A Bayesian framework with auxiliary particle filter for GMTI based ground vehicle tracking aided by domain knowledge

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1 A Bayesian framework wih auxiliary paricle filer for GMTI based ground vehicle racking aided by domain knowledge Miao Yu a, Cunjia Liu a, Wen-hua Chen a and Jonahon Chambers b a Deparmen of Aeronauical and Auomoive Engineering, Loughborough Universiy, UK, LE11 3TU ( {m.yu, c.liu5, w.chen}@lboro.ac.uk); b School of Elecronic, Elecrical and Sysems Engineering, Loughborough Universiy, UK, LE11 3TU ( J.A.Chambers@lboro.ac.uk) ABSTRACT In his work, we propose a new ground moving arge indicaor (GMTI) radar based ground vehicle racking mehod which explois domain knowledge. Muliple sae models are considered and a Mone-Carlo sampling based algorihm is preferred due o he manoeuvring of he ground vehicle and he non-lineariy of he GMTI measuremen model. Unlike he commonly used algorihms such as he ineracing muliple model paricle filer (IMMPF) and boosrap muliple model paricle filer (BS-MMPF), we propose a new algorihm inegraing he more efficien auxiliary paricle filer (APF) ino a Bayesian framework. Moreover, since he movemen of he ground vehicle is likely o be consrained by he road, his informaion is aken as he domain knowledge and applied ogeher wih he racking algorihm for improving he racking performance. Simulaions are presened o show he advanages of boh he new algorihm and incorporaion of he road informaion by evaluaing he roo mean square error (RMSE). 1. INTRODUCTION This paper focuses on he problem of racking a ground vehicle moving on he road by a GMTI radar sensor. A ground vehicle could manoeuvre wih differen movemen ypes (such as acceleraion, consan velociy and sop) consrained by he road. Measuremens from a GMTI radar are used o discriminae a moving arge from he saionary background, such as he range and azimuh angle of he racked objec. As menioned in [1], a GMTI radar sensor is capable of deecing arges moving on he ground for wide-area, day/nigh and all weaher condiions. Successful racking of single/muliple ground vehicle(s) moving on a road is imporan for raffic surveillance. The aim of vehicle racking is o esimae he vehicle s sae values (including posiions and velociies) according o he measuremens provided by he sensor. Various algorihms have been applied o solve he racking problem, he sandard ones include he Kalman Filer and is variaions, and he paricle filer [2]. As he racked ground vehicle could manoeuvre wih differen movemen ypes, muliple sae models are needed o describe he movemen of he racked vehicle and hen a sochasic hybrid sysem (SHS) can be consruced, which has differen modes (corresponding o differen sae models) governed by a discree sochasic process. Each mode is from a se of a finie number of possible modes and he ransiion beween one mode o anoher, beween consecuive ime insances, follows a cerain probabiliy. Sandard Kalman/paricle filers can only deal wih one sae model so ha hey are no applicable for solving he SHS problem in which muliple sae models are involved. In order o obain he sae esimaion in an SHS, E. Mazor e al. [3] proposed an ineracive muliple model (IMM) mehod. The KF/EKFs are applied in he IMM o obain he sae esimaes for every sae model, which are combined o obain he sae esimaion resul. The iniial condiions for he KF/EKFs corresponding o a paricular sae model are obained as a mixure of he esimaion resuls for all he sae models a he previous ime insance. Considering he fac ha he sae disribuion for every mode of an SHS could no be a single Gaussian and he measuremen model This work was suppored by he Engineering and Physical Sciences Research Council (EPSRC) Gran number EP/K1437/1 and he MOD Universiy Defence Research Collaboraion in Signal Processing.

2 may be non-linear (such as he GMTI based measuremen model), insead of Kalman filers, Y. Boers e al. in [4] implemened he paricle filering in he IMM framework o obain an IMMPF algorihm for solving he non-linear and non-gaussian problems. H. Blom e al. in [5] exended he IMMPF in [4] from a Markov-jump sysem o a non-markov sysem, considering he mode ransiion probabiliies are no consan bu depend on sae vecors in some realisic scenarios. Excep for he algorihm side, we also need o highligh ha informaion such as he road boundary, direcions, speed limis and so forh, can be fused o reduce he uncerainy of he arge moion o obain a more accurae sae esimaion resul. In [6], he vehicle sae is augmened wih a mode variable which indicaes wheher he vehicle is on he road or no, and a sampling imporance resampling (SIR) paricle filer is applied o esimae he augmened sae. A separae model is applied for describing he vehicle s on-road movemen; in his way he road informaion is exploied. M. Ulmke e al. in [7] made use of he road map informaion by considering a more complicaed road opography; besides, he road map errors were also aken ino accoun. Two schemes, including a framework of Gaussian sum approximaions and a paricle filering approach, were applied o incorporae he road informaion for sae esimaion. A hybrid mehod is applied in [8], in which he vehicle on he road moves in muliple modes. The arge sae space is regarded as a hybrid sae space which is pariioned ino he mode subspace and he arge subspace which conains oher sae values (such as he arge s posiion and velociy). And he hybrid sae is esimaed by a more efficien Rao-Blackwellizaion paricle filering scheme. For a realisic scenario, he changes of he vehicle s movemen modes are relaed o he vehicle s sae componens, such as posiions and velociies. For example, he vehicle is more likely o decelerae when i approaches a juncion and he vehicle is less likely o sop suddenly when i moves a a high velociy. So, a non-markov jump sysem in [5] which adops sae-dependen mode ransiion probabiliy (SDTP) should be applied o reflec he mode ransiion of he vehicle s movemen. Besides, i is more realisic o consider he widh of he road insead of simply modeling he road segmenaion as 1-D sraigh line ([6], [7] and [8]) wih he widh being zero, which conflics wih he realisic scenario. In his work, we propose a novel GMTI radar based ground vehicle racking mehod. The recursive Bayesian framework in [5] is adoped for a non-markov jump sysem. Bu insead of a generic paricle filer o implemen he Bayesian framework, an auxiliary paricle filering (APF) scheme is applied o make use of he measuremen informaion o obain a more accurae resul. Moreover, we exploi he road informaion as he domain knowledge o furher improve he racking resuls. This paper is organized as follows: Secion 2 proposes he general framework of he road informaion aided ground vehicle racking problem by applying a GMTI radar. The recursive Bayesian framework for a non-markov jump sysem is proposed in Secion 3. In Secion 4, he implemenaion of he recursive Bayesian framework by he auxiliary paricle filer is explained in deail. Secion 5 finally shows he simulaion resuls for GMTI racking of a vehicle moving on a road, which confirms he advanages of our proposed mehod. 2. GENERAL PROBLEM FORMULATION 2.1 Road informaion aided muliple model esimaion We assume ha he ground vehicle can move in differen modes (acceleraion, consan velociy and sop) and is movemen is consrained by he road. So, he sae model used o describe he movemen of he ground vehicle could be regarded as an SHS wih consrains. Generally, he vehicle dynamics on he road can be represened as: x = f(x 1, m, C, w 1 ) (1) where x R n denoes he arge sae (including he posiions and velociies of he racked vehicle) a ime insan, m is he discree mode sae used o indicae he movemen mode of he vehicle, w is he process noise and C represens he consrains of he arge sae x. Road informaion direcly deermines he consrain region C. For example, he road boundary and speed limis resric he arge s posiion and velociy o be wihin cerain inervals. By imposing he arge s sae vecor x in he region C insead of he whole R n space, he uncerainy of x is hereby reduced.

3 Addiionally, oher road informaion is relaed o he movemen mode ransiions. One example is ha he vehicle is more likely o decelerae when i approaches a bend in he road. This ype of informaion could be incorporaed ino he mode ransiion probabiliies by making hem dependen on he sae vecor. For insance, high ransiion probabiliy o decelerae mode is assigned when he vehicle s posiion is near he road urn. By adoping he sae-dependen mode ransiion probabiliy (SDTP), he SHS hen becomes a non-markov jump sysem. 2.2 Vehicle movemen model For describing he vehicle s movemen, wo ypes of coordinae sysems are considered. The firs is a global Caresian coordinae sysem OXY. The second coordinae sysem is a local curvilinear coordinae sysem associaed wih he road. The origin of he local curvilinear coordinae sysem is assigned o he saring poin of he road denoed as e r. The firs axis aligns wih he coninuous curve represening he cener line of he road, whereas he second axis is along he normal direcion of he curve. In he local curvilinear coordinae sysem, he vehicle posiion can be expressed by using he ravelled disance l r along he cener line from saring juncion and he deviaion d r from he cener line of he curren road. Examples of he global Caresian coordinae sysem and local curvilinear coordinae sysem are shown in Figure 1. These wo coordinaes could be convered o each oher by applying he corresponding ransformaion operaions Tg r ( ) and Tr g ( ) (which represen he ransformaion for he global coordinae o local road coordinae and local road coordinae o he global one respecively, and he deailed definiions er of hese funcions could be found in [9]). lr drp(lr,dr) Figure 1. The illusraion of wo ypes of coordinae sysems. The posiion of a poin P (represening a vehicle) on he road could be deermined by he ravelled disance along he cenerline l r (marked as red) and he normal disance o he cenerline d r. In his work, he vehicle movemen is modeled in he local curvilinear coordinae, in which he sae of he arge on he road r a ime is defined as x r = [ ] l l d d T, where ( ) T denoes vecor ranspose. Here l and d represen he posiions along and perpendicular o he cenerline, and l and d represen he corresponding velociies. One advanage of using he local curvilinear coordinae is he implici inegraion of curvaure informaion in he model wihou considering a urning moion. In addiion, he road widh consrain can be easily incorporaed in o he model as a d b, where [a, b] represens he feasible region for d, which is he disance perpendicular o he cenerline. And a normally chosen sae ransiion model is a consan velociy (CV) model [2] wih he following definiion: where x r = F x r 1 + G w r (m ) (2) 1 T T 2 /2 F = 1 1 T, G = T T 2 /2 (3) 1 T

4 and T is he sampling ime, w r (m ) is he Gaussian process noise represening he acceleraion componens conrolled by he movemen mode m, which has zero means and covariance marix Σ(m ) = diag{σ r l (m ), σ r d (m )}. 2.3 Measuremen model As menioned in [1], he sandard GMTI radar could measure he range and azimuh angle (denoed as r and θ a ime insance ) of he ground vehicle relaive o he posiion of he GMTI radar. By assuming hese measuremens are noise-corruped from he acual values, we have: y = [ r θ ] [ ] (xo x = ) 2 + (y o y ) 2 + (z o ) 2 arcan( y o y + n x o x ) (4) for a received measuremen y. Here (x, y ) represen he vehicle s posiion in he global coordinae sysem a ime, which is convered from he local road coordinae x r by Tr g ( ) (assuming he ground plane s z-coordinae in he global coordinae is zero), (x o, y o, z o ) represens he posiion of he observer (GMTI radar) and n represens he measuremen noises. Similar o he process noise, [ we assume ] ha i also follows a Gaussian disribuion σ 2 wih zero means and covariance marix denoed as r σθ IMM BAYESIAN ESTIMATION FRAMEWORK In his secion, we discuss he exac recursive Bayesian derivaions for a non-markov jump sysem. Given he measuremen sequence Z = {z 1,..., z } and he consrain C, he probabiliy densiy funcion p(x 1, m 1 Z 1, C 1 ) is updaed o p(x, m Z, C ) a by four main seps: mode mixing, sae inersecion, evoluion and correcion: 3.1 Mode mixing p(m 1 Z 1, C 1 ) Mixing p(m Z 1, C 1 ) (5) p(x 1 m 1, Z 1, C 1 ) Inerac p(x 1 m, Z 1, C 1 ) (6) p(x 1 m, Z 1, C 1 ) Evoluions p(x m, Z 1, C 1 ) (7) p(x m, Z 1, C 1 ) Correcion p(x, m Z, C ) (8) The evoluion of he condiional mode probabiliy from 1 o is characerized as he mode mixing ransiion. Using he law of oal probabiliy, i yields p(m Z 1, C 1 ) = p(m, m 1 Z 1, C 1 ) = m 1 M m 1 M p(m m 1, Z 1, C 1 )p(m 1 Z 1, C 1 ) where M represens he collecion of he mode indices and p(m m 1, Z 1, C 1 ) can be obained using he law of oal probabiliy as: p(m m 1, Z 1, C 1 ) = p(m m 1, x 1 ) p(x 1 m 1, Z 1, C 1 (1) ) dx 1 (9)

5 3.2 Sae ineracion The sae ineracion process is carried ou for each mode, o generae he iniial mode-condiioned densiy p(x 1 m, Z 1, C 1 ). Again, by using he laws of condiional/oal probabiliy, we can obain: p(x 1 m, Z 1, C 1 ) = p(x 1, m Z 1, C 1 ) p(m Z 1, C 1 ) p(m m 1, x 1 ) p(x 1, m 1 Z 1, C 1 ) = m 1 M p(m Z 1, C 1 ) (11) 3.3 Evoluion The sae evoluion sep is o propagae he mode-condiioned sae densiy p(x 1 m, Z 1, C 1 ) a 1 o p(x m, Z 1, C ). Given he iniial densiy provided in (11), he mode-condiioned sae densiy p(x m, Z 1, C ) can be calculaed via: p(x m, Z 1, C ) = p(x 1 m, Z 1, C 1 )p(x x 1, m, Z 1, C (12) )dx 1 where p(x x 1, m, Z 1, C ) is he sae ransiion probabiliy deermined by boh he sae model and consrains, which follows: { p(x x 1, m, Z 1, C p(x x ) 1, m, Z 1 ) if x C oherwise (13) when x is no wihin he consrain C, he probabiliy becomes zero; oherwise, i is calculaed by p(x x 1, m, Z 1 ), which is deermined by he dynamics of he sae model m. 3.4 Correcion Finally, he densiy funcion p(x, m Z, C ) condiioned on he curren measuremens and consrains is obained. According o he Bayesian formula, we have: p(x, m Z, C ) = p(z x ) p(x, m Z 1, C ) p(z Z 1 ) (14) By subsiuing he resuls of he mode mixing sep and evoluion sep, p(m Z 1 ) and p(x m, Z 1 ), ino (15) finally we can obain: p(x, m Z, C ) = p(z x ) p(x m, Z 1, C ) p(m Z 1, C 1 ) p(z Z 1 ) p(z x, m ) p(x m, Z 1, C ) p(m Z 1, C 1 ) (15) A his poin he IMM Bayesian filering cycle is finished and he probabiliy densiy funcion p(x 1, m 1 Z 1, C 1 ) a ime insance 1 is propagaed o p(x, m Z, C ) a ime. The mode probabiliy p(m Z, C ) and modecondiioned probabiliy p(x m, Z, C ) could be derived as: p(m Z, C ) = p(x, m Z, C )dx x p(x m, Z, C ) = p(x, m Z 1, C ) p(m Z, C ) (16)

6 where p(m Z, C ) represens he likelihood of a paricular mode. The mode wih he larges likelihood is regarded as he curren movemen ype of he racked objec (denoed as mode ). The maximum a poserior (MAP) or minimum mean square (MMS) esimaors of he sae vecor (Risic and Gordon (24)) for mode could be aken as he final sae esimaion, which could be obained from he mode-condiioned probabiliy p(x mode, Z, C ). 4. IMM AUXILIARY PARTICLE FILTER IMPLEMENTATION As here is no an exac analyical soluion for p(x, m Z, C ) due o he non-gaussian propery of he sae disribuion for a non-markov jump SHS and he non-linear propery of he measuremen model, o obain a more accurae probabiliy densiy funcion esimaion, a paricle filering scheme [2] is applied o implemen he IMM based Bayesian filering. However, insead of he generic paricle filering scheme, we apply he auxiliary paricle filering mehod, which akes he measuremens ino accoun o obain a more accurae esimaion. And we denoe he IMM based Bayesian filering implemened by he auxiliary paricle filer as IMMAPF. A ime 1, we assume ha N weighed paricles are allocaed o he corresponding mode-mached filer as: {x r,k 1, wr,k 1 ; k = 1,..., N} for each mode r M, which are used o approximae he join disribuion p(x 1, m 1 = r Z 1 ) as: where δ( ) is a Dirac dela funcion. p(x 1, m 1 = r Z 1, C 1 ) w r,k 1 δ(x 1 x r,k 1 ) (17) The approximaion in (17), could be applied o implemen he recursive IMM Bayesian derivaion framework. For he mode mixing sep, he prior mode probabiliy for mode m = s can be approximaed as: p(m = s Z 1, C 1 ) r M p(m = s m 1 = r, x r,k 1 ) wr,k 1 (18) Λ s 1 where he noaion Λ s 1 is defined o faciliae he remainder of he derivaion. The iniial condiion probabiliy for mode m = s could be obained by subsiuing (17) and (18) ino (11), such ha p(x 1 m = s, Z 1, C 1 ) p(m = s m 1 = r, x r,k 1 )wr,k 1 δ(x 1 x r,k 1 )/Λs 1 r M Assuming he number of he sae modes o be M, from equaion (19) we can see ha N M paricles are required o represen p(x 1 m = s, Z 1, C 1 ), which is he iniial densiy for he s-h mode-mached filer. Increasing he number of paricles from N o N M for probabiliy densiy funcion represenaion will make he number of paricles grow in an exponenial way as he ime increases ((5)). In order o solve his problem, a resampling sep is performed on (19) such ha N samples { x s,k 1, ws,k 1 } p(x 1 m = s, Z 1, C 1 ) are generaed for each mode o represen p(x 1 m = s, Z 1, C 1 ), which can be represened as: p(x 1 m = s, Z 1, C 1 ) (19) w s,k 1 δ(x 1 x s,k 1 ) (2) According o he obained samples { x s,k 1, ws,k 1 },...,N for a paricular mode s, he mode probabiliy p(m = s Z, C ) and he mode-condiioned probabiliy p(x m = s, Z, C ) could be calculaed.

7 4.1 Calculaing mode probabiliy p(m = s Z, C ) The mode probabiliy p(m = s Z, C ) could be obained similar o he derivaions in [5]. For each paricle x s,k 1, a new paricle xs,k can be generaed according o he sae proposal disribuion as defined in equaion (13). The generaed paricles {x s,k },...,N are hen applied o approximae p(x m = s, Z 1, C ) according o (12): p(x m = s, Z 1, C ) w s,k 1 δ(x x s,k ) (21) The obained p(x m = s, Z 1, C ) ogeher wih Λ s 1 in (18) could be insered ino (15) which approximaes he join disribuion p(x, m = s Z, C ) as: p(x, m = s Z, C ) p(z x s,k ) w s,k 1 δ(x x s,k = w s,k δ(x x s,k ) )Λ s 1 (22) where w s,k w s,k 1 p(z x s,k )Λ s 1 and normalized o saisfy N s M ws,k = 1. The mode probabiliy p(m = s Z ) could be obained by marginalizing p(x, m = s Z, C ) wih respec o x, which is approximaed by summing he weighs wih respec o mode s as: p(m = s Z, C ) = 4.2 Calculaing mode condiioned probabiliy p(x m = s, Z, C ) w s,k (23) For m = s, an auxiliary paricle filering echnique could be applied o esimae he corresponding mode condiioned probabiliy a ime insance. Iniially, some characerizaion u s,k (such as he mean or a sample) is generaed from he sae propagaion densiy funcion for every x s,k 1. The likelihood for he new sample, denoed as w s,k is calculaed as w s,k p(z u s,k ). We resample he obained sample se { x s,k 1 },...,N according o obained weighs {w s,k },...,N. And he resampled indices are kep as {i k },...,N. x s,k Predicion is performed according o he obained resampled indices. For every k {1,..., N}, a new sample could be generaed from he sae proposal disribuion p(x x s,ik 1, m = s, Z 1, C ) where x s,ik 1 is he ik paricle for he sample se { x s,k 1 },...,N. Finally, he weigh of every prediced paricle x s,k is updaed as: w s,k w s,ik 1 p(z x s,k = p(z x s,k ) p(z u s,ik ) )p(x s,k x s,ik 1, m = s, Z 1, C ) q(x s,k, i k Z, C ) (24) As menioned in [2], q(x s,k, i k Z, C ) represens he imporance densiy funcion wih he form: q(x s,k, i k Z ) w s,ik 1 p(z u s,ik )p(x s,k x s,ik 1, m = s, Z 1, C ) (25)

8 The resuling weighs {w s,k },...,N are normalized such ha N probabiliy could be approximaed by he new weighs and paricles as: ws,k = 1. And he mode condiioned p(x m = s, Z, C ) = w s,k δ(x x s,k ) (26) For all m M, p(m Z, C ) and p(x m, Z, C ) are calculaed. As discussed a he end of Secion 3, he mode wih he maximum mode probabiliy value is aken as he curren movemen mode a ime, denoed as mode. The MAP/MMSE esimaions of he sae vecor can be obained according o p(x mode, Z ), which is approximaed by he weighed paricles. 5. SIMULATION A vehicle is simulaed o move on a secion of road, which has a widh of 1m. Iniially, he vehicle moves a 7m/s for 5 seconds. When he vehicle is near he urn, i deceleraes is velociy wih a deceleraion of 4m/s 2 for 6 seconds. The vehicle hen increases is velociy o 7m/s again afer i passes he urn, wih an acceleraion of 4m/s 2. Afer he acceleraion, he vehicle reains he velociy of 7m/s for 6 seconds. A UAV is simulaed o fly a a heigh of 1m, moving wih a circular rajecory wih a radius of 1m and an angular speed of π/1 radians per second for cruising. An onboard GMTI radar is assumed o be applied o measure he relaive range and azimuh angle of he racked vehicle. The simulaed scenario is presened in Figure Z X Figure 2. The rajecories of he simulaed vehicle and UAV movemens. The red sar represens he rajecory of he UAV and he pink sar represens he rajecory of he vehicle Y Two sae models corresponding o he wo movemen ypes of consan velociy and acceleraion are applied. The sae evoluions of he wo sae models follow equaion (2). The process noise w in hese wo models follows a Gaussian disribuion wih zero means. For he acceleraion model, he covariance marix parameers σl r, σr d are se o σl r = 6 and σr d = 2. For he consan velociy model, a smaller value of σr l is assigned wih σl r = 1. As menioned previously, he sysem is a non-markov jump sysem and ransiion probabiliies beween differen sae models are no consan. When he vehicle s posiion is ouside he urn region (marked in he red recangle in Figure 5), he ransiion probabiliies o wo movemen modes are se o be equal o inroduce no bias owards any movemen ype as: p(m = CV m 1 = AC) =.5 p(m = AC m 1 = AC) =.5 p(m = AC m 1 = CV ) =.5 p(m = CV m 1 = CV ) =.5 (27) where CV represens consan velociy and AC represens he acceleraion. When he vehicle is in he urn region, we assume he movemen mode could only be acceleraion and he ransiion probabiliies are:

9 p(m = CV m 1 = AC) = p(m = AC m 1 = AC) = 1 p(m = AC m 1 = CV ) = 1 p(m = CV m 1 = CV ) = (28) y (meers) Turn region x (meers) Figure 3. The urn region in which he vehicle acceleraes/decceleraes. The x-coordinae range of he urn region is se o [3, 68] and he y-coordinae range of he urn region is se o [5, 48]. The parameers σ r and σ θ in he measuremen model defined in (4) are se o π/5 and 2. Three differen algorihms, including he boosrap muliple models paricle filer (BS-MMPF) [6], IMMPF ([5] and [11]) and our proposed auxiliary paricle filer aided IMMPF (IMMAPF) are compared. The firs comparison is made wihou any road informaion. The comparison resuls are presened as Figure 4, from which we can see ha beer performance is obained for he IMMAPF mehod wih lower roo-mean-squareerror (RMSE) m e e r s 4 2 m e e r s 4 2 m e e r s meers meers meers (a) (b) (c) IMMPF,RMSE=19.17 BS-MMPF, RMSE=21.76 IMMAPF, RMSE=13.36 m eers seconds 2 25 (d) Figure 4. The racking resuls of (a).bs-mmpf (b). IMMPF (c). IMMAPF on he ground plane wihou road informaion. The blue sar represens he vehicle s real posiions and he red sar represens he racked ones. (d) represens he corresponding RMSEs for he hree algorihms.

10 Nex we compare he hree algorihms by incorporaing he road informaion, and he resuls are presened in Figure 5. We can see ha for each algorihm, much lower RMSE is obained when he road informaion is applied. And our proposed IMMAPF mehod, sill achieves he bes performance among all he hree mehods afer he incorporaion of he road informaion. meers meers meers meers meers meers seconds (d) meers (a) (b) (c) BM,RMSE=2.51 BS-MMPF,RMSE=2.76 ABM, RMSE=2.17 Figure 5. The racking resuls of (a).bs-mmpf (b). IMMPF (c). IMMAPF on he ground plane wih road informaion. The blue sar represens he vehicle s real posiions and he red sar represens he racked ones. (d) represens he corresponding RMSEs for he hree algorihms. For a more comprehensive evaluaion, in each scenario (he road informaion is applied or is no), 3 Mone- Carlo simulaions were performed for each algorihm. The mean and variance of he calculaed RMSE are lised in Tables 1 and 2. From hese wo ables, we can see ha much beer performance could be achieved by incorporaing he road informaion wih much smaller mean and sandard derivaion of RMSEs for each algorihm. Besides, for he algorihm comparison, our proposed algorihm achieves beer performance han he oher wo algorihms for boh scenarios wih/wihou he road informaion. Table 1. Performances of differen algorihms wihou road informaion BS-MMPF IMMPF IMMAPF Mean Sandard deviaion Table 2. Performances of differen algorihms wih road informaion BS-MMPF IMMPF IMMAPF Mean Sandard deviaion

11 6. CONCLUSIONS In his work, we proposed a novel IMMAPF algorihm for he ground vehicle racking problem. The auxiliary paricle filer was incorporaed ino he IMM Bayesian filering framework, which can hereby achieve beer performance by aking accoun he measuremen informaion. The road informaion was also applied as domain knowledge o consrain he movemen of he racked vehicle for achieving a more accurae racking resul. From muliple Mone-Carlo simulaions, i was shown ha beer ground vehicle racking performance could be obained wih he aid of he road informaion. And our proposed algorihm achieved beer performance over he radiional ones. However, we need o consider a more realisic scenario for GMTI racking. For example, he vehicle will no be deeced by he GMTI radar if he measuremens are wihin he Doppler Blindness Zone [1]. Besides, some false alarm measuremens may occur due o he effec of cluer in he background. Afer inroducing such degraded measuremens and false alarms, he measuremen model becomes more complicaed and one possible soluion o deal wih i is random finie se heory [12], which akes he measuremens as a random finie se. References [1] Merens, M. and Ulmke, M., Precision gmi racking using road consrains wih visibiliy informaion and a refined sensor model, 28 IEEE Radar Conference, Rome, Ialy (28). [2] Risic, B., Arulampalam, S., and Gordon, N., Beyond he kalman filer: Paricle filers for racking applicaions, Norwood, MA: Arech House (24). [3] Mazor, E., Averbuch, A., Shalom, Y., and Dayan, J., Ineracing muliple model mehods in arge racking: a survey, IEEE Transacions on Aerospace and Elecronic Sysems 34(1), (1998). [4] Boers, Y. and Driessen, J., Ineracing muliple model paricle filer, IEE Proceedings on Radar, Sonar and Navigaion 15(5), (23). [5] Blom, H. and Bloem, E., Exac Bayesian and paricle filering of sochasic hybrid sysems, Aerospace and Elecronic Sysems, IEEE Transacions on 43(1), 55 7 (27). [6] Arulampalam, M., Oron, M., and Risic, B., A variable srucure muliple model paricle filer for GMTI racking, Proceedings of he Fifh Inernaional Conference on Informaion Fusion, Annapolis, MD, USA (22). [7] Ulmke, M. and Koch, W., Road-map assised ground moving arge racking, IEEE Transacions on Aerospace and Elecronic Sysems 42(4), (26). [8] Cheng, Y. and Singh, T., Efficien paricle filering for road-consrained arge racking, IEEE Transacions on Aerospace and Elecronic Sysems 43(4), (27). [9] Merwade, V., Maidmen, D., and Hodges, B., Geospaial represenaion of river channels, Journal of Hydrologic Engineering 1(3), (25). [1] Clark, J., Kounouriois, P., and Viner, R., A new Gaussian mixure algorihm for GMTI racking under a minimum deecable velociy consrain, IEEE Transacions on Auomaic Conrol 54(12), (29). [11] Liu, C. and Chen, B. L. W., Road nework based vehicle navigaion using an improved IMM paricle filer, 213 IFAC Inelligen Auonomous Vehicles Symposium, Gold Coas, Ausralia (213). [12] Vo, B., Vo, B., and Canoni, A., Bayesian filering wih random finie se observaions, IEEE Transacions on Signal Processing 56(4), (28).