Pursuit-Evasion with Acceleration, Sensing Limitation, and Electronic Counter Measures

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1 Cleson Uniersiy TigerPrins All Theses Theses 8-7 Pursui-Esion wih Accelerion, Sensing Liiion, nd Elecronic Couner Mesures Jing-en Png Cleson Uniersiy, Follow his nd ddiionl works : hps://igerprins.cleson.edu/ll_heses Pr of he Elecricl nd Copuer Engineering Coons Recoended Ciion Png, Jing-en, "Pursui-Esion wih Accelerion, Sensing Liiion, nd Elecronic Couner Mesures" 7. All Theses. 64. hps://igerprins.cleson.edu/ll_heses/64 This Thesis is brough o you for free nd open ccess by he Theses TigerPrins. I hs been cceped for inclusion in All Theses by n uhorized dinisror of TigerPrins. For ore inforion, plese conc kokeefe@cleson.edu.

2 PURSUIT-EVASION WITH ACCELERATION, SENSING LIMITATION, AND ELECTRONIC COUNTER MEASURES A Thesis Presened o he Grdue School of Cleson Uniersiy In Pril Fulfillen of he Requireens for he Degree Mser of Science Elecricl Engineering by Jing-En Png Augus 7 Acceped by: Richrd Brooks, Coiee Chir In Wlker Ad Hooer i

3 ABSTRACT The use of ge heory o nlyze he opil behiors of boh pursuers nd eders origined wih Isc s work he Rnd Corporion in he 95 s. Alhough ny riions of his proble he been considered, published work o de is liied o he cse where boh plyers he consn elociies. In his hesis, we exend preious work by llowing plyers o ccelere. Anlysis of his new proble using Newon s lws iposes n ddiionl consrin o he syse, which is he relionship beween plyers elociies nd llowed urning rdius. We find h nlysis of his relionship proides new insigh ino he eder cpure crieri for he consn elociy cse. We surize our resuls in flow chr h expresses he preer lues h deerine boh he ges of kind nd ges of degree ssocied wih his proble. Pursuiesion ges in he lierure ypiclly eiher ssue boh plyers he perfec knowledge of he opponen s posiion, or use priiie sensing odels. These unrelisiclly skew he proble in for of he pursuer who need only inin fser elociy ll urning rdii. In rel life, n eder usully escpes when he pursuer no longer knows he eder s posiion. We nlyze he pursui-esion proble using relisic sensor odel nd inforion heory o copue ge heoreic pyoff rices. Our resuls show h his proble cn be odeled s wo-person bi-rix ge. This ge hs sddle poin when he eder uses sregies h exploi sensor liiions, while he pursuer relies on sregies h ignore sensing liiions. Ler we consider for he firs ie he effec of ny ypes of elecronic couner esures ECM on pursui esion ges. The eder s decision o iniie is ECM is odeled s funcion ii

4 of he disnce beween he plyers. Siulions show how o find opil sregies for ECM use when iniil condiions re known. We lso discuss he effecieness of differen ECM echnologies in pursui-esion ges. Keywords: Pursui-Esion, ge heory, inforion heory, Elecronic Couner Mesures. iii

5 ACKNOWLEDGMENTS Firs, I would like o hnk y fily who proides e he opporuniy o sudy in US. Wihou he, I would no be where I ody. Second, I wish o hnk Dr. Brooks for oiing e, nd proiding e wih suppor nd guidnce during he ps wo yers. And I would lso like o hnk Dr. Wlker nd Dr. Hooer for king he ie fro heir busy schedules o reiew y hesis. Your effor is grely pprecied. Third, I would like o hnk ll y friends for heir consn suppor nd lwys being here for e. This eril is bsed upon work suppored by, or in pr by, he U. S. Ary Reserch Lborory nd he U. S. Ary Reserch Office under conrc/grn nuber W9NF i

6 TABLE OF CONTENTS Pge TITLE PAGE...i ABSTRACT...ii ACKNOWLEDGMENTS...i LIST OF TABLES...ii LIST OF FIGURES...iii CHAPTER I. INTRODUCTION... II. PURSUIT-EVASION BACKGROUND AND RELATED RESEARCH...5 III. MODELS FROM PHYSICS AND MATHEMATICS Vehicle Dynics Finding n Opil Ph o Poin on he Plne Finding he Mxiu Region for Vehicle Finding he Approxie Fron Boundry...6 IV. PURSUIT-EVASION WITH ACCELERATION...3 V. PURSUIT-EVASION WITH SENSING LIMITATION Sensor Models Inforion Theory Bsed Uiliy Funcion Sregy for Ges wih Perfec Inforion Experienl Resuls...5 VI. PURSUIT-EVASION WITH ELECTRONIC COUNTER MEASURES Effec of Elecronic Couner Mesures Experienl Resuls...58 VII. CONCLUSION AND FUTURE RESEARCH...67

7 Tble of Conens Coninued APPENDICES...73 A: Circle Inolue Bckground...74 B: Anlysis of Non-Criicl Ph Opil Phs...75 REFERENCES...77 Pge i

8 LIST OF TABLES Tble Pge 5. Four possible oucoes of he rge deecion rndo process Men, rince, sndrd deiion, upper nd lower 95% confiden inerl bounds for P s biliy o cpure E in four differen scenrios Cpure re for ech cobinion of sregy wih differen iniil condiions...56 ii

9 LIST OF FIGURES Figure Pge. The ehicle cnno urn ino he circulr region defined by is iniu urning rdius MTR Geoery of he ehicle dynics in he bsolue world coordine syse Geoery of criicl ph of he ehicle wih elociy no less hn s The ehicle reches poin W boe I-Q-Q fro poin I hrough T cse The ehicle reches poin W beween Q-Q nd Q-Q fro poin I cse The ehicle reches poin W beween Circle M, R nd I-Q-Q cse The circle inolue describes he cure rced by sring of gien lengh when wrpped ono circle. When he ehicle elociy is less hn s is oion is consrined only by is iniu urning rdius R, nd he cure describing he region MR when is less hn T s is defined by circle inolue. This gies he xiu region for he ehicle unil i reches s Unil ie T s, region MR is defined by he circle inolue. Afer T s o rech poins boe he line ngen o he iniu urning rdius circle he poin where i inersecs he circle inolue, he ehicle siply oes in srigh line The xiu region he ehicle cn rech by ie T The geoery of circle inolue for pursuer or eder The geoery of he iniil condiion of pursui-esion ge...8 iii

10 Lis of Figures Coninued Figure Pge 3. Cures show he possible posiions ie for pursuer Blue, boo nd eder Red, righ, boh sring fro he posiions shown by crosses ie. The doed line is Blue s effecie sensing rnge P s cpure region solid coers E s escpe region dsh. P cpures E P s cpure region solid doesn coer E s escpe region dsh. E escpes Clssificion of rdr redings P enropy sregy s. E enropy sregy. Lef: P found E; Righ: P los E P enropy sregy s. E perfec sensor sregy. Lef: P found E; Righ: P los E P perfec sensor sregy s. E enropy sregy. Lef: P found E; Righ: P los E P nd E boh perfec sensor sregy. P found E Resuls for he firs siulion using he noinl FP re Resuls for he second siulion using he noinl TP re Siulion resuls for represenie cobinions of TP nd FP res Geoery explins why sring disnce 85 hs significn low cpure re Siulion resuls for ECM where P nd E use perfec inforion sregies A flow chr h surizes he ges of kind for pursui-esion ges wih ccelerion...68 ix

11 Lis of Figures Coninued Figure Pge A- The circle inolue bold cure is fored by king line segen of fixed lengh nd wrpping i round circle of fixed rdius. Angle IOM, Line IO = R B- Use differen ccelerion o rech ngen poin T nd go srigh o poin W...75 x

12 CHAPTER ONE INTRODUCTION Consider he proble of one ehicle eping o oerke nd cpure ehicle belonging o n opponen. In ddiion o being he cenrl eleen of ny Hollywood oie scrips, his proble hs ny prcicl pplicions, for exple in he lw enforceen nd iliry doins. This proble is coonly known s pursuiesion ge; recen sureys of reled reserch re in [] nd [6]. Pursui-esion ges were originlly posed in he 95 s s he Hoicidl Chuffeur Proble in series of Rnd echnicl repors [9]. In h ersion of he proble, slow pedesrin eder E, h cn chnge direcion will, eps o oid being run oer by fs cr drien by hoicidl pursuer P. E nd P rel wih consn speeds e nd p, respeciely. P s iniu urning rdius, i.e. he ighes urn possible for he ehicle due o is seering echnis see Figure., is R p. Figure.: The ehicle cnno urn ino he circulr region defined by is iniu urning rdius. Depending on = e / p <, P cpures E when: R sin. p If his inequliy is reersed, E escpes fro P [9].

13 In 967, Cockyne [6] deried wo necessry nd sufficien condiions for P o be ble o cpure E. Using he noion inroduced in his chper, he condiions re: p > e, nd. p / R p e / R e.3 Noe h when elociies re consn nd he errin unifor, if cpure is possible, wih enough ie i is sypoiclly cerin. Anlyses of pursui-esion ges ypiclly ssue he pursuer hs fser elociy nd he eder hs sller iniu urning rdius. Bu hese odels re no esily pplied o os relisic siuions, since hey ssue he elociies of boh plyers re consn. This ignores he fc h ech ehicle s biliy o neuer esiely during pursui is lrgely consrined by is biliy o ccelere nd/or decelere. In ddiion, s nyone who hs drien n uoobile cn es o, he xiu sfe speed ehicle cn inin is funcion of is urning rdius. In his hesis, we sr wih he rin of he pursui esion ge: pursuer in n uoobile ries o cpure n inelligen eder in n uoobile where boh ehicles he liied ccelerion nd urning biliy []. In his work, we find he condiions h deerine he ges of kind he soluion is winning sregy for one of he plyers bu no he ges of degree he soluion is coninuous lue, ex. ie o cpure. This work lso shows how he resuls of our nlysis proide new insigh ino he physicl ening of he cpure crieri gien in [6]. In his hesis, we odified his proble by hing he pursuer nd he eder boh rely on sensors o deerine he locion of heir opponens [3]. In h conex, he

14 eder escpes when he pursuer no longer hs cionble inforion bou he eder s curren posiion. Our sensor odel considers he decy of he rge s signure oer disnce. The sensor deecion process in he odel lso hs ype I flse negie nd ype II flse posiie errors. Inforion heoreic nlysis of he sensor odel proides uiliy funcion h he eder uses o find is opil escpe sregy. Siulions confired h, by considering he pursuer s sensing liiions, he eder cn grely enhnce is biliy o escpe. In his hesis, we llow he eder o use Elecronic Couner Mesures ECM o odify he rue posiie nd flse posiie error res of he pursuer s rdr. In odern cob, elecronic couner esures ply n iporn role in his process. A nuber of elecronic couner esures exis where one pricipn cn inenionlly disrup he sensing cpbiliy of heir opponen. Coon couneresure echnologies include: Chff used o rigger lrge nuber of flse posiies. Aerosols used o enue signls nd lower he rue posiie re. Decepion nd blip enhnceen odifies he redings reurned by sensor o skew redings, increse heir rince, or odify rge clssificion. Flres nd decoys iic rge signures o produce uliple rcks, only one of which belongs o he rel rge. The res of his hesis is orgnized s follows: Chper Two reiews preious pursui-esion ge reserches; In Chper Three, reiew of ehicle dynics, criicl ph, xiu region, nd fron boundry re proided; Chper Four explins our pursui-esion ge wih ccelerion; Chper Fie explins our pursui-esion ge 3

15 wih sensor; Chper Six explins our pursui-esion ge wih elecronic couner esures; Chper Seen gies conclusions nd describes res for fuure reserch. 4

16 CHAPTER TWO PURSUIT-EVASION BACKGROUND AND RELATED RESEARCH In his Chper we reiew reled pursui-esion ges h he been presened in he lierure. The pursui-esion proble ws originlly clled he Hoicidl Chuffeur Proble in [9]. A ore gile bu slower eder E ries o oid being run oer by fser pursuer P. P s oion is consrined by iniu urning rdius R. E hs no iniu urning rdius nd cn chnge direcion will. A ore syeric rin of his proble, where boh P nd E drie crs wih fixed speeds nd respecie urning rdii R p nd R e, is presened in Merz s 97 PhD disserion []. He fully soled he ges of kind nd degree for his proble []. He lso considered he proble of finding which plyer, gien specific iniil condiions, is beer posiioned o be he pursuer [3]. This nlysis is ery useful for deerining eril cob sregies s shown in [] nd [4]. Vehicles h oe in 3-diensions he six degrees of freedo. In ddiion o oeen in he x, y, nd z direcions, here re he following degrees of freedo reled o heir orienion: pich, roll nd yw. If we consider ehicle cenered coordine syse where he y-xis is he ehicle s longiudinl fro il o nose xis, he x-xis is he horizonl perpendiculr o he y-xis prllel o he wings, nd he z-xis is perpendiculr o he xy plne. Then, he ehicle pich is roion bou he x-xis, roll is roion bou he y-xis nd yw is roion bou he z-xis. Mos conenionl ircrf nd high speed issiles he liied yw res nd herefore A ge of kind is quliie ge wih fixed se of oucoes; ypiclly he oucoe is siply which plyer wins. A ge of degree is quniie ge h hs n infinie nuber of possible oucoes, e.g. he ie o cpure. 5

17 perfor bnk-o-urn neuers o oid using he yw degree of freedo. The resuls in [4] use his insigh o show how he 3-D proble reduces o he -D proble for hese clsses of ircrf. They show how o clcule criicl lues for he difference beween he roll res of he pursuer nd eder. If he pursuer s roll re is no sufficienly lrger hn he eder s, hen he eder will escpe. If he pursuer s roll re is sufficienly greer, he pursuer hs opil neuers h llow i o consrin he eder o neuers wihin he se xy plne. In which cse, s long s i is sfe o ssue h he xiu llowble elociies for he ehicles re liied by he cenrifugl force h hey experience, hen our resuls should hold for 3-D probles s well. Noe h our resuls should pply o cob ircrf where neuerbiliy is liied by he oun of force he pilo cn susin wihou losing consciousness. One recen pursui-esion rin is he herding dog nd sheep proble, in which one dog eps o seer ny sheep o gien locion. The deerinisic cse of his proble cn be soled using dynic progring []. A sochsic rin is considered in [3]. Th work exends he pursui-esion proble by llowing uliple eders, bu i doesn consider soe rel world fcors such s rying speed, urning rdius, ehicle rolloer, ccelerion, ec. Mny on ny ges re considered in [3]. A e of unnned eril nd ground ehicles pursues e of eders in n unknown errin. They inegre uonoous gens wih heerogeneous cpbiliies ino n inelligen dpie syse nd propose disribued hierrchicl hybrid syse rchiecure h ephsizes he uonoy of ech gen nd llows for coordined e effors. 6

18 Very lile work hs been done on pursui esion probles wih iperfec inforion. Soe reserchers he considered pursui-esion ges in clsses of errins where he pursuer s field of iew is liied. A ph-plnning lgorih in [6] gurnees h ll eders will eenully be deeced in n enironen wih occlusions defined by rbirry cures. This exends he work in [5]. Occluded isions in polygonl enironens were considered in []. In h work, rndoized pursuer sregies were nlyzed. The only ECM reled pursui esion reserch o our knowledge is Ph.D. disserion [7] where he eder uses decoys o confuse he pursuer. To he bes of our knowledge his is he firs reserch o consider he clsses of ECM probles h we he used. 7

19 CHAPTER THREE MODELS FROM PHYSICS AND MATHEMATICS 3. Vehicle Dynics In our ge, he pursuing ehicle P chses he eding ehicle E. Boh ehicles follow Newon s lws for circulr oion long circle of rdius R, where R is liied fro below by R p nd R e, respeciely, s in Figure.. Our pproch deerines he noinl ph h ehicle should follow during pursui-esion ge. Mny probles wih rel ehicle oion, such s wheel slippge, re no explicily expressed in hese equions. Wheel slippge is ouside he scope of his hesis, since i is recion o n unforeseen een nd no soehing h ffecs he ehicle s pursui-esion sregy. I would be riil o exend our pproch o hndle slippge nd reled issues, by using our soluions s noinl conrol signl for drieby-wire feedbck conroller, like he one in [3] h senses he ehicle s inercions wih is enironen. Anoher proble wih ehicle oion is roll oer. As ehicle urns, cenripel force is genered on is cener of ss. This force in urn generes orque h y cuse he ehicle o rolloer. The srengh of his orque is esily prediced using Newon s second lw. The cenripel force on ehicle wih ss M oing wih urning rdius R nd elociy hs gniude F roll = M /R. The ehicle will roll oer if he cenripel force exceeds hreshold lue F roll h depends on is suspension nd design. The ehicle will rein sble when /R F roll /M. Since F roll nd M re properies of he ehicle, his consrin becoes: 8

20 R K roll F roll M 3. Noe h fcor K roll dds physicl insigh o he cpure condiion fro Cockyne [6]; ehicles ble o urn higher elociies he n dnge. Fro Equion 3., we derie he ehicle s sfe elociy s. A ehicle reling elociy slower hn s cn urn wih ny urning rdius greer hn or equl o is iniu urning rdius. If he iniu urning rdius is R nd he xiu elociy is, hen: s roll in, K R 3. We lso derie he sfe urning rdius R s. The ehicle cn rel ny elociy less hn is xiu elociy wih he urning rdius greer hn or equl o R s, i.e.: R s roll x R, K 3.3 Gien he curren elociy c, we derie he curren llowed iniu urning rdius R c. The ehicle cn urn wih ny urning rdius greer hn or equl o is R c, i.e.: R c c roll x R, K 3.4 Ech plyer hs wo conrol ribles, u nd. The use of conrol rible u, where u =R c /R, is described in [9]. The ehicle rels on circle of rdius R = R c /u, where u rnges fro - o. This les he plyer choose he insnneous urning rdius nd lso llows he ehicle o oe in srigh line i.e. u =, wihou explicily deling wih n infinie urning rdius. The oher conrol rible,, is he insnneous ccelerion ny poin in ie, which is bounded fro boe by ehicle dependen xiu ccelerion consn,. The insnneous ehicle elociy, which is lso bounded fro boe by ehicle dependen consn, cn be 9

21 clculed ny poin in ie fro he ehicle s iniil elociy nd he hisory of lues. These equions of oion fro [5] express Newon s equions for circulr oion wih ccelerion in ers of he wo conrol ribles see Figure 3.: x y R c sin c cos c u Rc x R, c c c d K roll 3.5 where x nd y represen he posiion wih respec o, ; represens he ehicles orienion wih respec o Y-xis; is he iniil elociy nd K roll is he rolloer consn. Figure 3.: Geoery of he ehicle dynics in he bsolue world coordine syse. 3. Finding n Opil Ph o Poin on he Plne In his chper, we define he opil ph o ny poin W in he plne s he ph h kes he ehicle fro is iniil posiion o W in he inil oun of ie. If he

22 ehicle cn rech W wihou urning, h ph is srigh line nd is biliy o rech W is liied only by is xiu elociy nd ccelerion. If W is on he curren llowed iniu urning circle wih rdius R c defined by Equion 3.4, he opil ph is n rc on h circle. If W is inside he ehicle s curren llowed iniu urning circle defined by R c, i is eporrily unrechble. The opil ph o hese poins is ore difficul o deerine, since he ehicle us rech posiion where W is ouside h circle see [9] for deiled nlysis of his proble wih he hoicidl chuffeur consrins. For he res of his Chper, we discuss only phs o poins o he righ of he ehicle. Equilen phs for poins o he lef cn be found by syery. For soe poins, finding he opil ph is srighforwrd. Le be he iniil elociy. If < s Equion 3., unil he ehicle reches s i cn ccelere wih he xiu re nd urn wih ny urning rdius greer hn or equl o R. For ny poin W ouside he urning circle wih rdius R, here is line ngen o he circle h psses hrough W. The ehicle rels firs long he iniu urning circle. If he ehicle reches he poin where he ngen line inersecs he circle before i reches s, he opil ph follows he circle wih rdius R o he ngen line nd hen follows he ngen line o W. While on his ph, he ehicle cceleres wih unil i reches. Once i reches, he ehicle us sop ccelering. This ph is clerly opil, since i is he shores disnce he ehicle cn follow beween he wo poins nd he ehicle is oing s quickly s possible ll poins on he ph. There is region in fron of he ehicle h conins ll hese poins.

23 The res of Chper 3 explins how o find he boundries of regions conining poins whose opil phs re found using siilr ehods. By finding he phs h re boundries beween regions, we deerine how o find he opil ph o ny gien poin. Since he opil ph fro ny iniil posiion o he poin where he ehicle reches s is clerly defined, we ignore his ph segen for he res of he discussion nd ssue h he ehicle srs fro posiion wih elociy s. We lso express circles s Circle cener, rdius for he res of he discussion. If he ehicle reches s before reching he ngen poin on is iniu urning circle, fro h poin on he urning rdius nd elociy re uully consrined by Equion 3.. Fro Equion 3.4, gien n insnneous elociy c, he llowed iniu urning rdius is R c = c /K roll R. We now consider he ph where he ehicle cceleres wih unil reching while urning s ighly s possible i.e., u= wihou rolling oer. In Figure 3., he ehicle srs poin I wih iniil elociy = s direced long he rrow. I urns round CircleM, R. Is elociy increses wih c = +, during which he llowed iniu urning rdius increses s c /K roll. This coninues unil he ehicle reches poin Q. The ehicle cn hen go srigh following he ngen line Q-Q or coninue urning long he circle wih rdius R s Equion 3.3 following rc Q-Q.

24 3 Figure 3.: Geoery of criicl ph of he ehicle wih elociy no less hn s. Ph I-Q-Q is he Criicl Ph. The iniu urning circle is CircleM, R nd he sfe urning circle is CircleS, R s. For ny poin T on ph IQ, when he ehicle is T i hs elociy nd llowed urning circle, CircleC, R. For ech poin T on ph IQ, he ehicle hs posiion x, y nd orienion giing:., / / / cos sin y x K R R y x s c c c c c c c 3.6

25 4 We sole hese equions by: log log,, K d K K K R R roll roll roll c c c c c log log cos log log sin K y K x roll roll 4 cos sin cos sin sin sin cos sin cos sin sin sin. / log, /, log c d cr e c d cr e x d cr e c d cr e dr d cr e c dr d cr e c d cr e c d cr e dr d cr e c d cr e dr d cr e x d cr e x Then d K c K r Le r r r r r r r r r r r r roll roll By subsiuing ribles nd pplying he se ehod o y, we ge:., 4 sin cos 4 4 cos sin 4 log log s roll roll roll roll roll roll roll roll gien K K K y K K K K x K 3.7 Since he iniil elociy is greer hn or equl o s, R = /K roll. The ehicle srs poin T see Figure 3., wih is heding gien by he doed rrow. The ph fro I or T o poin Q is uniquely defined by Equion 3.7. Poin Q is disn poin on he ngen line of his ph poin Q where he ehicle reches.

26 Anoher ph I-Q-Q occurs when he ehicle coninues urning wih u= fro Q. I urns wih rdius R s owrds poin Q. We now sepre he proble ino hree disinc cses: poins boe I-Q-Q i.e. poin W, Figure 3.3, poins beween Q- Q nd Q-Q Figure 3.4, nd 3 poins below I-Q-Q nd ouside CircleM, R Figure 3.5. The res of our discussion only considers he ph fer he ehicle reches elociy s, since he ph o h poin is lredy uniquely defined. For h reson, wihou loss of generliy, we will sy h he iniil elociy is greer hn or equl o s. 3.. Cse Theore 3.: The opil ph for poin W boe ph I-Q-Q lies beween phs I-T -W nd I-T -W. Figure 3.3: The ehicle reches poin W boe I-Q-Q fro poin I hrough T cse. 5

27 Proof: Suppose he ehicle srs poin I, oriened owrds Y, nd is gol is poin W loced boe I-Q-Q s in Figure 3.3. One wy for he ehicle o rech W is o rel long I-Q-Q unil reching ngen poin T nd hen ke he ngen line srigh o W. On ph I-T -W he ehicle cceleres wih unil i reches, i hen rels wih. I is no possible for i o oe ore quickly long ny oher ph. When copred o ny ph I-T -W boe i, ph I-T -W is shorer nd herefore quicker. So he opil ph cn no lie boe I-T -W. Alerniely, he ehicle cn rel long iniu urning circle CircleM, R o poin T where i inersecs is ngen line o W. The ehicle hen follows he ngen line srigh o W. This ph I-T -W is he shores possible ph o W. Any ph below I- T -W will ke longer o rech W, since i cnno ccelere ore quickly nd us rel greer disnce. The opil ph, which reches W in he shores ie, us herefore lie beween I-T -W nd I-T -W. QED The ph I-T -W hs he higher elociy nd I-T -W hs he shorer lengh. We noe fro Figure 3.3 h hese phs re ery siilr. Using rigonoery we see h s he disnce o W grows, he size of his difference shrinks sypoiclly. Since he ehicle cceleres ore quickly long I-T -W i hs higher elociy, so eenully ph I-T -W will becoe opil. I is he opil ph for los eery poin in his region. There y be sll se of poins beween phs I-T -W nd I-T -W where I-T - W is no opil. In Appendix B we describe how o find he opil ph nuericlly 6

28 for his se of poins. In prcice, we dise ignoring he resuls of Appendix B nd siply following he criicl ph. This is bsed on he following insighs: The disnce beween phs I-T -W nd I-T -W is los lwys inconsequenil, so h ny iproeen fro using he ruly opil ph is likely o be inil. The ie required o clcule his ph nuericlly is los cerin o be greer hn he perfornce iproeen ined. 3.. Cse Theore 3.: For ny poin W beween phs Q-Q nd Q-Q, he opil ph fro I o W is I-T -Q -W. Figure 3.4: The ehicle reches poin W beween Q-Q nd Q-Q fro poin I cse. Proof: In his cse Figure 3.4 he ehicle s gol poin W is loced beween line Q-Q nd cure Q-Q. We sr our nlysis by finding he poin T on CircleM, R h is he 7

29 firs poin h cn correspond o poin I in Figure 3.3. Ph T -Q -Q is he criicl ph fro T, where T nd Q correspond o I nd Q in Cse. W us now lie on line Q -Q. One wy o rech W is o rel long CircleM, R o T ; nd hen ccelere wih nd se u=. This is like reling long he criicl ph. The ehicle psses Q where i reches nd hen goes srigh o W, following ph I-T -Q -W. An lernie ph o W kes criicl ph I-Q o CircleS, R s, which i follows o poin T where he ngen line o W inersecs CircleS, R s. The ehicle hen kes he ngen line srigh o W. This is ph I-Q-T -W in Figure 3.4. Le s define = AngleIMT nd =AngleQST. In ph I-T -Q -W, he ehicle urns long CircleM, R for degrees nd hen rels he criicl ph. In ph I-Q-T -W, he ehicle rels he criicl ph nd hen long CircleS, R s for degrees. Using rigonoery, we see h =. The ie difference beween hese wo phs is due o he ie spen reling on he wo circles. The ie reling long CircleM, R is = R / s. The ie reling long CircleS, R s is = R s /. Since /R s = s /R nd > s, we he /R s < s /R or R s / > R / s, i.e. <. Ph I-T -Q -W is herefore lwys fser hn ph I- Q-T -W. The se rguen holds for ll phs beween he wo phs. The ehicle should herefore neer rel boe ph I-T -Q -W. A hird ph rels long CircleM, R o poin T where he circle inersecs is ngen line o W. I hen follows he ngen line srigh o W. This is ph I-T -W in Figure 3.4. This is he shores possible ph o W. Any ph beneh his us rel furher lower elociy. Ph I-T -Q -W nd ph I-T -W re idenicl unil T. 8

30 Fro cse, we know h ph T -Q -W is beer hn ny ph beween T -Q -W nd T -T -W. This deonsres h he opil ph cn no lie beneh ph I-T -Q -W. This proes h ph I-T -Q -W is he opil ph o ny poin W beween Q-Q nd Q- Q. QED 3..3 Cse 3 Theore 3.3: Ph I-T -cp-w is he ie opil ph fro poins I o W in Figure 3.5. Figure 3.5: The ehicle reches poin W beween CircleM, R nd I-Q-Q cse 3. Proof: In his cse see Figure 3.5 he ehicle s gol poin W is loced beween CircleM, R nd ph I-Q-Q. Noe h ges inoling poins inside CircleM, R re hndled in deil in [9]. As wih cse, we find poin T on CircleM, R h is he erlies poin wih criicl ph of he for T -Q -Q h reches W. Using he se logic s in cse, W us lie on ph T -Q -Q. If W is on cure T -Q, he ehicle will no be reling wih elociy W. If W is on line Q -Q, i will he elociy W. 9

31 One wy o rech W is ph I-T -cp-w, which follows CircleM, R o T nd hen follows he criicl ph o W. As wih cses nd ny ph boe I-T -cp-w will be subopil, since he ph will be longer hn I-T -cp-w nd he ehicle cn no be oing wih lrger elociy ny poin in ie. We now drw ngen line fro W h inersecs CircleM, R T nd consider ph I-T -T -W. Using he se logic s in cse, ny ph beneh I-T -T -W will require ore ie hn I-T -T -W nd ny ph beween I-T -T -W including I-T - T -W nd I-T -cp-w will require ore ie hn ph I-T -cp-w. Ph I-T -cp-w us herefore be he opil ph o ny poin W beween CircleM, R nd I-Q-Q fro I. QED 3..4 The Generl Cse Fro los ll iniil condiions, he shores ie ph fro n iniil sring posiion o gien poin in he plne cn be found following he se generl procedure. The wo excepions being: Poins inside he iniu urning circle require copliced neuers. We do no re his proble here, s i is hndled in deph in [9]. To rech hese poins, he pursuer neuers o posiion where he poin is no longer in is iniu urning circle. Wihou loss of generliy, we cn perfor he nlysis gien in his Chper he end of he neuer suggesed in [9]. Soe gol poins fro Cse re reched ore quickly using ph beween I-T -W nd I-T -W. These excepions re discussed in deph in Chper 3...

32 For ll oher cses, gien ny iniil elociy nd n rbirry gol poin on he plne, if is less hn s, ccelere wih unil he elociy is s or nd urn wih u being or - unil one of he following occurs: If he ngen line is reched, ke he ngen line direcly o W. When is greer hn or equl o s, find he insnneous criicl ph. If he poin is boe he criicl ph, ccelere wih unil is nd urn s uch s possible i.e. u= or - unil he line ngen o W is reched, hen follow he line o W. If he poin is below he criicl ph, use he curren elociy o urn long he curren iniu urning circle =, u= or -. When W is on he ehicle s criicl ph, follow he criicl ph o W = before, = fer ; u= or - before he ngen line, u= fer. Noe h he ehicle lwys ses is conrol lues o soe cobinion of = or nd u =,, or -. This cn be isulized using n elsic sring nd spool. The sring will follow he consrining spool i.e. u= unil here is srigh line o is oher end i.e. u=. The elsic sring nurlly confors o he shores ph beween is ends o iniize he ension. For he ccelerion, if he ehicle needs o hug he currenly llowed iniu urning circle, i ses o ; oherwise, i cceleres using he xiu ccelerion s long s i cn. This behior could he been prediced by noing h he chrcerisics of Equion 3.5 indice he Hilonin corresponding o he iniu ie proble hs degenere criicl poin in he conrols. Th is, he conrol is expeced o be bng-bng. This deriion is no used since i requires ore coplex heics nd ois ll inuiion gined in our preceding discussion.

33 3.3 Finding he Mxiu Region for Vehicle This chper shows how o find he xiu region MR, he region conining ll poins he ehicle cn rech by ie sring fro he iniil condiions. Alerniely, his could be considered he slles region gurneed o conin ehicle ie h sred fro known iniil posiion ie. The ehicle srs poin O, wih elociy. If he ehicle rels in srigh line, i y ccelere for os = in T, - / ie unis. Therefore, he longes disnce d i cn rel by ie T is: d T / T. 3.8 If is less hn s, he ehicle requires ie T s = s - / o rech s. As long s he curren elociy is less hn s, he ehicle cn urn wih ny urning rdius greer hn he iniu urning rdius R. To find MR for ny ie less hn T s, we firs clcule dt s. We hen find MR by cuing sring of lengh dt s, ching one end of he sring o he ehicle s iniil posiion, nd rcing he cure defined by he oher end of he u sring s i oes fro he consrining circle defined by R on he lef o he consrining circle defined by R on he righ. This cure is he inolue of circle of rdius R see Appendix A for deils. The circle inolue ogeher wih he iniu urning circles defines he xiu region for he ehicle up unil i reches s. The poin O is he iniil posiion see Figure 3.6. The wo dshed circles re he iniu urning circles CircleM, R nd CircleM, R. The solid cures indice he fron boundry of he xiu region differen lues of. The bold solid cure is he sfe-

34 elociy cure, which indices when he ehicle reches s nd inersecs he iniu urning circle poin S on he righ, S on he lef. Figure 3.6: The circle inolue describes he cure rced by sring of gien lengh when wrpped ono circle. When he ehicle elociy is less hn s is oion is consrined only by is iniu urning rdius R, nd he cure describing he region MR when is less hn T s is defined by circle inolue. This gies he xiu region for he ehicle unil i reches s. Once he ehicle reches elociy s, wo cses exis. The res of our discussion considers he righ hlf of he region; resuls for he lef hlf cn be found using syery. Our wo cses re sepred by line ngen o CircleM, R he poin S, where he inolue inersecs he circle Cse This is he region boe he ngen line. The ehicle cceleres unil i reches s nd hen coninues oing in srigh line. Is oion is no consrined by he fcor K roll defined in Equion 3.. To find he xiu region for his cse, clcule he xiu srigh disnce d h he ehicle cn rel. Cu sring wih lengh d nd ch one end on O 3

35 nd using he u sring o wrp CircleM, R unil i ouches S. The rck of noher end of he sring is he fron boundry of he x region boe he ngen line, which is circle inolue. The upper pr of he region, shown in Figure 3.7, is defined by his cure nd he ngen lines. Figure 3.7: Unil ie T s, region MR is defined by he circle inolue. Afer T s o rech poins boe he line ngen o he iniu urning rdius circle he poin where i inersecs he circle inolue, he ehicle siply oes in srigh line Cse Below he ngen line, he ehicle s biliy o urn is liied by is elociy Equion 3.3. Recll he opil ph defined in Chper 3.. In Chper 3..4, we show h ehicle reches gien poin in he shores ie by following he criicl ph s soon s i exiss. By using relie coordines cenered on he ehicle when i reches poin S, we p his proble o he one shown in Figure 3. wih equilen poin I. This diides he region ino wo sub-cses: boe nd below ph I-Q-Q. In he firs sub-cse, he ehicle rels long he criicl ph for ie nd hen srigh. By seing o T T s 4

36 including zero, we find h he region boe he criicl ph fro poin S is enclosed by he end poins of hese phs. In he second sub-cse, in he reining ie T T s, he ehicle rels long CircleM, R for ie nd hen follows he criicl ph once i exiss. By seing o T T s including zero, we find he se of poins enclosing he pr of he xiu region loced below he criicl ph fro poin S. By king he union of hese sub-cses wih he resuls fro cse, we find he xiu region see Figure 3.8 ehicle cn rech by ie T. Figure 3.8: The xiu region he ehicle cn rech by ie T. In Figure 3.8, we cn see h poins inside he lef righ iniu urning circle cn be reched by reling o he righ lef hen going srigh. Alerniely, he ehicle cn go srigh unil he poin is ouside he insnneous iniu urning circle hen urn round. 5

37 3.4 Finding he Approxie Fron Boundry In his ge, here is no discernble dnge o oing less hn he xiu elociy possible ny oen. Therefore, we ssue in his hesis h ech plyer uses he xiu ccelerion possible wihin consrins iposed by Equions 3. nd 3.4 ny poin in ie. This ens h ech plyer hs only one conrol rible, u he re of urn. We discuss his ssupion furher in he Conclusion Chper. If he disnce beween pursuer nd eder is fr, he difference beween urning wih R s nd R c see Equion 3. nd 3.3 could be ignored. Therefore, in opil ply wih perfec knowledge u is ypiclly se o eiher or lues corresponding o R s. Figure 3.9: The geoery of circle inolue for pursuer or eder. In Figure 3.9, plyer wih iniil posiion I is reling long he Y xis. The dshed circles he rdius R s. Up o ie, he plyer cn rech ny poin wihin region IDCBAFGHI. To rech h poin, he plyer urns s quickly s possible for is curren elociy unil i reches specific ngle hen i follows srigh line. The deriion of 6

38 7 his opil sregy is in []. We define he ngle h he ehicle urns s M ngle DSI or HTI in Figure 3.9. We refer o ngles CSD nd GTH s L, nd ngles CSI nd GTI s H. Since we ssue boh plyers ccelere wih heir xiu ccelerion, he plyer will be soewhere on rc BAF ny gien ie. The plyer cn go furhes by going srigh, nd we cll his disnce rel line IA in Figure 3.9. Gien curren elociy c, xiu elociy nd xiu ccelerion, rel is clculed s: /,, / in c c T T T rel T 3.9 If we cener he XY coordine syse on poin I, he equion for rc BAF is: cos sin sin cos cos sin sin cos s s s H H H H R R R y x 3. where L H, H = rel/r sfe, L = x, H - M. The lengh of BAF is: L H R s lengh 3. Figure 3.: The geoery of he iniil condiion of pursui-esion ge.

39 In our ge, P iniilly hs elociy p posiion, heding long he Y-xis. E iniilly hs elociy e posiion x E, y E heding long ngle wih respec o he Y-xis, s shown in Figure 3.. We now express he posiions for boh plyers s funcion of he sfe urning rdius, ccelerion, iniil posiion nd ie: For he pursuer: cosp P sinp sin cos RP xp cos PH sin PH RP s yp sin PH cos PH RP. s P P relp where P [ PL, PH ], PH, PL x, R.. s P. s x P xp y y 3. P P For he eder: xe cos EH sin EH RE. s cos E E sin E RE. s ye sin EH cos EH RE. s sin E E cos E rele where E [ EL, EH ], EH, EL x, EH EM R x y E E cos sin E. s sin x cos ye E x y E E PH P PM 3.3 Equion 3. expresses he pursuer s BAF cure, which we cll is fron boundry. Equion 3.3 is he eder s fron boundry, where is heir relie direcion nd x E, y E is he relie iniil posiion of he eder s shown in Figure 3.. So ie, he se of possible posiions for ech plyer is consrined o he solid boundry rcs shown in Figure 3.. 8

40 Figure 3.: Cures show he possible posiions ie for pursuer Blue, boo nd eder Red, righ, boh sring fro he posiions shown by crosses ie. The doed line is Blue s effecie sensing rnge. For ny gien posiions of pursuer nd eder, he Eucliden disnce beween he is: d PE x x y y 3.4 E P E P 9

41 CHAPTER FOUR PURSUIT-EVASION WITH ACCELERATION Assue P pursuer nd E eder oe in he plne wih consn speeds p nd e, respeciely. R p nd R e re heir respecie iniu urning rdii. The following wo condiions deerine he ge of kind: A B p p R e p e R e 4. Theore 4.: If nd only if A nd B re sisfied, P cn cpure E fro ny iniil posiion. The proof of Theore 4. cn be found in [6]. Insed of ssuing consn elociy for boh pursuer nd eder, we use he ehicle dynics presened in Equion 3.5 in Chper 3.. The plyers cn ccelere unil hey rech xiu elociies. Their elociies nd urning rdii consrin ech oher. In Figure 3., he pursuer P nd eder E is chrcerized by he following consns: xiu elociy p nd e, xiu ccelerion p nd e, iniu urning rdius R p nd R e nd rolloer coefficien K p.roll nd K e.roll, respeciely. P iniilly hs elociy p posiion, heding long he Y-xis. E iniilly hs elociy e posiion x, y heding long ngle wih respec o he Y-xis. Ech ehicle s equions of oion re defined by he iniil condiions nd Equion 3.5. We now nlyze he ge using relie coordine syse cenered on P. If P is poin x p, y p heding in direcion p nd E is poin x e, y e heding in 3

42 3 direcion e in he bsolue coordine syse. We le x r, y r nd r be E s posiion nd heding in he relie coordine syse. This gies: p e r p p e p p e r p p e p p e r y y x x y y y x x x cos sin sin cos 4. Or p e p e p e p p p p r r r y y x x y x cos sin sin cos 4.b The syse dynics in he relie coordine syse becoes: cos sin R u R u u x R y u y R x ec e e pc p p r r e p p r pc p r r e p r pc p r 4.3,, x, x.. u u if if if if d d K R R K R R e p e e e e e e e p p p p p p p e e e p p p roll e e e ec roll p p p pc

43 If he disnce beween boh plyers is bounded fro boe, we sy h he pursuer conrols he eder. If soe ie, he disnce beween P nd E goes o zero, we sy h he pursuer cpures he eder. If he disnce beween he plyers is unbounded oer ie, we sy h he eder escpes fro he pursuer. Le s consider his ge using region MR fro preious chper. If p > e, P s region will eenully include E s enire region. So h, for ny se of iniil condiions, P cn lwys conrol E. Bu if p < e, if he iniil condiions re forble, P cn conrol E for cerin period of ie nd een cpure E. We now deerine he crieri of boh cpure nd escpe. Theore 4.: If K p < K e, i.e. p /R p < e /R e, E cn oid cpure by P. Proof: Recll condiions A nd B fro [6] see Equion 4.. Alhough, in conrs o [6], elociies re no consn in our ge, ech ehicle s oion is consrined by condiion /R K roll. As shown in Chper 3., he opil phs o poins where elociy nd urning rdius require re uully consrined require /R = K roll for pr of he ph. A ny poin in ie, he ehicles curren elociy c liis he ehicle s effecie iniu urning rdius o R c c /K roll. Subsiuing his inequliy ino condiion B we see h, since K roll is consn for ech plyer, condiion B will eiher be uniforly rue or flse for ny insnce of our pursui-esion ge. When K p.roll < K e.roll, condiion B does no hold. Thus P will no be ble o cpure E. QED Theore 4.3: If K p.roll K e.roll nd p > e, P cpures E eenully. 3

44 Proof: We consider he ge s ie pproches infiniy. We he esblished h K p.roll K e.roll corresponds o condiion B fro Theore 4.. If he pursuer hs lrger xiu elociy, no er wh he iniil condiions, eenully p > e. Then condiion A is sisfied. Now since boh condiions of Theore 4. re sisfied. We conclude h P cn cpure E. QED For he res of his chper, we consider he cse where he pursuer hs lrger rolloer coefficien bu sller xiu elociy hn he eder, i.e. K p.roll K e.roll nd p < e. Under hese condiions, we need o ccoun for he ehicles iniil elociies, p, e, nd xiu ccelerion, p, e. For P o cpure E, i us he higher elociy hn E for soe period of ie. For E o escpe fro P, i needs only rech p before cpure. Wihou loss of generlizion, we ssue h P s ccelerion is greer hn E s if P s xiu elociy is less hn E s. Since he cpure condiions for boh plyers depend on which hs he higher elociy, here is no reson for he o eer use less hn he xiu ccelerion unil hey rech heir xiu llowed elociies, i.e.: p e p p e, e if if p p p nd ccelere oherwise e e nd ccelere oherwise by curren ssupion e 4.4 The following ies re criicl o our nlysis: T e is iniu ie for E reches is xiu elociy, T p is iniu ie for P reches is xiu elociy, nd T ed is iniu ie for E reches he P s xiu elociy. 33

45 T e e e p p p e ; Tp ; Ted 4.5 e p e Le 4.: Assuing p > e, p < e, nd T ed < T p. Le e = in e + e, e, nd p = in p + p, p. Then e > p for ll. Proof: Le F = e p. When > T ed, by definiion, E will he elociy greer hn p, which is he x elociy of P. Then F > nd e > p. Consider T ed. Since p > e, p < e, T ed < T p, i is esy o show e > p, i.e. e > p, nd e T ed > p T ed. Since p > e, F is coninuous nd non-incresing. By en lue heore, i follows h F >. QED Theore 4.4: Gien K p.roll K e.roll, p < e, p > e, E cn escpe fro P if p p / p > p e / e, i.e. T p > T ed. Proof: If T ed < T p, by Le 4., E will lwys he elociy greer hn P. Condiion A is neer sisfied. We conclude h P cn neer cpure E. Since E hs higher elociy, he disnce beween he is unbounded nd P cnno conrol E. E escpes fro P. QED When T ed > T p, nd P hs higher elociy hn E for soe period of ie. Le he erlies ie when P hs higher elociy be. Fro p + p > e + e, we he = e p / p e. If e < p, P hs higher iniil elociy, i.e. =. So, = x, e p / p e. 34

46 Le P he higher elociy no ler hn. E cn hen urn wih is curren llowed iniu urning rdius nd hen ccelere unil i reches P s xiu elociy. R ec =x R e, e /K e.roll, = R ec / e + p e / e. P s cpure region, denoed by PCR, is clculed by finding he xiu region beween nd. E s escpe region, denoed by EER is clculed by: Clcule he boundry of E s xiu region fro ie =. Rein he region which is no in PCR; Copue he fron boundry in discree poins nd clcule he xiu region for ie = + d fro he hese poins. 3 Repe unil ie =. The union of he regions kep in is E s escpe region. Theore 4.5: Gien K p.roll K e.roll, p < e, p > e nd p p / p < p e / e, we cn find he ie period fro o, when e < p : = x {, e p / p e } nd R ec =x R e, e /K e.roll, = R ec / e + p e / e. Using he opil ph o find he cpure region for P nd he xiu escpe region for E, if P s region coers E s, P cn cpure E. Proof: If P s region coers E s region, since P hs higher elociy nd higher rolloer consn, condiion A nd B re sisfied, i.e. P cpures E, shown in Figure 4.. Oherwise, h ens E hs ph o go ou of P s conrol nd cn iole condiion A. So E escpes fro P, shown in Figure 4.. QED 3 This is siply nuericl inegrion using firs order pproxiion. Iproed pproxiions cn be obined using higher order ehods. In our experience, his pproch is sufficien. 35

47 Figure 4.: P s cpure region solid coers E s escpe region dsh. P cpures E. Figure 4.: P s cpure region solid doesn coer E s escpe region dsh. E escpes. The cses where K p.roll K e.roll, p < e nd p < e, cn be soled in siilr nner. If P hs lower iniil elociy, E cn escpe. If P hs higher iniil elociy, R ec =x R e, e /K e.roll, = R ec / e + p e / e. I is srighforwrd o pply hese resuls o soling he ge of degree for his proble. If E cn escpe hen he lue of he ge is infinie nd E s sregy is o ke 36

48 ny rjecory h reins ouside of P s cpure region. If E cn no escpe fro P, bu is no conrolled by P, hen he lue of he ge is infinie nd E s sregy is o ede P in he finl sges of cpure. In boh cses P s sregy is irrelen, since i is dooed o filure. If E cn no escpe nd is conrolled by P, hen he lue of he ge is he ie when P s cpure region firs enelopes E s escpe region. E s opil sregy is o follow he opil ph o ny of he poins h re on he inersecion of he boundry s of boh regions h ie. P s opil sregy is o choose ph h sypoiclly conerges wih he obsered ph of E. 37

49 CHAPTER FIVE PURSUIT-EVASION WITH SENSING LIMITATIONS 5. Sensor Model Pursui esion reserch o de hs concenred eiher on syses wih perfec knowledge, or wih sensors whose sole liiion is geoeric rnge consrin. In rel life, he biliy of sensor o deec n objec is subjec o n rry of enironenl influences nd noise sources h ke sensor rnge unpredicble bes [, 3]. For iliry pplicions, sensor redings cn lso be subjec o elecronic couner esures ECM h re designed o odify sensor inpus in wys h deceie he deecion process. Trge presen Trge bsen Trge deeced TP: True Posiie FP: Flse Posiie Type II error Trge no FN: Flse Negie deeced Type I error TN: True Negie Tble 5.: Four possible oucoes of he rge deecion rndo process Insed of using cookie cuer odel [3], we consider sensing o be he process of deecing known rge signure obscured by bckground noise. For gien sensor inpu, his process oupus one of he four oucoes in Tble 5.. TP is rue posiie, he rge is deeced nd presen; FP is flse posiie, he rge is deeced bu bsen Type II error; FN is flse negie, he rge is presen bu no deeced Type I error; And 38

50 TN is rue negie, he rge is bsen nd no deeced. For TP nd TN, he sensor works correcly. FN FP corresponds o Type I Type II error in decision heory. The sensor rge pursuer or eder eis signl in noisy enironen. The sensor us differenie beween he rge signl nd rndo bckground noise. When he rge signl is wek, is specru is hidden in he noise. Typiclly, he sensor deecs rge only when he rge signl s power exceeds fixed hreshold. Receier Opering Chrcerisics ROC cures re used o find he opil hreshold s described in [3]. For signls subjec o Gussin noise nd propging in hoogeneous ediu, he signl srengh, nd herefore probbiliy of deecion, decys s n inerse exponenil of he disnce d beween he sensor nd he rge. This gies probbiliy of deecion: P TP Cd 5. where ries in prcice fro o 5 [36]. Noe h probbiliy should be bounded beween nd. And PTP is lredy greer hn zero bu y exceed one, so in fc, PTP = in, Cd -. In he following of his hesis, when we enion Cd -, for esy represenion, i ens in, Cd -. In his hesis, we use = 4, which corresponds o he rdr equion [3]. Since he flse lr re is dependen solely on bckground noise, we ke he coon ssupion of consn flse lr re, i.e. PFP = K FP = consn. This ssupion is enble s long s he bckground noise is uncorreled, which is he cse in hoogeneous enironen. Using Tble 5. nd he fc h when 39

51 rge is presen i is eiher deeced or no deeced, we ge PFN = PTP. Siilrly, n bsen rge is eiher deeced or no, giing PTN = PFP. We noe in pssing h, lhough he Gussin signl decy in free spce ssupion is ubiquious in signl processing, i does no lwys correspond o reliy. Occlusion, signl shdowing, uli-ph fding, scering, nd cluer re ofen responsible for signl decy h does no follow Equion 5.. A fuller discussion of hose issues, long wih heoreicl pproch h llows deriion of ore relisic hi res PTP nd disribuions cn be found in [34, 35]. For he ske of sipliciy, we use Equion 5. in he exples gien here. The conceps in [34, 35] cn be used o generlize hese resuls o ore coplex cluered enironens. We use decision heory o decide, gien sensor reding, wheher he reding is correc or in error. Therefore, we conclude h rge is presen only when PTP>PFP. The effecie sensing rnge of our sensor is herefore R sense where CR sense = PFP, i.e.: R sense P FP C 5. If no priori eidence is ilble, we beliee h deecions wihin ouside his rnge re rue posiies flse lrs siply becuse his is he os probble cse. The signl enuion fcor is ypiclly in he rnge 5 [36] depending on he sensing odliy. Of priculr ineres re he following cses: = for cousic nd seisic sensing. The signl propges in plne. Is power dissipes proporionl o he re of circle cenered on he signl source. = 3 for ige or ideo sensors [3]. Deecion re is ied o rge size in he 4

52 ige, which shrinks wih d 3. For signls propging in 3-diensionl spce power will dissipe proporionl o he olue of sphere cenered on he source. = 4 for rdr deecion, since i corresponds o he rdr equion [3]. Rdr is n cie sensor. The signl dissipes on is wy o he rge fro he sensor nd while reurning o he sensor. We use = 4 in our exples in his hesis. 5. Inforion Theory Bsed Uiliy Funcion We consider pursui esion ges like he ones surized in Chper 4 nd hndled in deil in []. The only difference is h in his chper he pursuer nd eder rely on sensors o ge inforion bou ech oher s posiion. Since his is pursui ge nd no serch ge [9], we ssue boh he pursuer nd eder sr wih priori inforion regrding ech oher s posiion. In ge wih perfec inforion i does no er how fr he eder ges fro he pursuer or how ofen i edes cpure, he pursuer lwys knows where he eder is. The eder cn neer elude he pursuer; i cn only ou run he pursuer. An eder wih fser xiu elociy will ede cpure een when he pursuer hs perfec inforion nd does no need o exploi he pursuer s sensing weknesses o win. Siilrly, o exploi sensing weknesses, he eder will need o be ble o rech region where he pursuer s sensing cpbiliy is wek. For hese resons, we ssue in his ge h he pursuer hs lrger xiu elociy bu he eder hs lrger xiu ccelerion nd iniil elociy. We lso ssue h he disnce beween he 4

53 wo is lrge enough for he eder o eporrily oe ouside he pursuer s rdr s effecie sensing rnge. We ssue h he pursuer nd eder he perfec priori inforion of ech oher s posiions he beginning of he ge. Bu fer h ie, boh plyers rely on heir sensor inpus. A sensor reding is uple consising of ie, deecion se nd rge coordines when rge is deeced. When flse posiie occurs, he sensor reurns rndo coordines wihin he effecie sensing rnge. As shown in Figure 5., ny poin in ie he pursuer s rdr hs wo possible ses: rge deeced or no rge deeced. If here is deecion, he pursuer needs o decide wheher he deecion is rue posiie TP or flse posiie FP. This clssificion is done by deerining which cse is os likely, gien he priori inforion. When no rge is deeced ND, he syse needs o deerine wheher he reding is rue negie TN or flse negie FN, gin using priori inforion. A no deecion een is rue negie if he eder is ouside he rdr s effecie sensing rnge, s defined in Equion 5. in Chper 5. Figure 5.: Clssificion of rdr redings. 4

54 Figure 5. illusres he clssificion process. There re hree deecion ses: TP, FP, nd ND nd for ech se he syse cn clssify i s eiher rue or flse. In Figure 5. D refers o he sensor reding nd C refers o he plyer s inerpreion clssificion of he sensor reding. A deecion een clssificion cn refer o one of four possible cobinions: rue posiie clssified s rue posiie TP TP, rue posiie clssified s flse posiie TP FP, flse posiie clssified s flse posiie FP FP or flse posiie clssified s rue posiie FP TP. Howeer, no deecion only hs wo possible belief clssificions: ND TN or ND FN. We explin he reson for his disincion ler in his chper. We denoe he six blocks on he righ o be cses hrough 6. Their ssocied probbiliies re p, p p 6. Since he P nd E ech sr wih priori knowledge of ech oher s locion, hey re wre h he fuure posiions boh plyers y he ie re consrined o he boundry cures in Figure 3.. Since boh ehicles oe wih he xiu llowble elociy, heir sregy is defined purely by he urning re u. This ens h for pursuer P nd eder E ech sregy corresponds o poin on heir boundry cure, nd for ech cobinion of P nd E sregies he disnce d beween he wo ehicles is esily copued. We now derie pyoff rix where ech eleen i, j of he rix corresponds o he oun of ceriny h P will he bou E s posiion if P uses sregy i corresponding o discree poin of P s boundry cure nd E uses posiion j corresponding o discree poin of E s boundry cure. This is done by: Clculing he disnce d beween P nd E s posiions, Clculing he likelihood p k of ech of he cses presened in Figure 6, 43

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