Correspondence Performance Evauaton for MAP State Estmate Fuson Ths paper presents a quanttatve performance evauaton method for the maxmum a posteror (MAP) state estmate fuson agorthm. Under dea condtons where data assocaton s assumed to be perfect, t has been shown that the MAP or best near unbased estmate (BLUE) fuson formua provdes the best near mnmum mean squared estmate (LMMSE) gven oca estmates under the near Gaussan assumpton for a statc system. However, for a dynamc system where fuson s recursvey performed by the fuson center on oca estmates generated from oca measurements, t s not obvous how the MAP agorthm w perform. In the past, severa performance evauaton methods have been proposed for varous fuson agorthms, ncudng smpe convex combnaton, cross-covarance combnaton, nformaton matrx, MAP fuson. However, not much has been done to quantfy the steady state behavor of these fuson methods for a dynamc system. The goa of ths wor s to present anaytca fuson performance resuts for MAP state estmate fuson wthout extensve Monte Caro smuatons, usng an approach deveoped for steady state performance evauaton for trac fuson. Two dfferent communcaton strateges are consdered: fuson wth wthout feedbac to the sensors. Anaytc curves for the steady state performance of the fuson agorthm for varous communcaton patterns are presented under dfferent operatng condtons. I. INTRODUCTION Mutsensor data fuson, the process of estmatng the state of enttes by combnng nformaton from mutpe sensors, has been wdey used n many appcatons such as surveance target tracng. The fuson process can be centrazed or dstrbuted. Centrazed fuson (CF), whch processes a sensor measurements at a snge ocaton, can theoretcay produce optma resuts, but may be mpractca due to communcaton or reabty consderatons. In dstrbuted fuson, on the other h, each sensor processes ts own measurements, communcates the oca trac estmates wth other sensors or processors to obtan goba trac estmates. Compared wth CF, dstrbuted fuson enjoys better reabty ower communcaton bwdth, but the fuson Manuscrpt receved September 10, 2002; revsed August 7, 2003; reeased for pubcaton November 25, 2003. IEEE Log No. T-AES/40/2/831394. Refereeng of ths contrbuton was hed by X. R. L. 0018-9251/04/$17.00 c 2004 IEEE 706 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 2 APRIL 2004
agorthms are typcay more compcated. Ths paper focuses on a herarchca verson of dstrbuted fuson. The fuson process nvoves three steps. 1) At each oca ste, a sensor updates ts oca trac estmates (prors) at every sampng tme usng the oca sensor measurements. 2) After every n oca sampng steps, where n 1 prescrbes the communcaton rate, a the oca stes send ther estmates to a fuson ste, where goba trac estmates are computed based on not ony the oca measurements, but aso the avaabe goba pror nformaton at the fuson ste. 3) If a feedbac mechansm s empoyed, compete or parta fuson estmates s sent bac to be shared 1 wth the oca sensors. The above three steps consttute one fuson teraton, the prors at each teraton are updated by the estmates obtaned at the prevous teraton. For a statonary target that s observed ony once by the sensors, the trac state estmate fuson needs to be carred out ony once, whch eads to the snge-teraton case. In contrast, a target n most practca tracng systems s modeed as a dynamc process observed over mutpe tmes. Thus, the trac estmates need to be updated through mutpe fuson teratons based on sensor observatons communcatons. The oca estmates/tracs to be fused are generay correated, not ony due to the common process nose from the target beng traced, but aso due to shared common pror nformaton from prevous communcaton or the modes. The desgn of a hgh-performance fuson agorthm s compcated by these two sources of correaton among oca sensors. Extensve research has been done to desgn fuson agorthms that address the oca trac state correaton 2 [1 13]. The nformaton matrx (IM) based method ams at reconstructng the centrazed estmate from the oca estmates [8, 14]. Ths agorthm s optma 3 when the underyng target dynamc systems are determnstc (.e., no process nose), or when fu-rate communcaton (n = 1) s empoyed. However, when the process nose s not neggbe due to target maneuver sensors communcate nfrequenty to save communcaton bwdth, the reconstructon becomes an approxmaton, the agorthm may ony be near optma. The cross covarance (CC) based fuson agorthm derves the fuson formua from the statstca correaton between the oca trac 1 There s n genera more than one type of feedbac. In ths paper, the fuson estmates are used to repace the oca ones. 2 Ths correaton refers to the statstca correaton n the state estmates for a snge target. We assume that data assocaton s perfect do not address the trac to trac correaton probem. 3 In addton to the typca near Gaussan assumptons, t s aso assumed that there s no mscorreaton, msdetecton, or merged measurement n the process. estmates [10 11]. However, t does not tae nto account the common pror estmates, thus the trac state estmaton performance s suboptma [9]. Recenty, a fuson agorthm based on the MAP probabty concept was proposed [1]. Smar to the BLUE (best near unbased estmaton) fuson rue proposed n [2], the MAP fuson agorthm provdes the near mnmum mean square estmate (LMMSE) estmate gven the atest oca estmates under the near Gaussan assumpton n a statc stuaton (.e., snge teraton). Ths optmaty has so far ony been proved when the target s statc, both the oca sensors the fuson processor have perfect nowedge of the trac prors [3]. In a dynamc trac fuson system, the optma fused estmate s the best estmate condtoned on a pror measurements. The MAP agorthm, on the other h, ony ensures that the pror estmates used n subsequent teratons are the LMMSE estmate gven the prevous oca estmates; t s not the condtona estmate gven a the pror sensor measurements. Thus the assumptons for the optma MAP agorthm (based on a pror sensor measurements) may not be met exacty. Partcuary, the mpact of propagatng ths suboptma nformaton s not obvous, a detaed study s needed to anayze that. Furthermore, as shown ater n ths paper, the MAP agorthm needs the past covarance hstory between communcatons whch may or may not be avaabe off ne. When they are not avaabe, the communcaton oad coud ncrease dramatcay. Severa performance evauaton methods have been proposed to compare dfferent fuson agorthms [1, 3, 4], but they are ether based on a snge teraton anayss or a snapshot anayss where the correaton between oca estmates s assumed to be gven. However, for dynamc targets that are observed over mutpe tmes, we need to underst the steady state performance of the agorthm. An anaytca steady state performance evauaton for the MAP fuson agorthm s provded, usng an approach smar to that deveoped for trac fuson wth IM [5 7]. We evauate the performance of MAP fuson under two dfferent communcaton strateges. Specfcay, herarchca fuson wth wthout feedbac s consdered. Theoretca curves for the performance of the fuson agorthm wth varous communcaton patterns are gven. The resuts are compared wth those obtaned by the IM approach as we as the CC approach. The rest of ths paper s organzed as foows. Secton II presents the anayss dervaton of the fuson performance measure n terms of the covarance of the fuson state estmate error. The steady state covarance for herarchca fuson s derved. Secton III examnes the anaytc performance of trac fuson for dfferent nondetermnstc target dynamcs by varyng the process nose. CORRESPONDENCE 707
II. STEADY STATE ANALYSIS OF MAP STATE ESTIMATE FUSION The wor presented here focuses on a smpe case where the two state estmates obtaned from two oca processng nodes (sensors) are to be fused together. Apart from the typca near Gaussan assumptons, t s aso assumed that there are no mscorreaton, msdetecton, or merged measurements. A. Trac State Estmate Fuson System Descrpton As n [7], consder a smpe trac state estmate fuson system n whch two sensors are tracng the same target. The target state s modeed by a dscrete near tme nvarant dynamca system as x 1 = Fx Gv, =0,1,2,::: (1) where v s a zero-mean whte Gaussan process nose wth covarance Q. The measurement vectors of the two sensors are modeed as z (j) = H (j) x, j =1,2, =0,1,2,::: (2) where s a zero-mean whte Gaussan measurement nose wth covarance R (j).the measurement noses for the two sensors are assumed to be uncorreated. It s assumed that each sensor empoys a Kaman fter to generate a oca trac state estmate for each target. At the end of every n sampng ntervas, each oca sensor transmts ts state estmate to the fuson ste where trac state estmate fuson s performed. The fused trac w then be sent bac to the oca sensors when feedbac s on. The oca state estmates at tme are assumed to be mnmum varance based on prevous measurements, Z (j) = z (j), = 0,1,2,:::,, j = 1,2, namey, = E[x Z(j) ] wth the assocated error covarance P (j). B. MAP State Estmate Fuson Let ˆx P represent the fused state estmate ts assocated error covarance, respectvey. The MAP approach for trac-to-trac fuson, descrbed n [1], combnes trac state estmate at the fuson center as foows: ˆx = ˆx n W (1) (1) (ˆx ˆx n )W (2) (ˆx (2) ˆx n ) (3) P = P n xẑ 1 ẑẑ xẑ (4) where the gan matrx s defned as W [W (1) W (2) ]= xẑ 1 ẑẑ (5) xẑ [ 1 2 ] s the CC between the target state x the jont oca estmates ˆx (1) ˆ 11 ˆ 12 ẑ ẑẑ ˆx (2) ˆ 21 ˆ 22 s the covarance matrx of ẑ. These formuae (3) (4) can aso be shown to be optma n the LMMSE sense [2] gven the two oca estmates. The n-step predctor of the fused state vector ts covarance are ˆx n = F n ˆx n (6) P n = F n P n F n F GQG F : (7) It can be shown that ([2, 4], Appendx A) wth fu rate communcaton (n = 1) when feedbac s on, xẑ [ 1 2 ]=[P 1 P (1) P 1 P (2) ] (8) 11 12 ẑẑ =1 21 22 P 1 P (1) P 1 P (1) P(2) = P 1 P (1) P(2) P(21) P P (2) P(12) where P (j) = P() P 1 P(j) s the CC between ˆx (). It can further be shown that n ths case (fu rate (9) communcaton wth feedbac), ths MAP fuson rue s agebracay equvaent to the IM fuson rue [5] s optma n the sense that the resut s the same as the one obtaned by the CF. However, when n>1, the MAP fuson resut s not the same as the CF resut. Ths s because at each MAP fuson teraton, the pror at the fuson ste s the LMMSE estmate gven the prevous oca estmates, not the condtona estmate gven a the pror measurements. To underst the mpact of propagatng ths suboptma nformaton, we need to derve the correspondng xẑ ẑẑ. It can be shown that (see Appendx B) = E[(x ˆx n )(ˆx () ˆx n ) ] = F n P n (F n A () 1 ) =1 j = E[(ˆx () ˆx n ˆx n ) ] =(F n A () 1 )P n (Fn A (j) 1 ) =1 =1 (F A () B () R j F GQG (F A () F 1 ) F 1 )GQG (F A (j) F 1 ) (10) (11) 708 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 2 APRIL 2004
where matrces A (), B (), =1,:::,n, =1,2are defned n Appendx B. Note that (10) (11) represent the MAP fuson estmate covarance wth communcaton nterva n feedbac on. When feedbac s off, smar but more compcated expressons can be obatned (see Appendx C for n>1 Appendx D for n =1). C. Steady State Error Covarance In the foowng performance anayss, we assume that the oca fters the fuson process have reached the steady state. Defne P o m P, P o m P n, m xẑ, o m ẑẑ, m, j m j. At the steady state, we have = F n P o (F n A () 1 ) =1 F GQG (F A () F 1 ) F n P o (12) j = P o j j E j (13) where,, j,e j are defned as beow, F n A () 1 (14) j E j Defne therefore, =1 =1 =1 F GQG (F A () F 1 ) (15) (F A () F 1 )GQG (F A (j) F 1 ) (16) B () 1 2 R j : (17), 1 2 11 12 E11 E 12, E 21 22 E 21 E 22 [ 1 2 ]=F n P o (18) 11 12 o = P o E (19) 21 22 P o = P 1 o o = P o (F n P o ) ( P o E) 1 ( P n o F ) (20) ˆx = ˆx n (F n P o ) ˆx (1) ( P o E) 1 ˆx n ˆx (2) : ˆx n (21) We further assume a the reguarty condtons, ncudng observabty controabty condtons, necessary to assure a unque exstence of a postve defnte souton to these matrx equatons. Note that (20) (21) represent a systematc recursve procedure to compute the MAP fuson estmate the assocated covarance for arbtrary communcaton nterva n. It can be easy seen that the souton to the steady state covarance satsfes the foowng matrx functon whch can be soved usng a successve approxmaton method P o = F n P n o F F GQG F (F n P o ) =1 ( P o E) 1 ( P n o F ): (22) Note that une the IM fuson where the estmated covarance s not the true covarance [7], n MAP fuson, we obtan the true covarance, namey III. x ()=E[(ˆx x )(ˆx x ) ]=P : (23) PERFORMANCE COMPARISON To evauate the MAP fuson performance, a smpe target dynamc mode same as the one used n [9] [11] s assumed, 1 T x 1 = 0 1 x T 2 =2 T v (24) wth sampng tme T = 1 zero-mean whte process nose v wth varance q. Note that for rea systems, a Brownan moton mode mght be preferred for sensors that are not synchronzed wth possby rom measurement tmes. The measurement of the two sensors are modeed as z m ()=[1 0]x w m, m = 1,2 (25) where the two measurement noses are assumed to be ndependent wth varance r m = 1. Ths mode assumes two synchronzed, fx measurement rate sensors, each generatng a 1-dmensona measurement wth dentca observaton matrx. Fgs. 1 3 show the theoretca resuts based on the anayss derved n Secton II for the herarchca MAP fuson wth feedbac. In these fgures, the ratos of the eements of the covarance matrx P P for a wde range of process nose q are gven, where p1 p2 P = p2 p3 CORRESPONDENCE 709
Fg. 1. Rato of P1 for MAP fuson wth feedbac. Fg. 3. Rato of P3 for MAP fuson wth feedbac. Fg. 2. Rato of P2 for MAP wth feedbac. Fg. 4. Ratos of error epses area for MAP fuson wth feedbac. s the steady state error covarance matrx of the fused state P s the optma steady state error covarance matrx wth a snge sensor. Fg. 4 shows the ratos of the areas of the epses of uncertanty, A/A, whch are proportona to the square root of the determnant of the covarance matrces. The dotted nes are the anaytca resuts based on the CC approach derved n [11], the sod nes are the anaytca resuts based on (23) wth varous communcaton rates. It can be seen from these fgures that the anaytca fuson resuts are reatvey senstve to the process nose (partcuary for p3, snce veocty s not observed drecty) the communcaton nterva. Fgs. 5 6 compare the resuts of CC, IM, MAP, CF for n =4n = 8 respectvey. It can be seen from the fgures that MAP performs the best among the frst three agorthms. However, wth arge process nose ess frequent communcatons (n = 8), both IM MAP converge to the CC agorthm. It s to be noted that among the three agorthms, MAP s the most compcated one whe IM s the smpest one. Further more, as shown n (10) (11), to compute the CC, MAP needs the past covarance hstory between communcatons whch may or may Fg. 5. CC, IM, MAP fuson wth n =4. not be avaabe off ne. When they are not avaabe, the communcaton oad coud ncrease dramatcay. IV. CONCLUSIONS Ths paper presents a quanttatve performance evauaton method for the MAP trac state estmate 710 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 2 APRIL 2004
The ast equaty foows from the orthogonaty prncpe. Wth, 11 12 ẑẑ 21 22 j = E[(ˆx () ˆx 1 ˆx 1 ) ] Fg. 6. CC, IM, MAP fuson wth n =8. fuson agorthm. In partcuar, the steady state performance for herarchca fuson archtecture wth or wthout feedbac s anayzed n deta. It has been nown that the MAP agorthm taes nto account the oca estmate correaton due to both common pror common process nose whe the IM agorthm ony consders the common pror. It has aso been shown that the MAP fuson provdes the best (LMMSE) estmates under the near Gaussan assumpton n a statc stuaton (.e., snge teraton). Ths paper presents a systematc anaytca procedure for a dynamc stuaton provdes the theoretca curves for the steady state performance of the MAP fuson agorthm wth varous communcaton patterns. The resuts are compared wth the one obtaned by the IM approach as we as the CC approach. APPENDIX A When feedbac s on, ˆx (1) 1 = ˆx (2) 1 = ˆx 1. To derve (8) (9), from (3) (4), wth xẑ [ 1 2 ], we have 4 = E[(x ˆx 1 )(ˆx () ˆx 1 ) ] = E[(x ˆx 1 )[(x ˆx 1 ) (x ˆx () )] ] = P 1 E[(x ˆx 1 )(x ˆx () ) ] = P 1 E[[(x ˆx () () )(ˆx ˆx 1 )](x ˆx () ) ] = P 1 P () () E[(ˆx ˆx 1 )(x ˆx () ) ] = P 1 P () : (26) 4 Note that for smpcty, we have removed the condtonng varabe Z from a equatons. = E[[(ˆx () x )(x ˆx 1 )] [( x )(x ˆx 1 )] ] = P 1 E[(ˆx () x (j) )(ˆx x ) ] E[(ˆx () x )(x ˆx 1 ) ] E[(x ˆx 1 x ) ] = P 1 P (j) P() P(j) : Obvousy, from (27), when = j, = P 1 P (). Aso, n (27) But P (j) () = E[(ˆx x (j) )(ˆx x ) ] = E[[(ˆx 1 x )(ˆx () ˆx 1 )] [(ˆx 1 x ˆx 1 )] ]: (ˆx 1 x )(ˆx () ˆx 1 ) =(ˆx 1 x ) K () H () (ˆx 1 x )K () w () =[I K () H () ](ˆx 1 x )K () H () w () =[IP () H() R ()1 H () ](ˆx 1 x )K () w () (27) = P () P1 1 (ˆx 1 x )K () w () (28) P (j) = P() P1 1 E[[(ˆx 1 x )(ˆx 1 x ) ] P 1 1 P(j) K() E[w () = P () P1 1 P(j) K () R (j) K (j) w(j) ]K (j) where K () s the Kaman gan. Notce that n (28), when the measurement noses from two sensors are ndependent, P (j) = P () P1 1 P(j). APPENDIX B Wth n>1, when feedbac s on, ˆx (1) n = ˆx (2) n = ˆx n. CORRESPONDENCE 711
It can be shown that = A(j) (j) 1 ˆx n z (j) n = A (j) (j) 1 ˆx n = A (j) (j) 1 ˆx n =1 =1 =1 =1 H (j) F x n H (j) x n h= =1 n h H(j) F h Gv n n (29) where A (j) n1 = I, A(j) = A (j) 1 P(j) n P(j) nn1 1 F, = A (j) 1 P(j) n H(j) R (j)1. Notce that x = F n x n F Gv n (30) z (j) n = H(j) x n n = H (j) F x n H (j) F h Gv nh n, h=1 =1 =1,:::,n: (31) Wth (30) (31), the equates h H(j) F h = F A (j) F 1 (32) h= =1 H (j) F = F n A (j) 1 (33) we have ˆx n =(x ˆx n x ) Therefore, =(x ˆx n )A (j) (j) 1 (ˆx n x n ) =1 A (j) F 1 Gv n =1 n =(x ˆx n )A (j) 1 (ˆx n x n ) A (j) F 1 Gv n n : =1 j = E[(x ˆx n ˆx n ) ] = F n P n (F n A (j) 1 ) =1 =1 (34) F GQG (F A (j) F 1 ) (35) j = E[(ˆx () ˆx n ˆx n ) ] =(F n A () 1 )P n (Fn A (j) 1 ) =1 =1 APPENDIX C (F A () F 1 )GQG (F A (j) F 1 ) B () R j (36) Wth n>1, when feedbac s off, at each oca sensor, when, t s possbe to wrte [7], = P(j) s = P (j) s. P (j)1 s1 P (j)1 s1 1 P(j) s F(P (j) s P (j) s H (j) R (j)1 z (j) P (j)1 s1 =1 H (j) R (j)1 z (j) 12 P(j) s H (j) R (j)1 z (j) 1 ) = A (j)n n z (j) n (37) P (j)1 where A (j) = P (j) s s1 F, From (37) (30) (31) = A(j)n n =1 =1 H (j) F x n = A (j) P (j) s H (j) R (j)1. H (j) F h Gv nh h=1 n (38) ˆx n =(x ˆx n x ) =(x ˆx n )A (j)n ( n x n ) A (j)1 F 1 Gv n n =1 (39) where n (39) the equates (32) (33) have been used. From (39), we have j = E[(x ˆx n ˆx n ) ] =1 = F n P n F F n P cf(j) n A(j)n F GQG (F A (j)1 ) (40) =1 712 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 2 APRIL 2004
where P cf(j) n = E[(ˆx n x n n x n ) ]: From (39), we aso have where j = E[(ˆx () ˆx n ˆx n ) ] = F n P n F F n P cf(j) n A(j)n A ()n P cf() n F A ()n P (j)) n A(j)n (F A (j)1 F 1 )GQG =1 (F A (j)1 F 1 ) =1 B () R j P (j) (j) = E[(ˆx n n x (j) n )(ˆx n x n ) ]: APPENDIX D Defne P cf(j) 1 := E[(ˆx 1 x )(ˆx () 1 x) ] = E[(Fˆx 11 Fx 1 Gv 1 ) (Fˆx () 11 Fx Gv 1 ) ] (41) (42) = FP cf(j) 11 F GQG : (43) When feedbac s off, xẑ [ 1 defnton = E[(x ˆx 1 )(ˆx () ˆx 1 ) ] 2 ], where by = E[(x ˆx 1 )[(x ˆx 1 ) (x ˆx () )] ] = P 1 E[(x ˆx 1 )(x ˆx ()) ]: (44) Note that ˆx () x = P() P()1 () 1 (ˆx 1 x )Kw() where K s the Kaman gan. Therefore E[(x ˆx 1 )(x ˆx () ) ], = E[(x ˆx 1 )(x ˆx () 1 ) P ()1 1 P() ] = P 1 P cf() 1 P()1 1 P() : (45) Susttutng (45) nto (44), we obtan = P 1 P cf() 1 P()1 1 P() : (46) Thecovarancematrxsgvenby 11 12 ẑẑ, 21 22 where j = E[(ˆx () ˆx 1 ˆx 1 ) ] = E[[(ˆx () x )(x ˆx 1 )][( x )(x ˆx 1 )] ] = P 1 E[(ˆx () x (j) )(ˆx x ) ] E[(ˆx () x )(x ˆx 1 ) ]E[(x ˆx 1 x ) ] = P 1 P (j) P() P()1 1 P() P(j) 1 P(j)1 1 P(j) (47) Equatons (46) (47) can be equvaenty derved from Appendx C by tang the speca case of n = 1. Usng (43) A (j) 1 = P(j) P(j)1 F for n =1, 1 (40) becomes j = E[(x ˆx 1 ˆx 1 ) ] = FP 11 F FP cf(j) 11 A(j) GQG (I A (j) ) =(FP 11 F GQG ) (FP cf(j) 11 A(j) GQG A (j) ) ) = P 1 P cf(j) 1 P(j)1 1 P(j) whch concdes wth (46). Smary, the equvaence between (42) (47) can be estabshed. REFERENCES K. C. CHANG Dept. of SEOR, 4A6 George Mason Unversty Farfax, VA 22030-4444 E-ma: (chang@gmu.edu) ZHI TIAN Dept. of Eectrca Computer Engneerng Mchgan Technoogca Unversty Houghton, MI 49931 SHOZO MORI ALPHATECH, Inc. San Dego, CA 92121 CHEE-YEE CHONG [1] Mor, S., Barer, W. H., Chong, C. Y., Chang, K. C. (2002) Trac assocaton trac fuson wth non-determnstc target dynamcs. IEEE Transactons on Aerospace Eectronc Systems, 38, 2 (Apr. 2002), 659 668. [2] L, X-R., Zhu, Y., Han, C. (2000) Unfed optma near estmaton fuson Part I: Unfed modes fuson resuts. In Proceedngs of Fuson 00, Pars, 2000. [3] L, X-R., Zhang, K. (2001) Optma near estmaton fuson Part IV: Optmaty effcency of dstrbuted fuson. In Proceedngs of Fuson 01, Montrea, Aug. 2001. [4] Chong, C. Y., Mor, S. (2001) Convex combnaton covarance ntersecton agorthms n dstrbuted fuson. In Proceedngs of Fuson 01, Montrea, Aug. 2001. CORRESPONDENCE 713
[5] Chang, K. C., Tan, Z., Saha, R. K. (1998) Performance evauaton of trac fuson wth nformaton fter. In Proceedngs of the Frst Internatona Conference on Mutsource-Mutsensor Informaton Fuson, LasVegas, NV, Juy 1998, 648 654. [6] Chang, K. C. (2000) Evauatng herarchca trac fuson wth nformaton matrx fter. In Proceedngs of Fuson 00, Pars, 2000. [7] Chang, K. C., Tan, Z., Saha, R. (2002) Performance evauaton of trac fuson wth nformaton fter. IEEE Transactons on Aerospace Eectronc Systems, 38, 2 (Apr. 2002). [8] Chong, C. Y., Chang, K. C., Mor, S. (1986) Dstrbuted tracng n dstrbuted sensor networs. In Proceedngs of the Amercan Contro Conference, Seatte, 1986. [9] Chang, K. C., Saha, R. K., Bar-Shaom, Y. (1997) On optma trac-to-trac fuson, IEEE Transactons on Aerospace Eectronc Systems, 33, 4 (Oct. 1997), 1271 1276. [10] Bar-Shaom, Y. (1981) On the trac-to-trac correaton probem. IEEE Transactons on Automatc Contro, 26, 2(Apr. 1981), 571 572. [11] Bar-Shaom, Y., Campo, L. (1986) The effect of the common process nose on the two-sensor fused-trac covarance. IEEE Transactons on Aerospace Eectronc Systems, 22, 6 (Nov. 1986), 803 805. [12] Drummond, O. E. (1995) Feedbac n trac fuson wthout process nose. In Proceedngs of the SPIE Conference on Sgna Data Processng of Sma Targets, Vo. 2561, 1995, 369 383. [13] Chong, C. Y., Mor, S., Chang, K. C., Barer, W. (1999) Archtectures agorthms for trac assocaton fuson. In Proceedngs of the Second Internatona Conference on Informaton Fuson, Juy 1999, 239 246. [14] Chong, C. Y. (1979) Herarchca estmaton. In Proceedngs of MIT/ONR Worshop on C 3, 1979. New Assgnment-Based Data Assocaton for Tracng Move-Stop-Move Targets We present a new assgnment-based agorthm for data assocaton n tracng ground targets empoyng evasve move-stop-move maneuvers usng grgound movng target Indcator (GMTI) reports obtaned from an arborne sensor. To avod detecton by the GMTI sensor, the targets deberatey stop for some tme before movng agan. The sensor does not detect a target when the atter s rada veocty (aong the ne-of-sght from the sensor) fas beow a certan mnmum detectabe veocty (MDV). Even n the absence of move-stop-move maneuvers, the detecton has a ess-than-unty probabty (P D < 1) dueto obscuraton threshodng. Then, t s of nterest, when a target s not detected, to deveop a systematc technque that can dstngush between ac of detecton due to P D < 1 ac of detecton due to a stop (or a near stop). Prevousy, ths probem was soved usng a varabe structure nteractng mutpe mode (VS-IMM) estmator wth a stopped target mode (VS-IMM-ST) wthout expcty addressng data assocaton. We deveop a nove two-dummy assgnment approach for move-stop-move targets that consders both the probem of data assocaton as we as fterng. Typcay, n assgnment-based data assocaton a dummy measurement s used to denote the nondetecton event. The use of the stard snge-dummy assgnment, whch does not he move-stop-move moton expcty, can resut n broen tracs. The new agorthm proposed here hes the evasve move-stop-move moton by ntroducng a second dummy measurement to represent nondetecton due to the MDV. We aso present a ehood-rato-based trac deeton scheme for move-stop-move targets. Usng ths two-dummy data assocaton agorthm, the trac correspondng to a move-stop-move target s ept ave durng mssed detectons both due to MDV due to P D < 1. In addton, one can obtan reductons n both rms estmaton errors as we as the tota number of trac breaages. I. INTRODUCTION Ths paper focuses on the probem of tracng ground targets usng ground movng target ndcator (GMTI) reports obtaned from an arborne sensor. The sensor measurements consst of range, azmuth, Manuscrpt receved September 26, 2002; revsed March 18, June 25, December 15, 2003; reeased for pubcaton December 15, 2003. IEEE Log No. T-AES/40/2/831395. Refereeng of ths contrbuton was hed by P. K. Wett. 0018-9251/04/$17.00 c 2004 IEEE 714 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 2 APRIL 2004