Comparison of Out-of-sequence Measurement Algorithms in Multi-platform Target Tracking

Transcription

1 Coparso of Ou-of-sequece Measuree Algorhs Mul-plafor Targe Tracg Mahedra Mallc a Sefao Coralupp a ad Yaaov Bar-Shalo b allc@alphaechco sefaocoralupp@alphaechco ybs@egrucoedu a ALPHATECH Ic 50 Mall Road Burlgo MA 0803 USA b Uversy of Coeccu Sorrs CT USA Absrac The ceralzed racg archecure usg ulple sesor plafors has prove o have he bes perforace aog all archecures [] Measurees ca arrve ou-of-sequece (OOS) a he ceral racer due o varyg daa preprocessg es a he plafors delays rassso ao ad coucao ewor execuo The ceral racer ca also receve daa ou-ofsequece fro a sgle sesor f he sesor operaes ulple odes such as he wde area survellace ad secor search odes A uber of OOS easuree (OOSM) flerg algorhs are ow he research leraure [2]-[6] These algorhs fall o four dsc classes () sgle-lag sgleodel () ulple-lag sgle-odel () sgle-lag ulple-odel ad (v) ulple-lag ulple-odel algorhs Sgle-lag sgle-odel algorhs are preseed [2]-[5] ad ulple-lag sgle-odel ad sgle-lag ulple-odel algorhs are preseed [6] I hs paper we copare ad aalyze he perforace of hese algorhs Keywords: Ou-of-sequece Measuree Flerg Newor-cerc Tracg Ou-of-sequece Measuree Tracg Daa Assocao Mulhypohess Tracg Iroduco A egraed ad global pcure of he balespace s crcal for coaders ad weapo syses Couous ad exeded coverage of he balespace requres ulple arbore ad spacebore sesors such as groud ovg arge dcaor (GMTI) hgh rage resoluo GMTI (HRR-GMTI) syhec aperure radar (SAR) ad elecro-opcal (EO) sesors for he deeco ad racg of saoary ad ovg arges Mulple plafor cofguraos ca reove daa gaps arsg fro sesor dow e hrough urs sesor raecory ad occluso Daa fro ulple plafors ca also sgfcaly prove he esaed rac accuracy arge dey ad rac couy whch are exreely pora for weapo syses Dffere archecures exs for daa fuso ad racg usg ulple plafors I hs paper we cosder a cooly used archecure whch a ceral racer receves raw sesor easurees fro ulple sesor plafors hrough coucao ewors as show Fgure Daa colleced by ulple plafors are ulcas o ulple groud saos usg coucao ls Sesor easurees a he ceral racer ca arrve ou-of-sequece due o varyg pre-processg es a he plafors ad varyg delays daa rassso usg coucao ewors Two pora probles arse processg he ou-of-sequece easuree (OOSM) a ul-arge ul-sesor racg syse: daa assocao (he proble of assocag sesor easurees wh racs) ad sgle arge rac flerg (he proble of esag he arge sae gve a se of sesor easurees) Soe cooly used OOSM processg approaches are () eglecg he OOSM () daa reprocessg or rollbac ad () daa bufferg Neglecg he OOSM all cases s o desrable If he accuracy of he OOSM s hgh wh sall e delay or he OOSM coas hghaccuracy arge classfcao forao he reeco of he OOSM ca lead o degradao racg for e crcal arges I he rollbac approach sesor repors are sored he eory ad he OOSM s used o reorder he sesor easurees a rac hypohess The flerg s doe for he rac hypohess usg he ordered se of sesor easurees For a large uber of arges he rollbac approach requres sgfca copuer resources for daa sorage reorderg ad flerg ad hus hs approach poses poeal probles for real-e argeg applcaos The daa bufferg approach holds he cog easurees a buffer wh he buffer sze greaer ha he axu expeced delay of arrvg easurees Daa assocao ad flerg are perfored usg e-ordered easurees exraced fro he buffer Ths approach also requres sgfca eory ad sorage aagee Sce he racer processg always lags behd he curre e hs approach s o suable for soe real e applcaos Exsg research ceral racg has addressed he flerg proble assocaed wh he OOSM z a easuree e [2]-[6] whe he OOSM les bewee he laes wo easurees as show

2 SENSOR DATA SATCOM (Ku & UHF) SPACEBORNE PLATFORM SCDL AIRBORNE PLATFORM SENSOR DATA AIRBORNE PLATFORM SENSOR DATA CENTRAL TRACKER SCDL: Saelle Coo Daa L Fgure Ths fgure llusraes a cooly used ul-sesor racg archecure whch a ceral racer receves easurees fro ulple sesors Sesor easurees a he ceral racer ca arrve ou-of-sequece due o varyg pre-processg es a he plafors ad varyg delays daa rassso usg coucao ewors Fgure 2 We refer o hs proble as he sgle-lag OOSM proble I realy he OOSM ca frequely le ulple lags behd he curre easuree as show Fgure 3 for hree lags We have developed ad esed ew OOSM algorhs ha hadle ulple lags wh sgle eac odel ad sgle lag wh ulple eac odels [6] To he bes of our owledge o algorhs exs he research leraure ha address he ulple-lag ulple-odel flerg ssues ad he daa assocao ssues Seco 2 preses he lear dyac odel ad easuree odel ad Seco 3 descrbes varous OOSM flerg algorhs We descrbe he sulao process ad prese uercal resuls Seco 4 for varous OOSM algorhs usg he early cosa velocy (NCV) oo oe deso wh poso ad velocy eaurees Seco 5 preses coclusos 2 Keac Model Mos racg syses use a dscree verso of he couous lear sochasc dffereal equao for he dyacs of he arge sae x R wh addve Gaussa process ose Ths s a cooly used dyac odel os racg syses Measurees z R fro oe or ore sesors are avalable a dscree es = 02 Measuree Recep Te Measuree Te z 4 z 3 z 2 z z Measuree Fgure 2 The ou-of-sequece easuree z observed a e arrves afer he las processed easuree z observed a e Sce z les bewee he easurees z 2 ad z z s a sgle-lag ou-ofsequece easuree by coveo Measuree Recep Te Measuree Te z 4 z z 3 z 2 z Measuree Fgure 3 The ou-of-sequece easuree z observed a e arrves afer he las processed easuree z observed a e Sce z les bewee he easurees z 4 ad z 3 z s a hree-lag ou-ofsequece easuree by coveo

3 The easuree odel ay be a lear or olear fuco of he sae The couous e lear syse dyacs for he sae x R s [7]-[9]: dx( ( 2) = F( x( + d where F( R ad R s a zero-ea whe Gaussa process ose wh power specral desy Q( R : ( 22) E{ } = 0 E{ τ )} = Q( δ ( τ ) The lear easuree odel for he dscree e easurees z R s [7]-[9] ( 23) z = H x + v = 0 where x : = x( ) ad H R s he easuree arx We assue ha he easuree ose v R s a zero-ea Gaussa whe-ose sequece: ( 24) v N (0 R ) E{ v v } = δ R R R The dscree-e olear easuree odel s descrbed by ( 25) z = h ( x ) + v = 0 where h : R R s a olear fuco of he sae Sce he easurees are avalable a dscree es { } s ecessary o have a dscree e syse dyac odel for sae esao ad predco The dscree e syse dyac odel obaed fro (2) s [7]: ( 26) x = Φ( ) x + ) = 2 (27) ) = Φ( d = 2 where Φ ( ) : = Φ( ) R s he sae raso arx ad w ( ) : = ) R s he egraed process ose For saoary syses wh e depede F (2) he sae raso arx ca be expressed a closed for ( 28) Φ( ) = exp[ F ( )] We oe ha { w ( )} for a zero-ea Gaussa whe sequece: ( 29) E{ )} = 0 = 2 K (20) E{ ) ) } = δ Q( ) = 2 where Q( ) R s he covarace arx of he egraed process ose w ( ) 3 Lear OOSM Flerg Before he ou-of-sequece easuree z s receved he las easuree updaed sae esae x ˆ ( + ) a e usg he lear Kala fler s gve by [7]-[9] ( 3) xˆ ( + ) = xˆ ( ) + K[ z H xˆ ( )] where x ˆ ( ) s he predced sae a e The Kala ga K R ad covarace for he ovaos process G R are gve by [7]-[9]: (32) ( ) K = P H G = 2 2 ( 33) G = H P ( ) H + R = 2 2 where ( x ) ad P ( ) R are he error xˆ ( ) ad covarace of x ( ) defed by ( 34) x ( ): = xˆ ( ) x = 2 ( 35) P ( ) : = E{ x ( ) x ( ) } = 2 Slarly he error x ˆ ( + ) ad he correspodg covarace P ( +) R are defed by ( 36) x ( + ): = xˆ ( + ) x = 2 (37) P ( + ) : = E{ x ( + ) x ( + )} = 2 The Kala fler algorh gves [7]-[9] (38) P ( + ) R P ( + ) = [ I K H ] P ( ) = P ( ) P ( ) H G H P ( ) = 2 2

4 3 Equvale OOSM Model The easuree equao for he OOSM z s ( 39) z = H x + v The OOSM z observed a e arrves a he ceral racg syse afer he easuree z observed a e I geeral he easuree e ay be l lags behd l < < l l =2 I Fgures 2 ad 3 l = ad 3 respecvely The ey dea s o express he OOSM z as a fuco of he sae x a he laes e [2]-[4] Usg (26) we ge ( 30) x = Φ( ) x + ) (3) ) : = Φ( d We ca wre ) (30) as { ) } 0 ( 36) E = Le Γ ( ; ) R deoe he covarace of w ( ; ) The ( 37) Γ( ; ) : = E{ ; ) ; ) } Sce ay wo of he egraed process oses (34) are ucorrelaed (38) Thus Q( ) = l Γ( l ; ) + Γ( ; ) = ( 39) ) N(0 Q( )) Usg (30) we ca express he sae of he sae x a he laes e x as a fuco : (32) ) = + l + Φ( l l Φ( d d + K + Φ( 2 d (320) [ x ) ] x = Φ ( ) = Φ( ) [ x ) ] Usg (320) (39) we ge he equvale OOSM odel Exag (32) ad (27) we oe ha he egraed process oses o he rgh had sde of (32) are dffere fro he egraed process oses (27) I (27) he upper l of egrao ad frs argue of Φ are he sae However (32) he upper l of egrao ad frs argue of Φ are dffere for all he egraed process oses excep he las er volvg he e erval [ 2 ] Ths suao arses whe he uber of lags l s greaer ha oe Therefore we defe a ew egraed process ose of he geeral for ( 33) ; ) : = Φ ( d > The (32) ca be wre as (34) ) = l ; ) l + ; ) = We oe ha for he sgle-lag proble (l=) ( 35) ; ) = ) Sce E{ ; ) } = 0 ( 32) z = A x + v wh he easuree arx easuree error e R defed by ( 322) A : = H Φ( ) ( 323) e : = v A ) A R ad Sce v ad ) are zero-ea ad Gaussa e s also zero-ea ad Gaussa wh covarace arx P e R ( 324) e N (0 P e ) where Pe s o be deered The easuree ose v ad ay egraed process ose ) are ucorrelaed Therefore ( 325) E{ v )} = 0 Usg (323)-(325) (24) ad (39) we ge (326) Pe : = cov( e ) = E{ e e } = AQ( ) A + R

5 Sce x ( + ) depeds o ) = -l - l+ - ad e depeds o ) x ( + ) ad e are correlaed Le Px + e R ( ) deoe he cross-covarace bewee x ( + ) ad e Sce x ( + ) ad e are zero-ea ad Gaussa ( 327) P : { ( ) ( ) x + e = E x + e } We eed o deere P x ( + ) e order o process he OOSM z for sae esao 32 Processg OOSM x ( + ) s he sae esao error afer processg he easurees Z : = ( z z2 z ) x ( + ) ad v are ucorrelaed Le P x + w R ( ) deoe he cross-covarace bewee x ( + ) ad ) Hlo Mar ad Blar [2] ad Bar-Shalo [4] ae o accou he cross-covarace P x ( + ) w for he sgle-lag proble However he algorhs [2] ad [4] do o accou for he ulple-lag OOSM proble The algorh [4] s a codoal u ea square error (MMSE) esaor codoed o he easurees Z ad cludes he codoal ea of ) gve Z We shall refer o hs opal algorh for he sgle-lag proble as he A algorh he suffx correspodg o he sgle-lag proble The algorh [2] for he sglelag proble o be called he B algorh does o clude he codoal ea of ) gve Z ad zes he ucodoal MMSE 33 The A Algorh The seps of he A algorh are (328) xˆ( + ) = xˆ ( + ) + KA[ z A ( x E{ ) Z })] (329) K A = CG ( 330) C = [ P ( + ) + P x ( + ) ] w A (33) G = P ( + ) + { Q( ) A Q( ) H G HQ ( )} A + R P x ( ) w P x ( + ) w (332) E{ Z } = wˆ ( κ ) = Q( ) H G z ( 333) z : = z zˆ The expresso for P ( + ) w for he sgle-lag OOSM proble [2] [4] [6] s x ( 334) P x = ( I KH ) Q( ) ( + ) w 34 The B l Algorh Nuercal resuls preseed [4] for lear easuree odels show ha he B algorh [2] s oly slghly sub-opal ad requres less copuao copared o he opal A algorh For praccal probles of eres where he easuree odel s olear hs dfferece s eglgble copared o errors approxaos used for olear easuree odels Therefore for praccal reasos we chose o geeralze he B algorh for he ulplelag proble [6] o oba he B l algorh The equaos for he sae ad covarace updaes usg he OOSM for he B ad B l algorhs are he sae The dffereces are due o he equaos for P x ( + ) w The sae ad covarace updae equaos for he Bl algorh are gve by [6]: ( 335) xˆ( + ) = xˆ ( + ) + K B[ z A xˆ ( + )] (336) K B = CC2 (337) C = A [ P ( + ) + P + ( P 2 x ( ) + w x ( + ) w + Pe C2 R ( 338) P x ( ) e = P x ( + ) w A + ) ] A ( 339) x ( + ) : = xˆ( + ) x ( 340) x ( + ) = ( I KA ) x ( + ) + Ke ( 34) E{ x ( )} 0 sce { + = E x ( + )} = 0 E{ e } = 0 ( 342) P( + ) : = E{[ x( + ) Ex ( + )][{ x( + ) E x ( + )]} ( 343) P ( + ) = P ( + ) [ CK B + ( CKB )' ] + KBC 2 KB P ( + ) w equals x Qe [2] [3]; he egave sg arsg due o he oppose sg coveos used for x ( + ) We oe ha C 2 s syerc ad posve defe Usg (336) we have: ( 344) C K = ( ) 2 = 2 = ( ) B C C C C C C C K B ( 345) K B C2 K B = CC2 C2 ( C2 ) C = CC2 C

6 Subsuo of (344) ad (345) (343) gves: ( 346) ( ) ( ) C C P + = P + 2 C P + s ow fro prevous Kala fler processg of z The oly quay o evaluae s he cross-covarace P x ( + ) w The expresso for P x ( + ) w for he sgle-lag OOSM proble ( B algorh) [2] [3] [6] s gve by (334) A s evaluaed by (322) ad ( ) The geeral expresso for x P ( + ) w for he ulplelag OOSM proble ( B l algorh) s derved [6] ad s gve by: (347) (348) P (349) M : = x ( + ) w = M lq( l ; l ) l M Q( ; ) = Q( ; ) : = E{ ; ) ; )'} B = CC 2 C + B ( 350) B = I K H ( 35) C = BΦ( 2) 4 Sulao ad Resuls = 23 l We cosder he early cosa velocy oo oe deso wh poso ad velocy easurees [6] The sae raso arx ad he easuree arx are gve respecvely by (4) Φ( ) = Φ( 0 ( 42) H = 0 ( ) ) = 0 We choose he easuree error covarace arx (43) R = s correspodg o oe- wo- ad hree-lag probles respecvely The fler copued covarace arces for he A B ad B l algorhs represe he rue covaraces Sce he esao proble s lear wh Gaussa dsrbuos he race of he covarace arx such a codo s a rue easure of he sae esao accuracy The algorhs A ad B are o srcly vald for ulple-lag OOSM probles Our obecve s o see f hey ca be used a approxae aer whou sgfca errors Tables ad 2 a he ed of hs paper prese he covaraces ad races of he covaraces of sae esao error for varous cases We observe ha he covarace of he ulple-lag algorh B s early equal o he opal covarace l The covarace of he A algorh degrades wh creasg uber of lags The covarace of he B algorh degrades ad becoes o-posve-defe for he hree-lag proble Nuercal resuls also show ha he B algorh fals for hgher lag probles Resuls Table 2 show ha as he uber of lags creases he corbuo of he OOSM o he sae esao accuracy decreases Therefore f he OOSM coas oly he eac easuree (eg rage azuh rage-rae for a GMTI repor he wheher o process a OOSM wll deped o he accuracy ad e delay of he OOSM pror rac accuracy ad syse requree for rac accuracy However f he OOSM coas boh eac ad arge classfcao easuree (eg GMTI-HRR) [0]-[2] ad he arge classfcao easuree s hghly accurae he s esseal o process he OOSM for aag couous arge ID eve f he eac easuree does o prove he eac rac accuracy sgfcaly 5 Coclusos Nuercal resuls preseed hs paper show ha he ulple-lag algorh B s very close o he opal algorh The covarace of he sgle-lag A l algorh degrades wh creasg uber of lags The covarace of he sgle-lag B algorh degrades ad becoes o-posve-defe for he hgher uber of lags We used a cosa revs e of q = 05 2 s 3 = s ad he sulao The easuree error covarace arx of he OOSM was he sae as ha of oher easurees Sx easurees cludg he OOSM were geeraed The e delay of he OOSM was chose as 05 5 ad 25 secods

7 Refereces [] H Che T Krubaraa ad Y Bar-Shalo Deceralzed vs Ceralzed Tracg for Ar-o-ar Scearos Sgal ad Daa Processg of Sall Targes: Proceedgs of SPIE vol 4048 pp Aprl 2000 [2] R D Hlo D A Mar ad W D Blar Tracg wh Te-Delayed Daa Mulsesor Syses NSWCDD/TR-93/35 Dalhgre VA Augus 993 [3] S Blaca ad R Popol Desg ad Aalyss of Moder Tracg Syses Arech House 999 [4] Y Bar-Shalo Updae wh Ou-of-Sequece Measurees Tracg: Exac Soluo Sgal ad Daa Processg of Sall Targes: Proceedgs of SPIE vol 4048 pp Aprl 2000 [5] J R Moore ad W D Blar Praccal Aspecs of Mulsesor Tracg Mularge-Mulsesor Tracg: Applcaos ad Advaces Volue III Y Bar-Shalo ad Wlla Dale Blar (ed) pp -76 Arech House 2000 [6] M Mallc S Coralupp ad C Carhel Advaces Asychroous ad Deceralzed Esao Proceedgs of he 200 IEEE Aerospace Coferece Bg Sy MT March 200 [7] Gelb A Ed Appled Opal Esao The MIT Press 974 [8] Bar-Shalo Y ad L X Rog Esao ad Tracg: Prcples Techques ad Sofware Arech House 993 (repred by YBS Publshg 998) [9] B D O Aderso ad J B Moore Opal Flerg Prece Hall 979 [0] R Wllas J Weserap D Gross ad A Paloo Auoac Targe Recogo of Te Crcal Movg Targes Usg D Hgh Rage Resoluo (HRR) Radar IEEE AES Syses Magaze Aprl 2000 [] R Popp N Sadell R Washbur H Maey B Hodges A Baley M Mallc ad B Johso MTE Groud Sao Tesbed 999 [2] T G Alle D A Casao I A Farber W C Karl ad M Predy Mulresoluo Fuso of MTI HRR ad SAR for Ehaced Targe Tracg ad Classfcao of Groud Targes TR-952 ALPHATECH Ic Jauary

8 Table Coparso of covaraces of sae esao error for varous OOSM algorhs Nuber of Lags Before Processg OOSM Processg OOSM Order Opal Covarace 2 2 s Processg OOSM wh Mul-lag Algorh B l [6] 2 s 2 2 s Processg OOSM wh Oelag Algorh A [4] Processg OOSM wh Oe-lag Algorh B [2] Table 2 Coparso of race of covaraces of sae esao error for varous OOSM algorhs Nuber of Lags 2 3 Before Processg OOSM Processg OOSM Order Opal Trace of Covarace Processg OOSM wh Mul-lag Algorh B l [6] Processg OOSM wh Oelag Algorh A [4] Processg OOSM wh Oelag Algorh B [2] Falure