ xˆk −1 − xiS )2 + ( yˆ k −1 − yiS )2 xˆk2−1 + yˆ k2−1 ( xˆω x − + ) ( , A Σk −1 −A1 i A Σ k −1b k −1 ⎦⎥(11) k −1= 0.5 i T T ⎡ ⎤ ˆ ˆ ˆ h u h u h u ∂ ∂ ∂ ( ) ( ) ( ) k −1 k −1 ˆ k −1 K k =S Pˆk −1 ⋅ Pk −⎤1 + σ v2 ⎥ ; (8) ⎢ yˆk −1 − yi yˆ ∂u ∂uk ∂uk ⎦ THE EFFECTIVENESS OF THE − 2 ⎣ k −−11 2k ; −1⎥ . S 2 S 2 − 1 − 1 T T ⎥ ˆxk −1 − xi ) + (ωyˆ k −=1 −0.5 ˆ ˆ yi ) A Σ xkA + y −1 Σ ⎦b , ALGORITHM ANALYSIS ⎡ ˆk −1 )⎤ ; (11) (9) uˆk = uˆk −−11 + KkA k ⎣rk − h(u ⎦

)( )

( (

)

(

(

)

)

The resulting algorithm (Eqs. 8 – 10) is non-linear and ∂h(uˆk −1 ) ˆ ˆ class belongs the the adaptive it along with Pˆk to =P − Kof Pk −ones 1 k − 1 k 1 , because(10) (11) b, ∂ u k the estimation of the RFS the coordinates, an estimation of the unknown error v0 is determined. The initial conditions of uˆ0 and Pˆ0 must be set for the implementation of adaptive The initial evaluation of the vectoryˆ kuˆ−T01 is−uˆyT0iS= (xˆ0, yˆ0, 0) xˆk −1 −filtering. xiS xˆk −1 ; − The initial estimates 2of RFS2coordinates xˆ0,Syˆ02 are determined xˆk −1 + yˆ k −1 ( xˆk −1 − xi ) + ( yˆ k −1 − yiS )2 ( xˆk −1 − xiS )2 + ( yˆ k −1 − yiS )2 based on the method of least squares (LS) in case of 3 difference measurement distances (Amar and Leus 2010) using the Eq. 12:

ω = 0.5 AT Σ−1 A

(

−1

)

AT Σ−1b ,

(11)

(12)

where: ωT = (xˆ0, yˆ0, Rˆ0) is the vector consisting of the RFS assessment;

x1Sx1S y1Sy1S d10d10

AA = =x2Sx2S y2Sy2S d 20d 20; x3Sx3S y3Sy3S d30d30

2 2 x1Sx+1S y+1Sy−1S d−10 d10

S S 2 2 ; ; ; b =b =x2 x+2S y+2 y−2S d−20d 20 2 2 x3Sx+3S y+3Sy−3S d−30 d30

⎡ 2σ ⎡ 2v2σ 2 σ v2σ 2 σ v2σ ⎤2 ⎤ ⎢ ⎢ 2 v 2 v 2 ⎥v ⎥ Σ =Σ ⎢=σ⎢vσ v2 2σ2vσ v2 σ vσ⎥v2. ⎥ . ⎢ ⎢ 2 2 2 2 2 ⎥ 2 ⎥ ⎣⎢ σ⎣⎢vσ v σ vσ v 2σ2vσ⎦⎥v ⎦⎥

(13) ...continue

−

The efficiency analysis of recurrent adaptive algorithm (Eqs. 8 – 10) and its comparison with the quadratic correction one is performed by statistical modeling. The quadratic correction algorithm provides the highest accuracy among those considered in Amar and Leus (2010). It consists of 2 stages: (i) evaluate the ⎤ which depends on the distance R substitute RFS yˆkcoordinates, 0, −1 ⎥ . ; 1 − in2 the initial functionality then re-solve the linear optimization 2 ⎥⎦ xˆk −1 + yˆ k −1 problem; (ii)–adjust the solution for a quadratic relation. The modeling of algorithms is performed for the sensor network configuration (Fig. 1a), which consists of 9 sensors to determine the RFS position at the coordinates: S0(0; 0), S1(0; 20), S2(20√2; 20√2), S3(20; 0), S4(20√2; –20√2), S5(0; −20), S6(–20√2; –20√2), S7(−20; 0) and S8(–20√2; 20√2). Figure 1b shows a sensor network with a minimal number of sensors to form the initial adaptive filtering conditions with the coordinates: S0(0; 0), S1(20√2; 20√2), S2(20√2; –20√2) and S3(−20; 0). The RFS is placed on a circle with a radius of 100 km relative to the reference sensor D0. The root mean

J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.489-494, Oct.-Dec., 2017

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492

square (RMS) of the measurement error is sv = 30 m. As an indicator of the efficiency, the circular standard deviation was used, σˆ = √σˆ 2x + σˆ 2y . Figure 2 shows the dependence of the actual σˆ MC LS (curve 1)

circular RMS of the RFS position estimation error with 4 sensors (Fig. 1b) obtained by Monte Carlo simulation using (a)

Y RFS

y S1

S8 S7

r1 r7

S0

r6

S6

r2 r S3

r5 r3

r4

X x

S4

S5

(b)

r8

S2

Y RFS

y r3 S3

S1

r1 r

r2

S0

X x

S2 Figure 1. Configuration of the sensor network with (a) 9 and (b) 4 sensors.

90

8

120

Eq. 12, which corresponds to the initial conditions of the adaptive filter. This figure also shows the dependence of the theoretical (curve 2) circular RMS of the RFS position estimation error, which is calculated based on relevant elements of the correlation matrix of estimation errors determined using Eq. 14. RMS values σˆ MC LS ​range from 2 to 8 km. Figure 3 shows the dependence of the actual σˆ MC FK (curve 1) ˆ and theoretical σFK (curve 2) circular RMS of the RFS position estimation error with 9 sensors (Fig. 1) for the recurrent algorithm. The received actual and theoretical RMS values correspond well, indicating the proper operation of the algorithm. MSE values σˆ MC FK range from 1.3 to 1.9 km. The application of recurrent adaptive algorithm reduces the circular RMS of the RFS location estimation error in 1.5 – 4.2 times (Fig. 3). Figure 4 shows the dependence of the actual σˆ MC QC (curve 1; ˆ where QC is the quadratic correction) and theoretical σFK (curve 2) circular RMS of the RFS location estimation error for the quadratic correction algorithm. Figure 5 shows a circular RMS of the RFS location estimation error, which corresponds to the lower limit of the Cramér-Rao bound and characterizes the potential accuracy of the possible RFS coordinates. The values of circular RMS error o ​ f the RFS positioning of the recurrent adaptive and quadratic correction algorithms, positioned close to the corresponding values ​of circular RMS of the Cramér-Rao bound lower limit, indicate their high efficiency. Figure 6 shows the change of the actual (curve 1) and theoretical (curve 2) circular RMS error estimation of the RFS locations using recurrent adaptive algorithm for fixed RFS 90

1 2

60

4

30

1

150

2

30

0.5

2

180

0

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1

60

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6 150

2

120

210

300

330 240

270 Figure 2. Circular RMS of the RFS position estimation error for the initial conditions.

0

300 270

Figure 3. Circular RMS of the RFS position estimation error for recurrent adaptive algorithm.

J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.489-494, Oct.-Dec., 2017

Recurrent Algorithm for TDOA Localization in Sensor Networks

90

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Figure 4. Circular RMS of the RFS position estimation error for the quadratic correction algorithm. 90

2

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210

2 1.5 1 0.5 0

0

0.5

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2.5

Sensors

3

3.5

4

4.5

5

Figure 6. Circular RMS dynamics of the RFS locations estimation error under sequential data arrival.

Using the mathematical apparatus of Kalman filtering, the algorithm is developed, which, after the formation of the initial conditions, is based on measurements of the receiving time of the signals from 4 sensors. This allows specifying the location of the recurrent RFS as the measurement proceeds from the other sensors. It belongs to the class of adaptive algorithms, because, along with the estimation of the RFS coordinates, it determines the estimation of unknown error of reference sensor measurement. According to the simulation results, the use of recurrent algorithm can reduce the circular RMS of the RFS location estimation error by 1.5 – 4.2 times, providing characteristics similar to the potentially achievable. The implementation of the recurrent adaptive algorithm requires 2.7 times less computational cost than the quadratic correction one. The resulting algorithm can also be easily extended to the case of RFS filtering trajectory at which its motion parameters are estimated (Chiang et al. 2012).

AUTHOR’S CONTRIBUTION

330 240

2.5

CONCLUSIONS

30

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1 2

3

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3.5

RMS error estimation of RFS locations [km]

coordinates (x, y). The use of recurrent adaptive algorithm reduces the circular RMS of the RFS location estimation error in 2.5 times relative to the initial conditions. It is interesting to compare the calculation of the costs required in the implementation of the recurrent adaptive and quadratic correction algorithms. They can be assessed by determining the number of required multiplications (divisions), since they are performed from 100 to 150 times slower than addition and subtraction. For the example with the implementation of the recurrent adaptive algorithm, 461 multiplications are required and, for the quadratic correction one, this value is 1,246. Thus, the application of the developed algorithm reduces the computational cost by 2.7 times.

493

300 270

Figure 5. Circular RMS of the RFS position estimation error of the Cramér-Rao bound lower limit.

Introduction, Tovkach IO and Zhuk SYa; literature review, Tovkach IO and Zhuk SYa; Development, Tovkach IO and Zhuk SYa; Discussion, algorithm analysis, Tovkach IO and Zhuk SYa; Conclusion, Tovkach IO and Zhuk SYa.

J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.489-494, Oct.-Dec., 2017

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REFERENCES International Telecommunication Union (2015) Comparison of methods of determining the geographic location of the signal source based on the time difference of arrival and angle of arrival of the signal Report ITU-R SM.2211-1. Geneva: ITU. Amar A, Leus G (2010) A reference-free time difference of arrival source localization using a passive sensor array. Proceedings of the IEEE Sensor Array and Multichannel Signal Processing Workshop; Jerusalem, Israel. Buzuverov GV (2008) Algorithms for passive location in a distributed network of sensors for the time difference of arrival method. In: Buzuverov GV, Gerasimov OI, editors. Information-measuring and control systems. Moscow: Radio Engineering.

Chiang CT, Tseng PH, Feng KT (2012) Hybrid Unified Kalman Tracking Algorithms for heterogeneous wireless location systems. IEEE Trans Veh Technol 61(2):702-715. doi: 10.1109/TVT.2011.2180939. Rullan-Lara JL, Sanahuja G, Lozano R, Salazar S, Garcia-Hernandez R, Ruz-Hernandez JA (2013) Indoor localization of a quadrotor based on WSN: a real-time application. Int J Adv Robotic Sy 10(1):1-9. doi: 10.5772/53748. Welch G, Bishop G (2006) An introduction to the Kalman Filter. University of North Carolina at Chapel Hill, Department of Computer Science;

[accessed

May

24,

2016].

edu/~welch/media/pdf/kalman_intro.pdf

J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.489-494, Oct.-Dec., 2017

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