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1 REPOR DOCUMEAIO PAGE Frm Apprved OMB Public reprting burden fr this cllectin f infrmatin is estimated t average 1 hur per respnse, including the time fr reviewing instructins, searching data surces, gathering and maintaining the data needed, and cmpleting and reviewing the cllectin f infrmatin. Send cmments regarding this burden estimate r any ther aspect f this cllectin f infrmatin, including suggestins fr reducing this burden t Washingtn Headquarters Service, Directrate fr Infrmatin Operatins and Reprts, 115 Jeffersn Davis Highway, Suite 104, Arlingtn, VA 0-430, and t the Office f Management and Budget, Paperwr Reductin Prject ( ) Washingtn, DC PLEASE DO O REUR YOUR FORM O HE ABOVE ADDRESS. 1. REPOR DAE (DD-MM-YYYY) APR ILE AD SUBILE. REPOR YPE Cnference Paper Pstprint CURVAURE OLIEARIY MEASURE AD FILER DIVERGECE DEECOR FOR OLIEAR RACKIG PROBLEMS 3. DAES COVERED (Frm - ) Jun 08 Jul 08 5a. CORAC UMBER In-Huse 5b. GRA UMBER 5c. PROGRAM ELEME UMBER 670F 6. AUHOR(S) Ruixin iu, Pramd K. Varshney, Mar Alfrd, Adnan Bubal, Eric Jnes and Maria Scal 5d. PROJEC UMBER 5e. ASK UMBER 459E E 5f. WORK UI UMBER 7. PERFORMIG ORGAIZAIO AME(S) AD ADDRESS(ES) AFRL/RIEA 55 Brs Rad Rme Y SPOSORIG/MOIORIG AGECY AME(S) AD ADDRESS(ES) AFRL/RIEA 55 Brs Rad Rme Y PERFORMIG ORGAIZAIO REPOR UMBER 10. SPOSOR/MOIOR'S ACROYM(S) 11. SPOSORIG/MOIORIG AGECY REPOR UMBER AFRL-RI-RS-P DISRIBUIO AVAILABILIY SAEME Apprved fr Public Release; Distributin Unlimited. PA# WPAFB SUPPLEMEARY OES Paper presented at the Infrmatin Fusin 008 Internatinal Cnference, Jun 30-Jul 3 in Clgne, Germany. One r mre f the authrs is a U.S. Gvernment emplyee wring within the scpe f their Gvernment jb; therefre, the U.S. Gvernment is jint wner f the wr and has the right t cpy, distribute and use the wr. 14. ABSRAC Experimental results shw that fr a weely nnlinear tracing prblem, the extended Kalman filter and the unscented Kalman filter are gd chices, while a particle filter shuld be used fr prblems with strng nnlinearity. quantitatively determine the nnlinearity f a nnlinear tracing prblem, we prpse tw types f measures: ne is the differential gemetry curvature measure and the ther is based n the nrmalied innvatin squared (IS) f the Kalman filter. Simulatin results shw that bth measures can effectively quantify the nnlinearity f the prblem. he IS is capable f detecting the filter divergence nline. he curvature measure is mre suitable fr quantifying the nnlinearity f a tracing prblem as determined via simulatins. 15. SUBJEC ERMS racing, extended Kalman filter, unscented Kalman filter, particle filter, nnlinearity measures 16. SECURIY CLASSIFICAIO OF: 17. LIMIAIO OF ABSRAC a. REPOR U b. ABSRAC U c. HIS PAGE U UU 18. UMBER OF PAGES 9 19a. AME OF RESPOSIBLE PERSO Mar Alfrd 19b. ELEPOE UMBER (Include area cde) /A Standard Frm 98 (Rev. 8-98) Prescribed by ASI-Std Z39-18

2 Curvature nlinearity Measure and Filter Divergence Detectr fr nlinear racing Prblems Ruixin iu and Pramd K. Varshney Depart. f Electrical Engineering & Cmputer Science Syracuse University Syracuse, ew Yr, 1344, U.S.A. rniu@ecs.syr.edu Abstract Several nnlinear filtering techniques are investigated fr nnlinear tracing prblems. Experimental results shw that fr a wealy nnlinear tracing prblem, the extended Kalman filter and the unscented Kalman filter are gd chices, while a particle filter shuld be used fr prblems with strng nnlinearity. quantitatively determine the nnlinearity f a nnlinear tracing prblem, we prpse tw types f measures: ne is the differential gemetry curvature measure and the ther is based n the nrmalied innvatin squared (IS) f the Kalman filter. Simulatin results shw that bth measures can effectively quantify the nnlinearity f the prblem. he IS is capable f detecting the filter divergence nline. he curvature measure is mre suitable fr quantifying the nnlinearity f a tracing prblem as determined via simulatins. Keywrds: racing, extended Kalman filter, unscented Kalman filter, particle filter, nnlinearity measures. 1 Intrductin Fr nnlinear target tracing prblems with nnlinear dynamics r nnlinear measurements, there exist many nnlinear filtering techniques. A cmprehensive tutrial f these nnlinear techniques is given in [1]. In this paper, we investigate several f the mst ppular nnlinear filtering techniques, including the extended Kalman filter (EKF) [], the unscented Kalman filter (UKF) [3] and the particle filter (PF) [4-5]. Being parametric filtering methds based n the classical Kalman filter framewr, the EKF and the UKF incur nly a mdest amunt f cmputatinal cst and lead t near-ptimal perfrmances in prblems with wea nnlinearity. Hwever, in many highly nnlinear prblems, the EKF r the UKF may diverge, as shwn later in this paper. he PF, a simulatin-based nnparametric algrithm, has better perfrmances than the EKF and the UKF in these difficult highly nnlinear cases. But its implementatin requires a significantly larger amunt f cmputatin. Cnsidering the advantages and disadvantages f these filters, naturally we can emply different filtering Mar Alfrd, Adnan Bubal, Eric Jnes, and Maria Scal Air Frce Research Labratry/RIEA Rme, ew Yr 13441, U.S.A Mar.Alfrd@rl.af.mil algrithms fr prblems with different degrees f nnlinearity. In this paper, we study tw types f nnlinearity measures t quantitatively determine the nnlinearity f a nnlinear prblem. he first measure is based n the curvature measure [6], which cmpares the magnitude f the secnd-rder term with that f the firstrder term in a aylr series expansin f the nnlinear prblem. Recently, the curvature measures have been adpted in indicating the nnlinearity fr trajectry estimatin prblems [7-9]. Here, we use them t determine the nnlinearity in the tracing prblem. We als prpse the nrmalied innvatin squared (IS) [] f the Kalman filter, as a measure f nnlinearity and f filter divergence. In Sectin, a brief intrductin t the three nnlinear tracing algrithms is prvided. racing examples with strng and wea nnlinearities t illustrate the necessity f nnlinearity measures are presented in Sectin 3. he curvature nnlinearity measure and the divergence detectr based n Kalman filter s innvatin are prpsed in Sectin 4. here, the effectiveness f these tw measures is verified by simulatin results. Cnclusins are drawn in Sectin 5. nlinear filtering techniques Fr a target tracing prblem with nnlinear target mtin mdel and/r nnlinear measurements, ne f the mst ppular filtering techniques is the EKF, which is a minimum mean square errr (MMSE) estimatr based n the aylr series expansin f the nnlinear dynamics r the nnlinear measurement []. It is very easy t implement and requires nly mdest amunt f cmputatinal pwer. In the first-rder EKF the state distributin is apprximated by a Gaussian randm variable which is then prpagated analytically thrugh the first-rder lineariatin f the nnlinear system r nnlinear measurement. herefre, it can be viewed as prviding ``first-rder'' apprximatins t the ptimal terms. hese apprximatins, hwever, can intrduce large errrs in the true psterir mean and cvariance f the nnlinearly 1

3 transfrmed (Gaussian) randm variable, which may lead t sub-ptimal perfrmance and smetimes divergence f the filter, especially in strngly nnlinear estimatin prblems. he UKF addresses sme apprximatin issues f the first-rder EKF. In a UKF, he state distributin is again represented by a Gaussian randm variable, but is nw specified using a minimal set f carefully chsen sample pints. hese sample pints cmpletely capture the true mean and cvariance f the Gaussian randm variable, and when prpagated thrugh the true nn-linear system, capture the psterir mean and cvariance accurately t the 3rd rder (aylr series expansin). Fr details and the implementatin f the UKF, the readers are referred t [3]. Anther very ppular strategy fr estimating the state f a nnlinear system as a set f bservatins becmes available nline is t use sequential Mnte-Carl methds, als nwn as the particle filter (PF). hese methds allw fr a cmplete representatin f the psterir prbability density functin (pdf) f the states, s that any statistical estimates, such as the mean, mde, variance and urtsis can be easily cmputed. hey can, therefre, deal with any nnlinearity r prbability distributins [5]. he basic idea f the particle filter begins with using a set f particles and their assciated weights t apprximate the state psterir pdf. hese particles are prpagated frm ne time t the next accrding t the target mtin mdel, and their weights are updated based n the incming measurements. It has been shwn in [5] that under mild cnditins, the apprximatin f the density functin cnverges t the true psterir state prbability density functin almst surely as the number f particles ges t infinity. In this paper, we adpt a very simple and ppular versin f the PF, the sequential imprtance resampling (SIR) particle filtering algrithm [5]. 3 Mtivating examples F = in which is the time interval between tw adjacent samples. v dentes the er-mean white Gaussian nise with cvariance matrix Q 3 /3 / 0 0 / 0 0 Q = q /3 / 0 0 / where q is a scalar. Fr a radar that reprts range-bearing measurements, the measurement mdel can be mathematically written as where = h( x ) + w (4) is the radar measurement, ( ξ ξs) + ( η η s) hx ( ) = 1 tan [( η ηs) /( ξ ξs) ] ( ξ, η ) is the target psitin at time, and ( ξs, η s ) is the psitin f the radar. w dentes the measurement nise, which is Gaussian with er mean and cvariance matrix R. As we can see, the mtin mdel is linear and Gaussian. But the measurements are nnlinear functins f the target state. () (5) (3) te that fr the EKF, Jacbian f measurement is given by 3.1 arget and measurement mdels We assume that the target mtin is mdeled as fllws x = Fx + v 1 (1) J ξ η 0 0 ξ + η ξ + η = ηj ξj 0 0 ξ + η ξ + η (6) where x = [ ξξηη & & ]' is the target state at time, cnsisting f psitins and velcities alng ξ and η axes in a tw-dimensinal Cartesian crdinate system. F is the state transitin matrix where ξ = ξ 1 ξ and s η = η 1 η. s 3. A single radar with strng nnlinearity In this experiment, we assume that there is a single radar sensr that is tracing the target. An example fr the target trajectry has been shwn in Fig. 1. First, we investigate a case with highly nnlinear measurements. We assume that the range measurements

4 are very accurate, with a range errr standard deviatin (s.d.) f 10 m, and the bearing measurements are relatively carse, with a bearing errr s.d. f 3. Given the gemetry in Fig. 1, it can be shwn that the uncertainty regin f the target psitin based n the range and the bearing measurements is part f a thin ring with its span prprtinal t the s.d. f the bearing measurement. 4 able I. te that in the PF, 10 particles have been used. As we can see, the cmputatinal csts fr the PF are much higher than thse f the EKF and the UKF. able I. Average number f flating pint peratins per iteratin fr different algrithms EKF UKF PF Fig. 1 Gemetry f the target trajectry and the radar sensr. 4 In the experiment, the PF uses 10 particles. he rt mean square errrs (RMSEs) are btained by taing an average ver 100 Mnte-Carl simulatin runs. In Fig., the RMSEs f psitin estimates are shwn. Since the RMSEs f velcity estimates exhibit similar behavirs t thse f psitin estimates, they are nt shwn t save space. Frm Fig., it is clear that the EKF s averaged RMSEs are much higher than thse f the ther tw algrithms. his is because in sme Mnte-Carl runs, the EKF diverges. Since an unscented transfrm has been used in the UKF, the UKF prvides significant imprvement ver the EKF. he PF utperfrms the UKF, at the cst f much mre cmputatinal cmplexity. 3.3 A single radar with wea nnlinearity Secnd, we investigate a case with less nnlinearity in the measurements. We assume that the range measurement errr has a s.d. f 100 m, and the bearing measurement errr has a s.d. f 1. As shwn in Fig. 3, all the three algrithms have perfrmances clse t each ther. his implies that fr a recursive estimatin prblem with wea nnlinearity, the EKF r the UKF shuld be used t save the cmputatinal cst, while at the same time t achieve the same perfrmances as that f the PF. 3.4 Cmputatinal cmplexities quantitatively cmpare the cmplexities f different algrithms, the average number f flating pint peratins per iteratin fr each algrithm is listed in Fig. RMSEs in psitin estimates by different tracing algrithms. riangle: EKF. Square: UKF. Circle: PF. Fig. 3 RMSEs in psitin estimates by different tracing algrithms. riangle: EKF. Square: UKF. Circle: PF. 4 nlinearity measures Frm the examples in the last sectin, we can see that under different circumstances, it is desirable t apply different filtering algrithms. Fr a tracing prblem with 3

5 wea nnlinearity, the EKF and the UKF are gd chices, since they prvide near-ptimal tracing perfrmances, while at the same time have much less cmputatin cmplexity than the PF. Hwever, in a highly nnlinear prblem, the PF shuld be used, since the EKF and the UKF are prne t divergence in such a prblem. As a result, it is imprtant t develp a metric t gauge the nnlinearity f the estimatin prblem. Fr the tracing prblem with linear dynamic mdel and nnlinear measurement mdel, the nnlinearity is brught by the nnlinear measurements. We prpse the parametereffects curvature and intrinsic curvature, similar t thse used in [7], t measure the nnlinearity in the measurements. Hwever, we als bserve that the curvature measures can nt prvide a direct and timely indicatin f the divergence f the EKF r the UKF. detect the divergence f a filter in a timely manner, we als develp a Chi-square detectr based n the innvatin f the Kalman filter, which checs the cnsistency f the filter, and detects the filter divergence when it ccurs. 4.1 Curvature nnlinearity measures We try t measure the nnlinearity in the measurements using curvature measures [6], as prpsed in [7]. In [7], the measures have been used in trajectry estimatin prblems. Here we apply them fr measuring and detecting the nnlinearity in target tracing prblems. he nnlinear measurement mdel (4) can be rewritten as = h( p) + w (7) where p = [ξ η ]' is the target s crdinates at time, since is nt a functin f the target s velcity at time. In a EKF, the nnlinear measurement is apprximated by the first rder aylr series expansin f hp ( ) abut the predicted target psitin : where p hp ( ) hp ( ) + H& ( p )( p p ) (8) hp ( ) H & n np ( p ) =, H & ( p ) R (9) 1 1 p p= p 1 where n and n p are the dimensins f the measurement and the target psitin, respectively. he first-rder EKF uses the linear apprximatin by ignring the higher rder terms in the aylr series expansin. he linear apprximatin in the EKF is valid nly if the measurement functin is relatively flat near p and the hence the tangent plane apprximatin in (8) is valid. aing the aylr series expansin f (7) up t the secnd rder, we have hp ( ) hp ( 1 ) + H& ( p 1 )( p p 1 + ( p p )' H&& ( p 1 )( p p where h ( p) ( p ) =, H( p ) R m && mij 1 1 pi p j p= p 1 n np np (10) m = 1, L, n, i, j = 1, L, np Accrding t the linear apprximatin (8), hp ( ) lies in the plane tangent t the measurement surface at the pint p. herefre, the lineariatin in the measurement mdel is equivalent t apprximating the measurement surface by the tangent plane at p. he tangent plane is a gd apprximatin t the measurement surface if the nrm f the quadratic term ( p p )' H&& ( p 1 )( p p is negligible cmpared with the nrm f the linear term H& ( p 1 )( p p. It is useful t decmpse the quadratic term n the right hand side f (10) int cmpnents in the tangent plane and rthgnal t the tangent plane., (11) In rder t define the curvature measures f nnlinearity, velcity vectrs and acceleratin vectrs [6] are intrduced, which frm the Jacbian and Hessian, respectively. he n 1 velcity vectrs are defined by h1( p ) h ( p ) h& n ( i p = L (1) pi p p= p i 1 p= p 1 i = 1,, Ln p and H& ( p = h& 1 ( p 1 ) L h& n ( p ) (13) p he n 1 symmetric acceleratin vectrs are defined by hp ( ) ( p ) =, h ( p ) R && ij 1 ij 1 pi p j p= p 1 i, j = 1, L, n p n 1 (14) he acceleratin vectrs are decmpsed int cmpnents in the tangent plane and rthgnal t the tangent plane. Let H&& np np m( p ) R dente the mth face f the acceleratin array 4

6 m11 ( p L m1 n ( p ) p H&& ( p m = L L L mn 1 ( p ) h ( p ) p L && mnpnp m= 1,, Ln ( 15) he prjectin matrix P which prjects an n 1 vectr int the tangent plane is defined by 1 P (16) = H& ( p 1 ) H& '( p ) H& ( p 1 ) H& '( p 1) herefre, we can decmpse an acceleratin vectr int cmpnents tangent and rthgnal t the tangent plane. Let h && ( ij p 1) and ( p ) dente the tangential and ij 1 rthgnal cmpnents f the acceleratin vectr ( p ), respectively. hen ij 1 ( p ) = P ( p ) p I P h p ( p ) ( p ) ( p ) ij 1 ij 1 ij ( 1 ) = ( )&& ij( ij 1 = ij 1 + ij 1 (17) f r i, j = 1, L, n. Use f (17) fr the quadratic term in p (10) gives ' Hp ( ) ' H && ( = && p 1 ) + ' H&& ( p (18) where = p p (19) Bates and Watts [6] define the parameter-effects curvature K and intrinsic curvature K as tw measures f nnlinearity, which cmpare the quadratic term with the linear term in the directin f the vectr in the parameter space. hese tw curvature measures are defined as 4. K K ' H&& ( p = Hp & ( ' H&& ( p = Hp & ( ) 1 Chi-square detectr based n innvatin (0) (1) chec if the EKF r UKF diverges, a very effective way is t calculate the nrmalied innvatin squared (IS) [] f the EKF r UKF and cmpare it with a threshld. he IS is defined as 1 IS = [ h( x )]' S [ h( x )] () 1 1 where h( x ) is the innvatin f the EKF r the UKF, and S = J P J + R (3) 1 ' is the cvariance matrix f the innvatin, which is prvided by the filter. When the linear and Gaussian assumptin is reasnably gd, the innvatin is a ermean Gaussian and its cvariance is given by S. In such a case, the IS fllws a Chi-square distributin with n degrees f freedm. Hwever, when the filter diverges, the IS will n lnger fllw a Chi-square distributin. Hence, we prpse t chec if the IS falls int the 1 α cnfidence regin fr a Chi-square randm variable with n degrees f freedm, t judge if the filter diverges. Here α is a very small number, which is actually determined by the false alarm rate f the Chi-square detectr. 4.3 Experimental results A single radar with strng nnlinearity In this experiment, we assume that there is a single radar sensr that is tracing the target. he gemetry f the radar and the trajectry is similar t that shwn in Fig. 1. First, we investigate a case with highly nnlinear measurements. We assume that the range measurements are very accurate, with a range errr standard deviatin (s.d.) f 10 m, and the bearing measurements are relatively carse, with a bearing errr s.d. f 4. Fr a single Mnte-Carl run, the target state estimatin errrs are pltted in Fig. 4 fr different algrithms. It can be seen that the EKF diverges, while the PF and the UKF cnverges and have very similar perfrmances. In Fig. 5, the parameter-effects curvature defined in (0) have been pltted fr the EKF and the UKF. It is clear that the maximum curvature measure fr the EKF (arund 100) is much greater than that f the UKF (belw 1), indicating that the EKF is nt wring prperly. Hwever, the curvature measures can nt indicate the divergence f the EKF in real time, since its pea value ccurs a lng time after the EKF diverges. 5

7 In Fig. 8, the curvature measures fr the EKF and the UKF are pltted. It is clear that due t the lw nnlinearity, the curvature measures are very small. Similarly, the ISs fr the EKF and the UKF are within the 98% cnfidence regin fr all the time, as illustrated in Fig. 9. Fig. 4 RMSEs in psitin by different tracing algrithms w bearing sensrs In this experiment, we assume that tw bearing-nly sensrs are used t prvide psitinal infrmatin t the tracer. he target trajectry and the psitins f the tw sensrs are shwn in Fig. 10. Each sensr has a measurement errr s.d. f 3. In Fig. 6, the ISs are shwn fr the EKF and the UKF. It is evident that fr mst f the time, the IS f the EKF is utside the ne-sided 98% cnfidence regin f the Chisquare distributin with tw degrees f freedm, crrectly indicating the EKF is diverging. On the cntrary, the IS fr the UKF is inside the 98% cnfidence regin at all f the time. It is clear that the Chi-square detectr based n the innvatin f the Kalman filter is effective in detecting the filter divergence in real time. Fig. 6 ISs fr the EKF and the UKF. Dashed line dentes the ne-sided 98% cnfidence regin. Fig. 5 Curvature measures fr the EKF and the UKF A single radar with wea nnlinearity We investigate a case with very wea nnlinearity in the measurements. We assume that the range measurement errr has a s.d. f 100 m, and the bearing measurement has a s.d. f 1. As we can see frm Fig. 7, bth the EKF and the UKF wr well and have perfrmances which are very clse t that f the PF. Fig. 7 racing perfrmances f different tracing algrithms. 6

8 regin. he pea values f its curvature measures are much greater than thse f the EKF. Fig. 8 Curvature measures fr the EKF and the UKF. In summary, bth the curvature measure and the IS are effective measures f the nnlinearity f the nnlinear tracing prblems. One prblem with the curvature measure is that its evaluatin requires the true value f the target psitin [ ξ η ] ', which is nt available n-line. One pssible slutin is t use the updated psitin estimate [ ξ ', which is generated by the filter, t η ] replace [ξ η ] '. his estimated value, hwever, is nt rbust, especially when the filter itself is diverging. On the ther hand, ne can certainly use the curvature measures ffline via simulatins, t evaluate the nnlinearity f a specific tracing prblem, with all the pre-specified parameters. he IS, hwever, des nt require the nwledge f the true [ ξη ] ', since the innvatin is simply the difference between the predicted measurement prvided by the filter, and the incming measurement. As we have bserved in the experiments, the IS is very effective in indicating the divergence f the filter, when there is significant mismatch between its estimatin errr and its calculated cvariance matrix. Fig. 9 ISs fr the EKF and the UKF. he dashed line dentes the ne-sided 98% cnfidence regin. Fig. 11 racing perfrmances f different tracing algrithms. Fig. 10 Gemetry f the target trajectry and the bearing sensrs. he tracing perfrmances, the curvature measures and the ISs are pltted in Figs , respectively. It can be seen that the UKF diverges in the first 50 secnds, during which time, its IS falls utside the 98% cnfidence 5 Cnclusin In this paper, we investigated different nnlinear filtering techniques, including the EKF, the UKF, and the PF fr nnlinear target tracing prblems under varius scenaris. heir tracing perfrmances were cmpared and their cmplexities were evaluated in terms f the numbers f flating pint peratins. It was shwn that fr a tracing prblem with wea nnlinearity, the EKF and the UKF are near-ptimal and it is nt necessary t 7

9 emply the cmputatinally expensive particle filter. Hwever, fr prblems with strng nnlinearity, particle filters shuld be used t get reliable tracing perfrmances. Acnwledgment his wr was supprted by the USAF AFRL under Cntract FA C We are grateful t Mar Ka and im Bumpus fr valuable discussins during the curse f this wr. References [1] Z. Chen, ``Bayesian filtering: Frm Kalman filters t particle filters, and beynd, Adaptive Syst. Lab., McMaster Univ., Hamiltn, O, Canada. [Online], [] Y. Bar-Shalm, X.R. Li, and. Kirubarajan, Estimatin with Applicatins t racing and avigatin, ew Yr: Wiley, 001. Fig. 1 Curvature measures fr the EKF and the UKF ver time. [3] S.J. Julier, J.K. Uhlmann, ``A ew Extensin f the Kalman Filter t nlinear Systems, Int. Symp. Aerspace/Defense Sensing, Simul. and Cntrls, Orland, FL,1997. [4]. Grdn, D. Salmnd, and A. F. M. Smith, vel apprach t nnlinear and nn-gaussian Bayesian state estimatin, in Prceedings IEE-F, vl. 140, pp , [5] A. Ducet,. de Freitas, and. Grdn, Sequential Mnte Carl Methds in Practice. Statistics fr Engineering and Infrmatin Science. ew Yr: Springer-Verlag, 001. [6] D.M. Bates and D.G. Watts, nlinear Regressin Analysis and its Applicatins, Jhn Wiley, Fig. 13 ISs fr the EKF and the UKF. he dashed line dentes the ne-sided 98% cnfidence regin. measure the nnlinearity f the tracing prblem, we explred tw types f measures. One type f measure is based n the curvature measures, which cmpares the magnitude f the secnd-rder term with that f the firstrder term in a aylr series expansin f the nnlinear prblem. he secnd measure calculates the nrmalied innvatin squared (IS) f the Kalman filter, which indicates filter divergence when it exceeds a certain threshld. Experiments shw that bth measures are effective in measuring the nnlinearity f the prblem. he curvature measures are mre suitable fr the ffline peratin while the IS can detect filter divergence nline in real time. 8 [7] M. Mallic, B. F. La Scala and S. Arulampalam ``Differential Gemetry Measures f nlinearity fr the Bearing-Only racing Prblem, SPIE Cnference n Signal Prcessing, Sensr Fusin and arget Recgnitin, Vl 5809, Orland, FL, USA, March 005. [8] M. Mallic, and B.F. La Scala, ``Differential Gemetry Measures f nlinearity fr Grund Mving arget Indicatr (GMI) Filtering, Prceedings f the 8th Internatinal cnference n Infrmatin Fusin, pp. 19-6, July 005. [9] M. Mallic and B. F. La Scala, ``Differential Gemetry Measures f nlinearity fr the Vide racing Prblem, SPIE Cnference n Signal Prcessing, Sensr Fusin and arget Recgnitin, Vl 635, Orland, FL, USA, April 006.