The Chase Problem (Part 2) David C. Arney

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1 The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind of chaing were given in he eample of he previou ecion: miile inerceping oher miile, ani-ank round eeking a ank, and orpedo racking an enem hip. In hi ecion, we eend and refine hi fir model, build a coninuou model for hi problem, and build more effecivene and ophiicaion ino our chae algorihm. The Problem Our problem i o deermine he movemen pah for he chaer, given ha we know he locaion and omeime more informaion of he arge. We ar wih he aumpion ha he chaer know he arge poiion eacl. The chaer poiion i repreened in wo-dimenional Careian coordinae b,. We alo aume ha he chaer move a a conan peed given b and he arge poiion in wo dimenion i given b he parameric relaionhip,. We ar wih he echnique ha he chaer move direcl oward he arge. Laer we ll allow for he chaer o lead he arge. A he locaion of he arge change, he chaer coninuall adju i pah o coninue o move direcl oward he arge. The Model We can model hi procedure wih a em of differenial equaion, one for each pace dimenion we model. Since we ll perform our modeling in wo pace dimenion and, we ll build em of wo differenial equaion. Our modeling proce for a coninuou ime changing even i o e up relaionhip ha epre he derivaive of he changing variable, in erm of funcion of variable. Le draw a diagram of wha happen during a ime inerval. We will ake he limi a in order o ge coninuou feedback and coninuou movemen for he chaer. Figure how he locaion of he chaer and he arge a ome ime. The movemen made b he chaer over ha inerval i indicaed. The final poiion of he chaer i given b,. 9

2 n, n o n, o n o o n, o n o Figure : Movemen b he chaer during he ime inerval from ep n o ep n. We ee wo imilar riangle in Figure. Like he dicree cae developed in Par, he relaion beween hee imilar righ riangle enable u o wrie our model. Recall ha we can e up proporional equaion relaing he ide of he riangle wih he hpoenue of he riangle. Fir le deermine formula for each of he ide of our wo riangle. The larger riangle in Figure ha i horizonal ide of lengh -. The verical ide ha lengh -. Therefore, b he Phagorean Theorem he hpoenue ha lengh. The maller riangle ha ide - and -. We need o deermine he lengh of he hpoenue in erm of value oher han and. We alo know ha he chaer move a peed. Therefore, he lengh of he hpoenue can be approimaed b he diance raveled during he ime inerval or. We wrie ou he equaion relaing he ide of he riangle wih he hpoenue of he riangle. Fir he horizonal ide and hpoenue of boh riangle produce he relaionhip: The verical ide and hpoenue produce: n n n n n n n n We now have difference quoien on he lef ide of Equaion and. We ake he limi a of and o produce he differenial equaion 3

3 lim lim d d d d 3 Thi i our chae/movemen model equaion 3 and, which will provide a mean of deermining he movemen of he chaer, when we know he movemen of he arge. Thi em of differenial equaion i nonlinear and mu be olved numericall. Therefore, we will need a numerical olver on a compuer or calculaor o deermine he pah of he chae. Man ofware package ha ue Euler mehod or he Runge-Kua mehod are available. I i alo poible o implemen hee algorihm b convering he differenial equaion o difference equaion and implemening ieraion on a pread hee. Remember our aumpion: he chaer move a a conan peed and he chaer alwa ee he arge. When we olve our differenial equaion wih a numerical mehod, we acuall model he chaer moving oward he arge for a e ime inerval,, ued in he numerical cheme. We uuall e he ime inerval o be ver mall o aure accurac of he oluion and o approimae accurael he coninuou movemen of he chaer. We need o deermine when o op our calculaion. In he previou ecion, we dicued everal of he facor involved in hi deciion. There i no need o coninue afer he chaer ha caugh he arge. We need a opping crieria ha reflec caching he arge. We will aume ha caching he arge mean ju being cloe enough or wihin he olerance of he opping crieria denoed b ε. We have choice for deermining hi olerance value. I could be a fied value or a funcion of he peed and ime inerval. We implemen opping crieria in our model b deermining he diance, denoed d, beween he chaer and arge afer each ieraion. The value of d i deermined b he diance formula beween wo poin, d We op when d < ε or afer a pecified amoun of ime ha epired > M. Le look a an eample o ee how hi work. Eample : Ani-ank Round 5 A oldier locaed a poin,3 launche a racking ani-ank round wih peed 3.5 a a ank a ime following a ellipical coure given b he following parameric equaion: 8 3co and in 6 We will deermine he round pah baed on i eeker uing our racking model of moving direcl 3

4 oward he ank. If he kill radiu of he round i.5 uni, our opping crieria i ε.5. Subiuing he known value and funcion ino our model, Equaion 3 and, produce d d co 7 8 3co in d d 3.5 in 8 8 3co in Saring wih our iniial condiion, and 3, we ue a Runge-Kua olver wih τ.6 unil we achieve our opping crieria of d < ε.5. Thi produce he oluion for he pah of he round. The graph of he pah for boh he round and he ank, unil heir impac a a ime lighl grea han 5 econd, are given in Figure. Noice how he round curve around o follow he ank and evenuall cache i. z n, b n 6 z, a n, n Figure. Graph of he pah of he round olid curve and he ank doed curve, from launch o impac. Doe our oluion make ene? Doe he chaer move in an efficien pah oward he arge? Doe he chaer op when he opping crieria i achieved? In general he anwer o hee queion are e. I appear ha we have a good model, bu i ma no be he be. I could help if he round wa able o lead he ank o i could cach i faer. We ll r implemening a lead algorihm for hi chae problem laer in hi ecion. 3

5 The Modeling Proce Le review our modeling proce for hi problem. Our behavior of inere, he movemen of he chaer, i coninuou in naure. We modeled hi movemen wih a coninuou differenial equaion. Our oluion mehod for hi model, he Runge-Kua numerical mehod, i dicree and give an approimae oluion o he coninuou model. We hen convered he dicree equence of locaion of he pah o a coninuou pah b connecing he poin in he graph of Figure. We how hi inerpla beween dicree and coninuou repreenaion in our modeling proce in Figure 3. Behavior Model Soluion Mehod Verificaion Mehod movemen differenial Runge-Kua graph in Figure of chaer equaion numerical cheme dicree coninuou coninuou coninuou Figure 3. Inerpla beween dicree and coninuou in he modeling proce of he differenial equaion chae model. The lead algorihm How do we ge he chaer o lead he arge? We need o ake ino accoun boh he peed and he veloci of he arge, hen ue ha informaion o predic where he arge will be when he chaer cache he arge. We ue he Talor polnomial o do hi. The Talor polnomial i an approimaion o a funcion. For he funcion f and uing he ar poin a a, we wrie he Talor polnomial of degree n a f f a f a a n f a f a f a a a 3... a! 3! n! n 9 We can ue hi approimaion for he wo funcion and repreening he wo dimenion of he arge pah. Fir, le ue he -degree polnomial approimaion, which ake ino accoun he locaion and he veloci bu no he acceleraion. We mu be epediiou in our elecion of a and in Equaion 9, and e n. To ge our approimaion in he proper form, we ue τ and a, and herefore, -a τ. Then, we can wrie τ τ and τ τ. 33

6 The value of τ i he value of he ime advance o he locaion where he arge i prediced o be, in order o have a proper lead. The phanom locaion o aim for i impl he poin τ, τ. Therefore, we modif he model in Equaion 3 and, uing hi phanom lead poin in place of, and he formula in Equaion o obain d τ d τ τ d τ d τ τ The geomer of hi lead algorihm i hown in Figure. lead poin τ, τ, Targe curren locaion Chaer movemen direcion o, o Chaer curren locaion Figure : Pah of chaer when heading for he lead poin. We need an algorihm o deermine he value for τ. How much hould we lead? We could do hi everal wa. One wa i o hink of τ a he ime needed o cach he arge. We will approimae hi cach ime b uing he ime for he chaer o reach he arge curren locaion. Therefore, he formula for τ i impl he curren diance beween he chaer and he arge given in Equaion 5 divided b he peed. We wrie hi a τ 3 3

7 35 Then our new lead model i formed b ubiuing Equaion 3 ino Equaion and : d d d d 5 Le r hi model in our previou cenario of an ani-ank round. Eample : Ani-ank Round reviied A oldier locaed a,3 launche a racking ani-ank round wih peed 3.5 a a ank a ime, following a ellipical coure given b: 8 3 co and in. 6 We deermine he round pah baed on i eeker uing our new racking model of leading he ank. We ue ε.5 and ubiue he known value and funcion ino Equaion and 5. Saring wih he iniial condiion, and 3, we ue a Runge-Kua olver wih τ.6 unil we achieve our opping crieria of d < ε.5. Thi produce he oluion for he pah of he round. The graph of he pah for boh he round and he ank unil heir impac are given in Figure 5. B comparing hi graph wih ha of Figure, we ee ha he lead algorihm ave ime and chae diance b urning ooner and more harpl o cach he ank.

8 .5.5 z n, b n.5 6 z, a n, n Figure. Graph of he pah of he round olid curve and he ank doed curve, from launch o impac. I appear ha we have developed a beer model b leading he arge. I i he be we can do? Are here beer lead model? Of coure, hee are leading queion. There probabl are beer wa o compue τ. And we could keep more han ju one erm of he Talor Polnomial Equaion 9. If we keep wo erm, n, we would ge new approimaion for our lead locaion of he arge. We would now be aking ino accoun he acceleraion of he ank, a well a i veloci. The new model i wrien a follow: τ τ and τ τ. 7 Subiuion of hee formula ino Equaion 3 and, along wih uing Equaion 3 for τ, creae a model wih ver large, me differenial equaion. We won r o how hem here, bu we ll how an eample which compare he variou model we have dicued. Eample 3: To Lead or no o Lead Thi ime he oldier firing he ani-ank round i locaed a he origin,,. He launche a racking round wih peed 5 a a ank a ime. The ank follow an ocillaing coure given b: 3 3 and in3 8 We deermine he round pah baed on i eeker uing hree differen racking model: moving direcl oward he ank, leading he ank b uing he veloci one derivaive erm in he Talor polnomial a in Equaion and 36

9 5, and 3 leading he ank b uing veloci and acceleraion including erm in he Talor Polnomial hown in Equaion 7. Our opping crieria i ε.5. The pah of he round racking direcl for he ank i given in Figure 5. The round cache he ank a 3.9 econd. The pah of he round when leading he arge uing he ank veloci i hown in Figure 6. Thi pah i more direc and cache he ank in.8 econd. Finall, he new lead algorihm uing boh veloci and acceleraion produce he graph in Figure 7. Thi model produce a cach a.7 econd. Thi la mehod i no much faer han he veloci onl model. Someime i doen help or ma even hinder o lead he arge, bu, in general, he more informaion ou ue he quicker ou can cach our arge. z n, b n z, a n, n 5 Figure 5: Pah of arge doed curve and chaer olid curve uing he model of moving direcl oward he arge. z n, b n z, a n, n 5 Figure 6: Pah of arge doed curve and chaer olid curve uing he model of leading he ank b uing he veloci. 37

10 z n, b n z, a n, n 5 Figure 7: Pah of arge doed curve and chaer olid curve uing he model of leading he ank b uing boh he veloci and acceleraion. In hi ecion, we have udied and olved a challenging problem wih man applicaion. Our model and i oluion have performed well in he eample we have olved. We now know wha happen when we lead he arge, inead of moving direcl oward i. There are ill man queion we have no addreed. Wha abou he maneuverabili of he chaer? Can i alwa urn fa enough o make he necear move of he algorihm? How hould he arge move o evade he chaer? Thee are difficul queion ha meri furher ud and more ophiicaed mahemaical model. Good luck o hoe who ud hi imporan problem wih numerou miliar applicaion. Eercie. An enem ank, currenl a locaion,, i moving in a zigzag paern awa from our locaion wih parameric equaion: 3 and in. You launch a ank racking round moving a a peed of from our locaion a,. The guidance em of he radar-conrolled round alwa move direcl oward he ank arge. a Wrie a em of differenial equaion ha model he movemen of he round oward he arge. b Ue a numerical cheme o olve he equaion and plo he oluion for <<. c Wha i he diance beween he ank and he round a? I he round cloer a or?. Uing he ame general cenario a eercie for an enem ank aring a, and moving wih equaion: 3 and in, ou launch a racking round moving a peed of from our locaion a,. Thi new improved round ha a guidance em ha lead he ank b conidering i veloci. a Wrie a em of differenial equaion b ubiuing ino Equaion 5 and 6 ha model he movemen of he round oward he arge. b Ue a numerical cheme o olve he equaion and plo he oluion for <<. 38

11 c Wha i he diance beween he ank and he round a? I he round cloer a or? 3. A hip locaed a,5 deec a orpedo a 5,6 and begin he evaive maneuver of moving direcl awa from he orpedo a a conan peed of 8. a Wha are he parameric equaion, uing ime a he parameer, for he pah of he hip wih repreening he ar ime of hi pah? b If he orpedo follow he hip wih a peed of, wha are he differenial equaion ha govern he moion of he orpedo?. A hip locaed a,5 deec a orpedo a 5,6 and begin an evaive maneuver defined b he equaion 3 and 5 3in. a If he orpedo follow he hip wih a peed of, wha are he differenial equaion ha govern he moion of he orpedo uing he 3 chae algorihm decribed in hi ecion direc inercep, lead uing veloci, lead uing veloci and acceleraion? b Solve he 3 model in par a and deermine which algorihm guide he orpedo cloe o he hip afer 3 econd. 5. Dicu he dichoom of dicree and coninuou mahemaic. Include in our dicuion eample of behavior and funcion ha are naurall dicree and behavior and funcion ha are naurall coninuou. Reference Dunbar, Seven R., Minimodule: Chae Problem, The UMAP Journal, vol. 5, no., 99, pp Fellman, Bruce, Gue Who Coming o Dinner: Mechanim which Help Inec Ecape Ba, Naional Wildlife, vol.3, Feb 993, pp. -5. Johnon, Elgin and Mahew, Jerold, Projec Baed Calculu a Iowa Sae, Compuer Algebra Sem in Educaion Newleer, no., Sepember 99, pp. -7. Simmon, George, Differenial Equaion wih wih Applicaion and Hiorical Noe, nd ediion, New York: McGraw-Hill, 99. Yae, Rober C., Differenial Equaion, New York: McGraw-Hill, 95 pp. 9-. Yavin, Y. and Pacher, M. P. edior, Purui-Evaion Differenial Game III, Compuer & Mahemaic wih Applicaion, vol. 6, no. 6, Sep 93, pp. -5 enire volume dedicaed o hi ubjec. 39