Performance comparison of EKF and particle filtering methods for maneuvering targets

Transcription

1 Digial Signal Processing 17 (2007) Performance comparison of EKF and paricle filering mehods for maneuvering arges Mónica F. Bugallo, Shanshan Xu, Pear M. Djurić Deparmen of Elecrical and Compuer Engineering, Sony Brook Universiy, Sony Brook, NY , USA Available online 25 Ocober 2006 Absrac Online racking of maneuvering arges is a highly nonlinear and challenging problem ha involves, a every ime insan, he esimaion no only of he unknown sae in he dynamic model describing he evoluion of he arge, bu also he underlying model accouning for he regime of movemen. In his paper we review and compare several sequenial esimaion procedures, ha use appropriae sraegies for coping wih various models ha accoun for he differen modes of operaion. We focus on he applicaion of he recenly proposed cos-reference paricle filering (CRPF) mehodology, which aims a he esimaion of he sysem sae wihou using probabiliy disribuions. The resuling mehod has a more robus performance when compared o sandard paricle filering (SPF) algorihms or he ineracive muliple model (IMM) algorihm based on he use of he well known exended Kalman filer (EKF). Advanages and disadvanages of he considered algorihms are illusraed and discussed hrough compuer simulaions Elsevier Inc. All righs reserved. Keywords: Maneuvering arge racking; Sequenial esimaion; Cos-reference paricle filering 1. Inroducion In recen years he problem of arge racking has araced significan aenion in he signal processing communiy [1 7]. Mos of he sandard algorihms for arge racking have been focused on arges ha do no change heir regimes of movemen while hey are racked. If he arges have several regimes of movemen, i.e., differen movemen models, hey can maneuver, which implies ha racking canno be successful if i is done wih one model. By he inclusion of several models, coping wih maneuvering becomes possible, bu he overall racking is hen much more challenging [8,9]. From a heoreical poin of view his is a very ineresing nonlinear problem, which includes he esimaion of he underlying model a every ime insan as well as is unknown saes and parameers [9,10]. In his paper, we review, analyze, and compare he applicaion of differen echniques o he problem of racking a single maneuvering arge. Maneuvering arge racking has ofen been addressed by he ineracive muliple model (IMM) algorihm [11 13] and combined wih differen echniques like he probabilisic daa associaion (PDA) [14] or he various versions of he Kalman filer [11,15]. These mehods have become almos he sandard approach o maneuvering arge racking * Corresponding auhor. address: monica@ece.sunysb.edu (M.F. Bugallo) /$ see fron maer 2006 Elsevier Inc. All righs reserved. doi: /j.dsp

2 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) [16,17]. However, hey have some major limiaions due o he fac ha heir performance depends on he elemenal filers hey are based on. If he underlying filers do no fi he considered problem well, he performance of he algorihms ges severely degraded. Maneuvering arge racking has also been ackled by paricle filering schemes [8,9,15,18 21]. The primary reason o use hese mehods is heir flexibiliy and accuracy in resolving very difficul nonlinear problems where he underlying models are represened by dynamic sae-space equaions. Exising sandard paricle filering (SPF) mehods require a mahemaical represenaion of he dynamics of he sysem evoluion ha includes assumpions of probabilisic models. Cos-reference paricle filering (CRPF), unlike SPF, aims a he esimaion of he sysem sae from he available observaions wihou a priori knowledge of any disribuion funcion [22]. The reference derived from he assumed probabiliy disribuions is subsiued by a user-defined cos funcion ha measures he qualiy of he sae signal esimaes according o he available observaions. The resuling echniques presen a more robus performance han he one achieved by SPF mehods whose heory is based on probabilisic assumpions. The basic CRPF mehod and is varians have already been successfully applied o he problem of racking maneuvering arges [10,20,23]. In his paper, we review he CRPF mehod which uses a deerminisic approach for model proposals, a problem ha is inheren o maneuvering arge racking [23]. The algorihm is compared wih he muliple model auxiliary SPF which uses he same sraegy for coping wih he muliple models in he sysem [20]. We consider wo differen formulaions of he CRPF, one of which has been proven o be equivalen o ha of he auxiliary SPF [24]. For performance comparison, we also include he IMM algorihm based on he exended Kalman filer (EKF). The remaining of his paper is organized as follows. Secion 2 inroduces he dynamic sysem ha describes he problem of racking of maneuvering arges moving along a wo-dimensional space. The fundamenals of he CRPF family and is applicaion o he considered problem are described in Secion 3. We discuss oher algorihms, he IMM- EKF and he muliple model auxiliary SPF, which are used for comparison wih he CRPF in Secion 4. Compuer simulaion resuls ha illusrae he performance of he differen algorihms are presened in Secion 5. Finally, brief concluding remarks are included in Secion A dynamic sysem describing a maneuvering arge A maneuvering arge rajecory is characerized by a consan velociy regime wih shor periods of urning ha correspond o maneuvers. The dynamics of his ype of objecs can be described by a Markovian swiching srucure of he form [8] x = A(ω 1 ) x 1 + Bu, where x =[x 1, x 2, ẋ 1, ẋ 2, ] R 4 conains he Caresian coordinaes of he arge posiion (m) and velociy (m/s) in he xy-plane. The sae ransiion marices are given by sin(ω 1 0 T s ) ω 1 cos(ω T s ) T 2 s ω A(ω ) = cos(ω 2 0 T s ) sin(ω T s ) ω ω 0 0 cos(ω T s ) sin(ω T s ), B = T 0 2 s 2 T s 0, 0 0 sin(ω T s ) cos(ω T s ) 0 T s where T s is he sampling period (s) and ω is a urn rae variable. The maneuver behavior depends on he velociy and is modeled as a ω = + u ω,, ẋ1, 1 2 +ẋ2 2, 1 where a represens he ypical maneuvering acceleraion a ime insan, which randomly swiches among hree possible values, A ={a (1),a (2),a (3) }, and defines he naure of he movemen, i.e., sraigh moion, righ urn or lef urn. Therefore, we have hree differen sysem models, and we use m {1, 2, 3} o denoe he sysem model a ime insan (e.g., m = 1 corresponds o a = a (1) ). Swiching he movemen regime occurs according o he ransiion probabiliy marix, [ ] h11 h 12 h 13 H = h 21 h 22 h 23, h 31 h 32 h 33

3 776 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) where h ij = p(m = j m 1 = i) is he probabiliy of he sysem o swich from model i a ime insan 1 o model j a ime insan, i, j = 1, 2, 3, and he iniial model probabiliies are se o guaranee ha 3 i=1 p(m = i) = 1. The overall sae vecor is given by x =[ x ω ], (1) wih sae noises u R 2 and u ω, R. The observaion equaion is defined as y = h(x ) + v, (2) where v is he observaion noise wih independen componens v j,, j = 1, 2, and he funcion h( ) has wo componens. An emier on he moving arge ransmis a signal wih power P 0 hrough a fading channel wih aenuaion coefficien α. The received signal power and he relaive angle beween he arge and a reference poin wih fixed locaion r =[r x r y ] are measured, i.e., ( P 0 h 1 (x ) = 10 log 10 r p α ), h 2 (x ) = (p r), where p =[x 1, x 2, ] R 2 denoes he posiion of he arge and z = z z is he norm of he vecor z. The objecive is he sequenial esimaion of he arge sae, x 0:, given he sequence of measuremens, y 1:. 3. The CRPF mehod The CRPF mehodology allows for recursive esimaion of unobserved saes of dynamic sysems wihou assumpions abou he probabiliy disribuions of he noises in he sae and observaion equaions [22]. The main feaure of hese filers is ha he saisical reference provided by he a poseriori sae probabiliy densiy funcion (pdf), p(x 0: y 1: ), is subsiued by a user-defined cos funcion ha measures he qualiy of he sae signal esimaes according o he available observaions. The cos-reference paricle filer a ime is described by a se of M sae samples or paricles, which represen he sae, x, and associaed coss up o ha ime insan. To accoun for he muliple regimes exising in he maneuvering arge racking problem, he filer also includes he maneuvering model, m. The cos-reference paricle filer a ime insan is given by he measure [10] Ξ = { x (i) The paricles x (i), C (i),m (i) } M i=1., i = 1,...,M, are generaed using a seleced propagaion densiy, which is consruced under a se of mild condiions [22]. In general, he cos funcion is formulaed by means of a recursive addiive srucure, C (i) = C ( x (i) 0: y 1:,λ ) = λc ( x (i) 0: 1 y 1: 1,λ ) + C ( x (i) y ), (3) where he erm on he righ-hand side of he expression represens he cos up o ime 1 weighed by a forgeing facor, λ, plus a cos incremen, C(x (i) y ), obained from he sae and observaion vecors a ime. Differen funcions for he cos funcion as well as for he cos incremen can be proposed under some defined design condiions [22]. In his paper, we use as cos incremen C ( x (i) ) y = y h ( x (i) ) q, (4) where q is eiher 1 or 2. The measure, Ξ, is randomly propagaed when y +1 is observed and an updaed measure, Ξ +1, is buil. The muliple model CRPF mehod for he saed maneuvering arge racking problem consiss of he following seps: (1) Selecion or resampling: GivenM paricles a ime, 3M predicions or risks are evaluaed a ime + 1 (one for each model and each paricle). These predicions are calculaed using he paricles from he previous ime insan, {x (i) } M i=1 and he curren observaion, y +1. They represen anicipaed coss of he propagaed paricles. A possible expression for he risks is [22] R (i,m +1) +1 = λc (i) + C ( x (i) y +1,m +1 ), m+1 = 1, 2, 3,

4 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) where C(x (i) y +1,m +1 ) is obained by firs predicing a new sae according o he model m +1, and hen applying he incremenal cos funcion given by Eq. (4) [20]. Once predicions are generaed, we replicae paricles wih low risks while high-risk paricles are discarded. To perform his selecion, we consruc a probabiliy mass funcion (pmf) from he risk R +1, i.e., ˆπ (i,m +1) +1 μ ( R (i,m ) +1) +1, where μ( ) :R [0, + ) is a monoonically decreasing funcion. As shown in [22], he choice of μ( ) has a direc impac on he performance of he algorihm. In his paper we chose μ( ) as μ ( R (i,m ) +1) 1 +1 = (R (i,m +1) +1 min i {R (i,m +1) +1 }+δ), β where δ = 10 1 var{r (i,m +1) +1 } 3M i=1 wih var( ) denoing variance, and β = 2. Oher mehods for performing paricle selecion have also been explored [22,24]. I is imporan o remark ha only M rajecories survive he resampling sep. Specifically, we selec ˆx (i) = x (k) wih probabiliy ˆπ (k,m +1) +1, and build an inermediae paricle filer, ˆΞ +1 = {ˆx (i), ˆ C (i),m (i) +1} M i=1, where C ˆ (i) = C (k) if, and only if, ˆx (i) = x (k). (2) Paricle propagaion: Using a seleced propagaion densiy [22], new paricles are generaed by x (i) +1 p ( +1 x+1 ˆx (i),m (i) +1), and he associaed coss are updaed according o C (i) +1 = λ C ˆ (i) + C ( x (i) +1 y +1). As a resul, we obain he updaed represenaion of he paricle filer, Ξ +1 = { x (i) +1, C(i) +1,m(i) +1} M i=1. (3) Esimaion: One possibiliy o obain an esimae of he sae is by assigning a pmf π (i) +1, i = 1, 2,...,M,ohe paricles in Ξ +1. This pmf can be consruced analogously as ˆπ (i) +1 in sep (1), subsiuing he risks by coss. Using his pmf, useful esimaes of he sae based on he mean value can be calculaed according o x mean +1 = M i=1 π (i) +1 x(i) +1. Oher alernaives for esimaion include obaining esimaes based on he model ha has maximum poserior probabiliy, assigning esimaes based on each of he hree models ogeher wih he poserior probabiliies of he models, or applying mehods ha do no require pmf calculaion [22,24]. The muliple model CRPF algorihm is oulined in Table 1. The iniializaion and updae of disribuion parameers for paricle generaion have been previously addressed in [22]. 4. Review of he IMM-EKF and he muliple model SPF mehods For comparison and benchmarking purposes, we have also considered he IMM algorihm based on he use of he well known EKF and he algorihm known as auxiliary SPF [8]. The laer has an algorihmic srucure (resampling, imporance sampling and sae esimaion) very similar o he CRPF family. (5)

5 778 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Table 1 The muliple model CRPF algorihm Iniializaion For i = 1,...,M, iniialize: The paricle x (i) 0 U(I x 0 ) The cos C (i) 0 = 0 The parameers for he proposal funcion, p +1 ( ) Recursive updae For = 1oT (1) Selecion or resampling (for i = 1,...,M) Compue: R (i,m +1) +1 = λc (i) + y+1 h ( x (i,m +1) ) q +1, m+1 = 1, 2, 3, where x (i,m +1) +1 is a prediced sae based on he model m +1 ˆπ (i,m +1) +1 μ ( R (i,m +1) ) +1 = 1 (R (i,m +1 ) +1 min i {R (i,m +1 ) +1 }+δ) β (i) Resample o obain ˆΞ +1 = {ˆx, C ˆ (i),m (i) M +1} i=1 (2) Paricle propagaion (for i = 1,...,M) Draw x (i) +1 p ( +1 x+1 ˆx (i),m (i) ) +1 If >10, updae parameers for he proposal funcion, p +1 ( ) Calculae C (i) +1 = λ C ˆ (i) + y +1 h ( x (i) ) q +1 (3) Esimaion Compue: π (i) π (i) = = μ ( C (i) π (i) Mj=1 π (j) x mean +1 = M i=1 π (i) +1 x(i) +1 ) = 1 (C (i) min i {C (i) (for i = 1,...,M) }+δ) β (for i = 1,...,M) 4.1. The IMM-EKF mehod The EKF is a subopimal esimaion mehod suiable for sysems wih nonlineariies [1]. I proceeds by coninually updaing a linearizaion around he previous sae esimae and by approximaing he sae densiies by Gaussian densiies. For he maneuvering arge racking problem described in Secion 2, he EKF is combined wih he IMM scheme in order o deal wih several models in he sysem [25]. Therefore, hree EKFs, one per model, are implemened in parallel. As a resul, a mixure densiy is obained and approximaed by a single Gaussian which maches he firs and second momens. The IMM algorihm is summarized in Table 2 and each of is cycles consiss of he following seps [1]: (1) Calculaion of mixing probabiliies: Given ha model j is in effec a ime insan and condiioned on he observaions up o ime insan 1, y 1: 1, he probabiliy μ i j ( 1 1), i = 1, 2, 3, ha he ih model was in effec a ime insan 1 is calculaed. (2) Combinaion of mixing probabiliies: Given he oupus of he EKFs a ime insan 1, ˆx i 1 1, i = 1, 2, 3, he mixed iniial condiion for he filer mach o model j, ˆx 0j 1 1, and he corresponding covariance, P0j 1 1,are obained. (3) Model-mached filering: The esimaes in sep (2) are used as inpus o he filer mached o model j, which calculaes he oupu ˆx j and updaes he covariance marix Pj using he observaion y. (4) Updae of model probabiliies: Thejh model probabiliy a ime, μ j (), is updaed using he likelihood of he jh filer. (5) Esimaion: The model-condiioned esimaes, ˆx j, j = 1, 2, 3, are combined o ge a sae esimae.

6 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Table 2 The IMM-EKF algorihm Iniializaion For j = 1, 2, 3 (each model), iniialize: The esimae ˆx j 0 0 = E[ ] x 0 The covariance marix P j 0 0 = E[ (x 0 E[x 0 ])(x 0 E[x 0 ]) H ] The model probabiliy μ 1 (0) = p(1), μ 2 (0) = p(2), μ 3 (0) = p(3) Recursive updae For = 1oT (1) Calculaion of mixing probabiliies (for i, j = 1, 2, 3) Compue μ i j ( 1 1) = 1 c h j ij μ i ( 1), wih c j = 3 i=1 h ij μ i ( 1) (2) Combinaion of mixing probabiliies (for j = 1, 2, 3) Calculae: ˆx 0j 1 1 = 3 i=1 ˆx i 1 1 μ i j ( 1 1) ɛ ij 1 1 = ˆxi 1 1 ˆx0j 1 1 P 0j 1 1 = 3 i=1 μ i j ( 1 1) ( P i ) ɛij 1 1 ɛij, 1 1 (3) Model-mached filering (for j = 1, 2, 3) Use ˆx 0j 1 1 and P0j 1 1 as inpu o EKF j Esimae ˆx j and Pj Obain he likelihood Λ j () = p ( y j, ˆx 0j 1 1, ) P0j 1 1 (4) Updae of model probabiliies (for j = 1, 2, 3) Compue μ j () = 1 c Λ j () c j, wih c = 3 j=1 Λ j () c j (5) Esimaion Obain ˆx = 3 j=1 ˆx j μ j () 4.2. The muliple model auxiliary SPF mehod The SPF uses samples and associaed weighs o approximae he poserior disribuion of he esimaed saes. Unlike he CRPF, samples/weighs are generaed/updaed based on he available probabilisic knowledge of he sysem of ineres. For he muliple model case, he paricle se is given by [9] Υ = { x (i),w (i),m (i) } M i=1, where w (i) denoes he ih weigh. We focus on he auxiliary SPF [26, Chaper 23], which has an analogous basic srucure as he original CRPF [22]. Alhough several sraegies can be applied o cope wih he muliple models in he maneuvering arge racking problem [9,20], we adop he same scheme as he one described for he CRPF in Secion 3. The resuling muliple model auxiliary SPF implemenaion is an exension of he auxiliary SPF algorihm, which uses he prior densiy as he proposal funcion and proceeds, a ime insan + 1, as follows: (1) Resampling: Using he paricles from he previous ime insan, {x (i) } M i=1, and he hree possible models, 3M paricles are proposed. Resampling is carried using a mulinomial disribuion obained from he weighs w (i,m +1) +1 = w (i) p ( m +1 m (i) ) ( p y+1 x (i,m ) +1) +1, m+1 = 1, 2, 3, where p(y x (i,m +1) +1 ) is obained by firs predicing a new sae according o he model m +1, x (i,m +1) +1, and hen obaining he likelihood. M new paricles are sampled ou of he 3M proposed paricles o obain he ransiional filer ˆΥ +1 =, ŵ (i),m (i) {ˆx (i) +1} M i=1. (2) Paricle propagaion: I is carried ou using he sae ransiion disribuion and he model sample, i.e., x (i) +1 p( x +1 ˆx (i),m (i) +1).

7 780 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Table 3 Muliple model auxiliary SPF algorihm Iniializaion For i = 1,...,M, iniialize: The model m (i) 0 p(m 0) The paricle x (i) 0 p(x 0) The weigh w (i) 0 = M 1 Recursive updae For = 1oT,fori = 1,...,M (1) Resampling (for i = 1,...,M) Calculae w (i,m +1) +1 = w (i) p ( m +1 m (i) ) ( p y+1 x (i,m +1) ) +1, m+1 = 1, 2, 3, where x (i,m +1) +1 is a predicion of he sae based on he model (i) Resample o obain ˆΥ +1 = {ˆx, ŵ (i),m (i) M +1} i=1 (2) Paricle propagaion (for i = 1,...,M) Obain: x (i) +1 p( x +1 ˆx (i),m (i) ) +1 w (i) +1 p(y +1 x (i) +1 ) p(y +1 ˆx (i) ) w (i) +1 = w (i) +1 Mj=1 w (j) +1 (3) Esimaion Compue x mean +1 = M i=1 w (i) +1 x(i) +1 The number of paricles mus be large enough o ensure an adequae represenaion of he poserior densiy. Once he paricles are propagaed, weighs are calculaed and normalized [21]. (3) Esimaion: The procedure is compleely analogous as ha of he CRPF. The mean square esimae of he sae, x mean +1, is calculaed using he paricles and heir weighs. A summary of his algorihm is provided in Table 3. I is well known ha paricle filering is a compuaionally expensive mehodology. The SPF and CRPF algorihms have approximaely he same compuaional complexiy whereas he Kalman filering based mehods are much less demanding. Also, he more paricles we use for racking wih paricle filering, he slower he processing of he daa is. However, paricle filering can be implemened in parallel and savings in compuing imes of he order of magniude can be achieved [27]. 5. Compuer simulaions In his secion, we presen compuer simulaions ha illusrae he performance of he various racking algorihms described in he previous secions. We compared he following mehods: (1) he IMM-EKF (labeled in he figures as IMM-EKF); (2) he muliple model auxiliary SPF wih correc knowledge of noise disribuions (labeled MM-SPF); (3) he muliple model auxiliary SPF wih mismached informaion abou he observaion noise disribuion (labeled MM-SPFM); (4) he CRPF wih parameers q = 1 and λ = 0.95 (labeled CRPF); (5) he CRPF wih parameers q = 1 and λ = 0 (labeled CRPF0); (6) he CRPF wih parameers q = 2 and λ = 0.95 (labeled CRPF2); (7) he CRPF wih parameers q = 2 and λ = 0 (labeled CRPF20). For all he paricle filers, he number of paricles was iniially se o M = 500. The CRPF wih λ = 0 is of special ineres since i has been proven o be equivalen o he muliple model auxiliary SPF [20,24].

8 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Fixed rajecory In he firs experimen we sudied he performance of he algorihms over a single fixed rajecory. We considered a scenario where he arge sared from an iniial posiion close o (0, 0), and wih an iniial consan velociy of 55 m/s. I moved for 5 min and measuremens were made wih a sampling ime of T s = 5 s. The magniude of he velociy was consrained wihin he inerval (50, 60) m/s for he whole moion course. The maneuvering model variable, a, swiched among hree discree values {0, 5, 5}. The model, m, was changing according o he Markovian ransiion probabiliy marix, ( ) 0.15 H = , and he sae noises were modeled as zero-mean whie Gaussian processes, u N ( 0,σ 2 u I 2), uω, N ( 0,σ 2 u ω ), where σ 2 u = 10 2 and σ 2 u ω = 10 4 were he respecive variances and I 2 represens he 2 2 ideniy marix [7, Chaper 6]. Measuremens were colleced a he reference poin, following Eq. (2) wih P 0 = 1. The observaion noise was modeled as a mixure Gaussian disribuion, v 0.7N (0, 4 I 2 ) + 0.3N (0, 25 I 2 ). The seleced arge rajecory and he reference poin are shown in Fig. 1. Figure 2 shows he sysem rajecory in a single run and he esimaes corresponding o he algorihms. For clariy in he presenaion, we ploed he resuls of he four CRPF mehods on he lef, and he resuls for one of he CRPF algorihms and he res of he algorihms on he righ. I can be seen ha all he CRPF algorihms and he muliple model auxiliary SPF mehod wih perfec knowledge of he disribuions of he noises performed similarly and remained locked o he rue rajecory of he arge during he whole racking inerval. However, he IMM-EKF, which had o use a Gaussian assumpion for he observaion noise and considered v N (0, I 2 ), and he muliple model auxiliary SPF, which also used ha same wrong Gaussian assumpion, were no able o rack properly. Noe ha he CRPF algorihms were assuming he same disribuion as he IMM-EKF for propagaion of paricles and his choice did no affec heir performance. Therefore he CRPF algorihms, which do no rely on any prior probabilisic informaion, showed a more robus performance. We also compared he algorihms by means of he roo-mean-square (RMS) error of he posiion and he percenage of rack loss. Le p =[x 1, x 2, ] denoe he rue posiion of he arge a ime insan, and ˆp =[ˆx 1, ˆx 2, ] be he Fig. 1. Targe rajecory and he reference locaion.

9 782 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Fig. 2. Trajecory esimaes. Lef: Differen CRPF algorihms. Righ: Comparison of he CRPF mehod wih he res of he mehods. Fig. 3. Lef: RMS error of he posiion. Righ: Percenage of rack loss for various values of γ. corresponding esimae obained by a paricular filer. The RMS error a ime insan was calculaed by averaging over N = 500 independen simulaion rials, RMS = 1 N [ ( ˆx1,,i x 1,,i ) N 2 + ( ˆx 2,,i x 2,,i ) 2], i=1 where he subscrip i denoed he rial number. For a paricular simulaion run, a rack loss was confirmed when he posiion error, e = ( ˆx 1, x 1, ) 2 + ( ˆx 2, x 2, ) 2, exceeded a hreshold γ wihin a leas 10 consecuive sampling periods. The percenage was obained by averaging over N independen runs. Figure 3 depics he resuls. Again he IMM-EKF and he muliple model auxiliary SPF wih incorrec noise disribuions showed much worse performance han he oher filers Effec of he reference poin locaion In he nex experimen, we moved he reference poin away from he rajecory area from ( 2741, 2161) o ( 6000, 5000). This is equivalen o reducing he signal-o-noise raio (SNR). To comba he low SNR, we increased he number of paricles for he paricle filering algorihms, and we se i o M = The resuls are shown in Fig. 4. We can draw similar conclusions as from he previous experimen.

10 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Fig. 4. Effec of locaion of reference poin. Top: Trajecory esimaes. Boom: RMS error of he posiion and percenage of rack loss for various values of γ Effec of he sampling period We also checked he sensiiviy of he algorihms o variaions in he sampling period, T s. Using he same parameers as in he firs experimen, we se T s = 10 s. The resuls for he RMS error of he posiion and percenage of rack loss are shown in Fig. 5. I can be seen ha he RMS error increased for all he mehods. The IMM-EKF performed he wors. A possible reason is ha he maneuver onse and/or offse insan could be missed, and he saisical relaion beween adjacen samples was weakened Effec of he incomplee informaion The performance of he muliple model auxiliary SPF srongly depends on he availabiliy of correc model informaion, which includes noise disribuions, model swiching probabiliies, and model parameers. We previously showed ha he performance of his algorihm severely deerioraed when incorrec informaion abou he noise disribuions was used. In anoher se of experimens we esed he robusness of he considered algorihms wih respec o he model ransiion probabiliies and he models parameers. Firs, we considered he following model swiching probabiliy marices: H 1 = ( ) , H 2 = ( )

11 784 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Fig. 5. Effec of he sampling inerval. Lef: RMS error of he posiion. Righ: Percenage of rack loss for various values of γ. Fig. 6. Effec of availabiliy of swiching probabiliy marices. Lef: RMS error of he posiion. Righ: Percenage of rack loss for various values of γ. The rajecory was generaed by using he marix H in Secion 5.1 bu he algorihms assumed he previous marices for calculaion of he likelihoods. Figure 6 shows he resuls. We can see ha he performance of he muliple model auxiliary SPF became worse in he absence of such informaion. We nex supposed ha he ypical maneuvering model value was unknown, and we analyzed he performance of he algorihms by considering wo ses of possible values for he acceleraion variable, a A 1 ={0, 3, 3} and a A 2 ={0, 7, 7} insead of { 5, 0, 5}. Figure 7 shows he resuls. Again, i is clear ha he CRPF is more robus when his informaion is no available Differen rajecories We applied he considered algorihms on 500 randomly generaed rajecories. The resuls shown in Fig. 8 are similar o he previous experimens. 6. Conclusions In his paper, we have discussed and compared he applicaion of differen echniques o he problem of racking a maneuvering arge in he wo-dimensional space. We have focused on he cos-reference paricle filering (CRPF) mehodology, which does no use probabilisic assumpions required by sandard paricle filering (SPF), leading o a more robus performance. The muliple model problem is solved by a deerminisic model proposal process ha

12 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) Fig. 7. Effec of availabiliy of model value. Lef: RMS error of he posiion. Righ: Percenage of rack loss for various values of γ. Fig. 8. Differen rajecories. Lef: RMS error of he posiion. Righ: Percenage of rack loss for various values of γ. allows for exploring he differen regimes of movemen in each sep of he algorihm. We have compared he resuling algorihm wih he exended Kalman filer (EKF) and he muliple model auxiliary SPF mehod. The advanages and disadvanages of he considered algorihms were illusraed and discussed hrough compuer simulaions. Fuure work includes he sudy of differen cos funcions and heir relevance o aain beer performance and he exension of he algorihms o deal wih he problem of racking of muliple arges. Acknowledgmens This work has been suppored by he Naional Science Foundaion under he Award CCF and by he Office of Naval Research under Award N References [1] Y. Bar-Shalom, X.R. Li, T. Kirubarajan, Esimaion wih Applicaion o Tracking and Navigaion: Theory, Algorihms, and Sofware, Wiley, New York, [2] N.J. Gordon, A hybrid boosrap filer for arge racking in cluer, IEEE Trans. Aerospace Elecron. Sys. 33 (1) (1997) [3] F. Gusafsson, F. Gunnarsson, N. Bergman, e al., Paricle filers for posiioning, navigaion, and racking, IEEE Trans. Signal Process. 50 (2) (2002) [4] C. Hue, J.P.L. Cadre, P. Perez, Tracking muliple objecs wih paricle filering, IEEE Trans. Aerospace Elecron. Sys. 38 (3) (2002) [5] D. Li, K.D. Wong, Y.H. Hu, A.M. Sayeed, Deecion, classificaion, and racking of arges, IEEE Signal Process. Mag. 19 (2) (2002)

13 786 M.F. Bugallo e al. / Digial Signal Processing 17 (2007) [6] X. Li, V. Jilkov, Survey of maneuvering arge racking Par I: Dynamic sysems, IEEE Trans. Aerospace Elecron. Sys. 39 (4) (2003) [7] B. Risic, S. Arulampalam, N. Gordon, Beyond Kalman Filer: Paricle Filer for Tracking Applicaions, Arech House Publishers, [8] R. Karlsson, N. Bergman, Auxiliary paricle filers for racking a maneuvering arge, in: Proceedings of IEEE Conference on Decision and Conrol, vol. 4, 2000, pp [9] S. McGinniy, G.W. Irwin, Muliple model boosrap filer for maneuvering arge racking, IEEE Trans. Aerospace Elecron. Sys. 36 (3) (2000) [10] M.F. Bugallo, S. Xu, J. Míguez, P.M. Djurić, Maneuvering arge racking using cos reference paricle filering, in: Proceedings of he 29h IEEE Inernaional Conference on Acousics, Speech and Signal Processing (ICASSP 2004), Monreal, Canada, [11] H.A.P. Blom, Y. Bar-Shalom, The ineracing muliple model algorihm for sysems wih Markovian swiching coefficiens, IEEE Trans. Auoma. Conrol 33 (8) (1988) [12] X. Li, V. Jilkov, A survey of maneuvering arge racking Par V: Muliple-model mehods, IEEE Trans. Aerospace Elecron. Sys. 41 (4) (2003) [13] E. Mazor, A. Averbuch, Y. Bar-Shalom, J. Dayan, Ineracing muliple model mehods in arge racking: A survey, IEEE Trans. Aerospace Elecron. Sys. 34 (1) (1998) [14] T. Kirubarajan, Y. Bar-Shalom, D. Lerro, Bearing-only racking of maneuvering arges using a bach-recursive esimaor, IEEE Trans. Aerospace Elecron. Sys. 37 (3) (2001) [15] B. Risic, M.S. Arulampalam, Tracking a maneuvering arge using angle-only measuremens: Algorihms and performance, Signal Process. 83 (6) (2003) [16] W. Blair, G. Wason, IMM algorihm for soluion o benchmark problem for racking maneuvering arges, in: Proceedings of SPIE, vol. 2221, SPIE, [17] V. Jilkov, D. Angelova, T. Semerdjiev, Design and comparison of mode-se adapive IMM algorihms for maneuvering arge racking, IEEE Trans. Aerospace Elecron. Sys. 35 (1) (1999) [18] M.S. Arulampalam, N. Gordon, M. Oron, B. Risic, A variable srucure muliple model paricle filer for GMTI racking, in: Proceedings of he 2002 Inernaional Conference on Informaion Fusion, Annapolis, MD, [19] M.R. Morelande, S.S. Challa, Manoeuvring arge racking in cluer using paricle filers, IEEE Trans. Aerospace Elecron. Sys. 41 (1) (2005) [20] M.F. Bugallo, S. Xu, P.M. Djurić, Comparison of EKF- and PF-based mehods in racking maneuvering arges, in: Proceedings of he Aerospace Conference 2006, Big Sky, MT, [21] A. Douce, N. Gordon, V. Krishnamurhy, Paricle filers for sae esimaion of jump Markov linear sysems, IEEE Trans. Signal Process. 49 (3) (2001) [22] J. Míguez, M.F. Bugallo, P.M. Djurić, A new class of paricle filers for random dynamical sysems wih unknown saisics, EURASIP J. Appl. Signal Process. 15 (2004) [23] S. Xu, M.F. Bugallo, P.M. Djurić, Maneuvering arge racking wih simplified cos reference paricle filers, in: Proceedings of IEEE Inernaional Conference on Acousics, Speech and Signal Processing (ICASSP), Toulouse, France, [24] M.F. Bugallo, J. Míguez, P.M. Djurić, Posiioning by cos reference paricle filers: Sudy of various implemenaions, in: Proceedings of he 2005 Inernaional Conference on Compuer as a Tool (EUROCON), Belgrade, Serbia and Monenegro, [25] X.R. Li, Y. Bar-Shalom, Design of an ineracing muliple model algorihm for air raffic conrol racking, IEEE Trans. Conrol Sys. 1 (3) (1993) [26] A. Douce, N. de Freias, N.J. Gordon (Eds.), Sequenial Mone Carlo Mehods in Pracice, Springer, [27] M. Bolić, P.M. Djurić, S. Hong, Resampling algorihms and archiecures for disribued paricle filers, IEEE Trans. Signal Process. 53 (7) (2005) Mónica F. Bugallo received he Ph.D. degree in compuer engineering from he Universiy of A Coruña, Spain, in From 1998 o 2000 she was wih he Deparameno de Elecrónica y Sisemas a he Universidade da Coruña, Spain, where she worked in inerference cancellaion applied o muliuser communicaion sysems. In 2001, she joined he Deparmen of Elecrical and Compuer Engineering a Sony Brook Universiy where she is currenly Assisan Professor and eaches courses in digial communicaions and informaion heory. Her research ineress lie in he area of saisical signal processing and is applicaions o differen disciplines including communicaions and biology. Shanshan Xu received her B.S. in elecrical engineering from Tsinghua Universiy, Beijing, China, in 2000 and he M.S. degree in elecrical engineering from Sony Brook Universiy, New York, in She is now a Ph.D. suden in he Deparmen of Elecrical and Compuer Engineering, Sony Brook Universiy. Her research ineress are in he field of saisical signal processing and is applicaions. Pear M. Djurić received his B.S. and M.S. degrees in elecrical engineering from he Universiy of Belgrade in 1981 and 1986, respecively, and his Ph.D. degree in elecrical engineering from he Universiy of Rhode Island in From 1981 o 1986 he was Research Associae wih he Insiue of Nuclear Sciences, Vinca, Belgrade. Since 1990 he has been wih Sony Brook Universiy, where he is Professor in he Deparmen of Elecrical and Compuer Engineering. He works in he area of saisical signal processing, and his primary ineress are in he heory of modeling, deecion, esimaion, and ime series analysis and is applicaion o a wide variey of disciplines including wireless communicaions and bio-medicine.