www.ietdl.org Published in IET Signal Processing Received on 26th September 2013 Revised on 22nd January 2014 Accepted on 12th May 2014 doi: 10.1049/iet-spr.2013.0393 ISSN 1751-9675 Track fusion in the presence of sensor biases Hongyan Zhu, Shuo Chen Automation Department, School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China E-mail: [email protected] Abstract: A computationally effective approach is developed in this study to deal with the problem of track fusion in the presence of sensor biases. Aiming at the case that sensor biases are implicitly included in the local estimates, a pseudo-measurement equation is derived based on the Taylor series expansion ﬁrstly, which reveals the relationship explicitly between local estimates and the sensor biases; and then, the bias estimates can be obtained in the rule of recursive least squares; ﬁnally, based on the derived pseudo-measurement equation, the sensor biases can be removed from the original local estimates and track fusion can be carried out directly and easily. Monte Carlo simulations demonstrate the efﬁciency and effectiveness of the proposed approach compared with the competing algorithms. 1 Introduction The potential advantages of fusing information from disparate sensor systems to achieve better surveillance has been recognised [1–3]. In the distributed fusion system, local estimates are generated from individual sensors. At the fusion centre, they are combined according to a certain criterion to yield an improved estimate than local ones. An important prerequisite for the successful fusion is the transformation of sensor reports into a common spatial reference frame [4]. The fusion process relies on the accurate registration of sensors which is regarded as a process to eliminate the effects caused by the sensor biases. If uncorrected, sensor biases would lead to large tracking errors and cause ghost tracks. Consequently, the fusion performance cannot be guaranteed to be optimal. The classical approach to deal with the problem is to augment the system state to include the sensor biases as part of the state vector, and then implement an augmented state Kalman ﬁlter (ASKF). Friedland [5] proposed the idea of implementing two parallel, reduced-order ﬁlters instead of using an ASKF. Ignagni [6] generalised the two-stage method of [5]. Van Doorn and Blom [7] gave an exact solution for the augmented Kalman ﬁlter problem, but then decoupled the equations using an approximation to make the implementation feasible. In [8], under a rather restrictive algebraic constraint, the optimal sate estimate can be obtained by combining local bias-ignorant estimate and the estimated sensor bias. Ignagni [9] derived the optimal form of the separate-bias estimator for the general case in which the bias vector is stochastic in nature. Okello and Ristic’s work in [10] presented a batch maximum likelihood (ML) registration algorithm for spatial alignment of multiple, possibly dissimilar sensors. An exact solution was provided for the multi-sensor bias estimation problem in [11] by constructing the pseudo-measurements equation of the 958 & The Institution of Engineering and Technology 2014 sensor biases with additive noises that are zero-mean, white, and with easily calculated covariance. Okello and Challa [12] proposed an equivalent measurement method for registration at the track level by augmenting the state vector with the sensor biases. Although it is easy to implement, the accuracy is limited by the parallel extended Kalman ﬁlter (EKF). In [13], track registration and fusion is modelled as an ML joint estimation problem. In [11, 13], to construct the pseudo-measurement equation of sensor biases, lots of information besides local estimates, such as the measurement matrix, Kalman gain, system matrix and so on, is required which is not always available because of the limitation of communication bandwidth in many distributed fusion systems. As we know, sensor registration is often performed at the measurement level, since the sensor biases are directly added to the sensor measurements in general. However, it is not suitable to perform sensor registration at the measurement level for distributed track fusion systems, in which local sensors only send extracted tracks to the fusion centre. In this respect, sensor registration at the track level is needed more. In this paper, we address the problem of track fusion in which the sensor biases are added to the original sensor measurements and implicitly included in the local estimates. The main difﬁculties of the problem lie on how to model and remove the sensor biases inherent in the local tracks. For this purpose, we construct a pseudo-measurement equation based on the Taylor series expansion, which reveals the relationship explicitly between local estimates and the sensor biases; after that, the bias estimates can be obtained in the rule of recursive least squares (RLS); Consequently, the biased track can be corrected by using the estimated sensor biases according to the pseudo-measurement equation and track fusion can be implemented directly and efﬁciently. IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 www.ietdl.org The main merits of the proposed approach lie on the following several aspects: Firstly, the proposed approach requires far less information than the competing algorithms when constructing the pseudo-measurement equation. Secondly, the proposed approach has the higher computational efﬁciency than the competing algorithms. Finally, the proposed approach enables the local biased track to be corrected easily by the estimated sensor biases, as contributes to implementing track fusion directly and efﬁciently. Selected simulation results demonstrate the main advantages and features of the proposed approach compared with the competing algorithms. 2 Problem formulation Fig. 1 depicts the conﬁguration geometry with two sensors and one target. The location of each sensor {xom, yom)}(m = 1, 2) is assumed to be known exactly. The local Cartesian coordinate systems (LCCS) have the rotation angles f1 and f2, respectively, with respect to the global Cartesian coordinate system (GCCS) denoted by O. It is assumed that the target T is detected by both of the two sensors (S1 and S2), respectively. The measurement process is implemented in the local polar coordinate system (LPCS). Since there exist systematic biases and random errors, the original measurement rkm , um from k sensor m at time k can be modelled as m m rkm = rm k + Drk + vk,r umk (1) m m = uk + Dum k + vk,u (2) m where rm k and uk denote the real range and angle about the observed target, Drkm,i and Dum,i are the systematic biases, k m vm k,r and vk,u are the random errors. m Furthermore, it is assumed that the biases Drm k and Duk are invariant across the overall tracking time, that is Drkm = Dr , m Dum k = Du In this way, the original measurement rkm , umk m m rkm = rm k + Dr + vk,r (4) umk = uk + Dum + vmk,u (5) m m m m T is The random measurement noise Ṽ k = vk,r vk,u modelled as independent white Gaussian noises in the range 2 m 2 and angle with zero-mean and variances of sm , su . r The bias vector of the two sensors is denoted by T T T T h = h1 , where hm = Drm Dum is the h2 bias for sensor m. Each sensor m produces itsown local tracks based on the biased measurements rkm , um k , and each local track at time m m k is represented by a two-tuples {x̂m k , P k } (m = 1, 2). x̂k m and P k mean the state estimate and error covariance, respectively. The goal of track fusion is to combine local estimates to yield a better estimate than the local ones. Note that, local trackers cannot produce the bias estimates based on their own local measurements. Therefore, local estimates are biased estimates. The direct fusion of biased local estimates will not produce a satisfactory result. 3 Review of existing main approaches to sensor registration 3.1 Exact multi-sensor dynamic bias estimation with local tracks In what follows, the ‘Exact method’ given in [11] for bias estimation is reviewed. Suppose that the dynamic model of the target is xk+1 = F k xk + G k vk (3) from sensor m (6) After transforming the measurements from LPCS into LCCS, the measurement equation for sensor m is m m m m zm k = H k xk + Bk h + vk m can be rewritten as (7) where m is the sensor index, Fk is the transition matrix, ωk and vm k are zero-mean, white Gaussian noises with covariance Qk m m and Rm k , respectively. H k and Bk are the measurement matrix and the weighted matrix for the bias vector ηm. Because the local tracker has no the information about the sensor biases, the measurement model assumed by local trackers is of the following bias-ignorant formulation m m zm k = H k xk + vk (8) From the local estimate x̂m k+1 from sensor m, one has m m m m x̂m k+1 = F k x̂k + K k+1 zk+1 − ẑk+1|k m m m = F k x̂m k + K k+1 H k+1 F k xk + G k vk + Bk+1 h +vm − H m F k x̂m k+1 m k+1 k m = I − K k+1 H m k+1 F k x̂k m m m m + Km k+1 [H k+1 F k xk + G k vk + Bk+1 h + vk+1 ] (9) Fig. 1 Sensor conﬁguration geometry IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 Consequently, the following pseudo-measurement for sensor 959 & The Institution of Engineering and Technology 2014 www.ietdl.org P h,k+1|k = F h,k P h,k F Th,k + Qh,k m is deﬁned by + m m m m zm x̂m h,k+1 = K k+1 k+1 − I − K k+1 H k+1 F k x̂k (10) m m m = Hm k+1 F k xk + G k vk + Bk+1 h + vk+1 + is the pseudo-inverse of the Kalman gain where K m k+1 Km k+1 . Then the pseudo-measurement equation of the sensor biases can be given by zh,k+1 = Hk+1 h + ṽk+1 where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Rh,k+1 3.2.2 State estimate and covariance update: The predictive state, covariance matrix and the predictive measurements are (11) + zh,k+1 = z1h,k+1 − H 1k+1 H 2k+1 z2h,k+1 + Hk+1 = [ B1k+1 −H 1k+1 H 2k+1 B2k+1 ] T h = h1 T h2 T + ṽk+1 = v1k+1 − H 1k+1 H 2k+1 v2k+1 + + T = R1k+1 + H 1k+1 H 2k+1 R2k+1 H 1k+1 H 2k+1 After having the above pseudo-measurement equation, the optimal bias estimates can be obtained in the rule of RLS when given an initial estimate ĥk , P h,k . (see (13)) (15) zk = H k xk + Bk h + vk z1 H 1k Bk = diag B1k , B2k , where zk = k2 , H k = 2 , zk Hk 1 1 2 vk vk = 2 , Rk = diag Rk , Rk . vk The covariance matrices for process noise ωk and ωη,k are Qk and Qh,k , respectively. The decoupled Kalman ﬁlter consists of the following steps. 3.2.1 Bias prediction: Given the initial bias estimation ĥk and its covariance Pη,k, the predictive bias and the corresponding covariance can be computed by ĥk+1|k = F h,k ĥk (16) P k+1|k = F k P k|k F Tk + G k Qk G Tk (19) ẑk+1|k = H k+1 x̂k+1|k + Bk+1 ĥk+1|k (20) + Rk+1 (22) The Kalman gain for the system state xk + 1 −1 K x,k+1 = P̃ xz,k+1|k P̃ zz,k+1 (23) The updated state estimates and the corresponding covariance are computed by x̂k+1 = x̂k+1|k + K x,k+1 [zk+1 − ẑk+1|k ] P k+1 = I − K x,k+1 H k+1 P k+1|k (24) (25) 3.2.3 Bias estimates and covariance update: The bias vector can be updated by ĥk+1 = ĥk+1|k + K h,k+1 [zk+1 − ẑk+1|k ] P h,k+1 = I − K h,k+1 Bk+1 P h,k+1|k T = P h,k+1|k − K h,k+1 P̃ hz,k+1 (26) (27) where the Kalman gain for the bias vector ηk can be computed by −1 K h,k+1 = P̃ hz,k+1 P̃ zz,k+1 (28) P̃ hz.k+1 ≃ P h,k+1|k BTk+1 (29) where 4 Proposed approach to sensor registration and track fusion Since the sensor biases {Δrm, Δθm} are implicitly included in m local estimates {x̂m k , P k }, the key problem is to construct a pseudo-measurement equation which can reveal the ⎧ −1 ⎪ ⎨ ĥk+1 = ĥk + P h,k HTk+1 Hk+1 P h,k HTk+1 + Rh,k+1 [zh,k+1 − Hk+1 ĥk ] −1 ⎪ T T ⎩ P Hk+1 P h,k h,k+1 = P h,k − P h,k Hk+1 Hk+1 P h,k Hk+1 + Rh,k+1 960 & The Institution of Engineering and Technology 2014 (21) P̃ zz,k+1 ≃ H k+1 P k+1|k H Tk+1 + Bk+1 P h,k+1|k BTk+1 The accumulated measurement process for two sensors is formulated by (18) P̃ xz,k+1|k ≃ P k+1|k H Tk+1 Decoupled Kalman ﬁltering for bias estimation The decoupled Kalman ﬁltering for sensor registration is earliest presented in [7], and is extended in [14] (called ‘KVDB’). The dynamic equations of the system state xk and bias vector ηk are formulated by xk+1 = F k xk + G k vk (14) hk+1 = F h,k hk + vh,k x̂k+1|k = F k x̂k where the covariance between the state and the measurements, and the measurement prediction covariance are (12) 3.2 (17) (13) IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 www.ietdl.org relationship explicitly between local estimates and the bias vector to estimate. 4.1 measurement xm k ym k in LCCS (see (34)) where Notations ym xm rm rm (35) k = k cos (uk ), k = k sin (uk ) ^m 2 ^m 2 2 m 2 m rk = x k − xom + y k − yom = xm + yk (36) k m Firstly, we introduce some notations for the sake of clarity. m m T vm xk vx,k ym xm : real state in LCCS. k y,k k = ^m ^m ^m ^m T ^m x k = x k v x,k y k v y,k : real state in GCCS. m m T vx,k ym vm xm : equivalent measurement in k y,k k = xk LCCS. m ^m ^ ^m ^m ^m T x k = x k v x,k y k v y,k : equivalent measurement in GCCS. m m T v̂x,k ŷm v̂m : state estimate in GCCS. x̂m k y,k k = x̂k In all the notations given above, the ﬁrst and third components mean the position, and the second and fourth components mean the velocity. The state estimate x̂m k is affected by both sensor biases and random error. We deﬁne ^m m the equivalent measurements xk in LCCS (or x k in GCCS) as one part of the state estimate, which is affected by sensor biases only. Several transformations about the position components between different coordinate systems are m introduced as follow. xk Conversion from equivalent measurement in LCCS ^m ym k x to the equivalent measurement ^km in GCCS yk ^m m xk xk xom + (30) = R m wm y ^m yom k y k where the rotation matrix is cos (wm ) −sin (wm ) = sin (wm ) cos (wm ) R wm Conversion from real state in LCCS ^m x k ^m y k (31) in GCCS to real state ^m x k − xom −1 xm k = Rwm ^m ym k) y k − yom xm k ym k (32) where the rotation matrix is R−1 wm cos (wm ) = −sin (wm ) Conversion from real state xm k ym k xm k ym k sin (wm ) cos (wm ) m 4.2 Construction of the pseudo-measurement equation m To establish the link between the local estimate x̂m k , P k and the sensor bias, we ﬁrstly decompose the local estimate into different parts (from the real state, the sensor bias and random error, respectively) with a simple and easily calculated formulation. After that, a pseudo-measurement equation of the bias vector is derived, which serves to further bias estimation. the meaning of the biased local estimate mBy analysing x̂k , P m , it is seen that it is affected by two kinds of k errors: sensor biases and random errors. Moreover, the covariance P m k only reﬂects the effect from random errors. Therefore, local estimate x̂m k can be represented by the sum ^m of the equivalent measurement x k (affected by sensor biases only) in GCCS and the random error em k ^m ^m m k , hm + em x̂m k = x k + ek = fm x k (37) ^m where fm x k , hm is the part inﬂuenced by the sensor biases ηm, and em k is the part inﬂuenced by random noise that can be regarded as a random vector with zero-mean and covariance P m k . The following theory provides a formulation for local estimate decomposition. Theorem 1: Any local estimate x̂m k in GCCS can be approximated by the following linear form x̂m k ^m = x k + Jhm Drm + em k Dum where Jhm is the Jacobian matrix deﬁned by ∂f T Jhm = mm m ∂h (x̂k , 0 0 ) (38) (39) (33) in LCCS to equivalent The Jacobian matrix can be expressed by components of local estimate x̂m k as follows (see (40) at the bottom of the next page). Proof: See Appendix m m m m m rk + Drm cos (uk ) cos (Dum ) − sin (uk ) sin (Dum ) rm cos uk + Dum k + Dr m = m = m m m rk + Drm sin (uk ) cos (Dum ) + cos (uk ) sin (Dum ) rk + Drm sin uk + Dum ⎡ ⎤ xm ym m m m m m m m k k xm cos (D u ) − y sin (D u ) + cos (D u )Dr − sin (D u )Dr k ⎢ k ⎥ rm rm k k ⎥ =⎢ m m ⎣ ⎦ yk xk m m m m m m m m yk cos (Du ) + xk sin (Du ) + m cos (Du )Dr + m sin (Du )Dr rk rk IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 (34) 961 & The Institution of Engineering and Technology 2014 www.ietdl.org If two local estimates x̂1k and x̂2k are assumed from the same ^1 ^2 ĥk , P h,k , the update process is given by ⎧ −1 ⎪ T T ⎪ ĥ = ĥ + P A A P A + P ⎪ k+1 k h ,k k+1 h ,k k+1 k+1 k+1 ⎪ ⎪ ⎪ ⎨ [Y k+1 − Ak+1 ĥk ] −1 ⎪ T T ⎪ P = P − P A A P A + P ⎪ h ,k+1 h ,k h ,k k+1 h ,k k+1 k+1 k+1 ⎪ ⎪ ⎪ ⎩A P k+1 h,k target, that means x k = x k . Thus, we have x̂1k − x̂2k = Jh1 2 Dr1 Dr − Jh2 + e1k − e2k Du1 Du2 (41) Rearranging the terms in (41) and recalling the deﬁnition h = [ Dr1 Du1 Dr2 Du2 ]T leads to x̂1k − x̂2k = Jh1 −Jh2 h + e1k − e2k (48) (42) 4.4 In this way, we obtain the following Corollary, which provides a pseudo-measurement equation revealing the relationship between local estimates and the bias vector. □ Corollary 1: Given two local estimates and the corresponding m covariance matrices x̂m , P k k (m = 1, 2) from the same target, the following pseudo-measurement equation of the bias vector can be obtained Correction and fusion of local estimates having the bias estimates ĥk , by Drm m ŷm based on (38), we have k = x̂k − Jhm Dum After ^m x k + em ŷm k = k letting (49) Y k = Ak h + ek (43) Equation (49) indicates that ŷm k is affected by the random error m em k only. As a result, ŷk can be regarded as the unbiased local estimate. If we ignore the effect resulting from the cross-correlation between the bias estimation errors and the state estimation errors, the covariance of ŷm k can be approximated by Y k = x̂1k − x̂2k (44) T m m P k = P k + Jhm Phm Jhm where Ak = Jh1 −Jh2 (50) (45) ek = e1k − e2k (46) The covariance of the noise vector ek is 12 T P k = P 1k + P 2k − P 12 k − Pk (47) where P hm is the sub-matrix of Pη, k with respect to the bias vector ηm. Consequently, trackfusion based on corrected local tracks 1 1 2k can be done according to the BC ŷk , P k and ŷ2k , P (Bar-Shalom and Campo) fusion formula [16] ŷ = ŷ1k ! T "−1 1 12 1 2 12 12 Pk + Pk − Pk − P + Pk − Pk (ŷ2k − ŷ1k ) k 1 A recursive formula of the cross-covariance P 12 k between ek 2 and ek can be found in [15]. (51) The corresponding covariance is 4.3 Bias estimation based on recursive least squares ! T "−1 1 2 12 1k − P 12 12 =P 1k − P P + P − P − P P k k k k k In this paper, the sensor biases are modelled as the unknown constants, hence, we can obtain the RLS estimator based on the pseudo-measurement (43). Given an initial bias estimate ! x̂m k − xom ⎢ m 2 m 2 ⎢ − x + ŷ − y x̂ om om k k ⎢ ⎢ m m m ⎢ ŷ − y 2 v̂m − x̂m − x om om ŷk − yom v̂y,k x,k k ⎢ k ⎢ 2 m 2 3/2 ⎢ + ŷk − yom x̂m ⎢ k − xom ⎢ m =⎢ ŷk − yom ⎢ ⎢ m 2 m 2 ⎢ ⎢ − x + ŷ − y x̂ om om k k ⎢ m m m ⎢ m ⎢ x̂k − xom 2 v̂m y,k − x̂k − xom ŷk − yom v̂x,k ⎢ ⎢ 3 ⎣ 2 m 2 m x̂k − xom + ŷk − yom 2 ⎡ Jhm T " 1 12 Pk − P k 962 & The Institution of Engineering and Technology 2014 (52) ⎤ − yom ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ m −v̂y,k ⎥ ⎥ ⎥ ⎥ m ⎥ ⎥ x̂k − xom ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ m ⎥ v̂x,k ⎥ ⎦ − ŷm k (40) IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 www.ietdl.org k between two estimates ŷ1k and where the cross-covariance P 2 ŷk can also be computed as in (47). 12 4.5 Discussions 4.5.1 Extension to the multi-target case or the case of detection probability less than one: Here, we only address the problem of bias estimation at the track level in the single-target scenario. It is of great importance and challenging to extend the proposed approach to the multi-target case or the case of the detection probability less than one. It is well known that one needs sensor reports from the common target to perform bias estimation. However, in the multi-target case, it is expected to implement data association to determine the correspondence between local sensor reports. Therefore, sensor registration is highly conditioned on the results of track-to-track association. Moreover, note that if sensor biases are estimated and removed from sensor reports, one can determine the correspondence more accurately and easily. In this sense, we can say that track-to-track association and sensor registration are highly coupled with each other. An integrated way may be promising to implement data association and bias estimation jointly. Related work about this topic can be found in [17–19], where the sensor biases are assumed to be a direct additive term on local estimates, and only relative biases can be estimated. In addition, in the case of detection probability less than one, the target sets detected by different sensors do not coincide. Here, we refer to a local track from one sensor as ‘outlier’ when the track has no the corresponding track Fig. 2 Real and estimated track from another sensor. Such ‘outliers’ further complicate an already difﬁcult track-to-track association problem. In this case, dummy tracks can be introduced which can be used to associate with the ‘outliers’. Aiming at the problem of performing sensor registration at the track level and track-to-track association jointly, the detailed construction and solution of the joint optimisation model are beyond the scope of the work, and will be covered in the future work. 4.5.2 Extension to asynchronous sensor registration and track fusion: In this work, it is assumed that local Fig. 3 Estimated sensor biases a Estimated range bias for sensor 1 b Estimated range bias for sensor 2 c Estimated angle bias for sensor 1 d Estimated angle bias for sensor 2 IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 963 & The Institution of Engineering and Technology 2014 www.ietdl.org Fig. 4 RMSE for track fusion a RMSE in position b RMSE in velocity c RMSE in position d RMSE in velocity tracks from different sensors are synchronous. In many practical applications, sensor reports are usually not timecoincident because of different data rates of local sensors. Hence, it is meaningful to implement bias estimation and track fusion to the case of asynchronous sensors [20, 21]. The key point is, based on asynchronous local tracks, to establish the pseudo-measurement equation of the bias vector (including the asynchronous pseudo-measurement, the asynchronous pseudo-measurement matrix, the asynchronous bias vector and pseudo-measurement noise), and evaluate the statistical property of the pseudomeasurement noise. Future work will include the extension of the proposed work to asynchronous sensor registration and track fusion. 5 Simulation results In this section, we provide some selected simulation results to illustrate the performance of the proposed approach. 5.1 Simulation scenario We consider a two-dimensional tracking scenario with two sensors and one target. The initial state of the target is at (35,135) km with a velocity of (0.2128,0.1786) km/s. The motion of the target follows the following dynamic equation xk+1 = F k xk + G k vk 964 & The Institution of Engineering and Technology 2014 (53) where the state xk = [xk , ẋk , yk , ẏk ]T means the position and velocity of the target at x–y plane, Fk = diag{Fk,1 , 2Fk,1}, 1 T T /2 , G k,1 = Gk = diag{Gk,1, Gk,1}, F k,1 = . 0 1 T The sample interval T is set to be 10 s, and the surveillance time period is 2000 s. The covariance matrix of the zero-mean process noise ωk is designed as Q = (0.0002 km/ s2)I, where I is an 2 × 2 identity matrix. Each sensor measures the range rkm and angle um k to the target by the measurement equation rkm ^m 2 ^m 2 x k − xom + y k − yom + Drkm + vm = k,r # $^m $ y − y m m om uk = arctg %^km − wm + Dum k + vk,u x k − xom (54) (55) ^m ^m where x k , y k means the target’s real state in GCCS, (xom, yom) is the sensor’s position. In this scenario, sensor 1 is located at (xo1, yo1) = (20, 50) km in GCCS and the corresponding LCCS has a rotation angle of f1 = 0.242 rad; sensor 2 is located at (xo2, yo2) = (400, 100) km in GCCS and the corresponding LCCS has a rotation angle of f2 = m 0.375 rad. The random measurement errors (vm k,r and vk,u ) for both of sensors are modelled as white Gaussian with IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 www.ietdl.org variances (0.01 km)2 and (0.001 rad)2, respectively. The range and angle biases for sensor 1 and sensor 2 are (−1 km, −0.0042 rad) and (1.2 km, 0.0035 rad), respectively. Both of the sensors employ EKF to obtain the local biased tracks based on their own measurement information. The fusion centre receives local tracks from different sensor, and implements sensor registration and track fusion. The measurements are supported by 200 Monte-Carlo runs performed on the same target trajectory but with independently generated measurements for each trial. The initial bias vector is given by ĥ0 = [ 0 0 0 0 ]T , Pη,0 = diag ([106, 104, 106, 104]). The simulation results are given by implementing the proposed method and the competing algorithms (exact method in [11] and ‘KVDB’ in [11, 14]). Fig. 2 shows the real trajectory of the target and local biased tracks by different sensors at a time interval, respectively. Fig. 3 shows the estimation results for the range and angle biases. It is shown that the proposed method and ‘Exact method’ perform similarly, and yield better estimates than ‘KVDB’. The average root mean-square error (RMSE) in the position and velocity component over the whole simulation time are shown in Fig. 4. Since the ‘Exact method’ cannot be used to implement track fusion directly, we illustrate the fusion result based on the proposed approach, the ‘KVDB’ and direct fusion of biased local tracks (named by ‘bias-ignorant fusion’). It is seen that the proposed method outperforms the competing ones. Among them, Figs. 4c and d eliminate the result given by ‘bias-ignorant fusion’ so as to make the performance comparison between the proposed approach and the ‘KVDB’ more clearly. 5.2 Computation cost The executing time of these algorithms is summarised in Table 1. It is evaluated based on the following computer conﬁguration: CPU: Intel (R) Core(TM) i5-2430M [email protected] GHz 2.40 GHz; RAM:2 GB; Operating system: Windows 7. From the above table, it is seen that that the proposed approach has a signiﬁcant improvement in computational efﬁciency. The reason why the proposed approach is so computationally effective will be summarised in the following subsection. 5.3 Algorithm analysis As shown in Fig. 3, it seems that the ‘Exact method’ and the proposed approach behaves similarly in estimating the sensor biases. However, the proposed approach has its own important merits. Compared with the ‘Exact method’, the advantages of the proposed approach lie on the following several aspects. Firstly, the proposed approach requires far less information than the ‘Exact method’ when constructing the pseudo-measurement equation. The ‘Exact method’ requires lots of information from local trackers, such as the Kalman gain, system matrix, measurement matrix, covariance of Table 1 Executing time of algorithms Proposed approach 0.0073 s Exact method KVDB 0.0268 s 0.0245 s IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 measurement noise, local estimates, and so on. However, the proposed method requires local estimates and its covariance only. Secondly, the proposed approach has the higher computational efﬁciency than the ‘Exact method’ (shown in Table. 1). Although the process of deriving the pseudo-measurement equation seems a little bit complex, the ﬁnal (43) is so easy. However, the ‘Exact method’ requires computing the pseudo-inverse of the matrix when constructing the pseudo-measurement equation, which is not necessary in the proposed approach. Thirdly, the proposed approach has a wider use scope. The ‘Exact method’ requires that local tracker is a linearised model, the proposed approach does not have this restriction. Finally, the proposed approach enables the local track to be corrected easily by the estimated sensor biases, as contributes to implementing the track fusion easily, but the ‘Exact method’ cannot. Compared with the ‘KVDB’, the advantages of the proposed approach lie on the following several aspects. Firstly, the proposed approach can deal with sensor registration at the track level, which is needed in the distributed information fusion system. Secondly, the proposed approach has a higher computational efﬁciency than ‘KVDB’(shown in Table. 1), since the latter requires a two-stage ﬁlter process. Thirdly, about the bias estimation, the proposed approach behaves more stable than ‘KVDB’. Finally, about the fusion performance, the proposed approach outperforms the ‘KVDB’ in RMSE for track fusion (shown in Fig. 4c–d ). 6 Conclusion In the distributed multi-sensor tracking system, sensor registration is expected to be performed at the track level rather than at the measurement level. However, it becomes quite difﬁcult to remove the biases implicitly included in local tracks. In this paper, a novel approach to bias estimation and track fusion is proposed based on the carefully devised pseudo-measurement equation. Simulation results demonstrate that the proposed approach is computationally efﬁcient than competing algorithms. Unlike ‘Exact method’, the proposed approach enables it easy to implement the track fusion. Moreover, the proposed approach outperforms the ‘KVDB’ in bias estimation and track fusion. Further work includes the extensions of the proposed method to some challenging cases, such as the multi-target case, asynchronous sensor case, the case of detection probability less than one or time-varying sensor biases and so on. 7 Acknowledgments The work is supported by the State Key Program for Basic Research of China (2013CB329405) and the National Natural Science Foundation of China (No. 61203220). 8 References 1 Bar-Shalom, Y., Blair, W.D.: ‘Multitarget-multisensor tracking: applications and advances’ (MA, Artech House, 2000, 1st edn.) 2 Bar-Shalom, Y., Li, X.R.: ‘Multitarget-multisensor tracking: principles and techniques’ (Storrs, CT, YBS Publishing, 1995, 1st edn.) 3 Bar-Shalom, Y., Li, X.R., Kirubarajan, T.: ‘Estimation with applications to tracking and navigation’ (Wiley, New York, 2001, 1st edn.) 4 Dana, M.: ‘Registration: a prerequisite for multiple sensor tracking’. In Bar-Shalom, Y. (Ed.): ‘Multitarget-multisensor tracking:advanced applications’. (Artech House, MA, 1990, 1st edn.) pp. 155–185 965 & The Institution of Engineering and Technology 2014 www.ietdl.org 5 Friedland, B.: ‘Treatment of bias in recursive ﬁltering’, IEEE Trans. Autom. Control, 1969, 14, (4), pp. 359–367 6 Ignagni, M.B.: ‘An alternate derivation and extension of Friedland’s two-stage Kalman estimator’, IEEE Trans. Autom. Control, 1981, 26, (3), pp. 746–750 7 van Doorn, B.A., Blom, H.A.P.: ‘Systematic error estimation in multisensor fusion systems’. Proc. of SPIE Conf. on Signal and Data Processing of Small Targets, Orlando, USA, April 1993, pp. 450–461 8 Alouani, A.T., Rice, T.R., Blair, W.D.: ‘A two-stage ﬁlter for state estimation in the presence of dynamical stochastic bias’. Proc. of the 1992 American Control Conf., Chicago, USA, June 1992, pp. 1784–1788 9 Ignagni, M.: ‘Optimal and suboptimal separate-bias kalman estimators for a stochastic bias’, IEEE Trans. Autom. Control, 2000, 45, (3), pp. 547–551 10 Okello, N., Ristic, B.: ‘Maximum likelihood registration for multiple dissimilar sensors’, IEEE Trans. Aerosp. Electron. Syst. , 2003, 39, (3), pp. 1074–1083 11 Lin, X., Bar-Shalom, Y., Kirubarajan, T.: ‘Exact multisensor dynamic bias estimation with local tracks’, IEEE Trans. Aerosp. Electron. Syst., 2004, 40, (2), pp. 576–590 12 Okello, N., Challa, S.: ‘Joint sensor registration and track-to-track fusion for distributed trackers’, IEEE Trans. Aerosp. Electron. Syst., 2004, 40, (3), pp. 808–823 13 Huang, D.L., Leung, H., Bosse, E.: ‘A pseudo-measurement approach to simultaneous registration and track fusion’, IEEE Trans. Aerosp. Electron. Syst., 2012, 48, (3), pp. 2315–2331 14 Kastella, K., Yeary, B., Zadra, T., Brouillard, R., Frangione, E.: ‘Bias modeling and estimation for GMTI applications’. Proc. of the 3rd Int. Conf. on Information Fusion, Paris, France, July 2000, pp. TUC1-7–TUC1-12 15 Bar-Shalom, Y.: ‘On the track-to-track correlation problem’, IEEE Trans. Autom. Control, 1981, 26, (2), pp. 571–572 16 Bar-Shalom, Y., Campo, L.: ‘The effect of common noise on the two-sensor fused track covariance’, IEEE Trans. Aerosp. Electron. Syst., 1986, 22, (6), pp. 803–805 17 Levedahl, M.: ‘An explicit pattern matching assignment algorithm’. Proc. of SPIE Symp. on Signal and Data Processing of Small Targets, Orlando, USA, April 2002, vol. 4728, pp. 461–469 18 Papageorgiou, D., Sergi, J.D.: ‘Simultaneous track-to-track association and bias removal using multistart local search’. Proc. of IEEE Aerospace Conf., Big Sky, USA, March 2008, pp. 1–14 19 Danford, S., Kragel, B., Poore, A.: ‘Joint MAP bias estimation and data association: algorithms’. Proc. of the SPIE, San Diego, USA, August 2007, pp. 66991E-1–18 20 Lin, X.D., Bar-Shalom, Y., Kirubarajan, T.: ‘Multisensor-multitarget bias estimation for general asynchronous, sensors’, IEEE Trans. on Aerosp. Electron. Syst., 2005, 41, (3), pp. 899–921 21 Rafati, A., Moshiri, B., Rezaei, J.: ‘A new algorithm for general asynchronous sensor bias estimation in multisensor-multitarget systems’. Proc. of the 10th Int. Conf. on Information Fusion, Que. Canada, July 2007, pp. 1–8 9 Appendix 9.1 computed by ^m m xk xom xk + = R wm ^m yom ym yk k ⎡ m m xk cos (Dum ) − ym k sin (Du ) m m ⎢ xk ⎢ + cos (Dum )Drm − yk sin (Dum )Drm ⎢ rm rm ⎢ k = R wm ⎢ m k m m ⎢ yk cos (Du ) + xk sin (Dum ) ⎢ ⎣ ym xm + mk cos (Dum )Drm + mk sin (Dum )Drm rk rk xom + yom ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (57) and since ^m − x x xm −1 om k , R−1 = Rwm ^km wm ym ) k y k − yom cos (wm ) sin (wm ) = − sin (wm ) cos (wm ) (58) Combining (36), (57) and (58) leads to ⎡ ^m ^m ⎤ x k − xom cos Dum − y k − yom sin Dum ⎢⎛ ⎥ ⎞ ⎢ ⎥ ⎢ ⎥ m ⎢⎜ ⎥ ⎟ Dr ⎢ ⎜1 + ⎟ + x ⎥ ⎢⎝ ⎥ om ⎠ 2 2 m m ⎢ ⎥ ^ ^ ^m ⎢ ⎥ − x + y − y x k k om om ⎢ ⎥ xk ⎢ ⎥ = ^m m ^m m ⎥ ⎢ ^m yk ⎢ y k − yom cos Du + x k − xom sin Du ⎥ ⎢ ⎥ ⎞ ⎢⎛ ⎥ ⎢ ⎥ ⎢ ⎥ m ⎟ ⎢⎜ ⎥ Dr ⎢ ⎜1 + ⎟ + yom ⎥ ⎣⎝ ⎦ ^m 2 ^m 2 ⎠ x k − xom + y k − yom (59) The velocity component can be computed by Appendix 1: proof of Theorem 1 ^m xk Note that the equivalent measurement written by ⎡ ^m xk in GCCS can be ^m = ∂x k ^m v x,k ^m ∂x k ^m + ∂x k ^m ∂y k ^m v y,k ^m = cos(Dum ) + [(y k − yom ) cos(Dum ) ⎤ ⎢ ⎥ ⎢ ^m ⎥ ⎢ ^m ⎢ v x,k ⎥ ⎥ ^m m ⎥ x k = fm x k , h = ⎢ ⎢ ^m ⎥ ⎢ y ⎥ ⎢ k ⎥ ⎣ ⎦ ^m v x,k ^m + (x k − xom ) sin(Dum )] ^m (56) ^m v y,k × ^m [(x k Drm (y k − Dum ) − xom )2 + , ^m (y k − yom )2 ]3/2 ^m ^m v x,k (60) + − sin(Dum ) − [(y k − yom ) cos(Dum ) ^m + (x k − xom ) sin(Dum )] Next, we aim to express each component of the equivalent ^m xk ^m ^m Drm (x k − xom ) , ^m v y,k in GCCS by that of x k in GCCS. According measurement ^m to (30) and (34), the position components of x k can be × 966 & The Institution of Engineering and Technology 2014 IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 ^m [(x k − xom )2 + ^m (y k − yom )2 ]3/2 www.ietdl.org ^m v x,k ^m ^m v y,k where and are the velocity components of the x k in x and y coordinate, respectively. Similarly, we have ^m v y,k ^m = ∂y k ^m v x,k ^m ∂x k ^m + ∂y k ^m v y,k ^m ∂y k ^m = sin(Dum ) − [(y k − yom ) cos(Dum ) ^m − (x k − xom ) sin(Dum )] , ^m × Drm (x k − xom ) ^m ^m v x,k ^m [(x k − xom )2 + (y k − yom )2 ]3/2 ^m + cos(Dum ) + [(y k − yom ) cos(Dum ) (61) ^m − (x k − xom ) sin(Dum )] , ^m × ^m [(x k Drm (y k − yom ) − xom )2 + ^m (y k − yom )2 ]3/2 ^m v y,k xk , h ) = fm ( m fm (x̂m k, [0 ^m ∂fm m · [x k − x̂k ] + m ∂h (x̂m ,[0 k ∂f T m 0] ) + ^m ∂x k (x̂m ,[0 k 0]T ) ·[ Drm Dum 0]T ) ∂fm T Jhm = m m ∂h (x̂k , 0 0 ) m ⎡ x̂k − xom ⎢ m 2 2 ⎢ x̂k − xom + ŷm ⎢ k − yom ⎢ ⎢ ŷm − y 2 v̂m ⎢ om k x,km m ⎢ ⎢ − x̂m − x k om ŷk − yom v̂y,k ⎢ 3/2 ⎢ ⎢ x̂m − x 2 +ŷm − y 2 ⎢ k om om k m =⎢ ⎢ ŷk − yom ⎢ ⎢ m 2 2 ⎢ x̂ − xom + ŷm ⎢ k − yom ⎢ k 2 ⎢ ⎢ x̂m − xom v̂m k y,k ⎢ ⎢ −x̂m − x ŷm − y v̂m ⎢ om om x,k k k ⎣ 2 m 2 3/2 m x̂k − xom + ŷk − yom (62) ] ^m T m m · x k − x̂m x̂m + Jhm k = fm (x̂k , 0 0 ) + J^ k xk m Dr + em · k Dum ∂fm ^m | m ∂x k x̂k , 0 0 T = I IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967 doi: 10.1049/iet-spr.2013.0393 ⎤ (65) T Since fm x̂m , 0 0 = x̂m k k , we have x̂m x k + Jhm k = The Jacobian matrix is computed by x k − yom ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ m ⎥ −v̂y,k ⎥ ⎥ ⎥ ⎥ m ⎥ ⎥ x̂k − xom ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ m v̂x,k ⎦ ŷm k In this way, the local estimate x̂m k can be represented by ^m J ^m = − (64) In general, the sensor biases are assumed to be small quantities. By making the ﬁrst-order ^m Taylor series m expansion about the function fm x k , h at the point T (x̂m k , 0 0 ), we have the following approximation ^m where I is the identity matrix, and Drm + em k Dum (66) (63) Thus, Theorem 1 is proven. □ 967 & The Institution of Engineering and Technology 2014

1/--страниц