Guaranteed Strategies to Search for Mobile Evaders in the Plane (Draft)

Transcription

1 Guaraneed Sraegie o Search for Mobile Evader in he Plane (Draf) Timohy G. McGee and J. Karl Hedrick Abrac Thi paper udie everal problem of earch in he wo-dimenional plane for a mobile inruder or evader. In each cae, boh he earcher and evader are aumed o have bounded velociie, wih he velociy of he earcher greaer han ha of he evader. The earcher ha a enor wih a circular fooprin which allow i o deec he evader if i diance i le han he enor radiu. Unlike much of he pa work done in earch heory, he propoed raegie make no aumpion abou he moion of he evader, and are guaraneed o ucceed for any rajecory of he evader. The hree problem formulaion udied include he earch for mobile arge hrough a linear corridor, he earch for or bounding of a mobile arge which ar inide of a circular region, and he prevenion of a mobile arge from enering a circular region. The exenion of he propoed raegie for muliple earcher i alo dicued. I. INTRODUCTION Thi paper udie everal problem of earch in he wo-dimenional plane for a mobile inruder or evader. I i aumed ha he earcher and he evading arge have maximum velociie, and, wih >. No rericion are placed on he maximum urning rae of eiher pary. The earcher i aumed o have a enor wih a circular fooprin of radiu R which will deec he arge if he arge i inide of he enor fooprin. Unlike much of he pa work done in earch heory, he propoed raegie make no aumpion abou he moion of he evader, and are guaraneed o ucceed for any rajecory of he evader. The problem formulaion given, alhough abraced, have poenial applicaion for a wide variey of pracical problem including border and harbor parol, convoy proecion, and earch and recue operaion. The fir problem involve he earch for mobile arge hrough a corridor bounded by parallel line. The econd problem explore how o earch for or rap a mobile arge whoe iniial poiion i known o be wihin a circle of known radiu greaer han he enor radiu. The final problem, which i very imilar o he econd, explore how o creae a perimeer around a fixed poin in he plane in order o preven inruder from approaching he poin wihou being deeced. The major conribuion of hi paper i a raegy for earching he circular region, which i an improvemen o a imilar raegy oulined in [15]. While he proof of opimal raegie i difficul for hee problem, i i argued ha he opimal raegy mu be cloe o he propoed oluion. Thi i done by reducing he Thi work wa uppored by ONR gran #N C-0187, SPO # , and a Berkeley Fellowhip for Mr. McGee T. McGee i a graduae uden in Mechanical Engineering, Univeriy of California, Berkeley, CA 9470, USA mcgee@me.berkeley.edu J.K. Hedrick i a profeor in Mechanical Engineering, Univeriy of California, Berkeley, CA 9470, USA khedrick@me.berkeley.edu earch problem o he impler problem of ravelling around an expanding circle for which he opimal oluion can be found. I i alo hown how hee raegie can be execued by eam of muliple earcher o improve performance. There ha a large amoun of reearch ino earch heory daing back o WWII, including everal book devoed o he opic [7], [], [11]. Many earch problem are alo cloely relaed o game heory [10]. One major group of problem involve he earch for aionary or quai-aionary arge. Baeza-Yae oulined opimal mehod for earching for aionary poin or line in planar grid [6]. Several reearcher have udied opimal rajecorie o compleely cover an area wih a enor or ool wih minimal overlap beween pae [9], [1], [5], [13]. Search problem wih mobile arge can be furher claified by he inenion and informaion of he arge. In he rendezvou earch problem, boh player cooperae o find one anoher [3]. In he helicoper and ubmarine problem, fir propoed by Dankin [8], a helicoper earche for an evading ubmarine ha doe no know he poiion of he earcher. Several oher reearcher have udied he ame problem [18], [16], [19]. Anoher udy examine guaraneed earch raegie for an evader in an complex encloed workpace by a earcher wih a line of igh enor [1]. More recenly, earch wih eam of cooperaing agen ha become an acive reearch opic [14], [17], [4]. II. SEARCH OF A CORRIDOR The fir planar earch problem conidered i he parol of a corridor beween parallel border eparaed by widh W. Thi problem wa olved by Koopman [11] in he 1940 in order o deermine opimal parol raegie for aircraf earching for hip in a channel. I i reviewed here ince he concep ued o olve hi problem are laer hown o be applicable o more difficul earch problem in he plane. The corridor earch problem aume ha a he iniial ime, he e of all poible poiion of he arge, which we hall refer o a he arge e, i he area o one ide of a line perpendicular o he border of he corridor. The earcher begin on one of he corridor border wih i circular enor angen o he boundary of he arge e a hown in Fig. 1a. Since he evader i mobile, he arge e grow normal o i boundary a he maximum velociy of he evader,. Thu, in order o keep i enor angen o he border of he arge e, he earcher mu ravel a an angle o he normal of he arge e (Fig. 1b), where i calculaed a: V = arcin (1)

2 V problem could have applicaion for embargoe, parolling jail perimeer, or earch for mobile arge. Auming he arge i iniially inide a circle of radiu R o, and he earcher i ouide of hi circle wih i enor angen o arge e, we explore how he earcher hould ravel o guaranee ha he arge e i bounded for he large poible iniial circle wih radiu R o. For all circle maller han hi, he bounding raegy hould alo provide a mehod o hrink he area arge e o zero in order o guaranee capure. Thi i analogou o he advancing barrier for he channel parol problem. Fig. 1. V Corridor Parol Sraegy. By following hi raegy, when he earcher reache he oppoie border, he boundary of he arge e i ill a line perpendicular o he border, alhough he earcher i now ouide of he arge e (Fig. 1c). The earcher nex ravel parallel o he border unil i i again over he arge e, and i enor i angen o he boundary of he arge e (Fig. 1d). The earcher hen repea he ame maneuver o reurn o he oppoie border, compleing one cycle of he maneuver. The ime required for he earcher o complee each half cycle, 1, i he um of he ime required o cro he corridor and he ime o ravel up he border. 1 = W co () + R + = W V + R + () If he ravel ime of he earcher for each half cycle i equal o he ime required for he evader o ravel he widh of he earch enor, he boundary of he arge e a he beginning of each half cycle will remain aionary. W + R = R (3) V V + Koopman referred o hi cae a a ymmeric barrier. Solving for he widh of he corridor in hi cae give he maximum widh of a corridor ha can be parolled: 1 1 V W = R V + V (4) If he corridor widh i larger han W, he ize of he arge e will grow afer each cycle, and i canno be guaraneed ha he arge will be deeced. If he widh i le han W, he arge e will hrink, allowing i o be earched. Koopman called hee cae rereaing elemen barrier and advancing elemen barrier repecively. III. CIRCULAR BARRIERS AGAINST FLEEING EVADERS Anoher problem o conider i how o earch for or bound an evader ha ar wihin a known circular region. Thi A. Upper Bound for R o Since he opimal oluion for he guaraneed earch of an expanding dik i unknown [7], a good fir ep o exploring hi problem i o e bound on R o. One uch bound can be found by auming ha he earcher can apply i maximum earch rae arbirarily in he plane. For he problem under conideraion, he maximum earch rae i he produc of he earch velociy, and he widh of he enor, R. The growh rae of he arge e can be calculaed a he produc of i perimeer, πr o, and he velociy of he arge,. The maximum radiu of he arge e for hi unconrained problem, which ac a an upper bound for our problem, can hen be found by eing he earch rae equal o he growh rae of he arge e. B. Circular Search Paern R o R π (5) One imple raegy o bound an evader, which ha been offered by everal reearcher [11], [16], i o ravel a imple circular paern. Boh of hee udie calculae he large circle ha can be bounded uing a circular paern o be: R o = R ( /π 1) (6) Thi equaion i found by eing he ime for he earcher o ravel around hi circle π(r o + R )/ o be equal o he ime i ake he evader o ravel acro he diameer of he earch enor R /. One limiaion of hi calculaion, however, i ha i aume ha he evader i ravelling radially from he cener of he circle. The opimal pah of he evader i acually o ravel a a ligh angle, γ op o he radiu of he circle a illuraed in Fig.. Thi opimal angle γ op Fig.. Searcher Pah γ op Evader Pah Opimal Evaion Sraegy Again Circular Search Pah.

3 can be calculaed by maximizing he quaniy dr /dθ rel : dr dθ rel = R θ p θ = co (γ) /R p in (γ)/r (7) where R i he radiu of he evading arge, R p i he radiu of he earcher, and θ rel i he relaive angle beween he earcher and evader. The opimal choice of γ o maximize dr /dθ rel i: V R p γ op = arcin (8) R Alhough ravelling a γ op reduce he radial velociy of he arge, i force he earcher o ravel more han a full circle o cach he arge. Thu if he evader follow he pah pecified by Eq. 8, i can ecape he circle defined by Eq. 6. Alhough he circular earch pah i imple, i i no opimal, and i doen lend ielf o a mehod of reducing he arge e for R o < Ro. C. Propoed Paern o Search Expanding Dik The goal of he earch of he expanding dik problem can be reexpreed a he reducion of he area of he arge e, A T S, o zero in he minimum ime. Since an analyical oluion o hi problem i difficul becaue of he parial informaion of he evader poiion, an approach i aken o develop a raegy baed on inigh from imilar problem. Since inuiion can ofen be wrong, he performance of he developed raegy i compared again he heoreical bound on he opimal raegy in Eq. 6. Fir, we explore he inananeou rae of change of he arge e area which can be calculaed a he difference beween he growh rae of he arge e reuling from poible arge moion and he reducion of he arge e area by he moion of he earcher: A T S = P T S L S (9) where P T S i he perimeer of he arge e and L S i he widh of he enor fooprin perpendicular o he velociy of he earcher ha i over he arge e. The maximum value of L S i he widh of he enor, R. While minimizing A T S i difficul becaue P T S and L S are complicaed funcion of he choen rajecory, everal imporan properie for a well deigned raegy can be inferred from Eq. 9. Propery 1: The well deigned earch raegy hould ravel along he perimeer of he arge e. In order o minimize he value of A T S, he earch raegy hould alo minimize he perimeer of he arge e, P T S. If he earcher were o ravel ino he arge e a oppoed o along he perimeer, i would creae a channel ha would grealy increae he perimeer of he arge e. Thi propery eliminae divide and conquer ype raegie from being conidered for earch of he circle. Propery : The arge e hould remain near circular when he well deigned earch raegy i applied. For any given planar area, he hape encloing ha area wih he malle perimeer i he circle. Thu earch raegie ha preerve he circular hape of he arge e will be uperior o hoe ha do no. Thi propery eliminae raegie uch a line weep from being conidered. Fig. 3. θ = 0 Minimum lengh pah around expanding circle. Propery 3: The ouer edge of he earch enor hould be angen o he boundary of he arge e during mo of he well deigned earch. From Propery 1, i wa found ha he earcher hould ravel along he perimeer of he arge e. If he earch enor i angen o he boundary of he arge e, L S will equal i maximum value. In order o develop a earch raegy which accoun for he above properie, we conider a implified problem. Conider a mobile agen, ravelling a a conan peed,, ha mu ravel around an expanding circle in he plane. The radiu of he circle expand linearly in ime a a rae of. A he iniial ime, he agen i a θ = 0, and i diance from he cener of he circle i greaer han he iniial radiu of he circle. The minimum lengh (and hu ime) pah ha he agen can ake o ravel around he expanding circle i o fir ravel a raigh line ha will inerec he expanding circle angenially and hen o ravel along he perimeer of he expanding circle. Thi pah i illuraed in Fig. 3. The oluion o hi implified problem can be lighly modified o produce he improved earch mehod, illuraed in Fig. 4. Thi earch raegy coni of alernaing raigh line and ouward piral maneuver of he earcher, which reul from a ingle rule ha govern he velociy direcion of he earcher. A he iniial ime, he earcher begin ouide of he arge e wih i enor radiu angen o he circular arge e. From hi poin forward, a each inance in ime, a line i drawn which i angen o boh he earch enor and he ouide of he arge e, a hown by he doed line in Fig. 4a. The earcher hen ravel a an angle o hi line, which i he ame lead angle calculaed in Eq. 1. During he fir porion of he earch, hi produce a linear moion of he earcher. Once he ouide of he enor fooprin become angen o he circular porion of he arge e (Fig. 4b), he angen line o he earch enor and he arge e i angen o boh a he ame poin. A he earcher coninue o ravel a a lead angle o hi angen line, i will ravel along an ouward piral uch ha he radial velociy of he earcher i equal o he arge velociy: R p = in () = (10) θ p = V co () V = (11) R p R p where R p and θ p are he radiu and angle of he earcher wih repec o he cener of he arge e.

4 A D R 3 B R 1 R C Fig. 5. Secion of arge e. P * P * Fig. 4. Propoed paern. Fig. 6. Marginal improvemen o raegy. The earcher coninue hi piral maneuver (Fig. 4c) unil i reache he mall arc which connec he circular and linear porion of he arge e (Fig. 4d). A hi poin, he earch ravel along a maller ouward piral uing he ame lead angle raegy. Following he earch raegy decribed above will produce a arge e whoe perimeer can be divided ino four ecion a hown in Fig. 5. The fir ecion, A-B, i a linear ecion. The econd ecion of he perimeer, B-C, define he arc of a circle whoe radiu, (R 1 ), i equal o he radiu of he earcher minu he enor radiu. The hird ecion of he perimeer, C-D, define he arc of a circle whoe radiu, (R ), i equal o he radiu of he earcher plu he enor radiu. The final ecion of he perimeer, D-A, i he arc of a maller circle which connec he linear ecion, A-B, and he larger circle, C- D. During he iniial age of a earch, while he arge e i large enough, he radiu of he D-A ecion, R 3, will be greaer han he radiu of he earch enor, R, when he earcher reache hi ecion of he perimeer. During hee cae, he ouide of he earch enor will alway be angen o he edge of he arge e and he earch pah will be differeniable. When he arge e ge maller however, he earch enor will compleely cover he D-A ecion of he perimeer. In hee cae, he rule decribed above will produce a harp change in he direcion of he earcher. When hi condiion occur, here i a modificaion o he earch raegy ha can lighly improve performance. The diameer of he enor which i perpendicular he line angen o he enor and arge e can be ued o define he poin P a illuraed in Fig. 6a. Whenever hi poin P i ouide he arge e, he earcher can improve performance by ravelling radially inward oward he cener of he arge e unil P ouche he arge e (Fig. 6b). Thi modificaion can alo be ued a he beginning of he earch effor (Fig. 4a). Alhough hi modificaion can provide a mall improvemen, i complicae he moion and will no conidered when calculaing he performance of he propoed mehod. In order o deermine he large iniial radiu of he arge e ha can be bounded, Ro, uing he propoed paern, i aumed ha R 3 in Fig. 5 i equal o zero. Thi conervaive aumpion reul in a lighly maller R o han i acually achievable, alhough i grealy implifie he calculaion. The pah of he earcher can now be divided ino wo porion: he raigh ecion and he ouward piral. Conider he ime,, required by he earcher o ravel he raigh porion of he earch, auming he earcher ar a R o + R. Thi ime can be calculaed a: = R o R co = R o R V (1) Thi ime i he raio of he diance R o R, hown in Fig. 7, o he velociy of he earcher parallel o hi line egmen. The radiu of he earcher immediaely afer ravelling he raigh maneuver, R p, can be calculaed: R p = R o R + (13) The radiu of he earcher afer he piral maneuver can hen be calculaed by inegraing Eq. 10 and 11 : (θ f β) R p (θ f ) = R p exp V (14) where θ f = π in he cae of a ingle earcher, and β, illuraed in Fig 7, i he angle of he arge e ravered during he raigh maneuver: Ro R β = co 1 (15) R o + R

5 R o R β R o +R R o R Fig. 7. Calculaing lengh of raigh ecion. The radiu of he large boundable circle, uing he propoed piral earch can hen be calculaed by enforcing ha he earcher finihe a cycle of he earch where i ared: R (θ f ) = R o + R. (16) Combining Eq yield an expreion ha can be olved for R o: (R o R + R or R o + R = V ( ) (θ f co 1 ((R o R )/(R o + R ))) exp (17) V For = 0, = 1, and R = 100, a circular paern can bound R o = 536, while he propoed raegy can bound R o = 65. The heoreical upper bound from Eq. 6 i 636. IV. CIRCULAR BARRIERS AGAINST INTRUDERS The hird problem conidered i how o preven an inruder from enering a circular region wihou being deeced. Thi problem i eenially he revere of he circular barrier problem decribed above. Auming he arge i iniially ouide he circle R o, and he earcher i inide of hi circle wih i enor angen o R o, we explore how he earcher hould ravel o guaranee ha he inruder minimum diance from he cener of he circle i bounded. The goal i o find a raegy o guaranee bounding for he large iniial circle Ro. For all circle maller han hi, he bounding raegy hould alo provide a mehod o expand he cleared area o i maximum. The raegy of Secion III can be revered o expand he cleared area. Thi i done by defining he lead angle oward he cener of he cleared region, which produce inward piral raher han ouward piral. Thi raegy i illuraed in Fig.8. The applicaion of hi raegy will reul in a able final rajecory, analogou o a able limi cycle, around he cleared area. For large iniial radii of he cleared area, R o, he cleared area will hrink ince he earcher canno clear he perimeer fa enough. For mall iniial radii of he cleared area, he cleared area will be enlarged ince he earcher can clear area faer han he poible invader moion fill i in. Thi exience of a able final rajecory i differen from he unable rajecorie of III. When earching for a arge on he inide of an expanding circular region, if R o i larger han Ro, he arge e will ) Fig. 8. Propoed paern for inruder. increae indefiniely, while if R o i maller han Ro, he arge e will hrink o zero area. Thi i analogou o an unable limi cycle. V. MULTIPLE SEARCHERS The raegie decribed above can be eaily exended o muliple earcher. For he earch of he corridor [11], muliple earcher can ravel abrea, effecively increaing he widh of he earch enor. Unlike he corridor cae, for he earch of he expanding or collaping circular region, muliple earcher canno ravel abrea unle hey are able o vary heir velociie o ha heir angular velociie around he perimeer are equal. Conidering he earch of he expanding circle for muliple earcher, Eq. 9 become: A T S = P T S n L S i (18) where, n i he number of earcher. Thi ugge ha he ame hree properie oulined in Secion III for a ingle earch apply o he muliple earcher cae. If he muliple earcher are equally paced around he perimeer of he arge e, he implified problem of ravering an expanding circle for a ingle earcher hown in Fig. 3 can be modified o find he opimal pah for a ingle earcher o ravel from θ = 0 wih repec o he expanding circle o θ = π/n. Thu, he ame conrol law from he ingle vehicle cae i direcly applicable o he muliple vehicle cae. Thi raegy will produce a arge e hape imilar o Fig. 5, only inead of a ingle raigh line ecion, here will be a raigh line for each earcher. Thi i illuraed in Fig. 9 which illurae hree earcher in he expanding circle cae. In order o calculae R o for muliple earcher, Eq. 14 can be modified by eing θ f = π/n. Thi impoe he conrain ha each earcher will finih he cycle where he earcher in fron of i ared. In order o furher udy he effec of i=1

6 VI. CONCLUSIONS Guaraneed raegie for everal problem of earch in he wo-dimenional plane for a mobile inruder or evader have been preened. The propoed raegie make no aumpion abou he moion of he evader, and guaranee ha he earcher will ravel wihin enor range of he evader for any rajecory of he evader. The hree problem formulaion udied include he earch for mobile arge hrough a channel bounded by parallel line, he earch for or bounding of a mobile arge which ar inide of a circular region, and he prevenion of a mobile arge from enering a circular region. While no opimaliy proof for earch of he circular region exi, he propoed raegie are hown o perform cloe o a heoreical bound on he problem. The applicaion of he raegy for muliple earcher i alo preened. κ(n) Fig Fig. 10. Search wih muliple earcher Number of Searcher Cooperaion gain, κ for muliple earcher. cooperaing earcher, we define a cooperaion gain, κ(n): κ(n) = R o(n) nr o(1) (19) where R o(n) i he maximum boundable radiu wih n earcher. Figure 10 illurae, ha κ(n) i monoonically increaing ignifying ha n earcher can bound a circle wih a radiu greaer han n ime he radiu ha can be bound by a ingle earcher. Furher, a n increae, κ(n) approache he raio of he heoreical upper bound in Eq. 6 o R o(1): lim n κ(n) = R [R πv o(1)] 1 (0) Thu, a he number of earcher increae,he maximum boundable radiu approache he heoreical upper bound. I hould alo be noed ha ince he radiu of he maximum boundable circle i roughly proporional o he number of earcher, he maximum boundable area i roughly proporional o he quare of he number of earcher. REFERENCES [1] V. Ablavky and M. Snorraon, Opimal earch for a moving arge: A geomeric approach, in AIAA Guidance, Navigaion, and Conrol Conference and Exhibi, 000. [] S. Alpern and S. Gal, The Theory of Search Game and Rendezvou. Boon, Maachue: Kluwer Academic Publiher, 003. [3] E. Anderon and S. Fekee, Two dimenional rendezvou earch, Operaion Reearch, vol. 49, 001. [4] A. Anoniade, H. J. Kim, and S. Sary, Purui-evaion raegie for eam of muliple agen wih incomplee informaion, in Proceeding of he 4nd IEEE Conference on Deciion and Conrol, 003. [5] E. M. Arkin, S. P. Fekee, and J. S. Michell, Approximaion algorihm for lawn mowing and milling, Compuaional Geomery, vol. 17, 000. [6] R. A. Baeza-Yae, J. C. Culberon, and G. J. Rawlin, Searching in he plane, Informaion and Compuaion, vol. 106, [7] D. Chudnovky and G. C. (ed), Search Theory: Some Recen Developmen. New York, New York: Marcel Dekker, [8] J. M. Dankin, A helicoper veru ubmarine earch game, Operaion Reearch, vol. 16, [9] W. H. Huang, Opimal line-weep-baed decompoiion for coverage algorihm, in Proceeding of he 001 IEEE Inernaional Conference on Roboic and Auomaion, 001. [10] R. Iaac, Differenial Game. Mineola, New York: Dover Publicaion, [11] B. Koopman, Search and Screening: General Principle wih Hiorical Applicaion. New York, New York: Pergamon Pre, [1] S. LaValle, D. Lin, L. Guiba, J. Laombe, and R. Mowani, Finding an unpredicable arge in a workpace wih obacle, in Proc. of he IEEE Inernaional Conference on Roboic and Auomaion, [13] B. M. More, M. Collin, J. Saia, and L. Yu, Ice rink and cruie mile: Sweeking a imple polygon, in Proceeding of he 1 Workhop of Algorihm Engineering, [14] M. M. Polycarpou, Y. Yang, and K. M. Paino, A cooperaive earch framework for diribued agen, in Proceeding of he IEEE Inernaional Sympoium on Inelligen Conrol, 001. [15] I. Shevchenko, Succeive purui wih a bounded deecion domain, Journal of Opimizaion Theory and Applicaion, vol. 95, [16] L. Thoma and A. Wahburn, Dynamic earch game, Operaion Reearch, vol. 39, [17] R. Vidal, O. Shakernia, H. J. Kim, D. H. Shim, and S. Sary, Probabaliic purui-evaion game: Thoery, implemenaion, and experimenal evaluaion, IEEE Tranacion on Roboic and Auomaion, vol. 18, 00. [18] A. Wahburn and R. Hohzaki, The dieel ubmarine flaming daum problem, Miliary Operaion Reearch, vol. 6, 001. [19] A. R. Wahburn, Search-evaion game in a fixed region, Operaion Reearch, vol. 8, 1980.