Geometric Factors in Target Positioning and Tracking

Transcription

1 th Intenatonal Confeence on Infomaton Fuson Seattle, WA, USA, July 6-9, 9 Geometc Factos n aget Postonng and ackng Chun Yang Sgtem echnology, Inc. San Mateo, CA 944 chunynag@sgtem.com Ek Blasch A Foce Reseach Lab WPAFB, OH ek.blasch@wpafb.afb.ml Ivan Kada Intelnk Scences Systems, Inc. Lake Success, NY 4 kada@systemsscences.com Abstact - In taget postonng and tackng, most sensos povde measuements ethe as ange o beang o both. he measuements ae used to update an a po estmate ethe va a lneazed least squaes method o an etended Kalman flte. In ethe case, the esultng soluton has two components, one s elated to the measuement pedcton eos and the othe s an obsevaton mat obtaned fom lneazng the nonlnea measuement equatons aound the a po estmate. hs pape studes the geometc factos eplctly and elates the obsevaton mat to the lne of sght (LOS vecto fo a angng senso and the decton pependcula to the LOS vecto of a beang-only senso. As a esult, the updatng of estmaton eo covaance wth ange and beang measuements can be ntutvely assessed va the shapng of estmaton eo ellpse along LOS dectons. It povdes a valuable means fo taget postonng and tackng pefomance modelng and pedcton and can thus be used n actve management of dstbuted senso esouces and senso path plannng. Keywods: Rangng & Beang-Only Sensos, Geomety, LOS, GDOP. Intoducton Passve and actve sensos ae wdely used n taget poston locaton and tackng [,, 3, 7, ]. Passve sensos nclude ( acoustc sensos that detect gound vehcle moton and vbaton and ( magng sensos that captue eflected vsble lghts o adated themal enegy fom a taget. In addton to sgnatues chaactestc to tagets fo possble dentfcaton, these passve sensos also measue the decton of aval of the mechanc o electomagnetc sgnals of the tagets n tems of the beang angles elatve to the senso platfoms fo postonng and tackng. Addtonal eamples nclude passve sona sensos fo undewate applcatons and passve ada sensos fo space obsevatons [8]. Actve sensos nclude ada, lase ada, and actve sona. By measung the ound tp tme of flght of an enegy pulse, actve sensos povde ange measuements to tagets. me dffeence of aval (DOA s a common technque fo dffeental ange calculatons. Some actve sensos also measue the angles of aval (AOA of etuned pulse o the decton n whch the tansmt beam s ponted. Monopulse and antenna aay technques povde moe accuate angula measuements than mechancally scanned antennas. Othe actve sensos ae capable of etactng the Dopple fequency shfts mpacted by taget moton, thus measung the ange ate. Accodng to the type of measuements, most sensos used n taget postonng and tackng can be vewed as measung ethe ange o beang o both (wthout consdeng ange ate and taget sgnatue measuements []. A taget s poston s detemned usng multlateaton wth angng sensos o tangulaton wth angula sensos o the mtue. Multlateaton o tangulaton can be done when a taget s vewed by multple dstbuted sensos o by a sngle senso movng along (multple vews. A batch pocessng mode may use the conventonal least squaes (LS method wheeas a sequental pocessng mode calls fo the Kalman flte [,, 3, 7, ]. Indeed, the senso measuements ae used to update an a po estmate ethe va a lneazed least squaes method o an etended Kalman flte. In ethe case, the esultng soluton has two components, one s elated to the measuement pedcton eos and the othe s an obsevaton mat obtaned fom lneazng the nonlnea measuement equatons aound the a po estmate. he obsevaton mat, whch detemnes the estmaton eo covaance, has a dstnct stuctue contanng the geometc nfomaton fo a gven taget postonng and tackng settng. It s well known that the obsevaton mat s elated to the sensng geomety. Howeve, ths pape elucdates the geometc factos fo both actve and passve sensos [4, 4, 5] and eplctly elates the obsevaton mat to the lne of sght (LOS vecto fo a angng senso and the decton pependcula to the LOS vecto of a beang-only senso. he geometc epesson povdes a mathematcal bass fo the well-known fact that the eo ellpses fo co-located passve and angng sensos ae pependcula. Futhemoe, the updatng of estmaton eo covaance (.e., the egen stuctue wth ange and beang measuements can be ntutvely assessed va the shapng of estmaton eo ellpse along LOS dectons. Fo an antcpated tackng scenao, ths povdes an effcent means fo taget postonng and tackng pefomance modelng and pedcton and can thus be used n actve management of dstbuted senso esouces and senso path plannng [5, 6, 9, 6]. he est of the pape s oganzed as follows. In Secton, the least squaes method s used to elate the poston soluton to the obsevaton mat and to the LOS vectos. In Secton 3, the updatng of eo covaance by LOS vectos s chaactezed wth smulaton eamples. Fnally, the pape s concluded n Secton ISIF 85

2 Geometc Scalng va LOS Vectos In ths secton, we fst pesent the poston soluton va the least squaes method. We then elate t to the obsevaton mat va lneazaton. Net we epess the obsevaton mat n tems of LOS vecto fo actve and passve sensos.. Least Squaes Solutons Consde a taget at and the -th senso at. he -th senso s measuement s gven by: z f (, + v ( whee f (, s a nonlnea measuement equaton and v s the senso measuement eo beng zeo-mean Gaussan N(,. Denote an ntal estmate of the taget state by. he nonlnea measuement, be a ange o a beang, can be lneazed aound the estmate: z f + v (a (, + h ( z z z h (b n In tems of measuement pedcton eo, the equaton can be futhe wtten as: ~ z z f h + v (3 (, + h Fo m sensos, the lneazed measuements can be put nto a vecto fomat as: ~ z H + v (4a ~ z [ ~ z ~ z ~ zm] (4b v [ v v vm ] (4c H h h (4d [ ] m An estmate of the taget state gven the measuements can be obtaned as the least squaes (LS soluton to the followng pefomance nde as: ag mn( ~ ˆ z H R ( ~ z H (5a whee R s the measuement nose covaance mat defned as: R E{ vv } (5b he least squae soluton s gven by: ˆ ( H R H H R ~ z (6a P E {( ˆ ( ˆ} ( H R H (6b Assume that, n addton to the ntal estmate of the taget state, the estmaton eo covaance, denoted by P, s also known. hen, gven the taget measuements as n (4a, we can obtan the weghted least squae (WLS soluton that mnmzes the followng pefomance nde as: ˆ ag mn( ~ ( ~ z H R z H + ( P ( (7 he least squaes soluton s gven by: ˆ ( ( ~ H R H + P H R z + P (8a P ( H R H + P (8b Altenatvely, we can teat the a po knowledge and P as an addtonal measuement and obtan the augmented measuement equatons as: ~ z H + v (9a + ε (9b v ~ N(, R, ε ~ N(, P, E{vε } (9c Now apply the Kalman flte s measuement updatng step to the augmented measuement [ ~ z ] as: ~ z R H [ ] R ˆ H I [ ] H I ( P I P he tems n font of [ ~ z ] n ( consttute the Kalman flte gan. Epandng the matces n tems of the elements leads to the soluton (8a. Smlaly, the Kalman flte s covaance updatng step can be wtten up fo the augmented equatons, leadng to the updated covaance (8b. Note that the nvese of the covaance mat n (8b s also called the nfomaton mat, denoted by Λ: Λ P H R H + P (. Actve Rangng Sensos Fst consde the case wth actve sensos (angng measuements n a two dmensonal settng. he nonlnea ange equaton ( s spelled out as (gnoe the nose tem: f (, ( + ( y y (a he coespondng lneazed equaton aound an ntal estmate of s:, + h ( (b f ( wth h y y e sn( θ cos( θ (c It s clea fom (b that h s the lne of sght (LOS vecto fom the -th senso to the taget, denoted by e, as shown n Fg...3 Passve Beang-Only Sensos Refeng to Fg., we now consde the case wth passve sensos (beang measuements stll n a two dmensonal settng. 86

3 Fg. Rangng and Beang-Only Sensos he nonlnea angula equaton s: y y f (, θ tan ( (3a whch can be lneazed aound an ntal estmate of as: θ, + h ( (3b f ( whee y y sn( θ sn( θ e h cos( θ cos( θ (3c Compang (3c to (c, t shows that the measuement mat h fo the beang-only senso s ange-dependent. Moe mpotantly, t s pependcula to the lne of sght vecto fom senso to taget. When the ange-dependence and angula eos ae combned, t povdes a poston eo of along the decton pependcula to the LOS (.e., along e..4 Geometc Dluton of Pecson (GDOP A scala value that chaactezes the poston soluton s the geometcal dluton of pecson (GDOP defned as [4, 3]: GDOP tace(( H R H (4a tace(( H H when R I (4b Fo the case wth two angng sensos, the GDOP can be wtten as: GDOP Range + (5a sn ( θ θ y n +, θ θ π / (5b, θ θ, π (, y e e n θ e n θ e n e (, y ( n, y n e θ (, y Fo the case wth two beang-only sensos, the GDOP can be wtten as: GDOP Beang + (6a sn ( θ θ +, θ θ π / (6b, θ θ, π Eqs. (5 and (6 ae consstent wth ou ntuton that two pependcula sensos can poduce the best estmate and two collnea sensos cannot poduce a soluton. Dffeent fom angng sensos, howeve, beang-only sensos have ange tems eplctly as n (3c. In fact, the eo ellpses fo passve and angng sensos ae pependcula ae well known. It s meanngful to eplctly show t fom the obsevaton mat n tems of LOS vectos. As shown net, the lne of sght vectos and measuement eo covaance matces ae elated to the egnevalues and egenvecto dectons of the estmaton eo ellpses. When used to update a poston soluton, they shape the eo ellpses by adjustng ts sze and oentaton va the egenstuctue. 3 LOS Updatng and Chaactezaton As shown n the last secton, the lneazed measuement equaton fo an actve angng senso o a passve beangonly senso s chaactezed by ( the lne of sght (LOS vecto o ts pependcula and ( the measuement eo vaance. A poston soluton s also detemned by these two factos as s evdent fom the followng equaton of estmaton eo covaance: P ( H R H + P (7 whee P s the a po covaance mat, P s the a posteo covaance mat, H s the measuement mat, and R s the measuement eo covaance. o smplfy the analyss, fst consde the case whee P I, H a [cosθ, snθ], whch s a ow vecto wth θ beng the angle elatve to the -as, and R. hs epesents the case whee the a po estmate s updated by a sngle measuement. he updated covaance s gven by: ( H R H + P ( I + aa I γaa (8a whee γ a a + +,, (8b It s easy to vefy that a s an egenvecto of the mat (I γaa wth the coespondng egenvalue ( γa a γ. Fo the D case, the othe egenvecto s pependcula to a, denoted by a, wth the coespondng egenvalue beng unty. 87

4 In tems of geomety ntepetaton, the unty ccle s shaped nto an ellpse wth the followng egenvalues and egenvectos: Λ dag([ γ ] (9a a a cosθ snθ V [a a ] (9b a a snθ cosθ he followng smulaton s used to llustate the chaactestcs of LOS updatng. Let the ntal covaance be a unt mat, P dag([, ], whch epesents a ccula eo as shown n Fg.. We vay the LOS vecto s oentaton θ elatve to the -as and the measuement eo covaance R. Fg. shows two updates at (θ, (3 o, and (7 o,.5, espectvely. Each LOS update squeezes the ccle nto an ellpse wth dffeent oentaton and dffeent eccentcty. he oentaton s such that the egenvecto coespondng to the small egenvalue ponts along the LOS decton. Fg. 3 shows egenvecto constcton whee the majo and mno egenvectos ae off by 9 o as epected and the mno egenvecto s angle φ changes lnealy as the LOS angle θ. he majo egenvalue emans constant but the mno egenvalue changes nvesely wth (accodng to (5b as shown n Fg. 4. he bette the senso pefomance (the smalle, the smalle s the mno egenvalue afte update and vce vesa. If a senso staes at a taget n the same decton, the eo ellpse s updated epeatedly by the same LOS vecto (assume the taget s not movng. hen the educton of the mno egenvalue ove tme (.e., the numbe of updates s shown n Fg. 5 fo dffeent senso qualty (.e., R. hs s equvalent to tme aveagng. Fo a sngle senso, anothe way to educe the eos n taget postonng s to ccle the taget as n a spot synthetc apetue ada (SAR opeaton. Update n one decton squeezes the eo n that decton. When t s done n all dectons, t makes a smalle eo ccle as shown Fg. 6 whee the eos at 5 dscete angles ae shown togethe wth the ntal lage ccle. Fg. 7 shows the educton of the egenvalues as a functon of LOS angle θ. As shown, the majo egenvalue deceases athe quckly ove the fst 9 o and then much slowly fo net 9 o and does not change much fo the emanng 8 o. hs ndcates that a sngle angng senso can sgnfcantly educe postonng eo by cclng the taget by 9 o o so, whch s consstent wth ou ntuton and can be used as a pactcal gudelne n senso placement and schedulng. Consde the case wth two LOS updates smultaneously. he two sensos have the LOS vectos a [cosθ, snθ ] and a [cosθ, snθ ] wth and, espectvely. he updated covaance s gven by [8]: ( I + a a + a a I γ ( γ aa + γ a a ργ γ aa ργ γ a a (a whee ρ aa (b γ + a a + (c γ + a a + (d γ ρ γ γ (e Now consde two specal cases. In the fst specal case, a a but thee s unequal vaance. hat s, a taget s detected n the same decton but at dffeent anges. It s easy to vefy that (a becomes: ( I + aa I γaa whee (a + (b + γ (c If, then / fo an addtonal update. hs s equvalent to educng the measuement eo vaance by n wth n addtonal updates. he coespondng egen value deceases accodng to /(n+ fo unty vaance (. he evaluaton of contnuously updated covaance s smla to unnng the covaance update equaton of a Kalman flte (.e., the Rccatt equaton. In the second specal case, a a, that s, they ae pependcula to each othe. hen ρ n (b and γ n (e and (a becomes: ( I + aa + aa I γ aa γ aa ( It s easy to vefy that ( has the followng egenvalues and egenvectos: ] Λ dag([ γ γ (3a V [a a ] (3b hs ndcates that fo two othogonal LOS vectos, smultaneous update wth two othogonal sensos s equvalent to two ndependent updates n tems of the fnal eo ellpse and the egen-stuctue. Fo the geneal case wth two smultaneous LOS updates, the updated covaance s gven by (. he calculaton of ts egenvalues and egenvectos s athe nvolved ecept fo the two specal cases gven n ( and (. he followng smulaton s used to analyze the geneal egenvalues and egenvectos. Agan let the ntal covaance be a unt mat, P dag([, ]. We f the fst LOS vecto at θ 3 o wth R. We then vay the second LOS vecto θ fom o to 88

5 eo ellpse e-shaped by los update eo ellpse e-oented by los update.8.6 ntal P θ3 o, θ7 o, mno eg φ majo eg φ y pncpal as angle φ los angle θ Fg. Update of Unt Ccle wth Dffeent and θ eo ellpse e-szed by los update Fg. 3 Egenvecto Dectons φ vs. LOS Angle θ eo ellpse e-szed by epeated updates egen values ma egen mn egen mn & ma egen values R update qualty, R update # Fg. 4 Egenvalues vs. Senso Qualty Fg. 5 Egenvalues vs. Numbe of Updates y eo ellpse e-shaped by contnuous ccula update ntal P updated 3 o updated 35 o updated 5 o updated 95 o updated 355 o ma, mn egen values eo ellpse e-shaped by contnuous ccula los update ma egen mn egen Fg. 6 Eos ove Ccula Updates los angle θ Fg. 7 Egenvalues 89

6 8 o and R fom. to. he majo and mno egenvalues as a functon of θ fo dffeent R ae showed n Fgs. 8 and 9, espectvely. Snce θ 3 o, the ma and mn values fo both majo and mno egenvalues appea at θ 3 o and θ o, espectvely, whch epesent the two specal cases gven n (8 fo θ θ - θ o and n (9 fo θ θ - θ 9 o. Agan wth the fst LOS vecto fed at θ 3 o wth R, Fgs. and show the majo and mno egenvecto angles as the second LOS vecto θ vaes fom o to 8 o and R fom. to. he two egenvecto angles have the same shapes ecept that they ae off by 9 o. Fo small R, the second update domnates. As a esult, the mno egenvecto ponts n ths LOS decton and the majo egenvecto s pependcula to t. Fo lage R, the second update becomes nconsequental and the fst update domnates. hat s why the cuves eman close to θ as θ vaes ts value. It s nteestng to note the cuves fo R R seemly have a jump of 9 o at θ o o θ -θ 9 o. hs s actually due to the swtch of labelng of majo and mno fo the same egenvectos. hs s best llustated n Fg. fo egenvalues and Fg. 3 fo egenvectos fo R R as a functon of θ. As shown n Fgs. and, when R and R ae compaable, the mno egenvecto angle les between θ and θ. As the second LOS vecto θ moves away fom the fst LOS vecto θ, the mno egenvecto also moves away fom θ. But the egenvecto angula dsplacement lags that of θ and stays somewhee between θ and θ. Based on the smulaton esults, heustc cuves ae shown n Fg. 4 fo analyss and desgn. Fg 4(a shows the appomate egenvalues as a functon of angula sepaaton between the two LOS vectos. Fg 4(b shows the appomate egenvecto angles as a functon of angula sepaaton between the two LOS vectos. Lnea ntepolaton can be used to fnd the egenvalues and egenvecto angles. In the fgues, the effects of R and R ae gnoed but could be added wth dffeent cuves n the same plot. Fo an abtay covaance mat A and ts nvese A -, the egen-decomposton s wtten as: A A UΛ U (4b whee UΛU (4a [ u ] U u n (4c Λ dag( [ λ λn ] (4d Λ ([ dag λ λ ] (4e n ae the egenvectos, egenvalues, and nvese egenvalues, espectvely. he mat A and ts nvese A - can be futhe wtten as: n A λ u u (5a n uu A λ (5b Snce the egenvectos ae othonomal, they can be consdeed as the LOS vectos as n (. As a esult, the update of a unty ccle wth a covaance mat (o a full ank obsevaton mat can be eadly obtaned fom (3, leadng to the updated ellpse, whch s e-oented by the ognal egenvectos (3b but e-szed by the updated egenvalues (3a. hs thus povdes a theoetcal justfcaton of ou ntuton about LOS updates, a geometcal ntepetaton of combnng two ellpses, and a computatonal pocedue fo fusng them. Snce the tace of a mat s the sum of ts egenvalues, we can evaluate the geometc dluton of pecson (GDOP dectly fom egenvalues. Fo the case of updatng wth two LOS vectos whee θ 3 o and R, Fgue 5 shows the taces as a functon of θ fo dffeent R. When θ θ, the tace s γ, whch s the lagest value. he smallest value occus at θ - θ 9 o, whch s - γ - γ. At othe θ, the tace les n between followng a snusodal cuve. Fg. 6 s a lnea appomaton. Fo lage R, the tace does not change much wth θ. But t s sgnfcant fo small R. As shown n Fg. 5, the change n tace ove θ 9 o s about the same fo R. to wth R. hs s a tadeoff between assgnng sensos wth good qualty vs. wth good geomety ove dffeent tme hozons. he above analyss has been appled to the geneal case wth an abtay ntal eo covaance []. Its elatonshp wth GDOP [3] as well as geometc measue of mets (GMOM [4] s pesented n the contet of senso esouce management fo layeed sensng []. 4 Conclusons In ths pape, the use of ange and beang measuements fo taget postonng and tackng was nvestgated fom the geometc pont of vew. As a scalng facto to the measuement pedcton eo vecto, the obsevaton mat was shown to be made of lne of sght (LOS vectos fom angng sensos to taget and unt vectos pependcula to the LOS scaled by ange fom beangonly sensos. As shown, the LOS vectos togethe wth the senso measuement eo covaance detemne the egenstuctue of poston estmaton eo covaance, shapng the eo ellpses. hs geometc nsght nto taget postonng and tackng wth ange and beang measuements s helpful n desgnng an effcent esouce management stategy fo dstbuted sensos. Acknowledgements Reseach suppoted n pat unde Contacts No. FA865-5-C- 88 and FA865-8-C-47, whch ae gatefully acknowledged. 9

7 eo ellpse e-shaped by two los updates.55 eo ellpse e-shaped by two los updates ma egen value R. R.5 R R 5 R mn egen value R. R.5 R R θ 3, R.5. θ 3, R R los angle θ Fg. 8 Ma Egen ove θ fo Dffeent los angle θ Fg. 9 Mn Egen ove θ fo Dffeent 5 eo ellpse e-shaped by two los updates R. R.5 R 8 6 eo ellpse e-shaped by two los updates R. R.5 R ma egen vecto angle 5 R 5 R θ 3, R mn egen vecto angle R 5 R θ 3, R los angle θ Fg. Ma Egenvecto ove θ fo Dffeent los angle θ Fg. Mn Egenvecto ove θ fo Dffeent. eo ellpse e-shaped by two los updates eo ellpse e-shaped by two los updates. 5 ma & mn egen values θ 3, R /R los angle θ ma & mn egen vecto angles 5-5 θ 3, R /R los angle θ Fg. Ma and Mn Egenvalues ove θ Fg. 3 Ma and Mn Egenvectos ove θ 9

8 Majo Egenvalue ma( γ, γ mn( γ, γ θ + 9 o (θ Mno Egenvecto Angle (Majo Egenvecto Angle R /R ~ R /R ~ θ (θ -9 o R Mno Egenvalue /R ~ γ θ θ θ θ θ θ o 9 o o 9 o (a Appomaton fo Egenvalues (b Appomaton fo Egenvecto Fgue 4 Lnea Intepolaton fo Egenvalues and Egenvecto as a Functon of θ.5 eo ellpse e-shaped by two los updates.4 tace:(ma + mn egenvalues R. R.5 R R 5 R γ ace Lnea Appomaton γ γ.6 θ 3, R los angle θ Fg. 5 ace vs. θ fo Dffeent o 9 o Fg. 6 Appomaton fo ace θ θ θ Refeences [] Y. Ba-Shalom and X.R. L, Multtaget-Multsenso ackng: Pncples and echnques, YBS Publshng, Stos, C, 995. [] S. Blackman and R. Popol, Desgn and Analyss of Moden ackng Systems, Atech House, Boston, 999. [3] A. Fana and F.A. Stude, Rada Data Pocessng (Vol. I & II, Wley, New Yok, 985. [4] I. Kada, Optmum Geomety Selecton fo Senso Fuson, Poc. SPIE Vol. 3374, p. 96-7, Sgnal Pocessng, Senso Fuson, and aget Recognton VII, Ivan Kada (Ed., Apl 998. [5] I. Kada, E. Blasch, and C. Yang, Invted Panel Dscusson: Issues and Challenges n Model-Based Pefomance Assessment of Mult-aget ackes, Poc. of SPIE Defense and Secuty 8: Sgnal Pocessng, Senso Fuson, and aget Recognton XVII, Olando, Fl, Mach 8. [6] B. Kahle and E. Blasch, Senso Management Fuson Usng Opeatng Condtons, Poc. of IEEE NAECON 8. Dayton, OH, July 8. [7] P. Maybeck, Stochastc Models, Estmaton, and Contol, Volume, Academc Pess, Inc, 979. [8] S.U. Plla, K.Y. L, and B. Hmed, Space Based Rada: heoy & Applcatons, McGaw Hll, New Yok, 8. [9] R. Popol, he Senso Management Impeatve, Ch. n Multtaget-Multsenso ackng: Applcatons and Advances, Vol., Y. Ba-Shalom (Ed., 35-39, Atech House, Nowood, MA, 999. [] B. Rstc, S. Aullampalam, & N. Godon, Beyond the Kalman Flte: Patcle fltes fo tackng applcatons, Atech House, 4. [] C. Yang, Pefomance Montong and Pedcton fo Actve Management of Dstbuted Sensos Fuson n aget ackng, Status Repot No. 4 (FA865-8-C- 47, Novembe 8. [] C. Yang, I. Kada, and E. Blasch, Pefomance-Dven Resouce Management n Layeed Sensng, Fuson 9, Seattle, WA, July 9. [3] R. Yalagadda, I. Al, N. Al-Dhah, and J. Heshey, GPS GDOP Metc, IEE Poc. Rada, Sona Navg, 47(5, Oct.. [4] A. Kelly, Pecson Dluton n angulaton Based Moble Robot Poston Estmaton, Intellgent Autonomous Systems, Amstedam, 3. [5] I. Kada, Passve Multsenso Multtaget Featue-Aded Unconstaned ackng: A Geometc Pespectve, Fuson, July. [6] M. Khaa and. Okada, A Satellte Selecton Method and Accuacy fo the Global Postonng System, Navgaton: Jounal of the ION, 3(,