Chalmers Publcaon Lbrary Exended Objec Tracng usng a Radar Resoluon Model Ths documen has been downloaded from Chalmers Publcaon Lbrary CPL. I s he auhor s verson of a wor ha was aeped for publcaon n: IEEE Transacons on Aerospace and Elecronc Sysems ISSN: 0018-9251 Caon for he publshed paper: Hammarsrand, L. ; Sandblom, F. ; Svensson, L. 2012 "Exended Objec Tracng usng a Radar Resoluon Model". IEEE Transacons on Aerospace and Elecronc Sysems, vol. 483, pp. 2371-2386. hp://dx.do.org/10.1109/taes.2012.6237597 Downloaded from: hp://publcaons.lb.chalmers.se/publcaon/161486 Noce: Changes nroduced as a resul of publshng processes such as copy-edng and formang may no be refleced n hs documen. For a defnve verson of hs wor, please refer o he publshed source. Please noe ha aess o he publshed verson mgh requre a subscrpon. Chalmers Publcaon Lbrary CPL offers he possbly of rerevng research publcaons produced a Chalmers Unversy of Technology. I covers all ypes of publcaons: arcles, dsseraons, lcenae heses, masers heses, conference papers, repors ec. Snce 2006 s he offcal ool for Chalmers offcal publcaon sascs. To ensure ha Chalmers research resuls are dssemnaed as wdely as possble, an Open Aess Polcy has been adoped. The CPL servce s admnsraed and mananed by Chalmers Lbrary. arcle sars on nex page
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 1 Exended objec racng usng a radar resoluon model Lars Hammarsrand, Fredr Sandblom, Lennar Svensson Senor Member, IEEE, and Joam Sörsed Absrac Ths paper concerns he problem of vehcle racng when mulple radar reflecon ceners could be resolved on each vehcle. For hs exended arge racng problem we propose a radar sensor model, capable of descrbng such measuremens, ncorporang sensor resoluon. Furhermore, we nroduce approxmaons o handle he nherenly complex daa assocaon problem. The evaluaon n erms of descrbng measured daa and resulng racng performance shows ha he model effecvely explos he nformaon n mulple vehcle deecons. Index Terms Radar, Sensor model, Exended arges, Tracng, Sensor resoluon. I. INTRODUCTION ADVANCED auomove acve safey sysems ofen use sensors, such as radar and camera, o gaher observaons on he raffc envronmen around he vehcle. Through a racng framewor, hese observaons are refned o esmaes of, e.g., poson of oher vehcles, pedesrans and he road. Based on he esmaes, dangerous suaons can be deeced and decsons of approprae acons are aen. For example, he sysem may warn he drver of an mpendng collson or nervene by brang or seerng n order o avod he collson or mgae s consequences. For he acve safey sysem o be able o mae effecve decsons, s of grea mporance ha he provded esmaes mee he requremens n erms of auracy and deal. To acheve hs wh a cos effcen sysem, he racng framewor needs o have an aurae descrpon of he sascal properes of he sensor observaons [2]. Many of he acve safey sysems on he mare oday are solely or parly radar based. Excep from beng robus agans dfferen weaher condons, he radar offers aurae measuremens of range and range rae o objecs. Furhermore, he radar has a long hsory of use n, e.g., arborne applcaons, and here exss a vas amoun of leraure on how o desgn a racng sysem based on radar observaons, see [3], [4] and he references heren. There are, however, mporan dfferences beween arge racng n arborne applcaons Ths wor was sponsored by he Swedsh Inellgen Vehcle Safey Sysems IVSS program, and s a par of he SEnsor Fuson for Safey Sysems SEFS projec. Ths arcle s a subsanally expanded verson of [1] presened a he IEEE symposum on Inellgen Vehcle Sysems, Isanbul, Turey, June 2007. L. Hammarsrand and L. Svensson are wh he Deparmen of Sgnals and Sysems, Chalmers Unversy of Technology, Gohenburg, Sweden lars.hammarsrand@chalmers.se., lennar.svensson@chalmers.se. F. Sandblom s wh Volvo 3P, Gohenburg, Sweden fredr.sandblom.2@volvo.com. J. Sörsed s wh Volvo Car Corporaon, Gohenburg, Sweden jsorsed@volvocars.com. and vehcle racng for acve safey sysems. In arborne radar applcaons he am s o rac arcrafs a dsances of ens of lomeers, whereas n auomove acve safey sysems he dsances o he objecs of neres are n he order of ens of meers. A such shor dsances, he radar resoluon s ypcally fner han he physcal exen of objecs. Where, n arborne radar applcaons, he arges behave as pon sources [3], [4], n auomove scenaros he radar s ypcally capable of deecng mulple feaures reflecon ceners on he same objec. In he radar leraure hs ype of arge s referred o as an exended or dsrbued objec/arge [5]. Recevng mulple deecons from a vehcle offers a possbly o exrac dealed nformaon abou he objec. For example, he spread of he ndvdual deecons gves nformaon regardng he physcal exen of he objec as well as s orenaon [2]. However, mulple measuremens per objec also nroduce some consderable dffcules compared o he pon source case. For one, he algorhms and models developed usng he pon source assumpon are no longer vald. Addonally, an aurae sensor model s more complex as he deecons are spread over large pars of he objec and no auraely descrbed as orgnang from a sngle pon. The sensor model also needs o consder he possbly ha a arge can generae mulple deecons n conras o a mos one n he pon source case. The uncerany n he number of arge deecons maes he daa assocaon problem more nrcae. The am of hs paper s o develop a compuaonally racable sensor model ha auraely descrbes he radar deecons from hs ype of objec vehcles. The ulmae purpose beng o mprove he racng of vehcles for auomove acve safey sysems. Alhough he classcal pon source assumpon does no hold for exended objecs, lle aenon has been gven o fnd a more suable racng formulaon. A good overvew of dfferen conrbuons up o 2004 can be found n [6], coverng exended objec racng and he closely relaed problem of racng groups of arges. More recen suggesons nclude, [7], [8] where a formal Bayesan racng framewor s proposed for esmang he cenrod of he exended arges or arge groups. The objec exenson s modelled as an ellpse and s assumed ha mulple measuremens can orgnae from each objec. The ellpcal shape of each objec s descrbed usng a posve defne random marx. By ncludng hese marces n he sae vecor, boh he arge cenrod and objec exenson are jonly esmaed from daa. Alhough he proposed approach shows promsng resuls whch are robus agans objec shape, s dffcul o explo objec specfc shape nformaon usng hs framewor, when such nformaon s avalable. Glholm e. al. [9] propose a parcle
2 HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL fler soluon where he deecons from he exended objec are modelled by a non-homogenous Posson pon process wh a nown bu arbrary spaal nensy. Usng hs descrpon, s possble o nclude nformaon abou he shape of he objecs, bu due o he lmed flexbly of he Posson dsrbuon, s ofen mpossble o ncorporae specfc nowledge regardng expeced number of arge reurns. The probablsc mul-hypohess racer PMHT [10] relaxes he pon source creron by modelng he measuremen o arge assocaons as sochasc and ndependen, and has been appled o exended objec racng n, e.g., [11]. Alhough he PMHT does no drecly provde covarance esmaes, he mehod s useful f he number of deecons orgnang from each arge canno be auraely modeled. For he problem of racng vehcles usng radar observaons, here are reasons o beleve ha boh he spaal dsrbuon and he number of deecons from a vehcle can be auraely modelled. For example, he sudy n [12] ndcaes ha radar reurn from vehcles manly orgnae from a number of specfcally srong reflecon ceners pon sources, such as he headlamps and he wheel housngs, see Fg. 3. Furhermore, f reflecon ceners are locaed whn a resoluon cell, he echoes from hese reflecors are merged no a sngle jon deecon cluser deecon. In [13] [16], a model ha capures he general behavor of a deecon from a cluser ncludng wo sources/arges s used n conjuncon wh a descrpon of he probably ha he wo arges are unresolved. Usng hs probablsc descrpon, he daa assocaon hypoheses and measuremen model are expanded o also consder a merged deecon from he wo arges. A smlar approach s proposed n [17], usng a Gaussan approxmaon of he wo-source cluser deecon densy orgnally derved n [18]. Alhough he soluons referred o here consder he nfluence of merged measuremens, hey are lmed o handle only wo sources, and he resul s no easly expanded o he more general case of mergng mulple sources. Inspred by he fndngs n [12], we propose a radar sensor model descrbng he spaal dsrbuon of vehcle deecons as well as a probablsc descrpon over he number of vehcle deecons. The proposed model also consders he effecs of mergng a general number of arge reflecons lmed resoluon. As such, we are able o boh ncorporae shape nformaon and expeced number of vehcle deecons, as well as descrbe he sascal behavor of he measuremens. More specfcally, he model famly descrbes he radar reflecons from a vehcle as orgnang from a se of reflecon ceners and, dependng on he resoluon of he sensor, reflecors lely o render a merged deecon are grouped. The number of deecons from each group s modelled as well as he dsrbuon of he resulng deecons. By assocang measuremens o reflecor groups, nsead of ndvdual reflecors or reflecor clusers, he number of assocaon hypoheses s sgnfcanly reduced. Furhermore, we derve a vehcle racng framewor based on our proposed sensor model. The framewor s based on a lnear mnmum mean square error LMMSE esmaor where he needed denses are esmaed usng he unscened ransform UT [19]. A generalzed verson of he jon probablsc daa assocaon JPDA echnque [3], [20] s used o handle daa assocaon unceranes. The proposed model s compared o he commonly used pon source model n wo aspecs: model lelhood and racng performance. The evaluaon clearly ndcaes ha he proposed model has sgnfcan benefs n boh aspecs. The paper s organzed as follows. In Secon II he racng problem s formalzed and he necessary noaon s nroduced. Secon III presens he radar sensor model, and n Secon IV we show how hs model can be used n a racng framewor. Fnally, Secon V presens evaluaon resuls of our proposed sensor model and he derved racng framewor. II. PROBLEM FORMULATION Ths arcle sudes he problem of racng vehcles wh nown physcal dmensons, usng mulple radars mouned on he hos vehcle. The objecve s wofold. Frs, o derve a famly of dealed sascal models descrbng he radar reurns from he vehcles. Second, o develop a vehcle racng framewor based on hs model wh he ulmae am o mprove he esmaes of,.e., he poson and he velocy of he vehcles. Ths secon s paroned as follows. The sae parameers o be esmaed are defned n Secon II-A ogeher wh a model of how hey evolve over me. Secon II-B descrbes he necessary bacground properes of he radar observaons, and Secon II-C dscusses n more deal he needed properes of he radar sensor model and he racng framewor for our specfc problem. A. Sae paramerzaon All he parameers of neres are colleced n he dscree me sae vecor z, where s he dscree me ndex correspondng o connues me nsance. The sae vecor conans boh he saes of he surroundng vehcles and he hos vehcle. Each vehcle, l, s descrbed by he sub-sae vecor z l = [ ζ l x, ζ l y, ψ l v l c l v l ] T, 1 where ζx, l, ζl y, s he poson of vehcle l expressed n a global Caresan coordnae sysem. As llusraed n Fg. 1, Ψ l s he headng angle and vl s he speed n he headng drecon of vehcle l and v l s s me dervave. The varable c l represens he curvaure of he curren pah of he lh vehcle. The sae vecors of all vehcles are saced o form he complee sae vecor z = [ z h T z 1 T z 2 T... z Nv T ] T, 2 where z h s he hos vehcle sae and N v s he number of surroundng vehcles. The sae vecor evolves over me as spulaed by he moon model, z = f 1 z 1, e 1, 3
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 3 ζ y 1 c l ζ l x, ζ l y ψ l v h, vh ψ h v l, vl ζ h x, ζ h y ζ x Fgure 1. Coordnae sysem and sae paramerzaon used n hs paper. where f 1 s a non-lnear funcon and e s a nose process ncluded o reflec boh model unceranes and he dynamcs of he vehcles. Assumng ha he vehcles move ndependenly, we can consder he moon of each vehcle separaely. To descrbe he moon of a vehcle we use a slghly modfed verson of he smplfed bcycle model derved n [1], where he dfference les n he use of curvaure nsead of yaw-rae, ψ = v c. B. Radar observaons For each dscree me a radar provdes M deecons, where M deecons orgnae from he raced vehcles and M c are cluer deecons. All deecons are sored n he unordered unlabeled measuremen vecor, [ y 1 T y = y 2 T ] T T.... 4 Each deecon s defned as y M y = [ r ṙ φ ] T, 5 where r s relaed o he range, ṙ o he range rae, and φ o he angle o he objec ha gave rse o he deecon relave o he sensor. Le us defne an ordered collecon of he deecons orgnang from he raced vehcles as y and a collecon of hose orgnang from cluer as y c. These vecors are relaed o he measuremen vecor, y, hrough an unnown permuaon marx, Π M p, wh dmenson [M xm ], [ y = Π M y p I c 3x3 y ]. 6 where s he Kronecer produc and I 3x3 s a hree-byhree deny marx. The purpose of Π M p s o descrbe mahemacally ha he measuremen orgn daa assocaon s unnown. In our model, all permuaon marces Π M p are equally lely, whch means ha he order of he deecons n y s compleely unnown random. The reamen of hs uncerany s an mporan par n he dervaon of he racng framewor and s furher dealed n Secon IV. However, le us frs defne y c and y n more deal. 1 Cluer deecons: I s commonly assumed, see e.g. [3], ha y c s descrbed by a homogenous Posson process n he observaon space aordng o y c, UnformV, M c PossonµV, 7 where y, c s he h cluer measuremen, µ s he cluer nensy and V s he volume of he observaon space. In addon, we assume ha he cluer deecons are ndependen from each oher and he arge deecons. 2 Targe deecons: Gven z, we assume s possble o paron he vsble reflecons ceners no N g well separaed groups, where each group can render mulple deecons. Furhermore, we assume ha he number of arge deecons for group n, M,n, has a probably mass funcon Pr { M,n z. 8 ha we can model. Condoned on M,n, he deecons from group n can be descrbed usng a sensor model y,n = h n z, w, M,n, 9 where w s a measuremen nose process capurng boh model unceranes and measuremen dsurbances. From 9 we can generae [ y y T ] =,1 y T T T,2... y,n g, 10 and he oal number of arge deecons s gven by C. Objecves N g M = M,n. 11 The man objecve of hs paper s o mprove racng performance by auraely modelng he radar response from he vehcles. To aomplsh hs, we need o derve an aurae model of he radar deecons whch s also suable o be used n a racng framewor. In hs secon we dscuss he objecves of he radar sensor model and he vehcle racng framewor separaely.
4 HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 1 Sensor model: The am of he sensor model s o descrbe he sascal behavor of he measuremens, gven z. The behavor of he cluer deecons s readly gven by 7, bu modellng he arge deecons 8 9 s more complcaed. I s crucal ha hese models capure he behavor of he vehcle deecons from dfferen aspec angles and a all ranges [21]. A vehcle radar response model,.e., expressons for 8 9, ha consders hese aspecs s derved n Secon III. 2 Tracng framewor: Assumng ha he number of vehcles s nown, he objecve of he racng fler s o recursvely calculae he poseror probably densy funcon pdf p z Y, where Y {y 1, y 2,..., y conans all he avalable observaons up o and ncludng me. From p z Y, s hen possble o compue esmaes and uncerany measures of z. The calculaon of p z Y s feasble f we have nowledge regardng wo specfc models [3], namely he moon model, defned n 3 and he sensor model, defned by 7-9. To arrve a a compuaonally racable soluon, we resrc our racng fler o an LMMSE esmaor of z. As such, only he frs wo momens of p z Y need o be calculaed,.e., ẑ = E { z Y, ˆP = Cov { z Y. 12 However, due o non-lneares n boh he process and measuremen model s dffcul o fnd an exac soluon o 12. Insead, flers whch approxmae hese momens are commonly used, e.g., he exended Kalman fler EKF [22] or he unscened Kalman fler UKF [19]. The laer s derved for he proposed sensor model n Secon IV, reang he uncerany n measuremen orgn daa assocaon modelled by he unnown permuaon marx Π M p. III. RADAR SENSOR MODEL Our proposed sensor model s based on he fndngs presened n [12], where he radar response from vehcles s modeled as orgnang from reflecon ceners feaures on he vehcles more lely o reflec he ncden radar wave. Due o lmaons n radar sgnal bandwdh, pulse duraon and anenna aperure sze, radar sensors are no capable of resolvng reflecon ceners ha are oo closely spaced. As such, no all of hese reflecors are always resolvable and he response from some mgh merge o form a jon deecon. In [12], a mappng s proposed for how o ransform he vehcle saes o a se of reflecor posons n observaon space. Addonally, a scheme s descrbed for how o form clusers of hose reflecon ceners ha are unresolvable and how o model deecons from hese clusers. As hs model was developed for smulaon purposes, raher han for use n a racng sysem, neglecs mporan probablsc descrpons needed n he racng conex. For example, he model requres he receved sgnal srengh of each ndvdual reflecon cener o be nown. Moreover, condoned on he sgnal srengh and z, boh whch reflecors are clusered and he posons of he resulng clusers are deermnsc. In he racng conex, s no realsc ha he sgnal srengh s nown a-pror and consequenly, we do no now whch reflecors are clusered or he poson of he resulng deecons from he vehcle. In hs secon, we derve a radar sensor model usng a sochasc descrpon of he receved sgnal amplude from each reflecor cener, arrvng a a model more suable n a racng framewor. The dervaon s conduced n four seps whch are shown n Fg. 2 and summarzed as follows. Frs, based on he model n [12], he posons of he reflecon ceners of he vehcles n z are mapped o he observaon space. Second, we form all possble clusers of reflecon ceners whch may generae merged deecons. Due o uncerany regardng whch reflecors are resolved and whch are no, he resulng probably densy of reflecor cluser could be hghly mul-modal. Thrd, o allevae hs mul-modaly, we group reflecors whch may belong o he same cluser, and approxmae he cluser densy by margnalzng over all cluser possbles. The resul s a descrpon of reflecor groups capable of generang mulple measuremens. Fnally, dependng on he probably of deecng he possble clusers n each group, we fnd an expresson for 8. Assumng ha he measuremen nose s addve and Gaussan, we now wre 9 on he form y,n = G n z, M,n + w, w N 0, W M,n 13 where W M,n = I M,n xm,n W. 14 The funcon G n maps z o he arge measuremen vecor for group n, gven nowledge regardng he number of measuremens generaed by he group, M,n. Noe ha alhough we condon on z and he number of deecons from he group, G n z, M,n s sll sochasc due o uncerany n whch reflecors ha are clusered we call hs cluserng uncerany. The followng secons presen he dervaon of he dsrbuon of G n z, M,n and M,n usng he seps descrbed above. To smplfy noaon, he me dependence s omed and all sochasc varables are condoned on z, even hough s no explcly saed. A. Reflecon cener model Aordng o he model n [12], he suded radar only receves reflecons from a dscree se of pons on a vehcle, so called reflecon ceners. The dfferen reflecon ceners are dvded no wo caegores: pon reflecors and plane reflecors. Fg. 3 dsplays he confguraon of pon reflecors suggesed n [12], where he reflecors are placed n he vehcle wheel houses and corners. Assocaed wh each reflecor s a vsbly regon, ndcaed by cones n Fg. 3; a reflecor can only render a reflecon f he sensor s whn hs regon. The plane reflecors are modelled as crcle secors ypcally descrbng he sdes of he vehcle. Furhermore, s assumed ha he radar only can receve a reflecon from hese plane reflecor f here s a pon on he surface whch normal pons drecly owards he sensor. The reflecng pon on a surface herefore depends on he poson of he sensor, and may change over me as he vehcle moves relave o he sensor plaform.
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 5 {c 1 L 1 =1 Group model g 1 M easurem en m odel y 1 z Reflecon cener model r Cluser model. {c n L n =1 Group model g n M easurem en m odel y n Fgure 2. Schemac vew of he measuremen generaon process n our proposed radar sensor model. The noaon used n he fgure s nroduced n subsequen secons. B. Cluser model In Secon III-A we presened a model for he vehcle response from a radar wh nfne resoluon hrough he mappng z R r. However, a sensor wh lmed resoluon canno resolve oo closely spaced reflecors. To model hs behavor, a resoluon cell s used d = [ r ṙ φ ] T 17 Fgure 3. Vehcle reflecon ceners wh assocaed vsbly regons. Gven he vehcles posons and physcal dmensons, each reflecor has a deermnsc poson n observaon space, denoed r = [r, ṙ, φ ] T and expeced sgnal power σ, expressed as [ r T ] T σ = R z, 15 where he mappng R s defned n Appendx A. Alhough he physcal dmensons of he observed objecs are assumed nown n hs paper, n a sensor daa fuson sysem, nformaon regardng objec exen could be provded by, e.g., a vson sensor and/or vehcle-o-vehcle communcaon. In addon o he poson of he reflecor n observaon space, s also mporan o model he sgnal amplude, A, of he receved reflecon. Ths s an mporan model feaure as he probably of deecng a reflecor depends on he srengh of he receved echo, and he poson of a merged deecon depends on he relave amplude of he ncluded echoes. The amplude model used n [12] s a deermnsc funcon of he radar anenna paern as well as he poson and vsbly of he reflecors. We nsead propose o use a Swerlng I model [23] for he amplude of he refleced sgnal, where he reflecon ampludes are modeled as flucuang aordng o he Raylegh dsrbuon, A Rayleghσ. 16 As s shown n he comng secon, usng a sochasc amplude model nsead of a deermnsc enables us o descrbe uncerany regardng number of vehcle deecons as well as her posons. and wo radar responses whch are no separaed more han d, n all hree dmensons, yeld a merged deecon. Unforunaely, he suaon s more complcaed for mulple reflecors, as unresolved clusers can be formed n several ways. 1 Cluser formaon: In [12], he followng algorhm s used o map reflecors no clusers, an operaon here denoes as r C c : Fnd he reflecor wh he sronges amplude, A. Form a cluser by denfyng he reflecors whch are whn he resoluon cell cenered a r. Repea and wh he remanng reflecors, unl no reflecors are lef. The cluserng algorhm above, can be used o dvde he se of all vsble reflecors no clusers and we refer o a descrpon of all resulng clusers as a cluser consellaon. However, s mporan o noe ha snce he ampludes of he reflecons are sochasc n conras o [12], several dfferen cluser consellaons may be possble, even for a gven z. For noaon, we consruc a ls of all possble consellaons, and nroduce he varable as a poner o he cluser consellaons n ha ls. The oal number of consellaons n he ls s denoed N and consequenly {1, 2,..., N, and he number of reflecor clusers n consellaon s L. Fg. 4 llusraes hese conceps by showng hree possble cluser consellaons n a smple example. Here, N = 3 wh L 1 = 1, L 2 = 2 and L 3 = 2 as he frs consellaon clusers all reflecors and he oher wo conans wo clusers each. Cluser n consellaon, conanng J reflecors 1 wh ndces 1,..., J, can a mos generae one deecon whch n ha case can be modelled as y w = c N 0, W. The sgnal componen, c + w where, s modelled 1 The noaon for he number of reflecors wll change as we can ge more specfc. In hs secon we use J o ndcae he number of reflecors n a generc cluser.
6 HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL r 1 r 2 r 3 r 1 r 3 r 1 r 3 r 1 r r 2 r 2 r 2 = 1 = 2 = 3 Fgure 4. Example of a vehcle wh hree reflecors op and hree dfferen cluser consellaons boom. The dashed lne corresponds o a reflecor cluser. as a weghed sum [12] of reflecor componens: where c = w l = J w l r l, 18 l=1 A l J l=1 A l. 19 Alhough hs s a raher smplfed model of he underlyng physcal phenomenon of merged measuremens such as arge gln [5], [24], serves he purposes for our radar sensor model. Snce he ampludes are sochasc, so are he weghs 19 and he sgnal componen of he cluser 18. As for he reflecor deecons, he receved amplude of a cluser s also Raylegh dsrbued bu wh he parameer σ = J σ 2 l. 20 2 Cluser densy: For a cluser, he dsrbuon of s poson s defned by 16, 18-19, whch s dffcul o evaluae. As he am s o use he proposed sensor model n a Kalman fler framewor, s convenen o approxmae p c z as a Gaussan densy wh he same frs wo momens as he underlyng dsrbuon. Le overscore denoe he expeced value of sochasc varables, such ha, e.g., Ā = E {A. Furher, le r l = r l l=1 c, w l = w l w l and se S = J, as gven by 18, s frs momen of c c = l=1 A l. The J w l r l 21 l=1 and afer some manpulaons an expresson for he covarance can be found as J C = r s r T E { w s w. 22 s, =1 The poson of each reflecor, r l, s gven by ransformaon 15, bu we also need o express w l and Cov {w s, w. As he momens of a Raylegh dsrbuon are well nown, approxmaons of hese quanes are readly found hrough Taylor expanson, w l = A l Ā l + A l S S S S Ā l S 2. 23 More deals on he dervaon of he mean and covarance of c Appendx B. as well as he approxmaons used are found n To summarze, we propose a sochasc mappng of reflecors no cluser consellaons, r {c N =1, where C N s deermnsc bu boh and c = [c 1,..., c L] are sochasc. The densy of each cluser n each consellaon, c, s approxmaed as a Gaussan densy, p c, z = N c ; c, C, 24 where c and C C. Group model are gven by 21 and 22, respecvely. In mul-arge scenaros, he oal number of possble clusers, N =1 L, can be sgnfcan. Hence, could be dffcul o fnd a compuaonally feasble soluon for assocang measuremens o ndvdual clusers. To mgae hs dffculy, we sugges o form reflecor groups conanng reflecors ha are lely o ge clusered, and descrbe he measuremen dsrbuon by margnalzng over he cluser consellaons. Le a group be a se of reflecors, formed such ha for every reflecor n he group, all oher reflecors belongng o one or more clusers wh reflecor are also ncluded. As a consequence, each reflecor n he group s posoned whn d o a leas one oher reflecor n he group. The number of groups, N g, s hen he oal number of such parons of he reflecors. Fg. 5 dsplays a scenaro where wo groups,.e., N g = 2, are formed; one conanng only one reflecon cener n he rear wheel and he oher composed of hree reflecon ceners n he fron. A subopmal, bu smplfed, soluon o he assocaon problem s obaned by assocang he deecons o he reflecor groups. By gnorng whch specfc cluser n a group ha gave rse o a deecon, he number of hypoheses are reduced subsanally. Each group s vewed as an eny whch can generae mulple and ndependen deecons. The number of deecons
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 7 g 1 Fgure 5. The formaon of wo reflecor groups sold lnes. from group n s denoed by Mn, and he sgnal componens [ he posons as g n = g n,1 T,..., ] T T g n,m n. Usng hs noaon, 13 can be wren as y n = g n + w, 25 As only reflecors whn each group can form clusers wh each oher, we can consder he cluser consellaons for each group ndependenly. For group n we can generae Nn cluser consellaons, and we le n hs case ndcae one specfc consellaon n hs group and L n denoe he number of clusers n hs cluser consellaon. A new ls of cluser consellaons s generaed for each group and s used o ndex one of he consellaons. Furher, le c denoe he sgnal componen of he l h cluser n consellaon for group n, and Pn l denoe he deecon probably of hs cluser - a probably easly compued from he Raylegh assumpon n 16 and 20. If we assume ha all possble cluser consellaons are equally lely, we can descrbe he pdf of g n by approxmang s componens g n, as ndependen and dencally dsrbued wh he densy 2 p g n, = 1 N n Nn n L =1 l=1 where he weghs q are defned as q = P n l g 2 q p c g n,, 26 L n m=1 P n m. 27 To furher smplfy he mplemenaon of a racng algorhm based on hs model we mae a Gaussan approxmaon, p g n, N g n, ; ḡ n,, C n. Usng he approxmaon n 24, he expeced value, ḡ n = E {g n, s gven by ḡ n = 1 N n Nn n L =1 l=1 q c 28 and he second momen, C n = E { g n, ḡ n g n, ḡ n T by C n = Nn L n =1 l=1 q Nn C + ḡ n c ḡn c T, 29 2 In 26, he noaon p c g n, should be nerpreed as he pdf of c evaluaed a g n,. where c and C s gven by 21 and 22, respecvely. Addonally, o complee he descrpon of he groups, we need o calculae he probably mass funcon Mn, he number of deecons orgnang from group n. By agan assumng ha all cluser consellaons are equally lely, we have Pr { Mn 1 = N n Nn =1 Pr { Mn, 30 where Pr { Mn s easly calculaed from Pn l. In summary; we group closely spaced reflecors and use he cluser descrpon o calculae he expeced sgnal componen each group, s covarance marx and he probably mass funcon for Mn, he number of deecons from group n. D. Targe measuremen model In Secons III-A, III-B and III-C we descrbe hree mappngs, R, C and G, relang he poson of he vehcles o he measuremen dsrbuon. The procedure can be depced as z R r C {c N G =1 g n, Mn, where he frs wo mappngs are deermnsc whereas he las wo are sochasc due o uncerany n he resoluon capables of he sensor. The ransformaon z R r descrbes he sgnal componens of srong vehcle relaed radar reflecors n observaon space. By modellng he resoluon capably of he sensor, C, we form a se of cluser consellaons, {c N =1. To reduce he complexy of he daa assocaon problem, we form groups of reflecors ha belong o he same cluser consellaons, G. The groups are descrbed by her spaal densy, p g n N g n ; ḡ n, C n, and he probably mass funcon Pr {Mn, descrbng he number of arge deecons from each group. The resul s a descrpon of arge measuremens orgnang from groups of reflecors, where each group n generaes Mn ndependen and dencally dsrbued measuremens. The h deecon from group n, y n, N ḡ n, C n + W, 31 s an ndependen of all oher deecons condoned on z. The complee arge measuremen vecor y can be generaed by drawng he number of arge deecons from he group, Mn, aordng o 30, for each group n = 1... N g. Subsequenly, consruc yn by generang Mn ndependen realzaons of yn, conformng o 31. The complee arge measuremen vecor s formed by concaenang all group measuremens as defned n 10. Ths concludes he dervaon of our sensor model. IV. TRACKING FRAMEWORK In hs secon we presen a racng framewor for recursvely calculang he poseror densy, p z Y, usng he proposed sensor model derved n Secon III. To handle uncerany n he number of arge and cluer deecons as well as he random permuaon marx, Π M p, n he calculaon of p z Y, s convenen o nroduce a daa
8 HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL assocaon hypohess vecor, λ. The purpose of hs vecor s o assocae deecon number j n y o a ceran group, n. Consequenly, λj = n, f measuremen j orgnaes from group n and, λj = 0, f s o be regarded as cluer. Usng hs descrpon, he sensor model can be wren as p y j { N y j λ, z = ; ḡ λj, C λj + W 1 V f λj 0 f λj = 0. 32 Ths formulaon maes possble o assocae measuremens o group n aordng o he suppor of Pr { Mn z. Tha s, each group can generae mulple deecons, each carryng nformaon regardng he sae of he vehcle. Ths dsngushes our sensor model from he classcal pon source model. By consderng all possble daa assocaon hypohess, he poseror densy can be formed as p z Y = p { z λ, Y Pr λ Y, 33 λ where p z λ, Y s he poseror densy whou daa assocaon uncerany and Pr { λ Y s he probably of ha assocaon. Due o, e.g., non-lneares n 15 and he dmensonaly of he daa assocaon problem, s dffcul o fnd an exac soluon o 33. Insead, we resor o an approxmae soluon. In he leraure s possble o fnd several possble approaches, such as parcle flers [25], Mulple Hypohess Tracng MHT flers [26] or he Probablsc Mulple Hypohess Tracer PMHT [10] o handle or smplfy hese ypes of problems. To mae he mplemenaon suable for real-me applcaons wh lmed capacy o bach measuremens, we propose a Kalman-le fler framewor [27] employng a generalzed verson of he Jon Probablsc Daa Assocaon JPDA algorhm [20]. The generalzaon of he JPDA algorhm for hs problem consss n allowng mulple measuremens o orgnae from he same group, n conras o a mos one n he orgnal JPDA formulaon. The condonal poseror densy, p z λ, Y, can be calculaed for each λ, and we approxmae hs dsrbuon by a Gaussan densy wh mean ẑ λ and covarance Pλ. Esmaes of ẑλ and P λ are found usng he UT [19] hrough ẑ λ = ẑ 1 + P λ zy P λ 1 yy y λ ŷ 1 λ 34 P λ = P 1 P λ zy P λ 1 yy P λ T zy, 35 where ẑ 1 and P 1 are esmaes of he mean and covarance of z Y 1, respecvely. The covarance, P λ yy, s he nnovaon covarance under he daa assocaon hypohess λ and, P λ zy s he correspondng cross covarance beween he sae and he measuremens. In aordance wh he JPDA dea, he resulng Gaussan mxure 33 s also approxmaed as a sngle Gaussan where he conrbuon from he ndvdual denses are weghed by her hypohess probably, Pr { λ Y. Gven a Gaussan pror densy, p z 1 Y 1, he proposed approach s brefly oulned below. 1 Sae predcon: Esmae ẑ 1 and P 1 by propagang z 1 Y 1 hrough he moon model 3 usng he unscened ransform. 2 Measuremen predcon: Transform z Y 1 o he observaon space, usng he seres of mappngs derved n Secon III and he unscened ransform o rean an approxmaon of he predced group measuremen ŷ,n = ĝ n = E { g n Y 1, 36 as well as he predced group covarances P n gg = Cov { g n,, g n, Y 1, 37 P nm gg = Cov { g n,, g m,j Y 1, 38 where j f n = m. The nnovaon covarance, P λ yy, s gven by 37 38 and he measuremen nose covarance, W. 3 Daa assocaon: Use ĝ n and P n yy = P n gg + W o perform measuremen gang. Then generae he se of all possble measuremen o group assocaon hypoheses and calculae her probables Pr { λ Y. 4 Measuremen updae: For all λ, form he needed enes n 34-35 and approxmae p z λ, Y. Fnally, p z Y, s found by margnalzng he daa assocaon hypoheses, see 33. A dealed descrpon of he dfferen seps s gven n he followng secons. A. Sae predcon The sae predcon s performed by calculang a Gaussan approxmaon of he predced densy p z Y 1 = p z z 1 p z 1 Y 1 dz 1. 39 Ths approxmaon s found by propagang p z 1 Y 1 hrough 3 usng he unscened ransform [19] as p z Y 1 N z ; ẑ 1, P 1. 40 Assumng he vehcles move ndependenly, he unscened ransform can be performed for each vehcle separaely. B. Measuremen predcon In Secon III we derved a radar sensor model condoned on he sae hrough a seres of mappngs. To form he expressons 34-35 we need esmaes of he frs and he second-order momens 36, 37 and 38. Agan we use he unscened ransform o fnd approxmaons of hese momens. However, for our proposed group measuremen model, he approxmaon s no as sraghforward as for he moon model. As such, requres some addonal dscusson. Usng he unscened ransform descrbed n [19], we choose 2n z + 1 deermnsc sgma pons wh assocaed weghs, where n z s he dmensonaly of z. The sgma pons and her weghs are chosen such ha hey capure he frs wo momens of 40 exacly. Le us denoe he se of sgma pons wh assocaed weghs as, { Z, u 2nz+1 =1. 41
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 9 Alhough no requred, we choose he weghs such ha u > 0, = 1... 2n z + 1. Ths s used o avod rss assocaed wh usng he unscened ransform n hgh dmensons, such as he rs of esmang non-posve defne covarance marces and a mean suaed far away from each propagaed sgma pon. By propagang each sgma pon hrough he mappng 15, we receve he sgma pon ses { R, u 2nz+1 =1, { Σ, u 2nz+1 =1 42 where R descrbe he reflecor posons of he h sgma pon n observaon space and Σ her expeced sgnal power. From 42 we can form esmaes of he frs wo momens of r Y 1 aordng o ˆr 1 P rr u R 43 u R ˆr 1 R T ˆr 1 44 where ˆr 1 s he esmae of he reflecor poson. The covarance marx, P rr, has he followng srucure P 11 rr P 12 rr... P 1nr rr P 21 rr P 22 rr... P 2nr rr P rr =........, 45 P nr1 rr...... P nrnr rr where n r s he oal number of reflecors on he vehcles n z. From Σ, he esmaed expeced sgnal power s smlarly aaned as ˆσ 1 u Σ, 46 whch s used n 30 o aoun for reflecor vsbly under sae uncerany, prmarly n arge headng. In pracce, we only need o consder hose reflecors whch are vsble,.e., for whch ˆσ 1 > 0. From 43, we deermne whch reflecors belong o he same group. As only reflecors whn each group are able o form clusers wh each oher, we consder he cluser consellaons for each group ndependenly. Followng he algorhm n Secon III-C, we form all possble cluser consellaons for each group. Usng he mappng 26 we can calculae wo of he sough momens, 36 and 37, as ĝ n = 1 N n P n gg = Nn Nn =1 l=1 n L =1 l=1 L n q N n q ĉ 47 P,n c l c l + ĝ n ĉ ĝn ĉ T. 48 Noe ha he weghs, q, are dependen on he esmaed expeced ampludes, ˆσ 1, hrough Pn. Values for ĉ and P,n c l c l are found usng he same approxmaons as n he dervaon of 21-22, dealed n Appendx B, and assumng ndependence beween w and r,, P,n c l c l,j=1 ĉ = E { J c, Y 1 =1 = Cov { c, Y 1 J T P llj rr + ˆr l 1 ĉ ˆr lj 1 ĉ w lˆr l 1 49 E { w l w lj, 50 where, l 1,..., l J are he ndexes o he reflecors n he cluser under consderaon. Usng 47, 48 we can form he Gaussan approxmaon of he h predced measuremen from group n as p y,n, Y 1 N y,n, ; ĝ n, P n gg + W. 51 The addonal sough covarance, P nm gg, assesses he covarance beween wo deecons from he same group or alernavely wo deecons from dfferen groups. For he fler o be able o perform he sae updae for all groups jonly, s mporan o auraely assess hese correlaons. Condoned on z, s assumed ha hese deecons are uncorrelaed. Hence, he correlaon only comes from uncerany n z and we can approxmae he group cross covarance as, Jn P nm gg = Jm j w n w nj P nmj rr, 52 where {n J n =1 and {m J m =1 are he ndces of all he reflecors n each group, respecvely. The weghs, w n and w n, are calculaed usng he assumpon ha all reflecors n each group are clusered as, for example, { w n = E A n J n j A nj. 53 The cross covarance beween deecons from he same group s smply found when n = m. C. Daa assocaon In dfference o sandard JPDA, he generalzed verson of he JPDA algorhm proposed here consders he possbly ha a sngle rac can generae mulple measuremens. To avod unlely daa assocaon hypoheses, we employ an ellpsodal gae [3] cenered a he group mean usng 51. From he gaed measuremens and nowledge regardng he maxmum number of deecons generaed by group n max M,n, s possble o consruc he se of all local hypoheses,.e., he se of all feasble assocaons beween y and group n. By combnng local hypoheses from all groups n an admssble fashon such ha each deecon n y s assocaed o precsely one group, or classfed as cluer we oban a global hypohess, descrbed by he vecor λ. The hypohess probably can be expressed as Pr { λ Y p y λ, Y 1 Pr { λ Y 1. 54
10 HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL The lelhood of he daa assocaon hypohess s found hrough he Gaussan approxmaon 51 and he cluer model 7 as p y λ, Y 1 = N y M c λ, ŷ 1 λ, 1 Pλ yy. 55 V where [ ĝλj1 ŷ 1 λ = T ĝλj2 T ĝλjm ] T T,,..., 56 [ T T ] T T y λ = y λj1,,..., 57 y λj2 y λjm for {j 1,..., j m = {j : λj 0, and smlarly P λ yy s consruced as Pgg λj1 + W... P λj1λjm gg P λ yy =...... 58 P λjmλj1 gg... P λjm gg + W The expeced sgnal componen of he group n, ĝ n, used n 56 s found n 47. Expressons for he covarance componens n 58 are found n 48 for he group covarance, P n gg, and n 52 for he cross covarances beween group n and m, P nm gg. The daa assocaon vecor, λ, provdes perfec nowledge regardng he number of cluer deecons, M c, and he number of deecons from group n, M,n. Hence, he pror probably for he assocaon vecor n 54 can be paroned as Pr { λ { Y 1 = Pr λ, M, c Y 1 = Pr { λ M, M c Pr {M c Pr { M Y 1, 59 [ ] T where M = M,1... M,N g and N g s number of groups. As M c s assumed o be Posson dsrbued, we have Pr {M c = µv M c exp µv /M c!. 60 Furhermore, Pr { λ M, M c s found usng combnaorcs, Pr { λ M M, M c g M n 1 = m=1 M,m M,n 1. 61 Fnally, he probably of he oal number of arge deecons, g Pr { N M Y 1 = Pr { M,n Y 1 62 { where Pr M,n Y 1 s approxmaed usng esmaed expeced sgnal amplude, ˆσ 1 n 30. D. Measuremen updae To perform he measuremen updae defned by 34-35, we frs need o consruc he ncluded enes. The predced mean and covarance are already gven n Secon IV-A and ŷ λ 1 and Pλ yy are gven by 56 and 58, respecvely. However, he cross covarance P λ zy needs o be esmaed. Usng he sgma pon ses n 41 and 42, P zr = Cov { z, r Y 1 can be esmaed as P zr u Z ẑ 1 R T ˆr 1. 63 From 63, P λ zy s calculaed n hree seps. Frs, approxmang he relaon beween reflecors and clusers as n 21 we fnd P,n zc l = N w l P l zr, 64 where l lss all reflecors n cluser l n cluser consellaon. Usng 64 and 28, s easy o fnd P n zg = 1 N n Nn n L =1 l=1 q P,n zc l, 65 from whch we can fnally consruc he sough cross covarance, P λ zy, as [ ] P λ zy = P λj1 zg,..., P λjm zg, 66 for {j 1,..., j m = {j : λj 0. The poseror mean and covarance esmaes under daa assocaon hypohess, λ, are found hrough nserng 56, 66 and 58 no 34-35. The poseror densy 33 s a weghed sum of he poseror denses for all λ weghed by 54. The mean and covarance of a Gaussan mxure model are readly calculaed hough momen machng, see e.g. [3]. Ths concludes he dervaon of he fler framewor for esmang he poson of vehcles usng possble unresolved radar deecons. V. EVALUATION In hs secon we compare he proposed exended arge model, denoed M 1, wh ha of a pon source model basc model, M 2, smlar o hose presenly used n he auomove ndusry. The evaluaon s performed n wo seps, Frs, we compare he ably he models o explan radar observaons. Second, we compare he esmaon error, e E { = z Y, M n z 2, of he racng sysem derved n Secon IV wh one based on M 2. The evaluaon [ s lmed z h T ] o sngle arge vehcle scenaros,.e. z =, z 1 T T. For boh evaluaons, radar observaons are colleced from hree sensors, one long-range radar a 77 GHz denoed s 1 and wo medum-range radars a 24 GHz denoed s 2 and s 3, mouned on he hos vehcle as llusraed n Fg. 6. Sensor s 1 has an updae rae of 10 Hz, a feld of vew of 16 o and a deecon range of approx. 150 m, whereas s 2 and s 3 cover a 150 o feld of vew up o approx. 70 m usng 13 ndependen receve beams, each delverng deecons every 40ms. The resoluon cell for he wo ypes of sensors are, s1 d = [2m,.5m/s, 3.5o ] and s2,3 d = [2m, 6m/s, ], respecvely, where s2,3 d s used o descrbe he resoluon n each of he receve beams of s 2 and s 3. The correspondng measuremen nose covarance for a pon arge s specfed as W s1 = dag [ ] 1π 2.4, 2, 180 and W s2,3 = dag [ ] 2..4,.5, 1.2π 180 Targe vehcle reference
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 11 s 2 s 3 = 6.2 s = 5 s = 0 s Fgure 6. Hos vehcle equpped wh hree radar sensors, one mechancally scanned 77 Ghz long range radar, denoed s 1, and wo medum range 24 GHz radars loong o he rgh and lef, denoed s 2 and s 3, respecvely. In he evaluaed racng scenaro he hos vehcle s ravelng a consan speed a a sragh pah. The arge vehcle drves a a crossng pah soppng n fron of he hos vehcle before mang a lef urn. poson, z, and hos vehcle poson measuremens are acqured usng aurae DGPS measuremens. We proceed by nroducng he pon source model, hen explan he wo comparsons and her resuls respecvely. A. Pon source model To evaluae he racng performance ganed n erms of esmaon error by consderng he vehcles as exended objecs, we compare our racng sysem wh one based on a pon source model. To mae he comparson as far as possble, we use he same sae paramerzaon and boh models explo nowledge regardng he physcal dmenson of he observed vehcle. Gven z 1 he model compensaes for offse errors by posonng he expeced arge measuremen, ŷm 1 2, on he nersecon beween he lne of sgh beween he radar sensor and ζx, 1, ζ1 x, and he vehcle frame. Usng hs model we desgn a probablsc daa assocaon fler PDAF [28], where only one measuremen may orgnae from he arge and he presence of mulple measuremens are modelled as cluer descrbed by a homogeneous Posson process. B. Sensor model comparson s 1 The ably o explan a se of gven observaons can be compared by evaluang he log-lelhood rao p y z, M 1 ly, z = log p y z, M 2 67 where he model specfc lelhood funcons can be paroned as p { y z, M = y λ, z, M P λ z, M. λ L p 68 In he followng secons, we presen expressons for p y λ, z, M and P {λ z, M for he dfferen models as well as he evaluaon of 67 for radar measuremens from wo ypes of radar sensors. 1 Lelhood funcon: The lelhood rao es s commonly used o compare hypoheses, n hs case whch model s more lely o have produced he radar measuremens. The es s reasonable f he number of unng parameers are he same for he compared models. For our proposed sensor model, M 1, 68 s gven by 55 and 59, wh he mnor dfference ha we do no have o negrae over z. The correspondng denses for M 2 are obaned smlarly as p { y N y j λ, z, M 2 = ; ŷ1 M 2, W 1 V M c : M = 1 1 V M c : M = 0 Pr { P D e µv µv M c λ z, M 2 = M M c! : M = 1 1 P D e µv µv M c M c! : M = 0. The probably of deecon s modelled n he same way for boh models and he cluer nensy, µ, s esmaed from daa usng nonparamerc PDA [29]. Noe ha he same parameers are used for unng boh models,.e., he measuremen nose covarance and he probably of deecon. All oher parameers are aen drecly from he sensor specfcaon. 2 Resuls: The log-lelhood rao 67 s evaluaed usng radar measuremens from wo sensors of dfferen ypes, s 1 and s 2. The daa s colleced whle he hos vehcle drves sragh owards he arge vehcle a an angle of 18 o offse from he sde of he vehcle sarng a a dsance of 40m. A scaer plo of he colleced daa s llusraed n Fg. 7a. Fgures 7b and 7c dsplay 67 evaluaed for measuremens delvered by sensor s 1 and s 2, respecvely. Boh fgures show a clear advanage n favour of our proposed model. Ths s especally clear for sensor s 1 whch has hgher resoluon han sensor s 2 and where we ofen receve mulple deecons nealy concenraed o he fron and rear wheel housngs, see Fg. 7a. For sensor s 2, he deecons are spread along he sde of he arge vehcle some even posoned ousde he vehcle frame. Ths behavor s also modelled n M 1 bu he advanage s no as domnan as n he case of sensor s 1. C. Tracng fler comparson The racng flers based on M 1 and M 2 are evaluaed usng daa from s 1, s 2 and s 3 n he scenaro depced n Fg. 6. Ths parcular scenaro s chosen, boh because s a relevan scenaro for acve safey sysems addressng nersecon adens [30] and because s challengng for a radar based racng sysem. In he evaluaon, boh flers are naed n he reference sae, z 1 0, provded by he DGPS sysem wh nal covarance P 0 = dag[1, 1, 3π 180,.5,.001, 1.5]2 and use he same process nose parameers, σ 2 v = 9 and σ2 ċ = 1 70. The fler mplemenaon for our proposed model s gven n Secon IV and for he pon source model we employ a sandard UKF. The resul of he comparson s shown n Fg. 8 n erms of absolue longudnal and laeral posonng error, e x and e y, n he reference vehcle coordnae frame as well as absolue velocy error, e v, and headng angle error, e ψ, n he global coordnae frame. The resul ndcaes a clear advanage for M 1 n erms of aurae and sable posonng of he arge vehcle, as well as velocy and headng esmaes. Worh
12 HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL ex [m] 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 2 ey [m] 1 a Scaer plo of he evaluaed arge measuremens from he s 1 red and s 2 blue radars. ly s1, z 10 5 0 0 2 4 6 8 10 12 14 16 b Log-lelhood rao s 1 ev [m/s] eψ [ o ] 0 0 1 2 3 4 5 6 7 8 9 10 4 2 0 0 1 2 3 4 5 6 7 8 9 10 20 10 0 0 1 2 3 4 5 6 7 8 9 10 [s] ly s2, z 8 4 0 4 0 2 4 6 8 10 12 14 16 c Log-lelhood rao s 2 Fgure 7. Log-lelhood rao comparson beween our model, M 1, and he smpler pon source model, M 2. nong s he laer par of he scenaro, [7, 10]s, where he wo vehcles are close and many of he feaures on he reference vehcle are resolved. In hs par of he scenaro he arge vehcle sars o urn, somehng ha confuses he smpler pon source model whle our model sll manages o poson he vehcle well. Addonally, he jump n he headng esmaon error a 9.8s s explaned by he flers only recevng measuremens from he rear par of he vehcle for some updaes. As measuremens agan appear from he fron of he vehcle, s clear ha our proposed model s able o ae more advanage of he new nformaon o deduce he headng of he vehcle more auraely han he pon source model. VI. CONCLUSION In hs paper we have proposed an aurae and racable radar sensor model capable of descrbng boh mulple deecons from a vehcle and her relaon o he lmed sensor resoluon. Furhermore, we have developed a framewor for racng vehcles based on hs model. The evaluaon of he sensor model shows ha our model s clearly beer han he reference model a descrbng he vehcle radar deecons from he wo evaluaed sensors. Addonally, he evaluaon of he racng performance ndcaes subsanal benefs usng our model compared o he reference model. The reference model s noably smpler han he proposed one, and clearly penalzed n he evaluaon when mulple feaures are resolved. However, s presenly used n many sysems and our comparson show ha a wdely used famly Fgure 8. Comparson of he absolue esmaon error from our racng framewor sold blue and he pon source model dashed red for longudnal and laeral posonng error n he reference vehcle coordnae frame, e x and e y, as well as absolue velocy error, e v, and headng angle error, e ψ, n he global coordnae frame. of racng framewors can be adaped o ncorporae he new measuremen model, wh mproved performance as a resul. By formally ncludng sensor resoluon n he model, can be used for a wde range of sensors and arges by changng approprae parameers aordng o he sensor specfcaon. Ths feaure s of mos mporance o he auomove ndusry as allows for sensors o be more easly replaced or updaed. APPENDIX A REFLECTOR MAPPING The reflecor mappng s dvded no wo pars; frs he reflecor s posoned n he observaon space, second, he he expeced sgnal amplude s modelled. A. Reflecor poson Assumng ha pon reflecor wh ndex s posoned a x = x, y n a local coordnae sysem wh he orgn n he cener of arge vehcle j. The global poson of he reflecor s hen gven by [ ζ j, x ζ j, y ] = [ ] [ ] ζ j x ζy j + Rψ j x y 69 where R s a 2x2 roaonal marx. Smlarly, assumng ha sensor, s, s mouned on he hos vehcle a x s = x s, y s and wh an angle of ψ s, he global poson of he sensor s defned as [ ] [ ] [ ] ζ s x ζ h ζ s = x y ζy h + Rψ h xs 70 y s Addonally, we defne he relave angle beween reflecor and sensor s aordng o ζ j, y ζy s α,s = arcan ζx j, 71 ζx s
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL 13 R Usng hese relaons, he mappng z r s defned as r = ζx j, ζx s 2 + ζy j, ζy s 2 72 ṙ = v j cosψl α,s + v j π cos 2 + argx α,s π v h cosψ h α,s + v h cos 2 + argx s α,s 73 φ = α,s ψ h ψ s 74 where v j = vj cj x and v h = vh ch x s are he velocy componen due o roaon of he arge and hos vehcle, respecvely. B. Sgnal amplude The sgnal power of he sensor s characerzed by wo funcons, he recprocal anenna gan paern, A a φ, and he sgnal aenuaon, A r r. Assocaed wh each reflecor s a vsbly funcon, νσα,s, ψ j dependen on he relave angle beween he observng sensor and he reflecor and he headng of he arge vehcle. Usng hese models, he expeced reurn amplude of reflecor s calculaed as, σ = A a φ A r r νσα,s, ψ j. 75 APPENDIX B GAUSSIAN CLUSTER DENSITY APPROXIMATION In hs secon we derve he gaussan approxmaon of he cluser densy n 21 and 22. Le a cluser conss of N reflecors posoned a r 1, r 2,..., r N n measuremen space. Aordng o 18 he sgnal componen of he cluser s gven by N c = w n r n, where n hs case he reflecor posons, r n, are nown whereas he weghs, w n, are sochasc. The weghs are expressed n erms of he receved sgnal ampludes of each reflecor, A n, A n w n = N m=1 A. m where A n Rayleghσ n. To fnd a Gaussan approxmaon of cluser densy we need o fnd he frs wo momens of c. A. Mean approxmaon The mean of c s c = E {c = N w n r n. 76 where w n = E {w n s no rval o express. However, mone- Carlo smulaons ndcae ha he approxmaon w n Ā n N m=1 Ām 77 where Ā n = E {A n = σ n π 2, 78 yelds a reasonable approxmaon of 76. B. Covarance approxmaon The remanng dffculy s o approxmae he covarance marx of c, { C = E c c c c T 79 The frs am s o fnd a represenaon whch more robus o approxmaons. For noaon, se r n = r n c and w n = w n w n. In he followng, we wll use he relaons N w n = 1, 80 whch follows from he defnon of w n, and N w n r n = 0, 81 whch s clear due o E {r r = 0. These relaons and noaons yeld N N T C = E w n r n r w n r n r N N T = E w n r n w n r n N N T = E w n r n w n r n = N m=1 N r n r T me { w n w m. 82 Recall ha he marces r n r T m are nown for all ndces n and m. Hence, we wll now srve o fnd approxmaons for he scalar facors Cov {w n, w m = E { w n w m. 83 To hs end, we use he Taylor approxmaon w n = A n S Ān S = Ān S + A n Ān S + A n S where S = N m=1 A m and S Ān S 2 S S Ān S 2 84 Ā n = E {A n 85 { N S = E A m. 86 m=1
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Journal Basc Eng., vol. 82, pp. 24 45, 1960. [28] Y. Bar-Shalom and E. Tse, Tracng n a cluered envronmen wh probablsc daa assocaon, Auomaca, vol. 11, Sepember 1975. [29] Y. Bar-Shalom and T. Formann, Tracng and daa assocaon. Academc-Press, Boson, 1988. [30] M. Bra nnsro m, E. Coelngh, and J. Sjo berg, Model-based hrea assessmen for avodng arbrary vehcle collsons, Inellgen Transporaon Sysems, IEEE Transacons on, vol. 11, no. 3, pp. 658 669, 2010. Lars Hammarsrand was born n Landveer, Sweden n 1979. He receved hs M.Sc. and Ph.D. degree n elecrcal engneerng from Chalmers Unversy of Technology, Gohenburg, Sweden, n 2004 and 2010, respecvely. From 2004 o 2011, he was wh he Acve Safey and Chasss Deparmen a Volvo Car Corporaon, Gohenburg, conducng research on racng and sensor daa fuson mehods for acve safey sysems. Currenly, Lars s a Posdocoral Research Fellow a he Sgnal Processng group a Chalmers Unversy of Technology where hs man research neress are n he felds of esmaon, sensor fuson and radar sensor modelng, especally wh applcaon o acve safey sysems. Fredr Sandblom receved he M.S. degree n elecrcal engneerng n 2004 and he Lcenae of Engneerng degree n 2008, boh from Chalmers Unversy of Technology, Gohenburg, Sweden. Snce 2005 he has been wh he Volvo group, worng wh acve safey sysem research as a par of hs Ph.D. sudes he wll defend hs hess n December 2011. Hs research neress concern objec racng and sensor daa fuson, parcularly sgma-pon mehods for esmang sascal momens and her applcaon o recursve flerng.
HAMMARSTRAND e al.: EXTENDED OBJECT TRACKING USING A RADAR RESOLUTION MODEL Lennar Svensson was born n A lva ngen, Sweden n 1976. He receved he M.S. degree n elecrcal engneerng n 1999 and he Ph.D. degree n 2004, boh from Chalmers Unversy of Technology, Gohenburg, Sweden. He s currenly an Assocae Professor a he Sgnal Processng group, agan a Chalmers Unversy of Technology. Hs research neress nclude Bayesan nference n general, and nonlnear flerng and racng n parcular. Joam So rsed was born n Sara, Sweden, on Aprl 3, 1976. He receved he M.S. degree n elecrcal engneerng n 2001 and he Ph.D. degree n 2007, boh from Chalmers Unversy of Technology, Gohenburg, Sweden. Snce 2007 he has been wh he Volvo Car Corporaon where he has wored on developng acve safey sysems. Hs research neress are n he area of sgnal processng; specfcally subspacebased esmaon mehods and objec racng. 15