PRINCIPLES OF RADAR TRACKING

Transcription

1 PRINCIPLES OF RADAR TRACKING Luke Adero, Akur Bakhi, Kareem Elahal, Joe Kell, Daid Kim, Vikram odi, Adam Patel, ad Joe Park, Ale Schader, Ale Sood, Adrew Weitraub ABSTRACT Adior: Rad Heuer Aitat: Karl Strohmaier Radar aloe gie oi etimate of a target locatio ad i icapable of directl meaurig elocit. To rectif thee hortcomig, we reearched liear etimator, a cla of algorithm that more accuratel etimate poitio ad elocit. We choe the Kalma filter becaue of it implicit, efficiec, ad low memor requiremet. We deeloped a Viual Baic.NET coole applicatio that retured the target poitio ad elocitie. The program tpicall proided poitio etimate withi a radiu of half a mile of the true poitio ad retured elocit etimate withi a rage of three mile per hour. INTRODUCTION Proce Whe mot people thik of radar, the thik of it depictio i moie a a foolproof wa of immediatel fidig a target eact locatio. Howeer, it i actuall much more comple tha impl boucig a wae off a target ad meaurig it retur. Due to ariou tpe of oie, or error, equatio ad algorithm mut be created to make the raw meauremet from the radar tem gie a more accurate etimate of the target poitio. Our project ioled creatig ad implemetig a algorithm kow a the Kalma filter. We ued the filter i fie differet ceario iolig either a igle or dual radar tem that meaured the target poitio. We ued the filter to help refie the meauremet ad brig them cloer to the target actual poitio b accoutig for both driig oie, the ariatio i the flight path of the target; ad meauremet oie, the accurac of the radar itelf. The ceario icreaed i difficult each time, moig from imple oe-dimeioal rage coordiate to Carteia coordiate ad the to polar coordiate. Polar coordiate iclude rage ad bearig from a et ormal lie poitig orth ad are cloer to what a true radar tem would meaure. After filterig the meauremet ad obtaiig our calculated data, we aalzed it o a comparatie bai with the true poitio ad elocit alue that were gie to u. To check the accurac of the filter, we ued a reidual graph a graph that how the differece betwee actual ad predicted poit at each poit i time. Whe doig thi, we looked to ee that the filter predicted alue icreaed i accurac a time progreed, that the filter did ot cotai a bia i a oe directio, ad that the filtered data were coitetl more accurate tha the raw meaured alue. Backgroud [6-]

2 Rudolf Kalma, a electrical egieer b traiig, i mot famou for hi co-ietio of the filter that ow bear hi ame, the Kalma filter. The Kalma filter i a digital mathematical igal proceig techique which ue recurio to etimate the tate of a damic tem from a erie of icomplete ad oi meauremet. The root of thi equatio ca be traced back to Carl Friedrich Gau 795 work. Kalma wa bor i Budapet, Hugar, o a 9, 93. He obtaied hi bachelor ad mater degree from IT i 953 ad 954, repectiel, ad hi doctorate from Columbia i 957 []. Kalma idea for the filter were firt met with o much reitace that he had to publih the reult i a mechaical joural rather tha a electrical oe. Howeer, after Kalma iited Stale Schmidt at the NASA Ame Reearch Ceter i 967, hi filter wa ued i trajector etimatio for the Apollo program aigatio tem []. Sice the, the Kalma filter ha gaied a wide ariet of ue i a diere rage of field. Some of the area i which it i ued iclude uclear power plat itrumetatio, demographic modelig, maufacturig, detectio of udergroud radioactiit, fuzz logic (a brach of logic i which truth i ot abolute), eural etwork traiig, ad ecoometric [3]. The umber of applicatio ha icreaed rapidl i recet ear with the adet of ew computer techologie, ad it ha ow etered ito the deelopmet of ophiticated weapo delier tem, atellite ureillace tem, ad o-militar trackig tem uch a Air Traffic Cotrol [4]. It i alo beig ued toda i three-dimeioal eiromet techologie to track the moemet of target. Sceario Oeriew Cae : Oe Dimeioal Trackig Cae : Two Dimeioal Trackig Cae 3: Polar Coordiate Trackig Cae 4: Dual Radar Trackig Cae 5: aeuerig Target Trackig We were gie fie differet cae i which to ue the Kalma filter. Each had a icreaig leel of difficult ad compleit. I the firt cae, we had to track a target moig ol i oe dimeio. For the ecod cae, the target wa moig i two dimeio, ad the meaured data wa gie to u i Carteia coordiate. I the third cae, we were agai told to track a target moig i two dimeio, but the meaured data wa gie to u i polar coordiate, which correpod more cloel to the data gie b real radar. The fourth cae itroduced the problem of haig multiple radar trackig a igle target. I the fial cae, we had to track a maeuerig target that witched elocit twice durig it coure. KALAN FILTER EQUATIONS True Poitio ad Radar Iput The true poitio of the object at time k +, gie the poitio at time k i: ( k ) = Φ( k) q( k) + + () [6-]

3 The tate ector ( k) i: ( k ) = ad the tate traitio model Φ i: I Φ = t I I I repreet the idetit matri. The tate ector keep track of the target poitio ad elocitie i differet dimeio (uuall the ad dimeio). The purpoe of the Kalma filter i to etimate the true tate ector gie a erie of dicrete radar meauremet. The tate traitio model update the tate ector each timetep. The tate traitio model update each poitio b addig the time iteral betwee each radar meauremet multiplied b the elocit i the ame dimeio. Becaue of mechaical ad pilot error, howeer, it i impoible for a flig object to maitai a cotat elocit. Thi i called driig oie ad i repreeted b q(k). It i added to the tate ector of each timetep to accout for uch driig irregularitie. athematicall, thi oie, while zero o aerage, i a radom Gauia oie proce with kow coariace matri: ar co Q = ( q ) L co( q, q ) ( q, q ) L ar( q ) O L O L co ar ( q ) L co( q, q ) ( q, q ) L ar( q ) I geeral, the coariace of a ector of radom ariable i defied a: L O L O ( ) T ( X) E ( X E( X) )( X E( X) ) co. () Driig oie betwee poitio ad elocitie i ucorrelated, which eplai the zero i the bottom left ad top right quadrat of Q. [6-3]

4 Becaue radar ca ol meaure poitio ad ot elocit, the tate ector mut be coerted ito a meauremet ector b the followig equatio. The meauremet ector i a fuctio of the tate ector plu a radom oie proce: ( k) H( k) r( k) = +, (3) where the meauremet ector ad the oberatio model H i: ( k) i: ( k) =, [ ] H =. I I order to coert the tate ector ito a meauremet ector, all the elocitie mut be elimiated ice the caot be meaured. Thi i accomplihed b multiplicatio with the oberatio model, which remoe eer elocit b effectiel cuttig the tate ector i half. Jut a driig oie wa added to the tate ector, meauremet oie mut be added to the meauremet ector. Ituitiel, thi oie repreet the iabilit of the radar trackig deice to preciel meaure the object poitio. Thi could be due to eeral techical problem, from limitatio i the radar cree reolutio to ibratio i the equipmet. athematicall, meauremet oie repreet the tadard deiatio σ betwee the poitio that hould be meaured ad the poitio that are actuall meaured. Therefore, r( k) i a Gauia radom proce that follow a multiariate ormal ditributio with coariace matri: σ R = σ σ L O L σ σ. σ Predictio The firt phae of each iteratio of the Kalma filter i the predictio tage, i which the algorithm gie both predictio of the object tate ector ad a etimate of how reliable the predictio i. Predictio of the object tate ector are gie uig the followig equatio: ˆ ( k k ) = Φˆ ( k k), (4) ˆ mea the predictio of ector at time m made at time.) The Kalma filter mut etimate both poitio ad elocit ee though radar ca ol track poitio. which i impl the predictio aalog of Eq. (). (The otatio ( m ) [6-4]

5 B Eq. (), the etimate of the predicted tate ector reliabilit i gie b P E ( ˆ )( ˆ ) ) T, which i the coariace of the differece betwee predicted ad actual tate. Thi differece hould be zero o aerage, but P itelf will eer be le tha Q. Coariace i a meaure of the degree to which umber ar. I other word, applied to the tate coariace matri, coariace meaure how pread out the error are. Epadig thi defiitio of the tate coariace matri gie: ar co P = ( ε ) L co( ε, ε ) ( ε, ε ) L ar( ε ) O L O L co ar ( ε ) L co( ε, ε ), ( ε, ε ) L ar( ε ) where ε ˆ. The term alog diagoal i the upper left ad bottom right quadrat deote the ariace of error i poitio, K, ad elocitie, K,. Thee umber hae a practical applicatio i that the gie the formula of a ellipe i which the error hae a certai probabilit of lig. I two dimeio, the equatio of a ellipe that ha ot bee rotated i: L O L O a + b =. (5) Rotatig the coordiate plae b agle θ gie the traformatio: ad ubtitutig thi ito Equatio (5) gie: = coθ + iθ, = coθ iθ co θ i θ + co i + θ θ a b a + b i a θ co + b θ =. (6) The ditace betwee a meaured poitio (, ) ad the predicted poitio (, ˆ ) meaured i uit of tadard deiatio quared, i gie b: T T ( σ ) = HPH, ˆ, where = ˆ ˆ. ultiplig thi out gie: [6-5]

6 ( σ ) ( ˆ ) ( ) ( ˆ )( ˆ ) ( ) ( ˆ ar co, + ) ar( ) T det( HPH ) =. (7) Howeer, thi i the equatio of a ellipe, o: det det ( ) co θ i T = + ( HPH ) a b ar ar ( ) i = T ( HPH ) a b co (, ) T ( HPH ) θ co + θ θ = iθ coθ a b det Fig. : Error ellipe gie b tate coariace matri If σ =. Solig for a, b, ad θ gie: a = ar ( ) + ar( ) + ( ar( ) ar( ) + ( co(, ) b = ar ( ) + ar( ) ( ar( ) ar( ) + ( co(, ) (, ) co ( ) ( ) θ = ta. ar ar Whe ( ) ar( ) ar =, the agle of rotatio i irreleat becaue the ellipe reduce to a circle. A graph of the ellipe how thi iformatio (Fig. ). The ditace gie b Eq. (7) i ued i determiig whether a object ha maeuered, or chaged coure while the Kalma filter i ruig. A certai tolerace leel (uuall σ or 3σ) i built ito a implemetatio of the filter. If the ditace betwee the meaured ad predicted poitio eceed the tolerace leel durig a et umber of cocurret timetep, the Kalma filter mut be reiitialized uig the Fig. : Coure of a maeuerig target with uperimpoed error ellipe ad Kalma predicted poitio lat two meauremet. Fig. how a graph of a maeuerig object with error ellipe ad predictio uperimpoed o the object coure. [6-6]

7 Etimatig iitial alue for the tate coariace matri i oe of the mot difficult part of ruig the Kalma filter algorithm. Reaoable alue are choe for each elemet i the iitial tate coariace matri P ( ) baed o what i kow about the tem. The are updated b the followig equatio: P ( k + k) = ΦP( k k) Φ T + Q (8) The matri will gie a better predictio of error a the algorithm goe through more iteratio. Update A the update tage begi, time k become k +. The meauremet reidual k, the differece betwee the actual meaured poitio ad the predicted poitio, i gie b: ad the coariace of ( k) i: ( k) = ( k) Hˆ ( k k ), (9) ( ( )) = HP( k k ) H T + R S = co k. () The reidual coariace matri i imilar to the tate coariace matri, ecept that it ol accout for poitio ad it meaure coariace betwee predicted ad meaured tate rather tha betwee predicted ad actual tate. Whe updatig tate ector etimate, the Kalma filter iclude a weightig factor kow a the Kalma gai matri, gie b: T ( ) = P( k k ) H S The tate ector etimate i the updated b the equatio: K k. () ( k k) = ˆ ( k k ) K( k) ( k) ˆ +. () Combiig Eq. () ad Eq. () how ituitiel that the greater the coariace matri S i, the le the Kalma gai matri i. Accordigl, the differece ( k) betwee meaured ad predicted tate i weighted le whe added to the ew tate ector predictio becaue there i a much higher poibilit of error, epeciall from meauremet oie. The tate coariace matri i updated b: ( k k) = ( I K( k) H) P( k k ) P. (3) ( ) [6-7]

8 Polar Traformatio Although the two dimeioal Kalma filter require meauremet to be i Carteia coordiate, radar tem meaure object poitio uig polar coordiate. To accommodate r thi problem, the meauremet ector ( k) = mut be coerted uig the traformatio: θ = r coθ. = r iθ Becaue the matrice Φ, H, Q, ad P do ot iole traformatio from polar to Carteia coordiate, the do ot chage from the form lited aboe. Howeer, becaue R meaure the coariace of meauremet error, which i gie i polar coordiate, a ew R i required. eauremet oie i ad poitio ca be etimated b takig differetial of the aboe traformatio: σ d = d ( r coθ ) = r d( coθ ) = r iθ dθ coθ dr rσ iθ σ coθ θ r + coθ dr σ d = d ( r iθ ) = r d( iθ ) = r coθ dθ + iθ dr rσ coθ + σ iθ θ r + iθ dr DEVELOPENT OF THE PROGRA Program Backgroud R σ ε σ ε σ σ σ. ε ε = ε σ ε The purpoe of the program wa to proide a geeral implemetatio of the Kalma filter. Iitiall, the program wa er imple ad ol worked with Cae. It wa etirel liear ad had o fleibilit. The ecod erio of our program wa impl a cop of the firt that wa modified to work with Cae. Thi program, too, wa er hard to modif. We rewrote the third erio of the program from cratch, i a attempt to deal with the modificatio iue. The code that ra the filter wa eparated from the code that wa ioled i the iitializatio ad put ito it ow fuctio. Although thi made it eaier to modif the code for Cae 3, mot of the code wa uable to be eail reued. It wa at thi poit that we witched to object orieted programmig tle. [6-8]

9 Breakig up the Program Itead of haig large chuk of u-reuable code, we broke dow each tak ito a et of related fuctio ad data, called clae. All of thee code egmet were eail reued, modified, ad eteded. The code became er modular, ad we dicoered that we could get all of the cae ito the ame program with little etra effort. The time eeded to add additioal cae alo dropped. The program wa diided up ito a umber of clae. The mai part were the cotrol loop; the KFilter cla, which hadled the filter operatio; the Startup module, which hadled iitializatio; the DataIterator, which read the file ad tore the data; ad atlib ad other utilit fuctio. Structure ad Clae The mot baic tructure ued i the program, called a Datum, tore a poitio ector ad the time. It i paed aroud betwee mot of the clae i the program. DataIterator i a iterface that proide two baic method for acceig data from a arbitrar iput ource. Thee two method are hanet() ad etdatum(). hanet() idicate if there i till more data. etdatum() gie the ew piece of iformatio to the callig fuctio if ew data eit. Thi iformatio i tored i a Datum tructure. DataIterator ha eeral implemetig clae that perform ariou operatio o the data before the are paed to the filter. The mot baic of thee clae i the FileReader. It impl read the data i Carteia coordiate (which ca cotai a umber of dimeio) ad place them ito the ector, alog with the time. The PolarFileReader cla eted the capabilitie of the FileReader cla. It read polar coordiate from the iput ource ad coert them ito Carteia o that the filter ca work with them. Ulike the FileReader, it ca ol accept two dimeio. The lat implemetatio of DataIterator i the PolarultiReader. It eted the capabilitie of the PolarFileReader cla b upportig iput data from a arbitrar umber of radar. The cla i iitialized with the coordiate of each radar, ad data i coerted to rectagular coordiate baed o the coordiate of the curret radar. The KFilter cla perform the filter mai calculatio b carrig out the filter operatio. Thee operatio ca be broke up ito two phae: predict ad update. After beig iitialized with the error coariace ad tate matrice, the filter predict method ca be called (which carrie out the predict tage of the filter). Time i paed a a argumet, ad the filter predict the et tate of the target baed o the time iteral. All data i tored withi the object itatiatio. Oce the predict tage i fiihed, the update tage commece. Thi ioled callig the update method of the KFilter itatiatio ad paig i the meauremet ector (which i retured b a DataIterator). The update method carrie out the update tage of the filter. The KFilter cla alo cotai a acceor method called getx(). Thi allow the curret tate of the filter to be obtaied. Latl, there i a reet() method that i ued whe the tate ad tate coariace matrice eed to be reet. [6-9]

10 The CommaWriter cla tore the reult of the filter ito a comma delimited tet file. Thi format wa choe becaue it i eail opeed i icrooft Ecel. CommaWriter i iitialized with the ame of the output file ad the umber of dimeio. Durig iitializatio, it write the appropriate header to the file. CommaWriter alo cotai the writelie() method which take a time ad ector a argumet, ad write them to the output file. Whe the output i fiihed, the cloe() method i called, which cloe the file. Additioal Fuctio The mai ub combie the fuctio of all of thee clae ito a coheret program. It begi b akig the uer which cae to ru. At thi poit, it perform cae-pecific iitializatio of the error coariace matrice ad other ariable. It alo et the correct iput ad output file, ad itatiate the correct implemetatio of DataIterator. It the begi the mai loop, which read a Datum from the DataIterator, call predict() with the time from the Datum, update the filter with the meauremet ector, ad write the output through the CommaWriter. Whe o more data i preet, the program eit. The iitialize method geerate the iitial tate ector. It i gie the firt two et of coordiate from the DataIterator, ad it retur a ector with the lat poitio ad the aerage elocit. Alo heail ued i the program i the atlib [5] librar. Thi librar cotai fuctio that ca perform baic matri arithmetic. The ol difficult i that equatio eed to be coerted ito prefi otatio (a oppoed to ifi otatio) to work with the matri librar. I geeral, thi iole lookig at the equatio ad recuriel goig through the order of operatio backward. All of the fuctio are ued heail i the KFilter ad iitializatio fuctio. I additio, a umber of fuctio that geerate matrice that are dimeio depedet were writte o that we would ot eed to hard-code the alue i for each cae. Fiall, oe of the cae require a fuctio to determie how ma tadard deiatio from the predictio the meauremet i. Thi i doe with the Sig() fuctio. Gie the (predictio), (meauremet), ad p (error coariace) matrice, it calculate (σ). Each cae require certai aumptio to ru properl. For all of the cae, we were gie the error of the radar ad the aumed driig oie. Thee alue were hard-coded ito the program. Each cae alo required certai adjutmet that allowed it to udertad the propertie of the data read from the file. Therefore, for each cae, we hard-coded the pecific propertie that were ecear. Puttig the Program Together To implemet the Kalma filter, we eeded a wa to iitialize mot of the matrice. Baed o ome guework, we were able to hard code mot of the alue ito the program. Oce the coariace matrice were iitialized, we were till left with the problem of obtaiig the iitial tate. The iitialize() fuctio doe thi b lookig at the firt two data poit that the program receie, ad ue them to make liear etimate of the poitio ad elocit. Thi [6-]

11 iitial tate ered a a platform o which to bae future tate etimate. Oe lat problem that we ra ito wa that the time iteral betwee data poit were ot cotat, although the were er cloe. Thi mattered whe we were calculatig the alue of Φ. To fid the chage i poitio, we eeded to kow the chage i time. To compeate for thi, we created a fuctio that retured the phi matri baed o the curret time iteral. Oce thi wa completed, we the moed o to ruig the actual algorithm. For each data poit read from the file, we predicted what the tate at the et time iteral would be. After thi, we recalculated the tate coariace ad moed o to readig the et actual data poit from the file. Oce we had the data poit, we updated the Kalma gai matri, the tate etimate, ad the tate coariace. The tate etimate, alog with the time, wa prited to file. The program wa the read to repeat the proce of readig the time iteral, predictig, ad correctig. SCENARIOS Cae Decriptio: Cae wa the implet of the problem we were gie. It aumed that the target wa a plae that wa flig directl oer the radar i oe dimeio. The plae meaured ditace from the radar wa gie to u at each timetep. Programmig Chage: Cae wa er baic. Therefore, we ol had to et the umber of dimeio to oe ad make ure that the DataIterator wa a FileReader. Reult: Figure 3 how two reidual (differece) graph. Oe how the reidual betwee the meaured poitio ad the actual poitio at each timetep, ad the other how the reidual betwee the predicted poitio ad the actual poitio of the target at each timetep. The poitio that were predicted b the Kalma filter were much cloer to the actual poit at almot eer poit. The radar-meaured poit had a aerage percet error of 4.4%, while the filterpredicted poit had a aerage percet error of.3%. I additio, the graph how how the performace of the Kalma filter improe oer time: the predicted poit get cloer to the actual poit, ad fewer predictio are er far off. Thi how the adatage of uig the Kalma gai matri i the algorithm, which decreae the ifluece of ew meauremet a the algorithm gai cofidece ad our etimate become cloer to the actual data. Figure 4 agai how how the performace of the Kalma filter improe oer time. While the predicted elocit origiall differed from the actual elocit b oer mph, it quickl corrected itelf to get cloer to the real alue. B the ed of the meauremet time the predicted elocit wa er cloe to the actual elocit, which i how b the lie approachig the ai. Though the iitial predictio were off, the filter adapted ad corrected the mitake after it read ome more accurate poit. Cae [6-]

12 Decriptio: Cae ioled a plae moig i two dimeio that paed b the radar. It wa moig i a traight path at cotat elocit. Programmig Chage: Reult: Cae wa idetical to Cae ecept that the umber of dimeio wa et to. I Figure 5, the rage (ditace from the radar) ad meaured reidual are compared. Becaue of oie, the radar meauremet were quite ditorted. Howeer, the predicted reidual were cloer to the actual poitio. The filter had a error of.99% i the directio ad.3% i the directio, while the radar aloe had a error of 3.% i the directio ad.9% i the directio. The filter i till quite powerful i two dimeio. Figure 6 how the reidual betwee the predicted elocit ad the actual elocit. I the begiig, gie ol a few poit, the filter oce agai howed a relatiel high error. Howeer, oce more poit came i ad time progreed, it become eidet that the filter become more ad more accurate i it predictio. Thi oce agai how the adatage of the Kalma filter oer time, ee whe aalzig data i two dimeio. Cae 3 Decriptio: Cae 3 ioled a target moig i two dimeio that wa tracked b a igle radar. Coordiate from the radar were i polar form. The rage repreeted the target ditace from the radar, ad the agle repreeted the target compa headig. Programmig Chage: The umber of dimeio remaied at. A PolarFileReader wa ued i place of a FileReader. The R matri had to be updated at each timetep betwee the predict() ad update() method accordig to the coerio of coariace matrice from polar to Carteia. Thi accouted for the chagig meauremet coariace matri. Reult: Figure 7 how how the Kalma filter help to improe the meauremet ad brig them cloer to the actual alue. The performace, agai improed oer time. The tartig coditio for thi cae were le tha ideal, a how b the firt few predicted poit, but b the ed of the target coure, the predicted rage wa much cloer to the actual rage tha wa the meaured rage. The meaured rage had a aerage percet error of 6.36%, while the predicted rage had a aerage percet error of 4.5%. [6-]

13 The elocit reidual for Cae 3 i how i Figure 8. A with preiou elocit reidual graph, the iitial elocit reidual wa er far from. Thi wa due to low elocit predictio for the iitial coditio. Howeer, the radar wa able to adjut to the coditio to gie fairl accurate predictio of the elocitie b the fial time tep. The predict ad update algorithm of the Kalma filter work well whe uig polar coordiate that are atie to radar tem. Cae 4 Decriptio: I cae 4, two radar tatio moitored the target, with ol oe recordig data at a gie poit i time. The meauremet were gie a polar coordiate relatie to the actie radar. Programmig Chage: Cae 4 wa idetical to Cae 3 ecept that it ued a PolarultiReader itead of a PolarFileReader. Reult: The oie aociated with two radar did ot throw the filter off. The predicted poitio remaied much better tha the meaured poitio (ee Figure 9). After the firt few poit, the filter became fairl accurate. It i oteworth that betwee 6 ad 6.5 miute, whe the radar witched, the filter wa ol off b.5 mile. A i the preiou 3 cae, the iitial elocit etimate were er iaccurate. Simpl aeragig the firt two poit did ot proide a accurate predictio. A the time moed o, howeer, the filter adapted ad corrected the error (ee Figure ). Cae 5 Decriptio: I Cae 5, we were attemptig to track a UFO that wa maeuerig to elude our radar. The UFO chaged elocit twice. Coordiate were gie i polar form. Programmig Chage: Cae 5 wa er imilar to Cae 3, ecept we eeded a wa to let the filter kow whe the target maeuered. I order for the filter to recogize the chage i elocit, it had to check the accurac of it model at each timetep. To do o, it checked how ma tadard deiatio the meauremet wa from the predictio baed o the error coariace matrice. Thi wa doe i the Sig() fuctio. If the meauremet wa at leat four tadard deiatio from the predictio for three cocurret timetep, the filter wa reet b callig the filter reet() method. Thi reiitialized the tate coariace matri, P, ad reet the tate to the preiou meauremet. [6-3]

14 Reult: Figure how that the predicted poitio were lightl wore after each tur, becaue the filter belieed that the object wa moig i a traight path. Howeer, oce the error wa too big, the filter reet ad wa able to make more accurate predictio. The elocit graph (Figure ) till follow the tred of fairl iaccurate iitial predictio. The predicted elocit approached the true elocit util the object tured, cauig a large error elocit reidual. The filter reet with aother fairl iaccurate predictio but oce agai approached the true elocit. The fial tur caued the ame problem but at a maller cale. 3 Predicted Reidual eaured Reidual Rage Reidual (mi) Time (mi.) Fig. 3: Rage Reidual for Cae [6-4]

15 4 Predicted Velocit Reidual Reidual Velocit (mi/hr) Time (mi) Fig. 4: Velocit reidual i Cae Rage Reidual(mi) Predicted Reidual eaured Reidual Time (mi) Fig. 5: Rage reidual i Cae [6-5]

16 Reidual Velocit (mi/hr) Predicted Reidual Time (mi) Fig. 6: Velocit reidual i Cae.5.5 Rage Reidual (mi) Time (mi) Fig. 7: Rage reidual i Cae 3 Predicted Reidual eaured Reidual [6-6]

17 Reidual Velocit (mi/h) Velocit agitude Reidual - Time (mi) Fig. 8: Velocit reidual i Cae 3.5 Predicted Reidual eaured Reidual.5 Rage Reidual (mi) Time (mi) Fig. 9: Rage reidual i Cae 4 [6-7]

18 Predicted Velocit Reidual 8 6 Reidual Velocit (mi/hr) Time (mi) Fig. : Velocit reidual i Cae Rage Y (mi) Rage X (mi) Fig. : Trajector i Cae 5 Predicted Trajector eaured Trajector Actual Trajector [6-8]

19 4 Predicted Reidual 3 Reidual Velocit (mi/hr) Time (mi) Fig. : Velocit reidual i Cae CONCLUSION The Kalma filter ue liear algebra to predict the poitio ad elocit of a target. Baed o the predictio ad the radar meauremet, the filter i able to correct error. The filter iteral correctio method ca be ued to adapt the filter to figure out whe a target i maeuerig b reettig the parameter ad begiig the predictio from a ew poit, a wa ee i Cae 5. Utilizig a powerful matri librar, the relatiel imple fuctio of the Kalma filter were eail adapted to create a Viual Baic.NET applicatio. Becaue radar aloe proide abmal etimate of poitio ad caot directl meaure elocit at all, a method for accuratel determiig both i eeded. Baed o our reult, the Kalma filter i adept at both. B aumig a ormal ditributio of error, it i able to quickl ad efficietl correct meauremet error ad geerate elocitie without torig large amout of data ad performig legth computatio. [6-9]

20 REFERENCES [] [IEEE] Ititute of Electrical ad Electroic Egieer. 3 Ja 3. Rudolf E. Kalma. IEEE hitor ceter. < html> Acceed 5 Jul. [] Abolute atroom. 5 Jul 7. Kalma filter. < ecclopedia/k/ka/kalma_filter.htm> Acceed 5 Jul 8. [3] Simo, Da. Jue. Kalma filterig. < Article.jhtml?articleID=9968> Acceed 5 Jul 8. [4] Blackma, Samuel S ultiple-target trackig with radar applicatio. Artech Houe, Ic. Norwood, A. [5] Aa SA. 3 Ja 8. atri operatio librar.net. < com/b/cript/showcode.ap?ttcodeid=97&lgwid=> Acceed 5 Jul. [6-]