Altitude Estimation for 3-D Tracking with Two 2-D Radars

Transcription

1 th Interntonl Conference on Informton Fuson Chcgo Illnos USA July -8 Alttude Estmton for -D Trckng wth Two -D Rdrs Yothn Rkvongth Jfeng Ru Sv Svnnthn nd Soontorn Orntr Deprtment of Electrcl Engneerng Unversty of Texs t Arlngton TX 769 USA E-ml: yothn.rkvongth@mvs.ut.edu orntr@ut.edu ARCON Corporton Wlthm MA USA E-ml: jfeng@rcon.com sv@rcon.com Abstrct Ths pper frst proposes method to estmte the lttude of n rcrft usng two -D rdrs for Ar Trffc Control ATC pplctons. Assumng tht the rcrft fles wth nerly constnt velocty nd nerly constnt heght we frst estmte ts velocty ntl poston n -D coordntes from the mesurements of two -D rdrs usng mxmum lkelhood estmtor. Next we ncorporte the estmtes s ntl vlues for -D flght trckng tht uses n extended Klmn flter EKF. Furthermore multple model pproch s dopted to further mprove the trckng performnce. The smulton results re shown to demonstrte the effectveness of the proposed method. Keywords: Alttude estmton prmeter estmton -D trget trckng extended Klmn flterng. I. INTRODUCTION Informton on the lttude of n rcrft s necessry for Ar Trffc Control ATC to mnge the r trffc sfely nd effectvely. Generlly for n rplne flyng bove ft the rcrft s requred to report the lttude to the ATC center [] through Mode C trnsponder reples to the secondry survellnce rdr nterrogtons. If the rplne s not equpped wth Mode C trnsponder ts lttude cnnot be observed drectly by the ATC center nd the rdrs cn provde only -D mesurements.e. trget s rnge nd zmuth. Ths results n the need for estmton of the rcrft lttude n such stutons. To perform -D trckng usng mesurements from -D rdrs t nturlly requres method of lttude estmton to ntlze the trckng lgorthm owng to the dmensonl ncompleteness of mesurements. To estmte the ntl lttude from set of mesurements we usully hve to mpose n ssumpton to be ble to fnd the soluton. A common nd prctcl ssumpton n ATC pplctons s tht the rplne s crusng wth constnt lttude velocty nd hedng. In [] method to estmte the lttude usng sngle -D rdr by fttng the rnge curve from few rnge mesurement dt s proposed but the lttude estmton errors re lrge []. Ths s cused by the low observblty of the trget sttes from sngle rdr whch suggests tht we use more rdrs. In [] -D trckng pproch usng mesurements from two or more -D rdrs n WGS-8 coordntes s proposed. However the RMSE of three dmensonl poston theren mkes t not very ttrctve for prctcl pplctons. In [] n lgorthm referred to s the heght-prmeterzed extended Klmn flterng HPEKF whch ncorportes the utonomous multple model AMM method [6] s presented. In prtculr the lttude ntervl of nterest s frst prttoned nto subntervls n ndependent extended Klmn flter EKF s run for ech of the ntervls nd fnlly the overll trget s stte estmte s obtned by combnng the EKF models stte estmtes. In [7] nother lttude estmton method usng -D rdr network s proposed. Ths method consders the erth surfce s curvture nd employs the mxmum lkelhood ML estmtor to fnd the trget s lttude usng the mesurements obtned from synchronous multple rdrs. Recently n pproch for lttude estmton nd mtgton of slnt rnge errors on -D trget trckng wth -D rdrs hs been proposed n [8] where t uses mesurements n polr coordntes nd stte vectors n geodetc coordntes. In ths pper we propose n pproch for lttude estmton nd -D trckng usng two -D rdrs wth the ssumpton of constnt velocty nd lttude. Even though the method proposed heren does not depend on the number of rdrs we consder the cse of two rdrs becuse n relty mny rspces re stll covered by just two rdrs. Also f the trget s covered by more thn two rdrs see [9] nd [] for exmple multlterton methods cn be used to estmte the lttude. We do not consder the cse of non-prmry rdrs tht cn provde other knds of mesurements n ddton to rnge nd zmuth. To further mprove the lttude estmton ccurcy we lso propose the use of the EKF bsed AMM method where ech model hs dfferent ntl lttude vlue. The pper s orgnzed s follows. Secton II presents the problem formulton of lttude estmton nd dscusses n tertve lgorthm to solve for the lttude estmte. Secton II shows the estmton performnce for severl scenros nd lttude levels. A Crmér-Ro lower bound CRLB of the lttude estmtor s vrnce s lso provded. As n pplcton of the proposed lttude estmton Secton IV uses the estmted lttude nd optmzed prmeters to ntlze the -D trckng lgorthm wth two -D rdrs usng the EKFbsed multple pproch method. The trckng results re lso dsplyed nd compred to the exstng method. Fnlly some concludng remrks re presented n Secton V ISIF

2 II. ALTITUDE ESTIMATION Ths secton ms t presentng the formulton of heght estmton nd how to estmte the heght by usng the mxmum lkelhood crteron. A. Problem Formulton To begn wth let ρ m nd θ m be the rnge nd zmuth mesurements obtned from the rdr system t tme t =...N respectvely. Assume the ntl tme t =. Let the velocty nd ntl poston of the rcrft t t v x nd x n the X drecton nd v y nd y n the Y drecton nd z be the rcrft s lttude. Therefore the true poston of the rcrft n the system plne t tme t xtytzt s gven by xt = v x t+x yt = v y t+y zt = z. The true rnge nd the true zmuth t tme t s gven by ρt = xt x R +yt y R +zt z R xt xr θt = rctn. yt y R where x R y R z R s the correspondng rdr s poston n the system plne. Note tht t s not necessry tht the two rdrs detect the trget smultneously. The reltonshp between the true dt nd the mesurement dt from rdr s obtned by ρ m = ρ m t = ρt + ρ nd θ m = θ m t = θt + θ where ρt θt cn be computed from nd the mesurement errors ρ nd θ re ssumed to be ndependent Gussn wth zero men nd vrnce ndσ θ respectvely. Both ρ s nd θ s re lso ssumed to be ndependent. Under the ssumpton of constnt velocty nd lttude the m s to estmte the trget lttude. Here we present the ML estmtor to fnd the estmtes of v x x v y y nd z. Specfclly the log-lkelhood functon correspondng to set of mesurement dt s gven by LLp=log N = fρ m θ m ;p N = where p [v x x v y y z ] T nd fρ m θ m ;p s the jont probblty densty functon pdf of ρ m nd θ m. Snce ρ nd θ re ndependent fρ m θ m ;p cn be wrtten s fρ m θ m ;p = fρ m ;pfθ m ;p where fρ m ;p = fθ m ;p = exp σ ρ π exp σ θ π = ρ m ρt σ ρ θ m θt σ θ ρ m ρt θ m nd θt re defned s n. As consequence mxmzng LLp s equvlent to mnmzng Fp whch s defned by N Fp = G p+h p = ρm ρt θm θt. where G p = σ ρ nd H p = σ θ Ths optmzton problem cn be solved by usng n tertve lgorthm. Note tht ths ML estmtor correspondng to s lso dentcl to the nonlner lest squre NLS estmtor. B. Mnmzng the Objectve Functon To fnd the vlue of p tht mnmzes Fp n we need to solve the mnmzton problem descrbed by ˆp = rgmn Fp. p It s rther strghtforwrd to show tht ths s not convex optmzton problem. However f you pck good ntl vlue suffcently lrge vlue of z [] n ths cse nd use the Newton-Rphson lgorthm [] wth vryng step sze to gurntee tht Fp decreses n every terton k we emprclly found tht ths tertve lgorthm yelds stsfctory results. Specfclly we use ths lgorthm: p [k+] = p [k] λ [k] [ Fp [k] ] [ Fp [k] ] T where λ [k] = λ j k for < λ <. The ndex j k s chosen t ech step s the smllest non-negtve nteger tht would mke the objectve functon Fp [k+] < Fp [k].e. the ndex j k s present to mke sure tht the objectve functon decreses t every terton. The nottons F nd F re the Jcobn nd the Hessn of F respectvely. See Appendx A for ther expressons. In ths pper the ntl vlue for the lttude s set to be postve constnt. The remnng ntl vlues v x [] x [] v[] y y [] re chosen by mnmzng N = [ρ msnθ m v [] x t +x [] x R + ρ m cosθ m v y [] t +y [] y R ]. Ths mnmzton yelds closed form soluton for these prmeters s shown n Appendx B. Hence the ntl vlue p [] s gven by p [] = [v x [] x [] v[] y y [] z[] ]T. The stoppng crteron s tht the terton stops when p [k+] p [k] / p [k] < ε for some smll postve constnt logfρ m θ m ;p ε. It s worth notng tht ths problem cn be lso solved usng recursve lgorthm such s n terted Klmn flter IKF dscussed n []. III. NUMERICAL RESULTS OF ALTITUDE ESTIMATION To vldte the method proposed n the prevous secton we provde exmples of estmtng the lttude of crusng rcrft wth two -D rdrs n ths secton. A. Scenro Settng For smplcty ssume tht the two rdrs A nd B Fgure re t the sme level nd locted t the poston nd ll n nutcl mles nm respectvely. For both rdrs the ntenn rottng perod re ssumed to be equl to.6 s. The stndrd devtons of the mesurement errors re ssumed to beσ ρ =.6 nm ndσ θ =. =.96 rd. The smulton durton s s. We test three flght scenros wth detls s follows.

3 Y xs nm Rdr A Rdr B Scenro Scenro Scenro nformton of the ML lttude estmtor from Secton II-A s gven by [] [ ] Iz = E logfρ m...ρ mn ;z z z where ρ m s re defned s n Secton II-A nd logfρ m...ρ mn ;z = N logσ ρ π N ρ m xt x R +yt y R +z z R = σ ρ. X xs nm Fgure. Flght scenros Scenro : The rcrft s flght trjectory strts t z wth the hedng wth constnt speed knot = 8. nm/s nd constnt lttude z where we try three cses of the true lttude vlue z = 7 nd ft. Scenro : The flght trjectory strts t - z wth the hedng 9. Other prmeters re the sme s n Scenro. Scenro : The flght trjectory strts t z wth the hedng. Other prmeters re the sme s n Scenros nd. The three scenros re llustrted n Fgure. The stoppng crteron error s chosen s ε = 6 whle λ =.9 nd the ntl vlue of the heght for the tertve method n s chosen to bez [] = ft =.687 nm whch s emprclly found to be lrge enough for the generl vton cse where the lttude s usully not more thn ft. B. Smulton Results We use the frst N number of smples of the mesurement dt to estmte the lttude where N =.... The smulton results re bsed on Monte Crlo runs. Fgures -c show the estmton performnce n terms of the root men squre error over trls for ech N whch s defned by RMSE = ẑ [n] z. n= As we expect the RMSE vlue tends to decrese when there re more smples used n the ML estmton. Moreover for ech scenro the estmton ccurcy degrdes when the vlue of the true lttude s smller. Ths cn be explned by consderng the Crmér-Ro lower bound CRLB s follows. For the estmton n ths work the CRLB of ll fve estmtes ˆv x ˆx ˆv y ŷ nd ẑ defne the lower lmt of ther estmton error covrnce mtrx. Nevertheless let us ssume n ths nlyss tht the other four qunttes v x x v y nd y re known becuse we wnt to focus only on the lttude estmte n order to see how the vrnce of the lttude estmtor chnges wth the true lttude vlue. The Fsher Note tht we do not hve to consder the zmuth mesurements for Iz snce θ m does not depend on z. After some mnpultons the Fsher nformton of the lttude estmtor s obtned by Iz = σ ρ N = z z R xt x R +yt y R +z z R. From the CRLB the vrnce of ny unbsed estmtor s bounded below by the nverse of Fsher nformton: Vrẑ Iz. Hence the vrnce of the ML estmtor ẑ s bounded below.e. Vrẑ N = σ ρ. 6 z z R xt x R +yt y R +z z R It s strghtforwrd to show tht ths CRLB on the rghthnd sde of 6 s decresng functon of z wth z. Therefore for good ML estmtor t s resonble to expect the RMSE to become smller when the true vlue of the lttude becomes lrger gven the other fctors re fxed. Note tht for smplcty we set z R = for ll n our smulton. The lttude estmton results for Scenro nd n Fgures b nd c follow the sme trend s n Scenro but wth lrger RMSEs. For exmple wth the true lttude ft the RMSE s round ft for N = n Scenro ft n Scenro nd round 7 ft n Scenro. It s becuse ths lttude estmton from two -D rdrs depends on the geometrcl confgurtons of the flght scenro where the trget sttes become unobservble due to dmensonl ncompleteness. From Fgure we cn see tht the ground rnges for the frst few ponts of Scenros nd re lrger thn those of Scenro whch results n lrger vlue of the CRLB n 6 for fxed vlue of the lttude. Next we wll use ths lttude estmte to ntlze -D trckng lgorthms. IV. AN APPLICATION IN -D TRAJECTORY TRACKING The gol of ths secton s to use the heght estmte obtned n the prevous secton long wth the other four estmted prmeters s ntl vlues for -D flght trckng wth two -D rdrs usng the EKF. A. -D Trckng wth -D Rdrs usng the EKF As exmples of usng the lttude estmte for -D trckng we perform the trckng lgorthm for Scenros nd s defned n Secton II. We use the frst ponts to clculte the heght estmte usng the ML method s descrbed n Secton

4 z = ft z = ft 6 z = ft z = 7 ft z = 7 ft z = 7 ft z = ft z = ft z = ft Alttude RMSE ft 8 Alttude RMSE ft Alttude RMSE ft Number of smples N RMSE: Scenro Number of smples N Number of smples N b RMSE: Scenro c RMSE: Scenro Fgure. The RMSE of the lttude estmton when N =... II-A nd then strt the trckng lgorthm usng the EKF from the pont = onwrd. Ths s chosen s trde-off between the computtonl tme nd the estmton ccurcy. Assume the rcrft dynmc model s gven by where x = Fx +Γv F = Γ = T T T T T T T T T nd x = [xt yy zt ẋt ẏt żt ] T s the stte vector T s the tme ntervl. The process nose vector v s Gussn wth zero-men nd covrnce mtrx Q = dg[q q q ] where q = q = 9 q =. The rdr mesurement s z = h[x]+w where h[x]=[ρt θt ] T the mesurement nose vector w s zero-men Gussn wth covrnce mtrx R = dg[σ ρ σ θ ]. The vlues of σ ρ nd σ θ re the sme s n Secton II. In ths pper the ntlzton s tken s follows. For the stte estmte vector we set ˆx9 9 = [ˆx 9 ŷ 9 ẑ ˆv x ˆv y ] T whch cn be computed strghtforwrdly from the ML estmton n Secton II-A. On the other hnd for the stte estmte covrnce mtrx we set P9 9 = dg[p p p p p p 66 ] where p = p = p = p = p 66 = nd p s chosen such tht the lgorthm performs stsfctorly. We cll ths method the AE method snce t uses n lttude estmton to ntlze sngle EKF. In ddton to trget trckng wth sngle EKF we lso propose multple model EKF pproch to mprove the trckng performnce n the Z- xs. Ths pproch s explned n the next secton. B. The Multple EKF Models The de behnd usng multple models s tht more models would ncrese the chnce tht the ntl lttude vlue of one of the models s closer to the true lttude vlue whch wll result n trckng performnce mprovement becuse of the senstvty of n EKF to the ntl vlue []. In prtculr we use the EKF-bsed multple model pproch where ech model dffers only by the ntl vlue of the lttude. Snce we ssume the trget fles wth constnt heght the utonomous multple model AMM method [6] s good ft n tht t ssumes the system mode does not chnge. The key de of the AMM s tht ech EKF models dfferent lttude trjectory nd the overll lttude estmte s obtned by combnng estmtes from ndvdul EKFs. Specfclly the overll stte estmte s computed from the sums of the stte estmte of ech flter weghted by ts mode probblty whch results from ts model lkelhood. We refer to [6] for the detls relted to the AMM lgorthm. We perform the trget trckng usng three models where ech model s dfferent only t the lttude ntl vlue.e. Model : lttude ntl vlue = mxẑ z ft Model : lttude ntl vlue = ẑ Model : lttude ntl vlue = mnẑ + z ft where ẑ s the ML estmte s descrbed n Secton II-A. The ncrement vlue z s chosen to be n djustble constnt. multpled by the lower bound of the estmtor s stndrd devton Iz n 6 ssumng tht we know the true vlue of the bound. We cll ths proposed method tht uses the lttude estmton to ntlze three EKF s s the method. C. Trckng Results In ths subsecton we compre the trckng results of the proposed AE nd methods wth those of usng the HPEKF method s n [] whch provdes nother method for ntlzng the lttude for -D trckng wth -D rdrs. For the HPEKF method we follow the method n [] to desgn HPEKF s where we use seven flters to cover the lttude rnge from - ft. Ths number of flters seven flters s selected such tht the coeffcent of vrton [] s %. Prtculrly the ntl lttude vlues for the flters re nd 66 ft. For the purpose of

5 AE..7 AE.8 AE XY Poston RMSE nm.. x ft XY Poston RMSE nm.8... x ft XY Poston RMSE nm.. x ft XY-poston: z = ft b XY-poston: z = 7 ft c XY-poston: z = ft.6..8 AE 7 AE Z Poston RMSE nm... AE.. x ft Z Poston RMSE nm.. x ft Z Poston RMSE nm.6.. x ft d Z-poston: z = ft e Z-poston: z = 7 ft f Z-poston: z = ft Fgure. Scenro : The trckng performnces of AE nd methods n terms of the XY-poston nd the Z-poston RMSE s. AE.. AE..6 AE.7... XY Poston RMSE nm... x ft XY Poston RMSE nm..7. x ft XY Poston RMSE nm...8 x ft XY-poston: z = ft b XY-poston: z = 7 ft c XY-poston: z = ft.6 AE. 6 AE 7 AE 6 Z Poston RMSE nm.... x ft Z Poston RMSE nm.7.. x ft Z Poston RMSE nm.8.6. x ft d Z-poston: z = ft e Z-poston: z = 7 ft f Z-poston: z = ft Fgure. Scenro : The trckng performnces of AE nd methods n terms of the XY-poston nd the Z-poston RMSE s comprson wth our proposed we lso use the HPEKF method wth three flters whch results n the ntl lttude vlues of nd 98 ft. These two methods re clled the nd methods respectvely. The trckng results over Monte Crlo runs for Scenros nd re shown n Fgures nd respectvely where n ech scenro trckng wth three vlues of the true lttude: z = 7 nd ft s tested. For ech fgure the top pnel shows the comprson of the XY-poston RMSE whle the bottom one llustrtes the Z-poston RMSE. Comprng the two proposed AE nd methods t cn be seen from the bottom pnel of ech fgure tht the method mproves the trckng performnce n terms of the lttude RMSE from the AE method for ll three lttude

6 Tble I COMPARISON OF RELATIVE COMPUTATIONAL TIME USED IN THE FOUR METHODS Method AE Reltve Tme vlues nd three scenros. Specfclly the lttude RMSE vlue s reduced by usng the method up to nd ft n Scenros nd respectvely dependng on the true vlues of the lttude. Note tht the AE nd methods perform better when the true lttude s hgher becuse when the vlue of the true lttude decreses the performnce of the ML estmton degrdes s explned n Secton II. Consderng the performnce of usng the nd methods n terms of the lttude RMSE we cn observe tht the nd methods yeld lrger lttude RMSE vlues thn the AE method nd much lrger thn the method for ll cses except n Scenro where the nd methods mprove the lttude RMSE over the method up to ft. Ths s becuse the nd methods use three nd seven fxed ntl vlues of the lttude whle the AE nd methods use one nd three ntl vlues vryng wth the ML estmte of the lttude. Nevertheless n these two cses f we check the performnce n terms of the XY-poston RMSE the AE nd the methods perform pproxmtely the sme whle the nd methods yeld very lrge RMSE vlues t the frst few dt ponts round - dt ponts. Ths behvor s lso observed n the correspondng lttude RMSE plots. Ths s one of the dvntges of usng the ML estmton for ntlzng EKFs from whch we obtn not only the estmte of the lttude but lso the estmtes of the poston nd the velocty n X nd Y xes when trckng wth the AE nd the methods. In terms of computtonl complexty the verge vlues of computtonl tme for the four methods n Scenro wth z = 7 ft re tbulted n Tble I. The computtonl tme s shown s reltve vlue compred to the computtonl tme of the AE method. As shown the method requres bout three tmes of computtonl tme used n the AE method. Comprng the two methods whch use the sme number of EKFs the method requres bout % more n terms of computtonl tme thn the method but yelds much better results n generl. Ths extr computtonl tme comes from the ML estmton to estmte the lttude. In ddton to producng worse results thn those from the method the method whch uses seven flters requres pproxmtely twce the computtonl tme of the method whch uses three flters. V. CONCLUSION In ths pper method of lttude estmton nd trckng of crusng rcrft usng two -D rdrs s proposed. Specfclly n estmton method usng the mxmum lkelhood crteron s utlzed to estmte the lttude wth the ssumpton of constnt velocty nd lttude. The CRLB of the lttude estmtor s vrnce when other prmeters re ssumed to be known s lso provded. The smulton results show tht the proposed method performs stsfctorly n terms of the lttude RMSE. We then use the estmted lttude s n ntl vlue for the -D flght trckng lgorthm usng the EKF. To further mprove the trckng performnce especlly n terms of lttude the pper proposes three prllel EKFs ech wth dfferent ntl lttude estmte: one wth the ML estmted lttude nd two wth estmted lttude perturbed by vlue proportonl to the CRLB of the estmtor n both drectons. APPENDIX A The expressons for F nd F re gven by F = [F F F F F ] F = [F kl ] where {F ; =...} nd {F kl ; l k l} re gven by N + c + + c b +d d t c = +d N ρm c d N F = ρm d t t b c = = +d F = N ρm + c + N d = b c = +d F = N ρm t + d N c = t b c = +d F = N ρm + d N c = b c = +d F = N ρm + e = F = N t ρm ++ ρ mc = + N t c b +d d c = +d F = N ρm t ++ ρ mc = N F = σ ρ F = σ ρ t = N t = N + σ θ + σ θ = N b c d c d t ρm c d = c +d t b c d c d c +d 6

7 AE... AE.. AE XY Poston RMSE nm... x ft XY Poston RMSE nm..7. x ft XY Poston RMSE nm x ft XY-poston: z = ft b XY-poston: z = 7 ft c XY-poston: z = ft AE 6 Z Poston RMSE nm... AE. x ft Z Poston RMSE nm.7. AE x ft Z Poston RMSE nm.8. x ft d Z-poston: z = ft e Z-poston: z = 7 ft f Z-poston: z = ft Fgure. Scenro : The trckng performnces of AE nd methods n terms of the XY-poston nd the Z-poston RMSE s F = σ ρ F = σ ρ N = N + σ θ = N = t ρm c e ρm ++ ρ mc c b +d d c +d = F = F F = N ρm c d = + N b c d c d c = +d F = N ρm c e = F = N t ρm ++ ρ md = + N t d b +c c c = +d F = N ρm t ++ ρ md = + N d b +c c t c = +d F = N ρm d e t F = N ρm ++ ρ md = + N d b +c c c = +d F = N ρm d e = F = N ρm ++ ρ me = where = c +d +e b = θ m +rctn c d c = v x t +x x R d = v y t +y y R nd e = z z R. v [] x APPENDIX B The soluton for s gven by N = A = t N = t N = t N N A + = t = A N N = t N = t N = B = t N = t N = t N N B + = t = B = N N = t A N = t N N x [] = N = t v [] y = N N = t B N = t N N y [] = N = t N N = t N = t wherea = ρ m snθ m +x R nd B = ρ m cosθ m +y R. 7

8 REFERENCES [] The Electronc Code of Federl Regultons e-cfr Ttle Prt 9 Subprt 9. ATC trnsponder nd lttude reportng equpment nd use July. [] S. Ngok Heght estmton of crusng rcrft v rdr for r trffc control Electron. nd Commun. n Jpn vol. 7 no. pp [] D.E. Mnolks nd C.C. Lefs Arcrft geometrc heght computton usng secondry survellnce rdr rnge dfferences n IEE Proc. - Rdr Sonr & Nv. Apr. 99 vol. pp. 9. [] T. L. Ogle W. D. Blr R. J. Levn nd K. W. Hrrgn Multpltformmultsensor trckng wth survellnce rdrs n Proc. of the 6th Southestern Symp. on Syst. Theory Atlnt GA pp [] M.-J. G X. Y Y. He nd B. Sh An pproch to trckng Dtrget wth D-rdr n IEEE Intl. Rdr Conf. Arlngton VA My pp [6] X.-R. L nd V.P. Jlkov Survey of mneuverng trget trckng. prt v multple-model methods IEEE Trns. Aerosp. Electron. Syst. vol. no. pp. Oct.. [7] Z. Luo nd J.-Z. He ML estmton of true heght n -D rdr network n Intl. Conf. on Inform. Fuson Quebec QC Cnd July 7 pp. 7. [8] E. H. Aok A generl pproch for lttude estmton nd mtgton of slnt rnge errors on trget trckng usng D rdrs n Intl. Conf. on Inform. Fuson Ednburgh UK July. [9] D.E. Mnolks Effcent soluton nd performnce nlyss of - d poston estmton by trlterton IEEE Trns. Aerosp. Electron. Syst. vol. no. pp. 9 8 Oct [] D.E. Rce Heght estmton by qudrlterton The Mrcon Rev. XLVI 8 pp [] S. Boyd nd L. Vndenberghe Convex Optmzton Cmbrdge Unversty Press Cmbrdge UK. [] F.W. Bell nd B.M. Cthey The terted Klmn flter updte s Guss-Newton method IEEE Trns. Automt. Contr. vol. 8 no. pp Feb. 99. [] S. M. Ky Fundmentls of Sttstcl Sgnl Processng: Estmton Theory Prentce Hll Upper Sddle Rver NJ 99. [] Y. Br-Shlom X.-R. L nd T. Krubrjn Estmton wth Applctons to Trckng nd Nvgton: Theory Algorthms nd Softwre John Wley & Sons New York NY. [] N. Pech Berngs-only trckng usng set of rnge-prmetersed extended Klmn flters IEE Proc.-Contr. Theory nd Appl. vol. no. pp. 7 8 Jn