Combining IMM Method with Particle Filters for 3D Maneuvering Target Tracking

Transcription

1 Combnng IMM Method wth Patcle Fltes fo D Maneuveng Taget Tacng Pe Hu Foo Depatment of Physcs Natonal Unvesty of Sngapoe Sngapoe g657@nus.edu.sg Abstact - The Inteactng Multple Model (IMM) algothm s a wdely accepted state estmaton scheme fo solvng maneuveng taget tacng poblems, whch ae geneally nonlnea. Dung the IMM flteng pocess, seous eos can ase when a Gaussan mxtue of posteo pobablty densty functons s appoxmated by a sngle Gaussan. Patcle fltes (PFs) ae effectve n dealng wth nonlneaty and non-gaussanty. Ths wo consdes an IMM algothm that ncludes a constant velocty model, a constant acceleaton model and a D tunng ate (DTR) model fo tacng thee-dmensonal (D) taget moton, usng vaous combnatons of nonlnea fltes. In exstng lteatue on combnng IMM and patcle flteng technques to tacle dffcult taget maneuves, a PF s usually used n evey model. In compason, smulaton esults show that by usng a computatonally economcal PF n the DTR model and Kalman fltes n the emanng models, supeo pefomance can be acheved wth sgnfcant educton n computatonal costs. Keywods: Maneuveng taget tacng, nteactng multple model, patcle flte. Intoducton Due to the nceasngly wdespead use and sophstcaton of mltay and cvlan suvellance systems, much nteest has been geneated n the development of algothms fo taget tacng. Taget tacng poblems can be solved by modelng dynamcal systems. Fo lnea Gaussan poblems, the Kalman flte (KF) [] can be appled to obtan optmal solutons. Howeve, nonlneaty and/o non-gaussanty often exst n taget tacng poblems. Nonlnea flteng technques ae equed to solve such poblems. Fo non-maneuveng taget tacng, an extended Kalman flte (EKF) o an unscented Kalman flte (UKF) [] s usually mplemented to povde Gaussan appoxmaton to the posteo pobablty densty functon (pdf) n the state space. The fome uses the fst-ode Taylo sees expanson of the nonlnea system equatons that descbe the gven poblem. The latte uses multple detemnstcally chosen ponts n the state space to Gee Wah Ng Advanced Analyss and Fuson Laboatoy DSO Natonal Laboatoes Sngapoe ngeewah@dso.og.sg appoxmate the state dstbuton. Anothe goup of popula technques s the class of sequental Monte Calo (SMC) methods, also nown as the patcle fltes (PFs) []. These methods do not have the lmtaton mposed by the Gaussan assumpton equed fo EKFs and UKFs. Hence, they can be appled to poblems wth abtay nonlneates n dynamcs and/o measuements, ncludng those wth non-gaussan pocess and/o measuement noses actng on the system. Random samples (o patcles) ae geneated fo the dect estmaton of the posteo pdf. It s geneally dffcult to use a sngle model to epesent the moton of a maneuveng taget, as the maneuves ae often abupt devatons fom pecedng moton. Hence, multple model (MM) based appoaches ae often used fo maneuveng taget tacng [9][]. The models n these methods un n paallel and descbe dffeent aspects of the taget behavo. In patcula, the Inteactng Multple Model (IMM) algothm [9][][] s wdely accepted as one of the most cost-effectve dynamc MM methods. It has been shown to acheve hgh pefomance wth elatvely low complexty. Howeve, an essental opeaton of the IMM flteng pocess, the mxng/nteacton step (detals gven n Secton.4), yelds a Gaussan mxtue of posteo pdfs. The IMM algothm appoxmates ths non-gaussan element by a sngle Gaussan, whch often esults n seous eos. In the ecent yeas, eseaches have developed technques fo solvng nonlnea taget tacng poblems by combnng MM based appoaches (to account fo mode swtchng) and PF vaants (to account fo nonlnea and/o non-gaussan chaactestcs of the posed poblem). A few nstances ae gven below. In [], a PF wth a swtchng/nteacton step of the same fom as that n the IMM algothm was developed fo stochastc hybd systems. A egulazed patcle flte (RPF) was adopted n evey model of the IMM algothm n []. Instead of esamplng, a Gaussan sum pdf was computed to appoxmate the condtonal posteo pdf fo the state n each mode. Ths method nvolved hgh computatonal complexty and addtonal appoxmatons. In an mpoved appoach [6], dect samplng fom each mode condtonal posteo pdf (a weghted sum of dstbutons) was mplemented, n place of appoxmaton wth Gaussan mxtues. An unscented patcle flte

2 (UPF) and a genec/standad patcle flte (SPF) wee appled n each model of the IMM algothm n [5] and [] espectvely. In ths wo, we mplement an IMM algothm that compses a constant velocty (CV) model, a constant acceleaton (CA) model and a D tunng ate (DTR) model, wth vaous combnatons of nonlnea fltes used fo the models. Exstng vaants usually use a KF o a PF n evey model. A new appoach whch uses a PF n the DTR model and KFs n the emanng models s consdeed. Smulaton esults show that new vaants that use computatonally economcal PFs n the DTR models pefom well fo a maneuveng taget tacng poblem. The est of the pape s oganzed as follows. Secton pesents the fomulaton of the dynamcal system model fo maneuveng taget tacng n ths wo. Secton gves a bef dscusson on the IMM algothm and the nonlnea fltes to be used fo the models. Secton 4 contans detals of the smulaton tests conducted. Analyss and compason of the numecal esults obtaned ae also epoted. Concludng emas ae gven n Secton 5. Poblem Fomulaton Consde a state-space model of a maneuveng taget tacng system [] gven by the pocess equaton and the obsevaton equaton X = f(x, w ), () + Z + = h(x, v ), () + + whee the system tanston functon f( ) and measuement functon h( ) ae usually nonlnea. The dscete-tme foms of equatons () and () ae gven by X + = F X G w () + and Z = H X + v (4) espectvely. Fo the -th /step, X s an n x state vecto, Z s an n z measuement vecto, w s an n w pocess nose vecto, v s an n z (addtve) measuement nose vecto, F s the n x n x tanston matx, G s the n x n w pocess nose gan matx and H s the n z n x measuement matx. Let the ntal pdf, p(x Z ) p(x ), of the state vecto (also nown as the po) be assumed to be avalable, wth Z beng the set of no measuements. The complete soluton to the state estmaton poblem s the tue posteo pdf, p(x : Z : ), whee X : {X,, X } and Z : {Z,, Z }. Equaton () defnes the pobablstc model of the state evoluton (tanston po), p(x X - ), whle equaton () defnes the lelhood functon of the cuent measuement gven the cuent state, p(z X ). In Bayesan tacng, t s of nteest to obtan the posteo pdf va the ecusve computaton of the po pdf (pedcton densty) of the state, p(x Z :- ), and the magnal posteo pdf (flteng densty) of the state, p(x Z : ). Fo most nonlnea poblems, the posteo pdf cannot be detemned analytcally. Hence, tactable flteng methods ae equed to obtan appoxmate solutons fo such poblems. Flteng Algothms The nonlnea fltes to be used fo the models n the IMM algothm n ths wo ae befly dscussed n ths secton. Detals on the fltes can be found n the espectve cted efeences.. Extended Kalman Fltes EKFs [][] ae commonly used fo solvng nonlnea taget tacng poblems. An EKF uses fst-ode Taylo sees expansons to appoxmate nonlnea system functons n equatons () and (). It povdes a Gaussan appoxmaton to the magnal posteo pdf of the system state though ts condtonal mean and covaance. Fo poblems wth mld nonlneaty, an EKF povdes satsfactoy esults wth much effcency. Howeve, when poblems ae hghly nonlnea and the effects of the hghe ode tems of the Taylo sees expansons ae not neglgble, appoxmaton esults obtaned wth an EKF ae pone to eos. In such stuatons, the flte s lely to pefom badly o even dvege. Thee exsts a second-ode EKF whch taes nto consdeaton second-ode tems n the Taylo sees expansons. Howeve, t s not wdely used because of nceased complexty and mplementaton costs. Thee s also no guaantee of mpoved esults.. Unscented Kalman Fltes A UKF [] uses a mnmal set of detemnstcally chosen sample ponts (o sgma ponts ) to obtan a Gaussan appoxmaton to the magnal posteo pdf of the system state. Fo an abtay nonlnea poblem, the posteo mean and covaance of the state can be accuately estmated to the second ode (thd ode fo a Gaussan po) of the Taylo sees expansons. UKFs have been epoted to outpefom EKFs n many poblems at no addtonal computatonal costs (the two goups of fltes have almost the same computatonal complexty). Howeve, UKFs also have the lmtaton that they cannot be appled to geneal non-gaussan poblems.

3 . Patcle Fltes Most PFs developed ove the past yeas ae based on sequental mpotance samplng (SIS) []. In pactce, the posteo pdf s usually epesented by a set of andom patcles dawn fom a nown mpotance densty functon (also nown as poposal dstbuton). Then a coespondng mpotance weght s computed fo each patcle. Posteo estmates ae computed based on the samples and the assocated weghts. As the sample sze becomes nceasngly lage, by the Stong Law of Lage Numbes, the appoxmaton conveges to the tue posteo pdf []. It s not easy to desgn an mpotance densty functon that povdes easonable appoxmaton of the posteo pdf. A convenent choce of mpotance densty functon s the tanston po, whch s smple and easy to mplement. The dea of esamplng [][] was ntoduced to elmnate patcles wth low mpotance weghts and multply patcles wth hgh mpotance weghts, wth the numbe of patcles, N s, ept unchanged. Ths s useful n allevatng the advese effects of the degeneacy phenomenon n whch all but vey few of the mpotance weghts become neglgble. Resamplng s to be caed out when the effectve sample sze N eff (a measue of degeneacy) falls below some pedetemned theshold N T (hee, N T = N s /). As t s not possble to evaluate N eff exactly, an estmate of N eff s used dung pactcal mplementaton. A vaety of esamplng schemes s avalable n the lteatue []. In ths wo, systematc esamplng [] s used. Thee ae seveal PF vaants that do not eque esamplng. An example, the Gaussan patcle flte (GPF), s evewed n Secton Standad Patcle Flte The SIS and esamplng technques dscussed above fom the coe composton of the SPF descbed by Algothm n []... Regulazed Patcle Flte The ntoducton of esamplng can esult n the loss of dvesty among the patcles, especally when the nose n the dynamcal system s small o even neglgble. In such a case, the esultant sample set would contan many epeated ponts n the state space, hence gvng a poo epesentaton of the posteo pdf. RPFs [] wee developed wth the am of povdng a emedy fo the above mentoned poblem, called sample mpoveshment. A egulazaton step s added when esamplng s beng conducted n the SPF. The enel method s used to obtan a egulazed empcal epesentaton of the the tue posteo pdf. In ths wo, the RPF used s adapted fom Algothm 6 n [], wth the Epanechnov enel eplaced by the Gaussan enel. The mpotance densty functon used s the tanston po... Extended Kalman Patcle Flte The mpotance densty functon used n the SPF, the tanston po, does not tae nto consdeaton the most ecent measuement data. Consequently, defcency may ase n PFs, especally when thee s lttle ovelap between the mpotance densty functon and the posteo pdf. To avod poblems that may ase fom usng the tanston po as the mpotance densty functon, eseaches have come up wth vaous lneazaton-based appoaches whch ncopoate the latest measuement data. One example s the extended Kalman patcle flte (EKPF) []. Fo each patcle n ths famewo, a sepaate EKF s used to geneate and popagate a Gaussan mpotance densty functon...4 Unscented Patcle Flte Fo each patcle n the famewo of the EKPF, use a UKF nstead of an EKF fo geneatng the mpotance densty functon. The esultant flte s nown as the UPF []...5 Gaussan Patcle Flte The GPF s an nstance of seveal patcle flteng technques that do not eque esamplng [7][8]. Le SIS based PFs, the GPF uses mpotance samplng to obtan patcles. It appoxmates the posteo pdf by sngle Gaussans, smla to Gaussan fltes such as EKFs. The GPF popagates only the mean and covaance of the posteo pdf. Howeve, hghe moments can be popagated due to the fact that all moments can be estmated usng mpotance samplng. The GPF usually outpefoms conventonal Gaussan fltes, especally fo solvng poblems wth nontval nonlneates. It also has lowe computatonal complexty than PFs whch eque esamplng, a pocess that may be computatonally expensve..4 The IMM Algothm The IMM algothm s bult fom seveal dynamc moton models that epesent dffeent taget behavoal tats. The models can swtch fom one to anothe accodng to a set of tanston pobabltes govened by an undelyng Maov chan. One complete cycle of the IMM algothm compses fou pats, namely, an nput mxe (nteacton), a flte fo each model, a model pobablty evaluato and an output mxe (combnaton). The flow dagam of an IMM algothm wth models s shown n Fgue. Hee, =. Table shows the vaous combnatons of fltes consdeed fo the espectve models.

4 An outlne of the -th ( ) cycle of a typcal IMM algothm vaant mplemented n ths wo s gven below [][]. Step : Input mxe (nteacton) Mode-condtoned state estmates and state eo covaances of the pevous step ae meged usng mxng pobabltes fo the ntalzaton of the cuent step. Fo model at tme, M (), compute the ntal state estmate, xˆ (- -) = xˆ ( ) µ ( ), = and the coespondng state eo covaance, Pˆ ( - -) = T µ ( ){Pˆ ( - -) + [ ~ x ( ) ][] }, = whee xˆ ( ) s the po state estmate, ( ) s the coespondng state eo covaance, Pˆ - µ ) = c p µ ( ) s mxng pobablty, ( = pµ ( ) ~ = x ( ) xˆ ( ) xˆ = c s a nomalzng constant, ( ), and p s the assumed tanston pobablty fo swtchng fom model (at tme -) to model (at tme ). Step : Flteng Detemne the elevant flte fom Table. Wth the ntal state estmates, the coespondng state eo covaances and an exogenous measuement data z() as nputs, model updates fo M () ae pefomed by computng the state estmate xˆ ( ), and the coespondng state eo covaance Pˆ ( ). Step : Mode pobablty update Fo M (), wth the use of flte esdual ~ z (), the coespondng flte esdual covaance Ŝ () and the assumpton of a Gaussan dstbuton, the lelhood functon s computed as ~ T Λ ( ) = exp(.5( z ()) (Ŝ ()) ~ z ()). πŝ () The mode pobablty of M () s then updated as - µ ) = c Λ ()c, whee ( c = Λ ()c. = Step 4: Output mxe (combnaton) All the state estmates and the coespondng state eo covaances output fom the ndvdual models ae utlzed fo the computaton of the combned state estmate, xˆ ( ) = µ ()xˆ ( ), = and the coespondng state eo covaance, T Pˆ ( ) = µ (){Pˆ ( ) + [xˆ ( ) xˆ( )][ ] }. = { xˆ ( ), Pˆ ( )} Mxng/Inteacton {xˆ ( ),Pˆ ( )} { xˆ ( ),Pˆ ( )} Combnaton {xˆ ( ), Pˆ ( )} Flteng M () M () { µ ( )} Fgue. The IMM algothm ( models). Table. Fltes fo the models n the IMM algothm. Notaton fo algothm vaant Model (CV) Model (CA) Model (DTR) EKF EKF EKF UKF UKF UKF IEE EKF EKF EKPF IEG EKF EKF GPF EKF EKF RPF EKF EKF SPF IEU EKF EKF UPF IUE UKF UKF EKPF IUG UKF UKF GPF UKF UKF RPF UKF UKF SPF IUU UKF UKF UPF IEP EKPF EKPF EKPF IGP GPF GPF GPF IRP RPF RPF RPF ISP SPF SPF SPF IUP UPF UPF UPF IEUE EKF UKF EKPF IEUG EKF UKF GPF EKF UKF RPF EKF UKF SPF IEUU EKF UKF UPF 4 Smulaton Tests and Results The IMM algothm vaants lsted n Table ae mplemented fo smulated taget taectoes depcted n Fgues to 5. Wth efeence to equatons () and (4), the z () { Λ ()} { µ ( )} Mode pobablty update

5 vaables fo the models and system equatons ae as follows. The samplng nteval s T = (second). At tme step, the state vecto s X = [ x y z x y z x y z ] T. The pocess nose vecto w s zeo-mean multvaate Cauchy-dstbuted (multvaate t dstbuton, wth one degee of feedom) wth coelaton matx Q fo model, =,,. The measuement matx s H = [I ], whee I and ae the dentty and zeo matces espectvely. The measuement nose vecto v s zeomean Gaussan wth covaance R = 5 I. The tanston pobablty matx s [.96..;..96.;...96] and the ntal mode pobablty s [.96..]. Fo each PF vaant used, the numbe of patcles s N s = 6. The total numbe of smulaton uns s C =. Model (CV model): I TI.5T I F = I TI, G = TI, Q = 5 I. I Model (CA model): I TI.5T I.5T I F = I TI, G = TI, Q = I. I I Model (DTR model) []: I ω sn( ωt)i ω ( cos( ωt))i = F cos( ω T)I ω sn( ω T)I, ω sn( ωt)i cos( ωt)i [ ].5T I x G = TI, Q = y z I, whee ω = I [ x y z ] s the tunng ate. Let L denote the total numbe of s fo the duaton of tacng a taget (that s, the numbe of ponts on a taget taectoy). The oot mean squae eos (RMSEs) n the estmaton of poston (espectvely, velocty and L C / acceleaton) s defned as RMSE: = { L C û u }, = = whee û s the poston (espectvely, velocty and acceleaton) estmate, and u s the tue taget poston (espectvely, velocty and acceleaton), at the -th n the -th smulaton un. Tables to 5 show the RMSEs n state estmaton fo the IMM algothm vaants used. Fo each of these tables, the last column dsplays the mean pocessng tme (n seconds) pe smulaton un fo each vaant. The entes dvege ndcate occuence(s) of dvegence of the vaant. The notaton O( n ) (wth n beng a elevant postve ntege) s used to epesent vey lage RMSEs n state estmaton. In addton, the RMSEs (n metes) n poston estmaton wth measuement data ae as follows: Taectoy : (to two decmal places (d. p.)), Taectoy : 776. (to two d. p.), Taectoy : (to two d. p.), and Taectoy 4: (to two d. p.). Fgue. Taectoy. Fgue. Taectoy. Fgue 4. Taectoy. Fgue 5. Taectoy 4.

6 Table. Estmaton eos fo Taectoy. Table 4. Estmaton eos fo Taectoy. Table. Estmaton eos fo Taectoy. Table 5. Estmaton eos fo Taectoy 4. Algothm RMSE (to d. p.) Mean vaant poston (m) velocty (m/s) acceleaton (m/s ) tme (s) pe un IEE dvege 7.69 IEG IEU O( ) O( ) 9.49 IUE IUG dvege IUU O( ) O( 4 ) 9.6 IEP dvege IGP dvege 5. IRP O( 5 ) O( ) ISP O( 5 ) O( 4 ) IUP O( ) 9.7 IEUE O( ) O( ) IEUG dvege IEUU 65.5 O( ) O( 4 ) 9.9 Algothm RMSE (to d. p.) Mean vaant poston (m) velocty (m/s) acceleaton (m/s ) tme (s) pe un IEE O( ) IEG IEU 74. O( 5 ) O( 6 ) 7. IUE O( 4 ) 48.8 IUG IUU 79.9 O( 5 ) O( 6 ) 7.57 IEP dvege 7.68 IGP dvege 5.7 IRP O( ).98 ISP IUP dvege 4.6 IEUE dvege 6.6 IEUG IEUU O( 5 ) O( 6 ) 7.7 Algothm RMSE (to d. p.) Mean vaant poston (m) velocty (m/s) acceleaton (m/s ) tme (s) pe un dvege. IEE dvege 9. IEG dvege O( ) IEU dvege 8.59 IUE O( ) 68.4 IUG IUU dvege 8.77 IEP dvege 8.67 IGP dvege 7.88 IRP O( 6 ) O( 5 ) O( 7 ) ISP O( 5 ) O( ) O( ) 56.5 IUP dvege 9.5 IEUE dvege 79.6 IEUG IEUU dvege 8.8 Algothm RMSE (to d. p.) Mean vaant poston (m) velocty (m/s) acceleaton (m/s ) tme (s) pe un IEE dvege IEG IEU 69.4 O( 4 ) O( 5 ) 56. IUE dvege 47.8 IUG IUU O( 4 ) O( 5 ) 55.8 IEP dvege.7 IGP dvege 4.67 IRP O( ) ISP O( ) IUP dvege 7.7 IEUE O( ) 4.97 IEUG IEUU O( 4 ) O( 5 ) 56.5 Let t(x) denote the mean pocessng tme (n seconds) pe smulaton un fo the algothm vaant wth notaton X n Table. The pocessng tmes ae elated by: t() t() << t(i*r), t(i*s) < t(i*g) < t(irp), t(isp) < t(igp), t(i*e) < t(i*u) < t(iep) < t(iup),

7 whee the symbol * s used to denote all possble cases. An obsevaton about the pocessng tmes s as follows: t(i*r), t(i*s): appoxmately ( t()); t(i*g): appoxmately ( t()); t(irp), t(isp): appoxmately (5 t()); t(igp), t(i*e): appoxmately (5 t()); t(i*u): appoxmately (6 t()); t(iep), t(iup): appoxmately ( t()). Computatonal complextes fo the vaants can be appoxmated by, : O(n x );,,,,,, IEG, IEUG, IUG: O(N s + n x ); ISP, IRP, IGP: O(N s ); IEE, IEUE, IUE, IEU, IEUU, IUU: O(N s n x ); IEP, IUP: O(N s n x ). Based on the RMSEs n state estmaton obtaned, t can be nfeed fom the numecal esults that,, and (hencefoth, collectvely called RS vaants) have bette oveall pefomance than the othe vaants. The man easons ae dscussed below. RS vaants ae elatvely consstent and stable. They usually yeld smalle RMSEs n state estmaton than the othe vaants. It s obseved that pefoms bette than RS vaants fo Taectoes, and 4. Howeve, dvegence occus fo Taectoy, whch suggests that s not vey stable. pefoms ust as well o bette than RS vaants n poston estmaton, but s less effectve n the estmaton of velocty and acceleaton. Each RS vaant has easonably modest computatonal load, whch s lage than those fo and but sgnfcantly smalle than those fo vaants whch use computatonally ntensve PFs (EKPF, UPF)., and I*G have smla computatonal complextes, but they usually pefom less satsfactoly than RS vaants. It s noted that when UKF s used n place of EKF n a vaant, bette esults (eflected by smalle RMSEs n state estmaton) s usually obtaned. Poo esults (dentfed by lage RMSEs n state estmaton, wth poston estmaton possbly wose than that done wth senso measuements) o dvegence obtaned n smulaton tests ae lely to be due to the accumulaton of oundng eos bought about by the lage numbe of mathematcal opeatons equed fo hgh-dmensonal poblems (hee, n x = 9) [4]. Ths poblem tends to be moe pomnent when a vaant uses a PF fo evey model o when t uses a computatonally ntensve PF n the DTR model and KFs n the emanng models. Implementng computatonally ntensve vaants may ncu pohbtve costs when solvng lage poblems, such as those wth N s o n x nceased. Fgues 6 to 9 povde gaphcal epesentaton of RMSEs n state estmaton at each fo,, I*R and I*S. The RMSE fo the -th s defned as C / RMSE(): = { û u }. C = RMSE ove all smulatons (m) RMSE ove all smulatons (m/s) RMSE ove all smulatons (m/s ) RMSE ove all smulatons (m) RMSE ove all smulatons (m/s) RMSE ove all smulatons (m/s ) Eos n estmaton of poston Eos n estmaton of velocty Eos n estmaton of acceleaton Fgue 6. Estmaton eos fo Taectoy. Eos n estmaton of poston Eos n estmaton of velocty Eos n estmaton of acceleaton Fgue 7. Estmaton eos fo Taectoy. 5 Concluson In ths wo, an IMM algothm that ncludes a CV model, a CA model and a DTR model s appled to the poblem of D maneuveng taget tacng. Vaous combnatons of nonlnea fltes ae consdeed fo the models used. Fo each algothm vaant, consde the aveage of RMSEs (ARMSEs) n state estmaton fo the fou sets of smulaton tests. Smulaton esults show that among the algothm vaants mplemented,,, and (gouped as RS vaants) acheve bette oveall pefomance. Wth espect to the standad IMM algothm (), ARMSEs n poston estmaton fo RS vaants ae

8 RMSE ove all smulatons (m) RMSE ove all smulatons (m/s) RMSE ove all smulatons (m/s ) Eos n estmaton of poston Eos n estmaton of velocty Eos n estmaton of acceleaton RMSE ove all smulatons (m) RMSE ove all smulatons (m/s) RMSE ove all smulatons (m/s ) Fgue 8. Estmaton eos fo Taectoy (: anomalous behavo due to dvegence). Eos n estmaton of poston Eos n estmaton of velocty Eos n estmaton of acceleaton Fgue 9. Estmaton eos fo Taectoy 4. wthn % of that fo. In addton, ARMSEs n velocty (espectvely, acceleaton) estmaton fo RS vaants ae about 5% (espectvely, 4%) less than that fo. RS vaants ae elatvely stable and ncu modeate computatonal costs. Geneally, an appoach chosen fo poblem solvng s a tade-off between computatonal complexty and accuacy n esults. When computatonal load has poty ove accuacy, t s possble fo to be pefeed ove RS vaants, due to sgnfcant savngs n computatonal costs (n tems of pocessng tme). Refeences [] S. Aulampalam, S. Masell, N. Godon, and T. Clapp, A tutoal on patcle fltes fo on-lne nonlnea/non-gaussan Bayesan tacng, IEEE Tansactons on Sgnal Pocessng, Vol. 5, No., pp , Febuay. [] H. A. P. Blom, and E. A. Bloem, Patcle flteng fo stochastc hybd systems, Poceedngs of the 4d IEEE Confeence on Decson and Contol (CDC 4), Atlants, Paadse Island, Bahamas, 4-7 Decembe 4, Vol., pp. -6. [] Y. Boes, and J. N. Dessen, Inteactng multple model patcle flte, IEE Poceedngs - Rada, Sona and Navgaton, Vol. 5, No. 5, pp , Octobe. [4] F. Daum, Nonlnea fltes: beyond the Kalman flte, IEEE Aeospace and Electonc Systems Magazne, Vol., No. 8 (Pat ), pp , August 5. [5] X.-l. Deng, J.-y. Xe, and H.-w. N, Inteactng multple model algothm wth the unscented patcle flte (UPF), Chnese Jounal of Aeonautcs, Vol. 8, No. 4, pp. 66-7, Novembe 5. [6] H. Dessen, and Y. Boes, Effcent patcle flte fo ump Maov nonlnea systems, IEE poceedngs - Rada, Sona and Navgaton, Vol. 5, No. 5, pp. -6, Octobe 5. [7] A. J. Haug, A tutoal on Bayesan estmaton and tacng technques applcable to nonlnea and non- Gaussan pocesses, Techncal Repot MTR 5W4, The MITRE Copoaton, Januay/Febuay 5. [8] J. H. Kotecha, and P. M. Duć, Gaussan patcle flteng, IEEE Tansactons on Sgnal Pocessng, Vol. 5, No., pp. 59-6, Octobe. [9] E. Mazo, A. Avebuch, Y. Ba-Shalom, and J. Dayan, Inteactng multple model methods n taget tacng: a suvey, IEEE Tansactons on Aeospace Electonc Systems, Vol. 4, No., pp. -, Januay 998. [] G. W. Ng, Intellgent Systems Fuson, Tacng and Contol, Baldoc, Hetfodshe, England: Reseach Studes Pess Ltd. and Phladelpha, USA: Insttute of Physcs Publshng,. [] R. van de Mewe, A. Doucet, J. F. G. de Fetas, and E. Wan, The Unscented Patcle Flte, Techncal Repot CUED/F-INFENG/TR 8, Cambdge Unvesty Engneeng Depatment, August. [] G. A. Watson, and W. D. Bla, IMM algothm fo tacng tagets that maneuve though coodnated tuns, Poceedngs of SPIE, Sgnal and Data Pocessng of Small Tagets 99, Olando, Floda, USA, - Apl 99, Vol. 698, pp [] N. Yang, W. Tan, and Z. Jn, An nteactng multple model patcle flte fo manoeuvng taget locaton, Measuement Scence and Technology, Vol. 7, Issue 6, pp. 7-, June 6.