A Practical Bias Estimation Algorithm for Multisensor Multitarget Tracking

Transcription

1 A Practical Bia Etimation Algorithm for Multienor Multitarget Tracking arxiv: v1 [tat.me 9 Mar 016 Ehan Taghavi, R. Tharmaraa and T. Kirubarajan McMater Univerity, Hamilton, Ontario, Canada {taghave, tharman, kiruba}@mcmater.ca Yaakov Bar-Shalom Univerity of Connecticut, Storr, Connecticut, USA yb@ee.uconn.edu Mike McDonald Defence Reearch and Development Canada Ottawa, Ontario, Canada mike.mcdonald@drdc-rddc.gc.ca Abtract Bia etimation or enor regitration i an eential tep in enuring the accuracy of global track in multienor-multitarget tracking. Mot previouly propoed algorithm for bia etimation rely on local meaurement in centralized ytem or track in ditributed ytem, along with additional information like covariance, filter gain or target of opportunity. In addition, it i generally aumed that uch data are made available to the fuion center at every ampling time. In practical ditributed multienor tracking ytem, where each platform end local track to the fuion center, only tate etimate and, perhap, their covariance are ent to the fuion center at non-conecutive ampling intant or can. That i, not all the information required for exact bia etimation at the fuion center i available in practical ditributed tracking ytem. In thi paper, a new algorithm that i capable of accurately etimating the biae even in the abence of filter gain information from local platform i propoed for ditributed tracking ytem with intermittent track tranmiion. Through the calculation of the Poterior Cramér Rao lower bound and variou imulation reult, it i hown that the performance of the new algorithm, which ue the tracklet idea and doe not require track tranmiion at every ampling time or exchange of filter gain, can approach the performance of the exact bia etimation algorithm that require local filter gain. Index Term Multitarget multienor tracking, regitration, bia etimation, tracklet, ditributed fuion. I. INTRODUCTION Bia etimation and compenation are eential tep in ditributed tracking ytem. The objective of enor regitration i to etimate the biae in enor meaurement, uch a caling and offet biae in range and azimuth meaurement of a radar, clock bia and/or uncertaintie in enor poition [1. In a ditributed multienor tracking cenario, each local tracker provide it own etimate of target tate for fuion. Local filter can be, e.g., Kalman filter or Interacting Multiple Model (IMM) etimator with different motion model. Thee local track, i.e., tate etimate vector and aociated covariance matrice, are ent to the fuion center for further proceing. Next, the fuion center carrie out track to track fuion. The fuion i done equentially ubequent to the etimation of biae baed on common target that are tracked by variou enor in different location. Uually, bia etimation i conidered a a two enor problem [, [3 where a tacked vector i aumed with all unknown biae and tate target. A drawback of thi approach i the computational burden due to increaed dimenion of the tacked vector. In addition, mot of the algorithm propoed for etimating the biae operate on the meaurement directly [4. That i, uch method perform filtering on the meaurement received from enor, which alo include the biae. In many practical tracking ytem, acce to meaurement before tracking at the enor level i not alway feaible. That i, enor may provide only proceed track to the uer for further proceing [1. Thu, method that can imultaneouly handle track to track fuion and bia etimation are needed. Although there are many different method in the literature for bia etimation and compenation, there i till a need for a method that require only the local track etimate and aociated covariance matrice for bia etimation. In [5 and [6 a joint track-to-track bia etimation and fuion algorithm baed on equivalent meaurement of the local track wa propoed. In [7, another approach baed on peudo-meaurement along with the Expectation-Maximization (EM) algorithm to perform joint fuion and regitration wa propoed. A different method that ue a multitart local earch to handle the joint track-to-track aociation and bia etimation problem wa introduced in [8. The concept of peudo-meaurement wa ued in [9 for exact Yaakov Bar Shalom wa upported by ARO Grant W991NF

2 bia etimation with further extenion in [10 and [11. In order to achieve exact bia etimation, the algorithm in [9 [11 require the Kalman gain from local tracker, which are not normally ent to the fuion center in practical ytem [1. Moreover, the previouly mentioned algorithm aumed that the fuion center receive the local track from all enor at every time tep, which i not realitic in ytem with bandwidth limitation [13. In addition, thee method require perfect knowledge about each local filter and it dynamic model. Alo, a the number of enor increae, the bia etimation problem uffer from the cure of dimenionality becaue of the commonly ued tacked bia vector implementation [. Finally, a the number of enor change over time, the algorithm in [9 [11 require appropriate peudo meaurement to be defined for the pecific number of enor. In thi paper, thee iue are addreed and a practical olution, which i mathematically ound and computationally feaible, i preented. The new approach i baed on recontructing the Kalman gain of the local tracker at the fuion center. In thi approach, the tracklet method [1, [14, [15 along with equential update a a fuion method i ued to provide a low computational cot algorithm for bia etimation. Alo, ome of the contraint that were dicued above are relaxed in the propoed algorithm. The main contribution of the new algorithm are: a) recontruction of Kalman gain at the fuion center, b) relaxing the contraint on receiving local track at every time tep, c) correcting local track at the fuion center and d) providing a fued track with low computational cot. The paper i tructured a follow: The bia model and the aumption for bia etimation are dicued in Section II. In Section III, a review of the exact bia etimation method [9 [11 i given. The new approach and it mathematical development are given in Section IV. Section V preent the calculation of the Cramér Rao Lower Bound (CRLB) for propoed algorithm. Section VI demontrate the performance of the new algorithm for ynchronou enor and compare it with that of the method in [9 and how comparion with the CRLB. Concluion are dicued in Section VII. II. PROBLEM FORMULATION Aume that there are M enor reporting range and azimuth meaurement 1 in polar coordinate of N target in the common urveillance region. Note that N i not exactly known to the algorithm and that it could be time varying. That i, bia etimation i carried out baed on time varying and poibly erroneou number of track reported by the local tracker. The model for the meaurement originating from a target with biae at time k in polar coordinate (denoted by upercript p) for enor i [9 [11 z p (k) = [ r p (k) θ p (k) = [ [1 + ɛ r (k) r (k) + b r (k) + w r (k) [ 1 + ɛ θ (k) θ (k) + b θ (k) + w θ (k) = 1,..., M (1) where r (k) and θ (k) are the true range and azimuth, repectively, b r (k) and b θ (k) are the offet biae in the range and azimuth, repectively, ɛ r (k) and ɛ θ (k) are the cale biae in the range and azimuth, repectively. The meaurement noie w(k) r and w(k) θ in range and bearing are zero-mean with correponding variance σr and σθ, repectively, and are aumed mutually independent. The bia vector β (k) = [ b r (k) b θ (k) ɛ r (k) ɛ θ (k) T can be modeled a an unknown contant over a certain window of can (non random variable). Conequently, the maximum likelihood (ML) etimator [16 or the weighted leat quare (LS) etimator [17 can be ued for bia etimation. On the other hand, a Gau-Markov random model [18 can alo be ued, in which cae a Kalman filter can be adopted for bia etimation. We model the meaurement a [ [ z p r (k) w r (k) = + C θ (k) (k)β (k) + (k) w(k) θ () where C (k) = [ 1 0 r (k) θ (k) Here, the meaured azimuth θ m (k) and range r m (k) can be utilized in (3) without any ignificant lo of performance [9 [11. Etimating the bia vector β (k) for all the enor i the main objective of thi paper. After bia etimation, all the biae can be compenated for in the tate etimate at the fuion center. Since target motion i better modeled and mot tracker operate in Carteian coordinate, the polar meaurement are converted into Carteian coordinate. It i aumed that thi doe not introduce biae [19; thi i verified in the imulation. Then, enor ha the meaurement equation (with the ame H (k) = H(k) for all ) z (k) = H(k)x(k) + B (k)c (k)β (k) + w (k) (4) where the tate vector x(k) = [ x(k) ẋ(k) y(k) ẏ(k) T and H(k) i the meaurement matrix given by [ H(k) = = H (5) While thi aume D radar, the extenion to 3 D radar i traightforward. (3)

3 Since ditributed tracking ytem may cover a large geographical area, the earth can no longer be aumed to be flat and coordinate tranformation need to include an earth curvature model like WGS-84 [0, [1. The matrix B (k) i a nonlinear function with the true range and azimuth a it argument. A contant B (k)c (k) alo reult in incomplete obervability a dicued in [11. Uing the meaured azimuth θ m (k) and range r m (k) from enor, B (k) can be written a [19 [ co θ m B (k) = (k) r m (k) in θ m (k) in θ m (k) r m (k) co θ m (6) (k) Finally, the new covariance matrix of the meaurement in Carteian coordinate (omitting index k in the meaurement for clarity) i given by ( ( ) ) r R (k) = σθ in θ + σr co θ σ (σ ) r rσ θ in θ co θ r rσ θ in θ co θ rσ θ co θ + σr in (7) θ where one can ue the oberved range and azimuth a well. III. REVIEW OF SYNCHRONOUS SENSOR REGISTRATION In thi ection, the bia etimation method introduced in [9 [11 for ynchronou enor with known enor location i reviewed. Further, the method in our previou work [ are examined in more detail and are extended in thi paper with variou imulation and the calculation of the lower bound for bia etimation in multienor multitarget cenario. Conider a multienor tracking ytem with the decentralized architecture [1. In thi cae, each local tracker run it own filtering algorithm and obtain a local tate etimate uing only it own meaurement. Then, all local tracker end their etimate to the fuion center where bia etimation i addreed. Only after bia etimation can the fuion center fue local etimate correctly to obtain accurate global etimate. The dynamic equation for the target tate i x(k + 1) = F (k)x(k) + v(k) (8) where F (k) i the tranition matrix, and v(k) i a zero-mean additive white Gauian noie with covariance Q(k). Becaue the local tracker are not able to etimate the biae on their own, they yield inaccurate etimate of track by auming no bia in their meaurement. Hence, the tate pace model conidered by local tracker for a pecific target t and enor i x t (k + 1) = F (k)x t (k) + v(k) (9) z t (k) = H(k)x t (k) + w (k) (10) The difference between (1) and (10) i that the latter ha no bia term and, a a reult, the local track are bia-ignorant [9 [11. Note that thi mimatch hould be compenated for. A. The peudo-meaurement of the bia vector In thi ubection, a brief dicuion on how to find an informative peudo-meaurement by uing the local track for the cae M = ynchronized enor i preented, baed on the method given in [9 [11. A in thee previou work, it i aumed that the local platform run a Kalman filter-baed tracker, although thi aumption may not alway be valid. However, a hown in the equel, multiple-model baed tracker can be handled within the propoed framework with ome extenion. In [9 [11 it wa aumed that one ha acce to the filter gain W 1 (k + 1) and the reidual ν 1 (k + 1) from the Kalman filter of local tracker 1 [3. Then, one can write ˆx 1 (k + 1 k + 1) = F (k)ˆx 1 (k k) + W 1 (k + 1)ν 1 (k + 1) = F (k)ˆx 1 (k k) + W 1 (k + 1) [ z 1 (k + 1) ẑ 0 1(k + 1 k) = [I W 1 (k + 1)H(k + 1) F (k)ˆx 1 (k k) + W 1 (k + 1) [H(k + 1) F (k)x 1 (k) + H(k + 1)v(k) + B 1 (k + 1)C 1 (k + 1)β 1 (k + 1) +w 1 (k + 1) (11) Note that the predicted meaurement ẑ 0 1(k + 1 k) i baed on the meaurement in which no bia i aumed by local tracker 1, i.e., tracker 1 ued a bia-ignorant meaurement model. Therefore, there i no term related to biae in the predicted meaurement.

4 Hence, if the local tate etimate i moved to the left hand ide of (11), and left-multiplied by the left peudo-invere [4 of the gain, one ha z 1 b (k + 1) W 1 (k + 1) [ˆx 1(k + 1 k + 1) (I W 1 (k + 1)H(k + 1))F (k)ˆx 1 (k k) = H(k + 1)F (k)x(k) + H(k + 1)v(k) + B 1 (k + 1)C 1 (k + 1)β 1 (k + 1) +w 1 (k + 1) (1) where the peudo-invere of the gain i W ( W T W ) 1 W T (13) Similarly, one can define z b (k + 1) W (k + 1) [ˆx (k + 1 k + 1) (I W (k + 1)H(k + 1))F (k)ˆx (k k) = H(k + 1)F (k)x(k) + H(k + 1)v(k) + B (k + 1)C (k + 1)β (k + 1) +w (k + 1) (14) It i worth mentioning that x(k) and v(k) in (1) and (14) are the ame. Thu, a peudo-meaurement of the bia vector, a in [9 [11, can be defined a follow: for the cae of uing imilar enor. Then, z b (k + 1) z 1 b (k + 1) z b (k + 1) (15) z b (k + 1) = B 1 (k + 1)C 1 (k + 1)β 1 (k + 1) B (k + 1)C (k + 1)β (k + 1) That i, one ha the peudo-meaurement of the bia vector +w 1 (k + 1) w (k + 1) (16) z b (k + 1) = H(k + 1)b(k + 1) + w(k + 1) (17) where the peudo-meaurement matrix H, the bia parameter vector b and the peudo-meaurement noie w(k + 1) are defined a and H(k + 1) [B 1 (k + 1)C 1 (k + 1), B (k + 1)C (k + 1) (18) b(k + 1) [ β1 (k + 1) β (k + 1) (19) w(k + 1) w 1 (k + 1) w (k + 1) (0) The bia peudo-meaurement noie w are additive white Gauian with zero mean, and their covariance i R(k + 1) = R 1 (k + 1) + R (k + 1) (1) The main property of (0) i it whitene, which reult in a bia etimate that i exact [9 [11. In thi approach, there i no approximation in deriving (17) (1) unlike the method previouly propoed in [5 [7. Thi wa one of the main contribution of [9. When the meaurement matrice H (k) are the ame for different local tracker, but only the econd enor ha a bia, the following implification reult: z b (k + 1) = z 1 b (k + 1) z b (k + 1) () b(k + 1) = β (k + 1) (3) H(k + 1) = B (k + 1)C (k + 1) (4) w(k + 1) = w 1 (k + 1) w (k + 1) (5) R(k + 1) = R 1 (k + 1) + R (k + 1). (6)

5 Input: ˆb 0 (k), Σ 0 (k), z 1 b,t (k), z b,t (k) Output: ˆb 0 (k + 1), Σ 0 (k + 1) 1: At time k : for t = 1,..., N do 3: Get the new peudo-meaurement uing 4: Compute the bia update gain and the reidual z b,t (k) z 1 b,t(k) H(k)H (k)z b,t(k) G t (k) = Σ t 1 (k)h t (k) T [ H t (k)σ t 1 (k)h t (k) T R t (k) 1 r t (k) = z b,t (k) H t (k)ˆb t 1 (k + 1) 5: Update the bia etimate and covariance 6: end for 7: return ˆb t (k) = ˆb t 1 (k + 1) + G t (k)r t (k) Σ t (k) = Σ t 1 (k) Σ t 1 (k)h t (k) T ) T [H t (k) Σ t 1 (k)h t (k) T + R t (k) 1 Ht (k)σ t 1 (k) ˆb 0 (k + 1) ˆb N (k) Σ 0 (k + 1) Σ N (k) Fig. 1: The Recurive Leat Square Bia Etimation (RLSBE) algorithm [9. B. The recurive leat quare bia etimator If the biae are contant over a certain window of can, one can contruct a Recurive Leat Square (RLS) etimator by uing the peudo-meaurement equation (17) [9. The recurion of the RLS etimator ha two tage: the firt i to update the bia etimate recurively for different target and the econd i to update it through different time can. Aume that at time k, one ha acce to the etimate of the bia vector and it aociated covariance matrix up to time k a ˆb t 1 (k) and Σ t 1 (k), form on the firt t 1 target and all previouly updated etimate. Now, the RLS method can be carried out a in Figure 1 to update the bia etimation at time k for all target [9 [11. Note that the covariance update equation in line 5 of Figure 1 may caue Σ t (k) to loe poitive definitene due to numerical error. To avoid thi problem, the Joeph form of the covariance update i ued [19 a Σ t (k) = [I G t (k)h t (k) Σ t 1 (k) [I G t (k)h t (k) T + G t (k)g t (k) T (7) C. Time-varying bia etimation: The optimal MMSE etimator In the cae of time-varying biae with the tandard linear white Gauian aumption one can implement the optimal MMSE etimator baed on the peudo-meaurement equation (17) and the dynamic model of the bia [9. f For the tacked bia vector, the dynamic model can be define a b(k + 1) = F b (k)b(k) + v b (k) (8) in which F b (k) i the tranition matrix of the tacked bia vector b, and v b (k) i the tacked proce noie of the bia vector, zero-mean white with covariance Q b (k). Aume that at time k one ha acce to the etimate of the bia vector and it aociated covariance matrix up to time k a ˆb t 1 (k k) and Σ t 1 (k k), repectively. Now, a Kalman filter can be ued a in Figure to update the bia etimate at time k for all target [9 [11. IV. THE NEW BIAS ESTIMATION ALGORITHM The algorithm given in Section III, i.e., Recurive Leat Square Bia Etimation (RLSBE, Figure 1) and Optimal MMSE Bia Etimation (OMBE, Figure ), are dependent on the Kalman gain provided by the local tracker, in addition to the tate etimate and the aociated covariance matrice at every time tep. Moreover, a the number of the enor increae, the above algorithm face an increae in computational requirement cubic in M. Thi i becaue a tacked vector of bia parameter i ued. In addition, for M >, it i challenging to extend (15) and (17). The extenion of (15) and (17) can be

6 Input: ˆb 0 (k k), Σ 0 (k k), z 1 b,t (k), z b,t (k) Output: ˆb 0 (k + 1 k + 1), Σ 0 (k + 1 k + 1) 1: At time k : for t = 1,..., N do 3: Get the new peudo-meaurement uing 4: Compute the bia update gain and the reidual z b,t (k) z 1 b,t(k) H(k)H (k)z b,t(k) G t (k) = Σ t 1 (k k)h t (k) T [H t (k)σ t 1 (k k) H t (k) T + R t (k) 1 r t (k) = z b,t (k) H t (k)ˆb t 1 (k k) 5: Update the bia etimate and covariance ˆb t (k k) = ˆb t 1 (k k) + G t (k)r t (k) Σ t (k k) = Σ t 1 (k k) Σ t 1 (k k)h t (k) T [ Ht (k)σ t 1 (k k)h t (k) T +R t (k) 1 H t (k)σ t 1 (k k) 6: end for 7: Update the bia etimate according to the model ˆb(k + 1 k) F b (k)ˆb N (k k) Σ(k + 1 k) F b (k)σ N (k k)f b (k) T + Q b (k) 8: return ˆb 0 (k + 1 k + 1) = ˆb(k + 1 k) Σ 0 (k + 1 k + 1) = Σ(k + 1 k) Fig. : The optimal MMSE bia etimation algorithm [9 (OMBE). done a in [8 by taking M 1 difference. Moreover, thee approache do not addre the joint fuion problem a well. With thi motivation, in thi ection, a new approach to relax the requirement of the Kalman gain matrice availability from the local tracker i given. In addition, the new algorithm alleviate the problem of the dimenionality by taking advantage of () (6) for a multienor multitarget cenario, and by olving the fuion problem a well. Finally, the algorithm i able to function properly with aynchronou local track update. In order to obtain the new algorithm, firt, a imple approach to calculate tracklet baed on [1 i dicued. Thi approach make it poible to obtain approximate equivalent meaurement of the local track directly and efficiently without any further proceing and it upport updating the bia etimate whenever a new local track i available at the fuion center. In addition, it can handle aynchronou update from different local tracker and map them to a common time [1, [15. Then, a equential update algorithm i propoed for the fuion tep. Although it i not an optimal approach for fuing the local track, it i computationally cheaper than parallel update [1. Finally, the complete algorithm baed on thee two approache with additional tep i preented. A. Equivalent meaurement computation uing the invere Kalman filter method baed tracklet The main goal in thi ubection i to contruct a et of approximately uncorrelated equivalent meaurement ( tracklet ) from the local track and the aociated covariance matrice for equential update in the fuion tep and alo to recontruct the local Kalman gain at the fuion center. It alo relaxe the requirement of receiving the local track at every time tep. To do o, the invere Kalman filter baed tracklet method from [1 i ued (for a clear derivation and the reaon for it uboptimality, ee [1, p. 577). Baed on thi method, the equation relating to the equivalent meaurement vector, u (k, k ), for a local track from platform at time frame k, given that the track data wa previouly ent to the global tracker for time frame k < k, are a follow: u (k, k ) = ˆx (k k ) + A (k k ) [ˆx (k k) ˆx (k k ) (9)

7 where { where Z k = z 1,..., z k }, and u (k, k ) = x(k) + ũ (k k) (30) [ E ũ (k, k ) Z k = 0 (31) A (k, k ) = P (k k ) [D (k, k ) 1 (3) D (k, k ) = P (k k ) P (k k) (33) U (k, k ) = E [ũ (k, k ) (ũ (k, k) ) T Z k = A (k, k )P (k k) = [A (k, k ) I P (k k ) (34) The information that the global tracker or the fuion center ue conit of the calculated equivalent meaurement vector u (k, k ) and it error covariance matrix U (k, k ). Note that in order to calculate ˆx (k k ) and P (k k ) one need the etimated target tate ˆx (k k ) and it covariance matrix P (k k ), in addition to the dynamic model the local tracker ued for filtering. Then one need to compute L = k k prediction tep without any new meaurement data to find ˆx (k k ) and P (k k ). Here, it i neceary to conider the L-tep prediction of tranition and proce noie covariance matrice a F (k, k ) and Q(k, k ), repectively. One can ue the concept of miing obervation in Kalman filter to find F (k, k ) and Q(k, k ) a in [9, pp It hould be mentioned that all thee computation require that P (k k ), P (k k ) and [ P (k k) 1 P (k k ) 1 be non-ingular. Thi method wa previouly ued in [6 for k = k 1 (for which the non-ingularity requirement doe not hold in general) and with a different approach for enor regitration. For the proof of (34) ee Appendix A. B. Sequential update a fuion method After calculating the equivalent meaurement of the tate for each local track at the fuion center, they can be ued a new meaurement for the etimation of fued tate and it covariance matrix. To do thi recurively, it hould be aumed that the fued tate etimate and it covariance matrix at time k a x f (k k ) and P f (k k ), repectively, are already computed. For k = 1, the parameter x f (k k ) and P f (k k ) are initialized with x(k k ) and P(k k ), repectively. Then thee two can be updated by following the tep in Figure 3. Although the equential update i ub-optimal [1, it ha the advantage of being computationally efficient to implement and, in addition, it i not dependent on the previou equivalent meaurement at time k. C. Multienor fuion and track-to-track bia etimation The firt tep for implementing a general bia etimation algorithm for radar ytem i to find the Kalman gain of each local track at the fuion center, by only uing the tate etimate and the aociated covariance matrice. To do o, firt, one mut calculate the equivalent meaurement and it covariance matrix a in (9) and (34) for enor, and at time frame k (the target index i omitted for implicity). Since the meaurement model here i linear, one ha R,k = H(k)U (k, k )H(k) T (35) W,k = P (k k )H(k) T [ H(k)P (k k )H(k) T +R,k 1 (36) y,k = H(k)u (k, k ) (37) Note that the reaon to keep the poition information only i that further in the new bia etimation algorithm we ue the corrected poition a new meaurement for equential fuion. Becaue the equivalent meaurement are ued for the fued track, the Kalman gain for it (denoted by ubcript f) at time frame k can be recovered a [ M 1 R f,k = H(k) (U i (k, k )) 1 H(k) T (38) i=1 W f,k = P f (k k )H(k) T [ H(k)P f (k k )H(k) T +R f,k 1 (39) Note that the noie/error in the equivalent meaurement are not white o uing a Kalman filter i not optimal. Thi amount to the ame approximation a in [1, p Alo in (39), a common coordinate ytem i ued for equivalent meaurement. A a reult, the ue of a common meaurement matrix H(k) for all the equivalent meaurement i feaible.

8 Input: x f (k k ) and P f (k k ), y,k and R,k for = 1,..., M Output: x f (k k) and P f (k k) 1: Compute x f (k k ) and P f (k k ) according to their dynamic model (prediction tep) : Aign: x temp = x f (k k ) P temp = P f (k k ). 3: for = 1,..., M do 4: Update x temp and P temp with new meaurement and it covariance matrix, i.e., y,k and R,k according to x temp = x temp + W temp ỹ W temp = P temp H T ( HP temp H T + R,k ) 1 ỹ = y,k Hx temp 5: end for 6: return P temp = (I W temp H) P [ temp H = x f (k k) = x temp P f (k k) = P temp Fig. 3: The Sequential Fuion Algorithm (SFA) with equivalent meaurement The enor regitration method propoed here ue the implified formula, i.e., () (6). To ue thee formula, an approximately bia-compenated fued track need to be found at the fuion center for the et {S} \, where S = 1,,..., M. The notation {S} \ tand for the et that contain all thoe element of S excluding element. Then, the bia etimation problem can be reduced to the cae of two enor. The firt one i an equivalent enor with fuion of bia-corrected meaurement of the et {S} \, and the econd i enor which ha bia. Then the bia in enor can be found with either the RLSBE (ee Figure 1) or OMBE (ee Figure ) algorithm. The next tep in thi approach i to correct the biae in the meaurement domain of the enor (except for enor ) and then fue them together. Thi can be done by going from the Carteian coordinate of equivalent meaurement to the polar coordinate of the radar and correct with the previouly etimated biae. Then by correcting the covariance matrix of the new bia compenated meaurement, they can be fued by equentially updating the fued track (excluding the track from enor ) by uing the Sequential Fuion Algorithm (SFA). Then, the Kalman gain for the fued and the now-corrected track can be calculated uing (39). For the equivalent meaurement, define u [ u x u x u y u y T (40) where time and target indexe are omitted for implicity. Then, to correct the biae in the meaurement domain, auming that the latet etimated biae are ˆɛ θ, ˆɛ r, ˆb θ and ˆb r, one ha 3 ( ) uy arctan θ b-c u x ˆb θ = (41) (1 + ˆɛ θ ) r b-c = (u x ) + (u y ) ˆb r (1 + ˆɛ r ) Now that the cale and offet biae are compenated for, one can go back to Carteian coordinate a follow: u b-c x = λ θ r b-c co ( θ b-c ) (43) u b-c y = λ θ r b-c in ( θ b-c ) (44) Y b-c = [ u b-c x u b-c T y (45) Here, bia-compenated mean a fued track, in which the bia-corrected equivalent meaurement by uing the latet etimated biae are ued for fuion. 3 Here, the upercript b-c i ued to denote the bia-corrected bearing and range. (4)

9 where ( ) λ θ = exp σ θ i the compenation factor for the bia in coordinate converion from polar to Carteian [30. The method from [30 i ued here becaue of the fact that bia correction and compenation along with the change in the covariance matrice and uncertaintie may violate the aumption made for the debaied converion in Section II. The next tep i to update the covariance matrix of the corrected equivalent meaurement. In addition to the term (35), the additional uncertainty in the bia etimate, i.e., their aociated covariance matrix and the uncertainty in the model of the radar, both in Carteian coordinate, mut be accounted for. The final formula for the covariance matrix with proper converion from polar to Carteian coordinate i [ R b-c = H(k)U (k k)h(k) T + B b-c σ (k) r 0 0 σθ B b-c (k) T +K b-c (k)p b, (k k)k b-c (k) T (47) where K b-c (46) (k) = B b-c (k)c b-c (k) (48) and P b, (k k) i the latet-updated bia covariance matrix at time k and for enor. Now that all the required formulation and variable are available, the new algorithm to find the bia etimate for all the enor i given in Figure 4. Input: input of SFA and RLSBE (defined within the correponding algorithm). Output: b (k) and Σ (k) for = 1,..., M. 1: At time k : for = 1,..., M do 3: Compute W,k a in (36) and 4: z b,t(k) W,k [ˆx(k k) (I W,kH (k))f (k, k L)ˆx(k L k L) 5: {1,..., M}\ 6: Call SFA with input x f (k k ) and P f (k k ), Y b-c (k k) and R b-c (k k). 7: Compute Wf,k a in (39) and zb,f (k) ( Wf,k ) [ˆx f (k k) (I Wf,kH f (k))f (k, k L)ˆx f (k L k L) 8: Call RLSBE with input zb,t (k), z b,f (k) and the lat update of the etimated biae and their aociated covariance matrix. 9: return b (k) and Σ (k) 10: end for Fig. 4: The Fued Bia Etimation algorithm (FBEA) A hown in Figure 4, one only need to call SFA and RLSBE with new input parameter. One of the advantage of thi approach i that in each for loop only a low dimenional Kalman filter that i independent of the ize of the tacked bia vector and number of the enor i needed. In addition, the fuion of local track can be done by only adding one equential update for the latet corrected meaurement of enor to the previouly fued track. It i alo important to note that the there i no contraint on the rate of receiving local track from the individual enor. To how how well thi new algorithm perform, in the next ection, the imulation reult on two different cenario are ued to compare it performance with thoe of the previouly propoed algorithm in [9 [11 for ynchronou enor. To better illutrate how the new bia etimation algorithm (FBEA) work, a block diagram repreentation of the method i hown in Figure 5 for a ingle time tep etimation of the biae for the firt enor. In Figure 5, by receiving the local track etimate from all available enor at time tep k, the firt tep i to calculate the tracklet for all of them uing (9) (34). Then, the equivalent meaurement of the firt enor i ent for Kalman gain recovery uing (35) and (36). At the ame time, the equivalent meaurement of all other enor are ent to bia correction to firt remove the bia from the equivalent meaurement by uing the previouly etimated biae at time tep k 1 uing (41) (45) and the Kalman gain recovery in (38) and (39). The next tep i to fue the track by uing SFA algorithm. Then, the fued and corrected etimate i ent to the peudo meaurement calculation block for each individual enor. At thi point one ha a two enor problem with only one enor having biae in the meaurement. The output i now ent to the RLSBE algorithm along with the previouly etimated biae for the firt enor o that the bia etimate can be updated at time tep k before proceeding to the next time tep.

10 Senor 1 Tracklet Calculation Kalman Gain Recovery Peudomeaurement Senor Tracklet Calculation Bia Correction Fuion Kalman Gain Recovery Senor M Tracklet Calculation Previouly Etimated Biae Fuion Center Bia Etimation Algorithm (Recurive Leat Square etimation) Peudomeaurement Fig. 5: Block diagram of the new offet and caling bia etimation algorithm. V. CRAMÉR-RAO LOWER BOUND FOR SENSOR REGISTRATION In thi ection, a tep-by-tep procedure i given for calculating the CRLB for enor regitration algorithm a the benchmark. Rewriting (4) for the cae of two imilar enor, one ha and z 1 (k) H(k)x(k) = B 1 (k)β 1 (k) + w 1 (k) (49) z (k) H(k)x(k) = B (k)β (k) + w (k) (50) If no biae exit, the enor mut point to the ame poition of the oberved target. Conequently, one ha z 1 (k) B 1 (k)β 1 (k) w 1 (k) = z (k) B (k)β (k) w (k) (51) and by reordering the meaurement term and uing the matrix form, one can rewrite it a [z 1 (k) z (k) = [ B 1 (k) B (k) [ β 1 (k) β (k) Further, for future ue, one can denote the term in (5) a where + w 1, (k) (5) Y (k) = B(k)b(k) + w 1, (k) (53) Y (k) = [z 1 (k) z (k) (54) B(k) = [ B 1 (k) B (k) (55) [ β1 (k) b(k) = (56) β (k) and w 1, (k) i additive white Gauian noie with covariance matrix equal to R 1 (k) + R (k) A. Calculation of the CRLB In the cae of having two enor and multiple target, the CRLB can be calculated a a batch proce. Taking all the (linearly independent) K pair of meaurement for N target in the urveillance region, one can write the meaurement equation a Y = gb + u (57) where Y, g and u are tacked vector given by [ (Y Y = (1) ) T ( Y N (1) ) T ( Y 1 (K) ) T ( Y N (K) ) T T (58) [ (B g = 1 (1) ) T ( B N (1) ) T ( B 1 (K) ) T ( B N (K) ) T T (59)

11 and u = [ (w 1 1,(1) ) T ( w N 1, (1) ) T ( w 1 1, (K) ) T ( w N 1, (K) ) T T Further, the covariance matrix of the noie vector u i R = diag ([ R 1 (1) R N (1) R 1 (K) R N (K) ) (61) where R i (k) = R1(k) i + R(k) i and the lower index indicate a pecific enor. A tated in [19, the covariance matrix of an unbiaed etimator ˆb i bounded from below a { ) ) } T E (ˆb b (ˆb b J 1 (6) In the above, J i the Fiher Information Matrix (FIM) given by { J = E [ b ln p(y b) [ b ln p(y b) T} b=btrue { = E [ b λ [ b λ T} b=btrue (63) where b true i the true value of the bia vector b, p(y b) i the likelihood function of b, λ = ln p(y b) and i gradient operator. From (57), one ha { 1 p(y b) = (π) K R exp 1 } [Y gbt R 1 [Y gb (64) and therefore λ = Cont. + 1 [Y gbt R 1 [Y gb (65) Uing the reult from [31 to implify the differentiation, b λ can be written a which yield (60) b λ = g T R 1 (Y gb) (66) Finally, when calculating the FIM, CRLB of deired element (here the diagonal) will be for i = 1,, 3, 4. B. Multitarget multienor CRLB J = g T R 1 g (67) CRLB {[b i } = [ J 1 ii (68) When number of the target i greater than two, the ame procedure a the propoed bia etimation algorithm to calculate the CRLB can be ued. In thi cae, one hould fue all the meaurement, except for enor i to create a ingle peudomeaurement for thi combined enor and then treat the problem a a two enor problem. Starting with the calculation of the combined meaurement and covariance matrix of the et Sı a in [1 yield Z comb = R comb (R j ) 1 z j (69) and R comb = j {S}\i R 1 j j {S}\i A in the bia etimation algorithm, it i aumed that only enor i ha bia. Thi mean that in the calculation of CRLB one hould have acce to the meaurement both with and without bia. The next i to calculate B comb (k). Similar to the bia etimation algorithm, one hould define H(k) = B comb (k) and ue the combined covariance and meaurement matrice a a bia free meaurement to find the CRLB of the bia etimation for enor i. Finally, R(k) can be calculated a 1 (70) R(k) = R comb (k) + R i (k) (71) By uing (69), (70) and (71), the formula for two enor and multiple target can be modified to calculate the CRLB for the cae of multienor multitarget cenario.

12 1.5 x 104 Target initial location Senor location 1 y (m) x (m) x 10 4 Fig. 6: Initial location of the target and enor A. Motion dynamic and meaurement parameter VI. SIMULATION RESULTS Here, a ditributed tracking cenario with five enor and ixteen target i conidered a hown in Figure 6. It i aumed that all enor are ynchronized. Without lo of generality and to eaily compare the reult of bia etimation for different enor, all the biae are aumed a β = [ 0m 1mrad T (7) The tandard deviation of meaurement noie are σ r = 10 m and σ θ = 1 mrad for target range and azimuth meaurement, repectively. In thi problem r According to [19, rσ θ σ r = = 0.4 which i the threhold for polar to Carteian converion to be unbiaed. Note that thi condition hold for r 00km a well. Scaling biae can be handled by the propoed method a well but ignored for implicity. The true dynamic of the target are modeled uing the Dicretized Continuou White Noie Acceleration (DCWNA) or nearly contant velocity (NCV) model with q x = q y = 0.1 m and contant turn rate model with rate ω = 0.1 deg 3. A for the filtering in local tracker, DCWNA and Continuou Wiener Proce Acceleration (CWPA) are ued with variou etting to be able to create different cenario for the imulation (for detail ee [19, p. 68 and p. 467). B. A two-enor problem To compare the reult of the new approach with thoe of the previouly propoed algorithm [9 [11, the cae of two enor (placed at (0m, 0m) and (5000m, 0m) in Carteian coordinate) i conidered with the ame kinematic and filter etting a in [9. The new algorithm that ue the recontructed Kalman gain detailed in (35) and (36) i denoted a EXL while the previou algorithm in [9 [11 i denoted a EX. Note that there i no fuion tep in thi cae and the two method only differ in the Kalman gain calculation, which in the EXL 4 cae i recontructed approximately. The etting of the variable are a follow. To handle the L = 1 and non ingularity problem, the tracklet with decorrelated tate etimate 5 that can be ued with only one meaurement in the tracklet interval [1 mut be elected intead of tracklet computed uing invere Kalman filter. The ampling interval are T = 1. The lag between each update at the fuion center i L = k k = 1. The initial tate etimate are the converted meaurement from polar coordinate to Carteian coordinate with zero velocity and covariance matrix [9 [11 [ P (0 0) = diag (00m) ( ) 0m/ (00m) ( ) 0m/ (73) Finally, the initial bia parameter etimate of all the enor are zero with initial bia covariance Σ (0 0) = diag [ (0m) (1mrad) = 1, (74) In the imulation, 100 Monte Carlo run are ued over 0 frame. The reult of the Root Mean Squared Error (RMSE) in logarithmic cale for offet bia etimate are hown in Figure 7 and 8. From Figure 7 and 8, it can be een that the performance of the EXL method, which recover the Kalman gain at the fuion center by taking advantage of tracklet 4 The EXL algorithm ha the luxury of operating without the local Kalman gain. 5 Thi i equivalent to the information matrix fuion method, which, for L = 1, i algebraically equivalent to the Kalman filter (ee [1, eq. ( )).

13 calculation, i very cloe to the accuracy of the EX method. The mall variation in the reult are due to the fact that the logarithmic cale i ued to how the convergence rate clearly. The CPU time for one iteration of bia etimation for a ingle target i 4.84 for the EX method and 6.0 for the EXL method, which repreent a 0% increae in CPU time for the Kalman gain recontruction. The CPU time over all iteration and target for bia etimation with the above method are and , repectively. All imulation are done on a computer with Intel R Core TM i7-370qm.60ghz proceor and 8GB RAM. Senor #1 Senor # RMSE (m) Senor # RMSE (m) EXL EX Senor # Fig. 7: RMSE of the bia parameter 10 b 10 r for enor 1 and in logarithmic cale (comparion between the previou (EX) and the propoed (EXL) algorithm) RMSE (m) Senor # Senor #1 RMSE (m) 10 Senor # EXL EX EXL EX Senor # EXL 5 10 EX Fig. 8: RMSE of the bia parameter b θ for enor 1 and in logarithmic cale (comparion between the previou (EX) and the propoed (EXL) algorithm)) C. A five enor problem with nearly contant velocity (NCV) Kalman filter a local tracker In the econd cae, a cenario with five enor with the ame ixteen target, L = k k = 10 and k = 1,..., 100 i imulated. The motivation for uing 100 time tep i to have enough update tep to demontrate the convergence reult in term of RMSE. Here the Fued Bia Etimation Algorithm (FBEA, Figure 4) i ued to etimate the biae at the fuion center. The reult of thi imulation are hown in Figure 9. From Figure 9, it can be een that the propoed algorithm work well in a cenario with five enor and with tracklet update at every L = 10 tep. Note that the FBEA i uing only a two-dimenional tate pace for the bia etimation tep for each enor. For the ame cenario, the previou EX algorithm would required a ten dimenional tate pace model. At it core, FBEA i a recurive leat quare (RLS) etimator. The computational complexity of RLS i of the order of O(n ), where n i number of parameter to be etimated. With n = 10 in the imulation, the computational complexity of the EX algorithm will be ubtantially higher. To how the performance of the new algorithm in the fuion tep, the reult in term of RMSE of the local track etimate for a pecific enor (enor 1) and the fued etimate are preented in Figure 10 in logarithmic cale. To compare the reult, the RMSE value of the local track and fued track with no biae are alo included. Figure 10 how that the equential fuion tep ued in the propoed bia etimation algorithm (FBEA) i a viable olution to the fuion problem. Clearly, the RMSE of the fued track with corrected meaurement i between thoe of the local track and the fued track of meaurement with no biae. Thi obervation indicate that the bia correction and fuion tep work well, which i another feature in the new algorithm, a thi correction i done at the fuion center and not at the local tracker. In thi cae, there i no need for a feedback channel. Thi reduce the communication requirement. In order to further evaluate the performance of the propoed algorithm, one can aume that all enor have caling and offet biae. Let the value of biae be β = [ 0m 1mrad T (75) The reult in term of the RMSE of the local track etimate for a pecific enor (enor 1) and the fued etimate are hown in Figure 11 in logarithmic cale. Figure 11 how that the propoed algorithm can handle offet and caling biae at the ame time and fue the corrected track in order to achieve better etimate of the target tate.

14 RMSE (m) RMSE (m) RMSE (m) RMSE (m) RMSE (m) Reult for b r Reult for b θ Fig. 9: RMSE of the bia parameter b r (left column) and b θ (right column) for all 5 enor from enor 1 (top) to enor 5 (bottom) in logarithmic cale. Note that reidual bia RMSE i an order of magnitude below the meaurement noie tandard deviation, i.e., it become negligible Target #3 Local track (uncorrected) Local track (no bia) Suboptimal fuion (corrected) Suboptimal fuion (no bia) RMSE x (m) Time Step k Fig. 10: RMSE of local track (enor 1) and the output of the fuion algorithm including offet biae for all enor in logarithmic cale D. A five enor problem with a two NCV IMM a local etimator Previouly, the noie level of local tracker filter were aumed to be known. To demontrate how well the propoed algorithm work when thi information i not available, the IMM etimator i ued in the next two example a local tracker filter. To tart with, an IMM etimator with two nearly contant velocity (NCV) or DCWNA Kalman filter with different noie intenitie are ued. The firt filter ue q x = q y = 10 m 3 while the econd one ue q x = q y = m 3 a intenitie in the

15 10 3 Target #3 Local track (uncorrected) Local track (no bia) Suboptimal fuion (corrected) Suboptimal fuion (no bia) RMSE x (m) Time Step k Fig. 11: RMSE of local track (enor 1) and the output of the fuion algorithm including caling and offet biae for all enor in logarithmic cale eat and north direction, repectively. Note that the noie intenity parameter q x and q y have the ame meaning a in [10, i.e., power pectral denitie. To enure accurate bia etimation, parameter of the RLSBE algorithm hould be changed to handle the mimatch in the model between local tracker and fuion center which ue only an NCV model for data proceing. Although there i no ytematic way to elect the intenitie for the NCV model at the fuion center, q hould be the value of the higher intenity in each coordinate. Figure 1 how the RMSE reult for the bia parameter that are etimated in the cae of having NCV NCV IMM etimator a local tracker and only one NCV model at the fuion center with inflated intenity level. Figure 1 how how well the bia etimation can be handled by EXL even when it i not poible to recover the exact Kalman gain that are ued the at local tracker. The difference in convergence between the previou example (Figure 9) and Figure 1 i negligible, which how the effectivene of the new algorithm in different ituation. E. A five enor problem with nearly contant acceleration nearly contant velocity (NCA NCV) IMM a local etimator Next, we demontrate the effectivene of the new algorithm in recovering the Kalman gain and etimating the biae and how how well the new algorithm work in the cae of mimatch in the model at the fuion center and local tracker. In thi example it i aumed that local tracker are uing an IMM etimator with one nearly contant acceleration (NCA) and one NCV Kalman filter with q x = q y = 10 m for the NCA model and q 3 x = q y = m for the NCV model. The iue in 3 thi cae i that the local tracker end only the combined output from the IMM etimator without any information about the acceleration. Figure 13 how the reult on the cenario of thi ubection with q x = q y = 00 m at the fuion center. 3 Simulation reult how convergence a in the previou example (Figure 9 and 1). Thi demontrate the robutne of the new algorithm even in the preence of uncertainty about the local tracker. F. Lower bound and convergence reult The calculation of the CRLB wa dicued in Section V. Three different example are ued to demontrate the performance of the propoed algorithm with repect to the CRLB. The firt example i the cenario implemented in Subection VI-C. The comparion i between the quare root of the diagonal element of the CRLB ( CRLB {[b i }), the quare root of the diagonal element of bia etimation covariance matrix ( Σ ii ) and the RMSE of the etimated biae. Figure 14 how that both Σ ii and the RMSE follow the CRLB {[b i }. The reult for the example in Subection VI-D and VI-E are hown in Figure 15 and Figure 16, repectively. Thee are approximately within the 95% probability region around the CRLB [3. Once again, the figure how that the etimation error follow the CRLB. G. Conitency of bia etimation algorithm In thi ection, a brief analyi of conitency of the propoed bia etimation algorithm i given. The analyi i baed on Normalized Etimation Error Squared (NEES) [19. Firt, the reult for EXL algorithm are hown in Figure 17 for 100

16 RMSE (m) RMSE (m) RMSE (m) RMSE (m) RMSE (m) Reult for b r Reult for b θ Fig. 1: RMSE of the bia parameter b r (left column) and b θ (right column) for all 5 enor from enor 1 (top) to enor 5 (bottom) in logarithmic cale. The local tracker ue IMM etimator with NCV NCV model. TABLE I: Offet bia b r1 related RMSE, CRLB {[b i } and Σ ii at final time-tep for three different local filter Kalman filter NCV NCV NCA NCV RMSE Σii CRLB { } [b i Upper 95% confidence interval Lower 95% confidence interval Monte Carlo run. The bound are for the 95% probability interval which how that the EXL algorithm i conitent at each time tep. To further examine the conitency of the propoed algorithm, the NEES for FBEA are computed and hown in Figure 18 for three different type of local etimator, i.e., Kalman filter, NCV NCV IMM and NCA NCV IMM for 100 Monte Carlo run. Here we ued one ided 95% probability interval. The reult how that FBEA i a peimitic filtering approach. Thi i motly due to the fact that the ue of the peudo meaurement in a Kalman filter fahion i an approximation becaue it error and the tate prediction error at the fuion center are correlated becaue of the common proce noie. Finally, in Table I and II the RMSE, CRLB {[b i } and Σ ii are compared for both offet bia parameter at their lat update for enor 1. A can be een from Table I and II, both RMSE and Σ ii are within the 95% confidence region of CRLB {[bi }, which indicate that the parameter etimate are unbiaed. TABLE II: Offet bia b θ1 related RMSE, CRLB {[b i } and Σ ii at final time-tep for three different local filter Kalman filter NCV NCV NCA NCV RMSE Σii CRLB { [b i } Upper 95% confidence interval Lower 95% confidence interval

17 RMSE (m) RMSE (m) RMSE (m) RMSE (m) RMSE (m) Reult for b r Reult for b θ Fig. 13: RMSE of the bia parameter b r (left column) and b θ (right column) for all 5 enor from enor 1 (top) to enor 5 (bottom) in logarithmic cale. The local tracker ue IMM etimator with NCA NCV model Σii RMSE CRLB{[bi } 95% Confidence Interval (meter) (radian) Fig. 14: Comparion between the quare root of diagonal element of CRLB ( CRLB {[b i }), quare root of diagonal element of the covariance matrix of bia etimation algorithm ( Σ ii ) and RMSE of the bia etimation for the cae of 5 enor with Kalman filter a local tracker (only the reult for the firt enor are hown).

18 (meter) 10 1 Σii RMSE CRLB{[bi } 95% Probability Interval (radian) Fig. 15: Comparion between the quare root of diagonal element of CRLB ( CRLB {[b i }), quare root of diagonal element of the covariance matrix of bia etimation algorithm ( Σ ii ) and RMSE of the bia etimation for the cae of 5 enor with NCV NCV IMM etimator a local tracker (only the reult for the firt enor are hown) Σii RMSE CRLB{[bi } 95% Probability Interval (meter) (radian) Fig. 16: Comparion between the quare root of diagonal element of CRLB ( CRLB {[b i }), quare root of diagonal element of the covariance matrix of bia etimation algorithm ( Σ ii ) and RMSE of the bia etimation for the cae of 5 enor with NCA NCV IMM etimator a local tracker (only the reult for the firt enor are hown). VII. CONCLUSIONS In thi paper, a new bia etimation algorithm that work with only the tate etimate and their aociated covariance matrice from ynchronized local tracker at varying reporting rate wa preented. The algorithm doe not require the tacking of the bia vector of all the enor together, which i a problem for previou algorithm with a large number of enor in the urveillance area. Alo, the new algorithm work without the local filter gain, which are not available at the fuion center in practical ytem. In addition, it give a olution to the problem of joint fuion and bia etimation. The reult from imulation how that the algorithm work reliably in different cenario with variou number of enor. Furthermore, thi algorithm

19 NEES NEES Senor # Senor # NEES 95% Probability Interval Fig. 17: NEES for EXL algorithm with Kalman gain recovery intead of uing original Kalman gain. NEES Upper Bound KF NCV NCV NCA NCV NEES NEES NEES NEES Fig. 18: NEES for FBEA and three different local tracker etimator (Kalman filter, NCV NCV IMM and NCA NCV IMM) for enor 1 (top) to enor 5 (bottom) compared to the upper bound of 95% probability interval. can work with low data rate between the enor and the data fuion center. Finally, the CRLB for multienor multitarget cenario with bia etimation wa preented and the RMSE reult matched well with the CRLB. Thi demontrate the tatitical efficiency and the veratility of the new algorithm. APPENDIX A

20 and Uing the propertie E DERIVATION OF THE EQUIVALENT MEASUREMENT COVARIANCE (34) [ x (k k) x (k k ) T Z k U (k, k ) can be expanded a U (k, k ) = E E [ũ (k, k ) Z k E [ ũ (k, k ) Z k = E [E [[x (k) ˆx (k k) = 0 (76) 0 (77) [x (k) ˆx (k k) + ˆx (k k) ˆx (k k ) T Z k [ [ = E E [x (k) ˆx (k k) [x (k) ˆx (k k) T Z k Z k Z k +E E [ [x (k) ˆx (k k) Z k [ˆx (k k) ˆx (k k ) T Z k }{{} =0 = P (k k) (79) [ũ (k, k ) ũ (k, k ) T Z k = A P (k k) A T + [A I P (k k ) [A I T A P (k k) [A I T [A I P (k k) A (78) = [A I P (k k ) [A I T A P (k k) A T + A P (k k) + P (k k) A T where the argument of A (k, k ) are dropped for clarity. By uing the property [ [A I P (k k ) = P (k k ) [P (k k ) P (k k) 1 I = P (k k ) P (k k ) [ [P (k k ) P (k k) 1 P (k k ) 1 P (k k ) = P (k k ) [P (k k ) P (k k) 1 [ I [P (k k ) P (k k) P (k k ) 1 P (k k ) (80) U (k, k ) can be further implified a = A [I I + P (k k) P (k k ) 1 P (k k ) = A P (k k) (81) U (k, k ) = A P (k k) [A I T A P (k k) A T + A P (k k) + P (k k) A T = A P (k k) A T A P (k k) A P (k k) A T + A P (k k) + P (k k) A T = P (k k) A T which yield (34). Note that (81) and (8) are tranpoe of each other, but, ince U (k, k ) i ymmetric, they are equal to each other. (8) REFERENCES [1 Y. Bar-Shalom, P.K. Willett, and X. Tian. Tracking and Data Fuion: A Handbook of Algorithm. YBS Publihing, Storr, CT, 011. [ B. Friedland. Treatment of bia in recurive filtering. IEEE Tranaction on Automatic Control, vol. 14, no. 4, pp , [3 M. P. Dana. Regitration: A prerequiite for multiple enor tracking. Multitarget-Multienor Tracking: Advanced Application, vol. 1, pp , Artech Houe, Norwood, MA, [4 N. Nabaa and R. H. Bihop. Solution to a multienor tracking problem with enor regitration error. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 35, no. 1, pp , [5 N. Okello and B. Ritic. Maximum likelihood regitration for multiple diimilar enor. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 39, no. 3, pp , 003. [6 N. Okello and S. Challa. Joint enor regitration and track-to-track fuion for ditributed tracker. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 40, no. 3, pp , 004. [7 D. Huang, H. Leung, and E. Boe. A peudo-meaurement approach to imultaneou regitration and track fuion. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 48, no. 3, pp , 01. [8 D. J. Papageorgiou and J. D Sergi. Simultaneou track-to-track aociation and bia removal uing multitart local earch. Proc. IEEE Aeropace Conference, BigSky, MT, March 008.

21 [9 X. Lin, Y. Bar-Shalom, and T. Kirubarajan. Exact multienor dynamic bia etimation with local track. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 1, no. 40, pp , 004. [10 X. Lin, Y. Bar-Shalom, and T. Kirubarajan. Multienor-multitarget bia etimation of aynchronou enor. Proc. SPIE, vol. 549, pp , 004. [11 X. Lin, Y. Bar-Shalom, and T. Kirubarajan. Multienor multitarget bia etimation for general aynchronou enor. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 41, no. 3, pp , 005. [1 O. E. Drummond. Track and tracklet fuion filtering. Proc. SPIE, Signal and Data Proceing of Small Target, vol. 478, pp , 00. [13 H. Chen and X. R. Li. On track fuion with communication contraint. International Conference on Information Fuion, Quebec, QC, Canada, July 007. [14 O. E. Drummond. A hybrid enor fuion algorithm architecture and tracklet. Proc. SPIE, Signal and Data Proceing of Small Target, vol. 3163, [15 O. E. Drummond. Tracklet and a hybrid fuion with proce noie. Proc. SPIE, Signal and Data Proceing of Small Target, vol. 3163, pp , [16 A. Balleri, A. Nehorai, and J. Wang. Maximum likelihood etimation for compound-gauian clutter with invere gamma texture. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 43, no., pp , 007. [17 T. Aparouhov and B. Muthén. Weighted leat quare etimation with miing data. Available at: GtrucMiingReviion.pdf, 010. [18 H. Rue and L. Held. Gauian Markov Random Field: Theory and Application. CRC Pre, Boca Raton, FL, 005. [19 Y. Bar-Shalom, X.R. Li, and T. Kirubarajan. Etimation with Application to Tracking and Navigation: Theory, Algorithm and Software. Wiley, New York, NY, 001. [0 I. Li and J. Georgana. Multi-target multi-platform enor regitration in geodetic coordinate. Proc. International Conference on Information Fuion, vol. 1, pp , Annapoli, MD, July 00. [1 M. Yeddanapudi, Y. Bar-Shalom, and K. Pattipati. IMM etimation for multitarget-multienor air traffic urveillance. Proc. IEEE, vol. 85, no. 1, pp , [ E. Taghavi, R. Tharmaraa, T. Kirubarajan, and Y. Bar-Shalom. Bia etimation for practical ditributed multiradar-multitarget tracking ytem. International Conference on Information Fuion, pp , Itanbul, Turkey, July 013. [3 G. Welch and G. Bihop. An Introduction to The Kalman Filter. Univerity of North Carolina: Chapel Hill, North Carolina, [4 R. A. Horn and C. R. Johnon. Matrix Analyi. Cambridge univerity pre, Cambridge, UK, 01. [5 K. Katella, B. Yeary, T. Zadra, R. Brouillard, and E. Frangione. Bia modeling and etimation for GMTI application. International Conference on Information Fuion, pp. TUC1 7, Pari, France, July 000. [6 P. J. Shea, T. Zadra, D. Klamer, E. Frangione, R. Brouillard, and K. Katella. Preciion tracking of ground target. Proc. IEEE Aeropace Conference, pp , BigSky, MT, March 000. [7 B. A. van Doorn and H.A.P. Blom. Sytematic error etimation in multienor fuion ytem. Proc. SPIE Conference on Aeropace Sening, vol. 1954, pp , [8 Y. Bar-Shalom and H. Chen. Multienor track-to-track aociation for track with dependent error. Journal of Advance in Information Fuion, 1(1), July 006. [9 J. Durbin and S. J. Koopman. Time Serie Analyi by State Space Method. no. 38, Oxford Univerity Pre, Oxford, UK, 01. [30 M. Longbin, S. Xiaoquan, Z. Yiyu, S. Z. Kang, and Y. Bar-Shalom. Unbiaed converted meaurement for tracking. IEEE Tranaction on Aeropace and Electronic Sytem, vol. 34, no. 3, pp , [31 F. A. Graybill. Theory and Application of The Linear Model. Duxbury Pre, Belmont, CA, [3 D. Belfadel, R. W. Oborne, and Y. Bar-Shalom. Bia etimation for optical enor meaurement with target of opportunity. International Conference on Information Fuion, pp , Itanbul, Turkey, July 013. Ehan Taghavi received the M.Sc. degree in communication engineering in 01 from Chalmer Univerity of Technology, Gothenburg, Sweden, where he worked on particle filter moother. He i currently puruing the Ph.D. degree in computational cience and engineering at McMater Univerity, Hamilton, Canada. Hi reearch interet include etimation theory, cientific computing, ignal proceing, parameter etimation, mathematical modeling and algorithm deign. Ratnaingham Tharmaraa wa born in Sri Lanka in He received the B.Sc.Eng. degree in electronic and telecommunication engineering from Univerity of Moratuwa, Sri Lanka in 001, and the M.A.Sc and Ph.D. degree in electrical engineering from McMater Univerity, Canada in 003 and 007, repectively. From 001 to 00 he wa an intructor in electronic and telecommunication engineering at the Univerity of Moratuwa, Sri Lanka. During he wa a graduate tudent/reearch aitant in ECE department at the McMater Univerity, Canada. Currently he i working a a Reearch Aociate in the Electrical and Computer Engineering Department at McMater Univerity, Canada. Hi reearch interet include target tracking, information fuion and enor reource management.

22 Accepted for publication in IEEE Tranaction on Aeropace and Electronic Sytem, June 015 Thiagalingam Kirubarajan (S 95M 98SM 03) wa born in Sri Lanka in He received the B.A. and M.A. degree in electrical and information engineering from Cambridge Univerity, England, in 1991 and 1993, and the M.S. and Ph.D. degree in electrical engineering from the Univerity of Connecticut, Storr, in 1995 and 1998, repectively. Currently, he i a profeor in the Electrical and Computer Engineering Department at McMater Univerity, Hamilton, Ontario. He i alo erving a an Adjunct Aitant Profeor and the Aociate Director of the Etimation and Signal Proceing Reearch Laboratory at the Univerity of Connecticut. Hi reearch interet are in etimation, target tracking, multiource information fuion, enor reource management, ignal detection and fault diagnoi. Hi reearch activitie at McMater Univerity and at the Univerity of Connecticut are upported by U.S. Miile Defene Agency, U.S. Office of Naval Reearch, NASA, Qualtech Sytem, Inc., Raytheon Canada Ltd. and Defene Reearch Development Canada, Ottawa. In September 001, Dr. Kirubarajan erved in a DARPA expert panel on unattended urveillance, homeland defene and counterterrorim. He ha alo erved a a conultant in thee area to a number of companie, including Motorola Corporation, Northrop-Grumman Corporation, Pacific-Sierra Reearch Corporation, Lockhead Martin Corporation, Qualtech Sytem, Inc., Orincon Corporation and BAE ytem. He ha worked on the development of a number of engineering oftware program, including BEARDAT for target localization from bearing and frequency meaurement in clutter, FUSEDAT for fuion of multienor data for tracking. He ha alo worked with Qualtech Sytem, Inc., to develop an advanced fault diagnoi engine. Dr. Kirubarajan ha publihed about 100 article in area of hi reearch interet, in addition to one book on etimation, tracking and navigation and two edited volume. He i a recipient of Ontario Premier Reearch Excellence Award (00). Yaakov Bar Shalom wa born on May 11, He received the B.S. and M.S. degree from the Technion, Irael Intitute of Technology, in 1963 and 1967 and the Ph.D. degree from Princeton Univerity in 1970, all in electrical engineering. From 1970 to 1976 he wa with Sytem Control, Inc., Palo Alto, California. Currently he i Board of Trutee Ditinguihed Profeor in the Dept. of Electrical and Computer Engineering and Marianne E. Klewin Profeor in Engineering at the Univerity of Connecticut. He i alo Director of the ESP (Etimation and Signal Proceing) Lab. Hi current reearch interet are in etimation theory, target tracking and data fuion. He ha publihed over 500 paper and book chapter in thee area and in tochatic adaptive control. He coauthored the monograph Tracking and Data Aociation (Academic Pre, 1988), the graduate text Etimation and Tracking: Principle, Technique and Software (Artech Houe, 1993; tranlated into Ruian, MGTU Bauman, Mocow, Ruia, 011), Etimation with Application to Tracking and Navigation: Algorithm and Software for Information Extraction (Wiley, 001), the advanced graduate text MultitargetMultienor Tracking: Principle and Technique (YBS Publihing, 1995), Tracking and Data Fuion (YBS Publihing, 011), and edited the book MultitargetMultienor Tracking: Application and Advance (Artech Houe, Vol. I, 1990; Vol. II, 199; Vol. III, 000). He ha been elected Fellow of IEEE for contribution to the theory of tochatic ytem and of multi target tracking. He ha been conulting to numerou companie and government agencie, and originated the erie of MultitargetMultienor Tracking hort coure offered via UCLA Extenion, at Government Laboratorie, private companie and overea. During 1976 and 1977 he erved a Aociate Editor of the IEEE Tranaction on Automatic Control and from 1978 to 1981 a Aociate Editor of Automatica. He wa Program Chairman of the 198 American Control Conference, General Chairman of the 1985 ACC, and CoChairman of the 1989 IEEE International Conference on Control and Application. During he erved a Chairman of the Conference Activitie Board of the IEEE Control Sytem Society and during wa a member of the Board of Governor of the IEEE CSS. He wa a member of the Board of Director of the International Society of Information Fuion ( ) and erved a General Chairman of FUSION 000, Preident of ISIF in 000 and 00 and Vice Preident for Publication in In 1987 he received the IEEE CSS Ditinguihed Member Award. Since 1995 he i a Ditinguihed Lecturer of the IEEE AESS and ha given numerou keynote addree at major national and international conference. He i corecipient of the M. Barry Carlton Award for the bet paper in the IEEE Tranaction on Aeropace and Electronic Sytem in 1995 and 000 and recipient of the 1998 Univerity of Connecticut AAUP Excellence Award for Reearch. In 00 he received the J. Mignona Data Fuion Award from the DoD JDL Data Fuion Group. He i a member of the Connecticut Academy of Science and Engineering. In 008 he wa awarded the IEEE Denni J. Picard Medal for Radar Technologie and Application, and in 01 the Connecticut Medal of Technology. He ha been lited by academic.reearch.microoft (top author in engineering) a number 1 among the reearcher in Aeropace Engineering baed on the citation of hi work. Michael McDonald received a B.Sc (Hon) degree in Applied Geophyic from Queen Univerity in Kington, Canada in 1986 and a M.Sc. degree in Electrical Engineering in 1990, alo from Queen Univerity. He received a Ph.D in Phyic from the Univerity of Wetern Ontario in London, Canada in He wa employed at ComDev in Cambridge, Canada from 1989 through 199 in their pace cience and atellite communication department and held a pot-doctoral poition in the Phyic department of SUNY at Stony Brooke from 1996 through 1998 before commencing hi current poition a Defence Scientit in the Radar Sytem ection of Defence Reearch and Development Canada, Ottawa, Canada. Hi current reearch interet include the application of STAP proceing and nonlinear filtering to the detection of mall maritime and land target a well a the development and implementation of paive radar ytem.