31 Itertol Jourl of Cotrol, Yog-Shk Automto, Km d Keum-Shk d Systems, Hog vol. 2, o. 3, pp. 31-318, September 24 A IMM Algorthm for Trckg Meuverg Vehcles Adptve Cruse Cotrol Evromet Yog-Shk Km d Keum-Shk Hog* Abstrct: I ths pper, usceted Klm flter () for curvler motos terctg multple model (IMM) lgorthm to trck meuverg vehcle o rod s vestgted. Drvg ptters of vehcles o rod re modeled s stochstc hybrd systems. I order to trck the meuverg vehcles, two kemtc models re derved: A costt velocty model for ler motos d costt-speed tur model for curvler motos. For the costt-speed tur model, s used becuse of the drwbcks of the exteded Klm flter oler systems. The suggested lgorthm reduces the root me squres error for ler motos d rpdly detects possble turg motos. Keywords: Adptve cruse cotrol, costt-speed tur model, exteded Klm flter, hybrd estmto, terctg multple model, oler flterg, usceted Klm flter. 1. INTRODUCTON Recetly, the mjorty of utomoble compes re developg vrous drver ssstce systems to crese vehcle sfety d llevte drver worklod. The drver ssstce systems clude dptve cruse cotrol (ACC), le-keepg support, collso wrg d collso vodce, d sssted le chges. The effectveess of these drver ssstt systems depeds o the terpretto of the formto rrvg from sesors, whch provde detls of the surroudg vehcle evromet d of the drver-sssted vehcle tself. I prtculr, ll these systems rely o the detecto d subsequet trckg of objects roud the vehcle. Such detecto formto s provded by rdr, ldr, d vso sesor. The ssstce systems metoed bove hve cert objectves tht ther cotrollers try to meet. Before cotroller c mke decso tht ebles the drver to feel turl, the moto of the surroudg object must be properly terpreted from the vlble sesor formto [2]. Muscrpt receved Mrch 1, 24; revsed July 1, 24; ccepted July 2, 24. Recommeded by Edtor-- Chef Myug J Chug. Ths work ws supported by the Mstry of Scece d Techology of Kore uder progrm of the Ntol Reserch Lbortory, grt umber NRL M1-32--39-3-J--23-1. Yog-Shk Km s wth the Deprtmet of Mechcl d Itellget Systems Egeerg, Pus Ntol Uversty, S 3 Jgjeo-dog Gumjeog-gu, Bus 69-735, Kore (e-ml: mmpdf@pus.c.kr). Keum-Shk Hog s wth the School of Mechcl Egeerg, Pus Ntol Uversty, S 3 Jgjeodog Gumjeog-gu, Bus 69-735, Kore (e-ml: kshog@pus.c.kr). * Correspodg uthor. Fg. 1 shows the cofgurto of ACC system. The ACC system cossts of drver terfce, rdr sesor whch mesures both the dstce d speed of precedg vehcles, cotroller whch cotrols both throttle d brke ctutors, d ctutors [17]. The blty to ccurtely predct the moto of precedg vehcles the ACC evromet c mprove the cotroller s blty to dpt smoothly to the behvor of those vehcles precedg t. Ths blty to predct motos s depedet o how well the rdr of ACC vehcle c trck other vehcles. I order to trck other vehcles usg the object formto obted from multple sesors, trckg techques bsed o the Byes pproch re usully used [1]. The trckg of meuverg trget s lredy wellestblshed topc the trget trckg lterture. Techques for trckg meuverg trgets re used my trckg d survellce systems s well s pplctos where relblty s the m cocer [1,6,13,14]. I prtculr, trckg meuverg trget usg multple models c provde better performce th usg sgle model. A umber of multple model techques to trck meuverg trget hve bee proposed the lterture: the multple-model lgorthms [13], the terctg multple model (IMM) lgorthm [1,14,, 23], the dptve IMM [7,18], the fuzzy IMM [3,16], Sesor (MMW Rdr) Reltve Velocty Reltve Dstce ACC Vehcle Speed Trget-Trckg Techque (IMM Algorthm) Drver Iterfce (Swtch/LCD) ACC Vehcle Cotroller Brke Actutor Fg. 1. Cofgurto of the ACC system. Throttle Actutor
A IMM Algorthm for Trckg Meuverg Vehcles Adptve Cruse Cotrol Evromet 311 d others. Geerlly, trget moto models c be dvded to two subctegores: the uform moto model d the meuverg model. A meuverg trget movg t costt tur-rte d speed s usully modeled s meuverg model, d clled coordted tur model [1,4,5,7,14,18]. For pplcto to r trffc cotrol, fxed structure IMM lgorthm wth sgle costt velocty model d two coordted tur models ws lyzed [14]. Ad, for the trckg of meuverg trget, vldto method of ew type of flght mode ws preseted [19]. Nb d Bshop [19] vldted o-costt speed coordted tur rcrft meuver model by comprg ther model wth the clssc Sger meuver model d costt-speed coordted tur model usg ctul trjectores. Semerdjev d Mhylov [22] dscussed vrble- d fxed-structure ugmeted IMM lgorthms, wheres fxed-structure lgorthm oly ws dscussed [14], d ppled to meuverg shp trckg problem by ugmetg the tur rte error. The drwbcks of the terctg multple model lgorthms usg exteded Klm flters (IMM-) re s follows. Frst, the pproxmtes o- Guss desty wth Guss desty [24]. Secod, the IMM pproxmtes the Guss mxture wth sgle Guss desty. If these ssumptos brek dow, the IMM- my dverge. I ths pper, becuse of these drwbcks of the IMM-, usceted Klm flter () [8,21], replcg the, s used for the curvler model. The lgorthm tself uses the sme IMM logc, but the modelmtched s replced by the model-mtched. The objectve of ths pper s to desg for curvler motos IMM lgorthm to trck meuverg vehcle for the drvg of ACC vehcle o rod. The cotrbutos of ths pper re s follows. Frst, the IMM lgorthm s provded s drvg lgorthm for ACC vehcle drvg o rod. Secod, two kemtc models for the possble drvg ptters of vehcles re derved: A costt velocty model for ler motos d costt-speed tur model for curvler motos re dscussed. Thrd, for the costt-speed tur model, s used becuse of the drwbcks of the. Fourth, the suggested lgorthm reduces the root me squres error the cse of rectler motos d detects the occurrece of meuverg quckly the cse of turg motos. Ths pper s orgzed s follows: I Secto 2, we provde the vrous drvg ptters of vehcles. A stochstc hybrd system s formulted, d two kemtc models re dscussed. I Secto 3, we compre wth for costt- speed tur model IMM lgorthm. I Secto 4, we evlute the performce of these flters usg Mote Crlo smulto uder the vrous drvg ptters. Secto 5 cocludes the pper. 2. PROBLEM FORMULATION I ths secto, fter lyzg the drvg ptters of vehcle o rod, stochstc hybrd system the form of IMM lgorthm for trckg the precedg vehcle usg sesors (rdr, ldr, sor, vso, etc) s formulted. Also, two kemtc models represetg the lyzed drvg ptters re troduced. 2.1. Drvg ptters Fg. 2 depcts the vrous drvg ptters of vehcle: strght le d curve, cut-/out, u-tur, d terchge. All of these ptters c be represeted by combto of costt-velocty rectler moto, costt-ccelerto rectler moto, costt gulr velocty curvler moto, d costt gulr ccelerto curvler moto. As kemtc models for descrbg these motos, two stochstc models wll be vestgted: oe for rectler moto d the other for curvler moto. These typcl drvg ptters re descrbed brefly s follows: ) Strght le d curve: I ths stuto, the ACC vehcle trcks precedg vehcle tht follows strght les d curves o curved rod [1,2]. ) Cut-/out: The cut-/out dctes the stuto whch meuverg vehcle cuts (or out) to (or from) the le whle the ACC vehcle s trckg other vehcle. I ths stuto, the trckg of up to three surroudg vehcles s ssumed: oe frot, oe to the left, d oe to the rght. I ths cse, the trget vehcle chges ts moto from rectler moto to curvler moto d the bck to rectler moto. ) U-tur: Ths stuto occurs whe the trget vehcle chges ts drvg drecto by 18. The u- tur cossts of three routes s follows: The trget vehcle moves rectlerly, udergoes uform crculr turg of up to 18 wth costt yw rte, d the coverts to rectler moto the opposte drecto. v) Iterchge: Whe the ACC vehcle s pssg through terchge, the trget vehcle udergoes 3-dmesol moto. The trget vehcle moves rectlerly, udergoes uform crculr turg of up to 27 wth costt yw rte, d the coverts to rectler moto. I ths pper, pssg terchge wll be smplfed by 2-dmesol moto. It wll be show the sequel tht costtvelocty model wll cpture both costt velocty d ccelerto rectler motos wthout d wth
312 Yog-Shk Km d Keum-Shk Hog ACC vehcle Meuverg vehcle #2 ACC vehcle Pssg vehcle ACC vehcle ACC vehcle Pssg vehcle Meuverg vehcle () strght le d curve Pssg vehcle Pssg vehcle Meuverg vehcle Pssg vehcle (b) cut-/out (c) u-tur Meuverg vehcle (d) terchge Fg. 2. Vrous drvg ptters of vehcles. Meuverg vehcle #1 27. Pssg vehcle ddtol ose term, respectvely. O the other hd, costt-speed tur model wll cover both costt gulr velocty d gulr ccelerto curvler motos wthout d wth ose term, respectvely. 2.2. Stochstc hybrd system Followg the work of L d Br-shlom [14], stochstc hybrd system wth ddtve ose s cosdered s follows: x( k) = f [ k 1, x( k 1), m( k)] + g[ k 1, x( k 1), ν[ k 1, m( k)], m( k)] (1) wth osy mesuremets z( k) = h[ k, x( k), m( k)] + w [ k, m( k)], (2) where xk ( ) R x s the stte vector cludg the posto, velocty, d yw rte of the vehcle t dscrete tme k, m(k) s the sclr-vlued modl stte (drvg mode dex) t stt k, whch s homogeeous Mrkov ch wth probbltes of trsto gve by Pm { j( k+ 1) m( k)} = πj, m, mj M, (3) where P{} deotes the probblty d M s the set of modl sttes, tht s, costt velocty, costt ccelerto, costt gulr rte turg wth costt rdus of curvture, etc. The cosdered system s hybrd sce the dscrete evet m(k) ppers the system. I the drvg of ACC vehcle, m(k) deotes the drvg mode of the precedg vehcle, effect durg the smplg perod edg t k, tht s, the tme perod ( tk 1, tk]. The evet for whch mode m s effect t tme k s deoted s j m ( k) = { m( k) = m }. (4) j j zk ( ) R z s the vector-vlued osy mesuremet from the sesor t tme k, whch s mode-depedet. ν[ k 1, m( k)] R ν s the mode-depedet process ose sequece wth me ν [ k 1, m( k)] d covrce Q[k-1, m(k)]. w [ k, m( k)] R z s the mode-depedet mesuremet ose sequece wth me wk [, mk ( )] d covrce R[k, m(k)]. Flly f, g, d h re oler vector-vlued fuctos. 2.3. Two kemtc models The cocept of usg ose-drve kemtc models comes from the fct tht oses wth dfferet levels of vrce c represet dfferet motos. A model wth hgh vrce ose c cpture meuverg motos, whle model wth low vrce ose represets uform motos. The multple-models pproch ssumes tht model c mmedtely cpture the complex system behvor better th others. Two kemtc models for rectler d curvler motos re ow derved. Frst, ssumg tht ccelertos the stedy stte re qute smll (brupt motos lke sudde stop or collso re ot covered), ler ccelertos or decelertos c be
A IMM Algorthm for Trckg Meuverg Vehcles Adptve Cruse Cotrol Evromet 313 resobly well covered by process oses wth the costt velocty model. Tht s, the costt velocty model plus zero-me ose wth pproprte covrce represetg the mgtude of ccelerto c hdle uform motos o the rod. I dscretetme, the costt velocty model wth ose s gve by 1 2 1 T T 2 1 T xk ( ) = xk ( 1) + ν ( k 1) 1 T 1 2, (5) T 2 1 T where T s the smplg tme (.e.,.1 sec ths pper), x(k) s the stte vector cludg the posto d velocty of the precedg vehcle the logtudl (ξ ) d lterl (η ) drectos t dscrete tme k, tht s, x( k) = [ ξ( k) ξ( k) η( k) η ( k)] (6) wth ξ d η deotg the orthogol coordtes of the horzotl ple; d ν s zero-me Guss whte ose represetg the ccelertos wth pproprte covrce Q. If ν ( k) s the ccelerto cremet durg the k th smplg perod, the velocty durg ths perod s clculted by ν ( kt ), 2 d the posto s ltered by ν ( kt ) /2. Secod, dscrete-tme model for turg s derved from cotuous-tme model for the coordted tur moto [1, p. 183]. A costt speed tur s tur wth costt yw rte log rod of costt rdus of curvture. However, the curvtures of ctul rods re ot costt. Hece, frly smll ose s dded to costt-speed tur model for the purpose of cpturg the vrto of the rod curvture. The ose the model represets the modelg error, such s the presece of gulr ccelerto d o-costt rdus of curvture. For vehcle turg wth costt gulr rte d movg wth costt speed (the mgtude of the velocty vector s costt), the kemtc equtos the ( ξ, η ) ple re ξ () t = ωη () t, η () t = ωξ () t, (7) where ξ () t s the orml (logtudl) ccelerto d η ( t) deotes the tgetl ccelerto, d ω s the costt yw rte ( ω > mples couterclockwse tur). The tgetl compoet of the ccelerto s equl to the rte of chge of the speed, tht s, η( t) = d η( t) / dt = d( ωξ( t)) / dt, d the orml compoet s defed s the squre of the speed the tgetl drecto dvded by the rdus of the curvture of the pth, tht s, 2 2 2 ξ () t = η ()/ t ξ() t = ω ξ ()/ t ξ() t where η ( t) = ωξ () t. The stte spce represetto of (7) wth the stte vector defed by x() t = [ ξ() t ξ() t η() t η ()] t becomes x () t = Ax() t, (8) where 1 ω A =. 1 ω The stte trset mtrx of the system (8) s gve by sωt 1 cosωt 1 ω ω At cosωt sωt e = 1 cosωt sωt. (9) 1 ω ω sωt cosωt It hs bee remrked tht f the gulr rte ω (7) s tme-vryg, (9) would be o loger true. I the sequel, followg the pproch [1, p. 466], erly costt- speed tur model dscrete-tme dom s troduced. I ths pproch, the model tself s motvted from (9), but the gulr rte s llowed to vry. A ew stte vector by ugmetg the gulr rte ω ( k) to the stte vector of (7) s defed s follows: x ( k) = [ ξ( k) ξ( k) η( k) η( k) ω ( k)], (1) where superscrpt deotes the ugmeted vlue. The, the erly costt-speed tur model s defed s follows [1, p. 467]: s ω( k 1) T 1 cos ω( k 1) T 1 ω( k 1) ω( k 1) cos ω( k 1) T s ω( k 1) T x ( k) = 1 cos ω( k 1) T s ω( k 1) T 1 ω( k 1) ω( k 1) s ω( k 1) T cos ω( k 1) T 1 2 T 2 T ( 1) 2 x k + ( k 1) T ν. (11) 2 T T Evdetly, both (5) d (11) re specl forms of (1). I ddto, t s resoble to ssume tht the trsto betwee the drvg modes of ACC vehcle hs the Mrkov probblty govered by (3). Cosequetly, the kemtc behvors of ACC vehcle c be sutbly descrbed the frmework of the stochstc hybrd systems.
314 Yog-Shk Km d Keum-Shk Hog 3. IMM ALGORITHM FOR TRACKING 3.1. The IMM lgorthm I order to ccurtely trck the moto of precedg vehcles the ACC evromet, IMM lgorthm s used ths pper. The cocept (structure) of the IMM lgorthm durg oe cycle s gve [1, p.454] d [14]. I ths pper two models of the IMM lgorthm re used: oe for rectler moto d the other for curvler moto. The trckg procedure of the vehcle rectler moto usg (5) s crred out by the stdrd Klm flter, whch s ot dscussed ths pper. However, for trckg curvler motos, whch requres the estmto of ω wth ew ugmeted model (8) Secto 2, s used. () Remrk 1: Whe trget dymcs re descrbed by multple-swtchg models, the posteror desty of the stte vector s mxture desty [1]. The pproxmtes the mxture compoets wth the Guss probblty desty fucto. The gol of the IMM lgorthm s to merge ll mxture compoets to sgle Guss dstrbuto such wy tht the frst d the secod momets re mtched. The m pot s tht for ech dymc model seprte flter s used. I ths pper, we use two Klm- bsed flters for two stochstc models: oe for rectler moto d the other for curvler moto. The results of these two model-mtched flters re mxed before flterg. The outputs of the model-mtched Klm-bsed flters t tme t k clude: the stte estmte xˆ ( k k ), covrce P ( k k ) d the model probblty µ ( k). The overll output of the IMM lgorthm s the clculted usg the Guss mxture equtos. The drwbcks of the IMM lgorthm usg re s follows. Frst, the pproxmtes o-guss desty by Guss desty [24]. Secod, the IMM lgorthm pproxmtes the Guss mxture by sgle Guss desty. If these ssumptos brek dow, the IMM lgorthm usg my dverge. 3.2. The for the costt-speed tur model The erly costt-speed tur model of (11) c be rewrtte s follows: x ( k) = f [ x ( k 1), ω( k 1)] + G( k 1) ν ( k 1), (12) where the fucto f ( ) s kow d rems uchged durg the estmto procedure. The ose trsto mtrx G(k-1) s the sme form s tht gve (11). Becuse of the well-kow drwbcks of the, the for the costt-speed tur model s used [8,21]. Smlrly to the, the s recursve mmum me squre error estmtor. But ulke the, whch oly uses the frst-order terms the Tylor seres expso of the o-ler mesuremet equto, the uses the true mesuremet model d pproxmtes the dstrbuto of the stte vector. Ths stte dstrbuto s stll represeted by Guss desty, but t s specfed wth set of determstclly chose smple (or sgm) pots. The smple pots completely cpture the true me d covrce of the Guss rdom vector. Whe propgted though y o-ler system, the smple pots cpture the posteror me d covrce ccurtely to the secod order. The m buldg block of the s the usceted trsform, descrbed below. The usceted trsform s method for clcultg the sttstcs of rdom vector whch udergoes o-ler trsformto. Let x R x be rdom y vector, p : R x R o-ler trsformto d y = p(x). Assume tht the me d the covrce of x re x d P x, respectvely. The procedure for clcultg the frst two momets of y usg the usceted trsform s s follows [8]. 1) Compute ( x + 1) sgm pots weghts W : κ χ = x, W = + κ, =, x χ d ther 1 χ = x + ( ( x + κ) Px), W =, 2( x + κ ) = 1,, x, 1 χ = x ( ( x + κ) Px), W =, 2( x + κ ) = x + 1,,x, (13) where κ s sclg prmeter for fe tug the hgher order momets of the pproxmto d ( ( x + κ ) Px) s the th row or colum of the mtrx squre root of ( x + κ ) Px. 2) Propgte ech sgm pot through the oler fucto ζ = p( χ) ( =,,2 x). (14) 3) Clculte the me d covrce of y s follows: x y = Wζ, x y ζ ζ P = W ( y)( y). () Usg the usceted trsformto, the equtos for the costt-speed tur model re gve Fg. 3. Note tht the requres the computto of mtrx squre root (13), whch c be performed usg the Cholesky fctorzto.
A IMM Algorthm for Trckg Meuverg Vehcles Adptve Cruse Cotrol Evromet 3 Remrk 2: I the usceted trsformto, o whch the s bsed, set of weghted sgm pots re determstclly chose so tht cert propertes of these pots mtch those of the pror dstrbuto. Ech pot s the propgted through o-ler fucto d the propertes of the trsformed set re clculted. Wth ths set of pots, the usceted trsform gurtees the sme performce s the tructed secod order Guss flter, wth the sme order of clcultos s but wthout the eed to clculte Jcobs. 4. SIMULATIONS RESULTS As descrbed ths secto, we cosdered stte estmto problem of vehcle two dmesos. Smultos were executed to compre the performce of both IMM lgorthms wth the d the, respectvely, for curvler motos. The performce of the two lgorthms ws compred wth the use of Mote Crlo smultos. The meuverg vehcle trjectores were geerted usg the vrous ptters metoed Secto 2.1. Two kemtc models were used to trck the meuverg vehcle: A costtvelocty model for rectler moto d costtspeed tur model for curvler moto. We the compre the performce of two dfferet IMM lgorthms wth these two models. 4.1. The drvg sceros It ws ssumed tht the vehcle moves rectlerly the begg. The trget tl postos d veloctes were dfferetly set for ech scero. The sgle-trget trck of the meuverg vehcle ws lso ssumed to hve bee prevously tlzed d tht trck mtece ws the gol of the IMM lgorthms. The results for 4 selected sceros re preseted, ccordg to the drvg ptters, Fg. 2. ) Scero for strght le d curve: The trget tl postos d veloctes were ( x = m, y = m, x = 28 m/s, y = 28 m/s, ω = ). Its trjectory ws costt velocty betwee s d 19 s wth speed of 28 m/s; tur wth costt yw rte of ω = 3.74 /s betwee 2 s d 59 s; costt velocty betwee 6 s d 89 s; tur wth costt yw rte of ω = 3.74 /s betwee 9 s d 129 s; costt velocty betwee 13 s d 149 s; tur wth costt yw rte of ω = 3.74 /s betwee s d 2 s. ) Cut-/out scero: The trget tl postos d veloctes were ( x = m, y = 2 m, x = 28 m/s, y = m/s, ω = ). Its trjectory ws strght le betwee s d 19 s wth speed of 28 m/s; tur wth costt yw rte of ω = 3.74 /s betwee 2 s d 39 s; costt velocty betwee 4 s d 41 s wth speed of 28 m/s; tur betwee 42 s d 63 s wth yw rte of ω = 3.74 /s ; strght le betwee 64 s d 134 s wth speed of 28 m/s; tur wth costt yw rte of ω = 3.74 /s betwee 135 s d 4 s; costt velocty betwee 5 s d 9 s wth speed of 28 m/s; tur betwee 16 s d 179 s wth yw rte of ω = 3.74 /s d strght le betwee 18 s d 2 s. ) U-tur scero: The trget tl postos d veloctes were ( x = 1 m, y = 1 m, x = 28 m/s, y = m/s, ω = ). Ths scero cluded Itlzto: x ˆ ( ), P ( ) wth k = 1 Sgm pots d weghts: χ ( k 1 k 1), W ( =,,) Tme-updte equtos: predcted sgm pots χ( k k 1) = f [ χ( k 1 k 1), ω( k 1)] predcted me d covrce xˆ ( k k 1) = Wχ ( k k 1), P ( k k 1) = Q + W[ χ ( k k 1) x ( k k 1)][ χ ( k k 1) x ( k k 1)] predcted mesuremet sgm pots I( k k 1) = h[ χ( k k 1), ω( k)] predcted mesuremet zˆ ( k k 1) = WI ( k k 1) Mesuremet-updte equtos: mesuremet covrce zz P = R( k) + W[ I ( k k 1) zˆ ( k k 1)] xz χ I [ ( k k 1) zˆ ( k k 1)] P = W[ ( k k 1) xˆ ( k k 1)] [ I ( k k 1) zˆ ( k k 1)] flter g 1 K ( k) = Pxz ( Pzz ) me d covrce xˆ ( k k) = xˆ ( k k 1) + K ( k)[ z( k) zˆ ( k k 1)] P ( k k) = P ( k k 1) K ( k) P K ( k) Fg. 3. equtos for the costt speed tur model. zz
316 Yog-Shk Km d Keum-Shk Hog o-meuverg drvg mode durg scs from 1 s to 6 s wth speed of 28 m/s, 18 -tur, lstg from sc 61 s to 145 s wth yw rte of ω = 3.74 /s, d o-meuverg drvg mode from sc 146 s to 2 s. v) Iterchge scero: The trget tl postos d veloctes were ( x = m, y = m, x = 28 m/s, y = m/s, ω = ). Ths scero cluded o-meuverg drvg mode durg scs from 1 s to 4 s wth speed of 28 m/s, 27 -tur, lstg from sc 41 s to 168 s wth yw rte of ω = 3.74 /s, d o-meuverg drvg mode from sc 169 s to 2 s. The meuverg vehcle speed ws 28 m/s. 4.2. Prmeters used the desg The prmeters used the desg re lsted here. Subscrpts CV d CST std for costt velocty d costt speed tur, respectvely. The tl yw rte of the drvg sceros ws ω () = 3 /s. The error covrces of the tl stte d covrces of process ose were s follows: CV mode: P KF () = dg{1 1 1 1}, KF 2 Q = (.1) I, CST mode: P () = P () 2 = dg{1 1 1 1 }, σ ω 2 2 2 2 2 Q = Q = dg{(.25) (.25) (.25) (.25) σ ω } where σ ω = (.1) /s. The mesuremet ose covrce mtrx ws clculted s σ = 1m d σ η = 1m. The trsto probbltes for the IMM lgorthms usg the d the, respectvely, were represeted the Mrkov ch trsto mtrx.95.5 πj = πj = πj.5.95. The tl mode probblty vectors µ were chose s follows: µ = µ =.5.5. 4.3. Performce evluto d lyss The RMSE of ech stte compoet ws chose s the mesure of performce. The performce of the IMM lgorthm wth d tht of the IMM lgorthm wth re show Fg. 4 - Fg. 11, where the RMSE the posto d the velocty re plotted. The results preseted here re bsed o 1 Mote Crlo rus. Frst of ll, t s evdet tht the suggested lgorthm hs lmost equl posto d velocty estmto ccurcy for ll sceros. The ξ RMS posto error (m) RMS velocty error (m/s) 5 45 4 35 3 25 2 1 5 1 2 Fg. 5. Comprso of the velocty errors the cse of strght les d curves. RMS posto error (m) 14 13 12 11 1 14 13 12 11 1 9 8 7 6 5 4 3 9 8 7 6 5 4 3 5 1 2 Fg. 4. Comprso of the posto errors the cse of strght les d curves. 5 1 2 Fg. 6. Comprso of the posto errors the cse of cut-/out. RMS velocty error (m/s) 4 35 3 25 2 1 5 1 2 Fg. 7. Comprso of the velocty errors the cse of cut-/out.
A IMM Algorthm for Trckg Meuverg Vehcles Adptve Cruse Cotrol Evromet 317 RMS posto error (m) RMS posto error (m) 14 13 12 11 1 9 8 7 6 5 4 3 5 1 2 Fg. 1. Comprso of the posto errors the cse of terchge. RMS velocty error (m/s) 14 12 1 8 6 4 2 4 35 3 25 2 5 1 2 Fg. 8. Comprso of the posto errors the cse of u-tur. RMS velocty error (m/s) 4 35 3 25 2 1 5 1 2 Fg. 9. Comprso of the velocty errors the cse of u-tur. 1 5 1 2 Fg. 11. Comprso of the velocty errors the cse of terchge. posto RMSE of the IMM wth s evdetly superor to tht of the IMM wth. Ths s becuse, ulkely, does ot pproxmte oler fuctos but drectly propgtes me d covrce through the oler system equto. I ddto, the IMM lgorthm wth s chrcterzed by lower-pek dymc errors d shorter respose tme. These coclusos were cofrmed by the RMSE plot preseted Fgs. 4-11, respectvely. 5. CONCLUSIONS I ths pper, terctg multple model lgorthm wth, s trckg lgorthm, to trck meuverg vehcles o rod ws desged. As models to trck the meuverg vehcles, two kemtc models were derved: The costt velocty model for ler moto d the costt-speed tur model for curvler moto. For costt-speed tur model, usceted Klm flter ws used becuse of the drwbcks of the exteded Klm flter oler systems. The suggested lgorthm reduced the root me squre error for ler motos, d t could rpdly detect possble turg motos. REFERENCES [1] Y. Br-Shlom, X. L, d T. Krubrj, Estmto wth Applctos to Trckg d Nvgto, Joh Wley & Sos, INC, New York, Chpter 11, pp. 421-49, 21. [2] D. S. Cveey, Multple Trget Trckg the Adptve Cruse Cotrol Evromet Usg Multple Models d Probblstc Dt Assocto, M. S. Thess, Uversty of Clfor, Berkeley, U. S. A., 1999. [3] Z. Dg, H. Leug, K. Ch, d Z. Zhu, Model-set dptto usg fuzzy Klm flter, Mthemtcl d Computer Modellg, vol. 34, o. 7-8, pp. 799-812, October 21. [4] F. Dufour d M. Mrto, Pssve sesor dt fuso d meuverg trget trckg, : Y. Br-Shlom (Ed.), Multtrget-Multsesor Trckg: Applctos d Advces, Artech House, Norwood, MA, Chpter 3, pp. 65-92, 1992. [5] J. P. Helferty, Improved trckg of meuverg trgets: The use of tur-rte dstrbutos for ccelerto modelg, IEEE Trs. o Aerospce d Electroc Systems, vol. 32, o. 4, pp. 1355-1361, October 1996. [6] A. Houles d Y. Br-Shlom, Multsesor trckg of meuverg trget clutter, IEEE Trs. o Aerospce d Electroc Systems, vol. 25, o. 2, pp. 176-189, Mrch 1989. [7] V. P. Jlkov, D. S. Agelov, TZ. A. Semerdjev, Desg d comprso of mode-set dptve
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