Approximation Schemes for two-player pursuit evasion games with visibility constraints

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1 Rootics: Scinc and Systms 2008 Zuich, CH, Jun 25-28, 2008 Appoximation Schms fo two-play pusuit asion gams with isiility constaints Souah Bhattachaya Sth Hutchinson Dpatmnt of Elctical and Comput Engining Unisity of Illinois at Uana Champaign Uana, Illinois {shattac, Astact In this pap, w consid th polm in which a moil pusu attmpts to maintain isual contact with an ad as it mos though an nionmnt containing ostacls. This suillanc polm is a aiation of taditional pusuitasion gams, with th additional condition that th pusu immdiatly loss th gam if at any tim it loss sight of th ad. W psnt schms to appoximat th st of initial positions of th pusu fom which it might al to tack th ad. W fist consid th cas of an nionmnt containing only polygonal ostacls. W po that in this cas th st of initial pusu configuations fom which it dos not los th gam is oundd. Moo, w poid polynomial tim appoximation schms to ound this st. W thn xtnd ou sults to th cas of aitay ostacls with smooth oundais. I. INTRODUCTION Tagt tacking is an intsting class of motion planning polms. It consids motion statgis fo a moil oot to tack a moing tagt among ostacls. In cas of an antagonistic tagt, th polm lis in th famwok of pusuit-asion which longs to a spcial class of polms in gam thoy. Th two plays in th gam a th pusu and th ad. Th goal of th pusu is to maintain a lin of sight to th ad that is not occludd y any ostacl. Th goal of th ad is to scap th isiility gion of th pusu (and ak this lin of sight) at any instant of tim. This polm has som intsting applications. In scuity and suillanc systms, tacking statgis nal moil snsos to monito moing tagts in cluttd nionmnts. In hom ca sttings, a tacking oot can follow ldly popl and alt cagis of mgncis. Tagt-tacking tchniqus in th psnc of ostacls ha n poposd fo th gaphic animation of digital actos, in od to slct th succssi iwpoints und which an acto is to displayd as it mos in its nionmnt [16]. In sugy, contollal camas could kp a patints ogan o tissu und continuous osation, dspit unpdictal motions of potntially ostucting popl and instumnts. In this wok, w addss th polm of a singl pusu tying to maintain isiility of a singl ad in a plana nionmnt containing ostacls. Th pusu and th ad ha oundd spds. W addss th following qustion: Gin th initial position of th ad, what a th initial positions of th pusu fom which it can tack succssfully? W us th tm dcidal gion to f to th st of initial positions of th pusu at which th sult of th gam is known. Similaly, w us th tm undcidal gion to f to th st of initial positions of th pusu at which th sult of th gam is unknown. Th main contiutions of this wok a as follows. Fist; w po that in an nionmnt containing ostacls, th initial positions of th pusu fom which it can tack th ad is oundd. Though this sult is tiially tu fo a oundd wokspac, fo an unoundd wokspac it is intiguing. Scond; In this wok, w poid polynomial-tim appoximation schms to ound th st of initial positions of th pusu fom which it might al to tack succssfully. If th initial position of th pusu lis outsid this gion, th ad scaps. Th siz of th gion dpnds on th gomty of th nionmnt and th atio of th maximum ad spd to th maximum pusu spd. Thid; w addss th polm of tagt tacking in an nionmnt containing non-polygonal ostacls. In th past, sachs [15] ha addssd th polm of saching an ad in non-polygonal nionmnts. How, w do not know of any pio wok that addsss th polm of tacking an ad in non-polygonal nionmnts. Fouth; although, w do not poid a complt solution to th dcidaility [5] of th tacking polm in gnal nionmnts, w psnt patial solutions y poiding polynomial tim algoithms to ound th undcidal gion. Th st of th pap is oganizd as follows. Sction II poids th latd wok. Sction III psnts th polm fomulation. Sction IV psnts polynomial tim appoximation schms to comput th dcidal gion. Sction V xtnds th appoximation schms to nionmnts containing nonpolygonal ostacls. Sction VI psnts th conclusions and futu sach dictions. II. RELATED WORK Som pious wok has addssd th motion planning polm fo maintaining isiility of a moil ad. In [4], an algoithm is psntd that opats y maximizing th poaility of futu isiility of th ad. In [14], algoithms a poposd fo disct-tim psntations of th systm in dtministic and stochastic sttings. Th algoithms com computationally xpnsi as th num of stags of th

2 gam is incasd. In [8], th authos tak into account th positioning unctainty of th oot pusu. Gam thoy is poposd as a famwok to fomulat th tacking polm, and an appoach is poposd that piodically commands th pusu to mo into a gion that has no localization unctainty in od to -localiz and tt tack th ad aftwad. In [5], th polm of tacking an ad aound a singl con is addssd. Th f wokspac is patitiond accoding to th statgis usd y th plays to win th gam. Th authos ha shown that th polm is compltly dcidal aound a singl con. How, in ality, w sldom ncount nionmnts haing singl con. Hnc th sults aout a singl con ha limitd application in al scnaios. In [18], th authos show that th polm of dciding whth o not th pusu is al to maintain isiility of th ad in a gnal nionmnt is at last NP-complt. This motiats th ncssity to us andomizd o appoximation tchniqus to addss th polm sinc any dtministic algoithm would computationally infficint. Som aiants of th tacking polm ha also n addssd. [7] psnts an off-lin algoithm that maximizs th ads minimum tim to scap fo an ad moing along a known path. In [9][3], a tagt tacking polm is analyzd fo an unpdictal tagt and an os lacking pio modl of th nionmnt. It computs a isk facto asd on th cunt tagt position and gnats a fdack contol law to minimiz it. [2] dals with th polm of stalth tagt tacking wh a oot quippd with isual snsos tis to tack a moing tagt among ostacls and, at th sam tim, main hiddn fom th tagt. Ostacls impd oth th tacks motion and isiility, and also poid hiding placs fo th tack. A tacking algoithm is poposd that applis a local gdy statgy and uss only local infomation fom th tacks isual snsos and assums no pio knowldg of tagt tacking motion o a gloal map of th nionmnt. In [19], th polm of tagt tacking has n analyzd at a fixd distanc twn th pusu and ad. Optimal motion statgis a poposd fo a pusu and ad asd on citical nts. Rsach has n don to tack on o mo ads using multipl pusus. [12] psnts a mthod of tacking sal ads with multipl pusus in an uncluttd nionmnt. In [11] th polm of tacking multipl tagts is addssd using a ntwok of communicating oots and stationay snsos. A gion-asd appoach is intoducd which contols oot dploymnt at two lls, namly, a coas dploymnt contoll and a tagt-following contoll. III. PROBLEM FORMULATION In this pap w consid a moil pusu and ad on a plan. Thy a point oots and mo with oundd spds, p (t) and (t). Thfo, p (t) : [0, ) [0, p ] and (t) : [0, ) [0, ]. W us to dnot th atio of th maximum spd of th ad to that of th pusu = p. Th wokspac contains ostacls that stict pusu and Fig. 1. d Sta gion d p p Sta Rgion associatd with th tx ad motions and may occlud th pusus lin of sight to th ad. Th initial position of th pusu and th ad is such that thy a isil to ach oth. To pnt th ad fom scaping, th pusu must kp th ad in its isiility gion. Th isiility gion of th pusu is th st of points fom which a lin sgmnt fom th pusu to that point dos not intsct th ostacl gion. Th ad scaps if at any instant of tim it can ak th lin of sight to th pusu. Visiility xtnds unifomly in all dictions and is only tminatd y wokspac ostacls (omnidictional, unoundd isiility). Now w psnt a sufficint condition of scap fo an ad in gnal nionmnts. W us it to po som impotant sults in th nxt sction. Th sufficint condition is asd on th th concpt of a sta gion. Th sta gion associatd with a tx is dfind as th gion in th f wokspac oundd y th lins suppoting th tx of th ostacl. Th shadd gion in Figu 1 shows th sta gion associatd with th tx. Th concpt of sta gion is only applical fo a conx tx(a tx of angl lss than π). Using th ida of th sta gion, a sufficint condition fo scap fo th ad can statd as follows. Sufficint Condition: If th tim quid y th pusu to ach th sta gion associatd with a tx is gat than th tim quid y th ad to ach th tx, th ad has a statgy to scap th pusus isiility gion. Th sufficint condition aiss fom th fact that if th ad achs th con fo th pusu can ach th sta gion associatd with th con, th ad may scap fom th sid of th ostacl hiddn fom th pusu. This is illustatd in figu 2. In th figu, th ad,, is at th con whil th pusu, p, is yt to ach th sta gion associatd with th con. If th pusu appoachs th sta gion fom th lft sid as shown y th solid aow, th ad can scap th isiility gion of th pusu y

3 p Sta gion Sta gion Fig. 2. Sufficint condition fo scap moing in th diction of th solid aow. On th oth hand, if th pusu appoachs th sta gion fom th ight sid as shown y th dottd aow, th ad can scap th isiility gion of th pusu y moing in th diction of th dottd aow. Th lation twn th tim takn y th pusu and ad can xpssd in tms of th distancs tald y th pusu and th ad and thi spds. Rfing to Figu 1, if d is th lngth of th shotst path of th ad fom th con, d p is th lngth of th shotst path of th pusu fom th sta gion associatd with th con and is th atio of th maximum spd of th ad to that of th pusu, th sufficint condition can also xpssd in th following way SC: If d < d p, th ad wins th gam. Fo th sak of conninc, w f to th sufficint condition as SC in th st of th pap. IV. APPROXIMATION SCHEMES FOR POLYGONAL ENVIRONMENT In this sction, w show that in any nionmnt containing polygonal ostacls, th st of initial positions fom which a pusu can tack th ad is oundd. Fist, w po th statmnt fo an nionmnt containing a singl conx polygonal ostacl. Thn w xtnd th sults to po in cas of a gnal polygonal nionmnt. This lads to ou fist appoximation schm. Thn w psnt two mo appoximation schms to ound th st of initial positions of th pusu fom which it might al to tack th ad. Th sults psntd in this sction hold fo unoundd as wll as oundd nionmnts. Consid an ad,, in an nionmnt with a singl conx polygonal ostacl haing n sids. Th dgs of th polygonal ostacl a 1, 2 n. Ey dg i is a lin sgmnt that lis on a lin l i in th plan. Lt = (x, y ) and p = (x p, y p ) dnot th initial position of th ad and th pusu spctily. Lt {h i } n 1 dnot a family of lins, ach gin y th quation h i (x, y,, ) = 0. Th psnc of th tms and in th quation imply that th quation of th lin dpnds on th initial position of th ad and th spd atio spctily. Each lin h i diids th plan into two half-spacs, namly, h + i = {(x, y) h i (x, y,, ) > 0} p h _ i h h i + a d a i d ################### $$$$$$$$$$$$$$$$$$$ ################### $$$$$$$$$$$$$$$$$$$ d a #################### $$$$$$$$$$$$$$$$$$$$ ##################### $$$$$$$$$$$$$$$$$$$$$ l d ###################### $$$$$$$$$$$$$$$$$$$$$$ a # $ i %%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&& %%%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&&& %%%%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&&&& %%%%%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&&&&& l %%%%%%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&&&&&& %%%%%%%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&&&&&&& %%%%%%%%%%%%%%%%%%%%%%%%%%%% &&&&&&&&&&&&&&&&&&&&&&&&&&&& Fig. 3. Poof of Lmma 1 and h i = {(x, y) h i (x, y,, ) < 0}. Now w us th SC to po an impotant popty latd to th dgs of th ostacl. Lmma 1: Fo y dg i, th xists a lin h i paalll to i and a cosponding half-spac h + i such that th pusu loss th gam if p h + i. Poof: Consid an dg i of a conx ostacl as shown in Figu 3. Sinc th ostacl is conx, it lis in on of th half-spacs gnatd y th lin l i. Without th loss of gnality, lt th ostacl li in th half-spac low th lin l i. Lt d a and d th lngth of th shotst path of th ad fom tics a and of th dg i spctily. Sinc th ostacl lis in th low half-spac of l i, th sta gion associatd with tics a and a in th upp half-spac of l i as shown y th gn shadd gion. Lt l a and l th lins at a distanc of da and d spctily, fom th lin l i. If th pusu lis at a distanc d gat than min( da, d ) low th lin l i, thn th tim takn y th pusu to ach th lin l i is t p d p min( da, d ) p. Th minimum tim quid y th ad to ach con a o, which is na, is gin y t = min(da,d ). Fom th xpssions of t p and t p w can s that t p > t. Hnc th pusu will ach th na of th two cons fo th ad achs lin l i. Hnc fom SC, w conclud that if th pusu lis low th lin h i paalll to i at a distanc of min( da, d ), thn th ad wins th gam y following th shotst path to th na of th two cons. In Figu 3, sinc d > d a th lin h i coincids with lin l a. Gin an dg i and th initial position of th ad, poof of Lmma 1 poids an algoithm to find th lin h i and th cosponding half-plan h + i as long as th lngth of th shotst path of th ad to th cons of an dg is computal. Fo xampl, in th psnc of oth ostacls, th lngth of th shotst path of th ad to th cons can otaind y Dijkstas algoithm. Now w psnt som gomtical constuctions quid to po th nxt thom. Rf to Figu 4. Consid a conx ostacl. Consid a point c stictly insid th ostacl. Fo l i

4 3 c Fig i i+1 i+1 a (a) i i c A polygon and its sctos l i h i h i i+1 Fig. 5. Poof of thom 1 5 i i i+1 ach i, xtnd th lin sgmnt i c to infinity in th diction i c to fom th ay c i. Dfin th gion oundd y ays c i and c i+1 as scto i c i+1. Th sctos possss th following poptis 1) Any two sctos a mutually disjoint. 2) Th union of all th sctos is th nti plan. W can xtnd th ao ida to any n sidd conx polygon. W us th constuction to po th following thom. Thom 1: In an nionmnt containing a singl conx polygonal ostacl, gin th initial position of th ad, th initial positions of th pusu fom which it can win th gam is a oundd sust of th f wokspac. Poof: Rf to Figu 5. Consid an dg i of th conx ostacl with nd points i and i+1. WLOG, th ostacl lis low l i. Lt c a point stictly insid th conx polygon. Extnd th lin sgmnts i c and i+1 c to fom scto i c i+1. By Lmma 1, using th initial position of th ad, w can constuct a lin h i paalll to i such that if th initial pusu position lis low h i, th ad wins th gam. In cas th lin h i intscts th scto i c i+1, as shown in Figu 5(a), th ad wins th gam if th initial pusu position lis in th shadd gion. In cas th lin h i dos not intsct th scto i c i+1, as shown in Figu 5(), th ad wins c () i l i th gam if th initial pusu position lis anywh in th scto. Hnc fo y scto, th is a gion of finit aa such that if th initial pusu position lis in it thn it might win th gam. Ey dg of th polygon has a cosponding scto associatd with it. Sinc ach scto has a gion of finit aa such that if th initial pusu position lis in it, th pusu might win th gam, th union of all ths gions is finit. Hnc th poposition follows. Figu 6 shows th ad in an nionmnt consisting of a hxagonal ostacl. Th polygon in th cnt oundd y thick lins shows th gion of possil pusu win. In th poof of thom 1, w gnat a oundd st fo ach conx polygonal ostacl such that th ad wins th gam if th initial position of th pusu lis outsid this st. In a simila way, w can gnat a oundd st fo a nonconx ostacl. Gin a non-conx ostacl, w constuct its conx-hull. W can po that Lmma 1 holds tu fo th conx-hull. Finally, w can us Thom 1 to po th xistnc of a oundd st. Du to limitations in spac, th poof is omittd. Fom th pious discussions, w conclud that any polygonal ostacl, conx o non-conx, sticts th st of initial positions fom which th pusu might win th gam, to a oundd st. Moo, gin th initial position of th ad and th atio of th maximum spd of th ad to th pusu, th oundd st can otaind fom th gomty of th ostacl y th constuction usd in th poof of Thom 1. Fo any polygon in th nionmnt, lt us call th oundd st gnatd y it, as th B st. If th initial position of th pusu lis outsid th B st, th ad wins th gam. Fo an nionmnt containing multipl polygonal ostacls, w can comput th intsction of all B sts gnatd y indiidual ostacls. Sinc ach B st is oundd, th intsction is a oundd st. Moo, th intsction has th popty that if th initial position of th pusu lis outsid th intsction, th ad wins th gam. This lads to th following thom. Thom 2: Gin th initial position of th ad, th st of initial positions fom which th pusu might win th gam is oundd fo an nionmnt consisting of polygonal ostacls. Poof: Th oundd st fd in this thom is th intsction of th B sts gnatd y th ostacls. If th initial pusu position dos not li in th intsction it implis that it is not containd in all th B sts. Hnc th xists at last on polygon in th nionmnt fo which th initial pusu position dos not li in its B st. By Thom 1, th ad has a winning statgy. Hnc th thom follows. Th intsction of th B sts gnatd y all th ostacls poids an appoximation of th siz of th dcidal gions. Fo any initial position of th pusu outsid th intsction, th ad wins th gam and hnc th sult is known. But w still do not know th sult of th gam fo all initial position of th pusu insid th intsction. How, w can find tt appoximation schms and duc th siz of th gion in which th sult of th gam is unknown. In th nxt susction, w psnt on such appoximation schm.

5 h 5 h h 6 c h h 2 h 3 Bounday of B st h 5 h h 6 4 h 1 5 c 6 h 2 h 3 Bounday of U st Fig. 6. B st fo an nionmnt consisting of a gula hxagonal ostacl and = 0.5. A. U st Now w psnt anoth appoximation schm that gis a tight ound of th undcidal gion. Fom Lmma 1, th ad wins th gam if p h + i fo any dg. W can conclud that if p n i=1 h+ i, th ad wins th gam. Sinc ( n i=1 h+ i )c = n i=1 (h+ i )c = n i=1 h i, wh Sc dnots th complmnt of st S, if p lis outsid n i=1 h i, th ad wins th gam. Hnc th st of initial positions fom wh th pusu might win th gam is containd in n i=1 h i. W call n i=1 h i as th U st. An impotant point to not is that th intsction can takn among any num of halfspacs. If th intsction is among th half-spacs gnatd y th dgs of an ostacl, w call it th U st gnatd y th ostacl. If th intsction is among th half-spacs gnatd y all th dgs in an nionmnt, w call it th U st gnatd y th nionmnt. Th nxt thom pos that th U st gnatd y a singl ostacl is a sust of th B st and hnc a tt appoximation. Thom 3: Fo a gin conx ostacl, th U st is a sust of th B st and hnc oundd. Poof: Consid a point q that dos not li in th B st. Fom th constuction of th B st, q must long to som half-plan h + j. If q h+ j, thn q / h j = q / n i=1 h i. This implis that th complmnt of th B st is a sust of th complmnt of th U st. This implis that th U st is a sust of th B st. Figu 7 shows th B st and U st fo an nionmnt containing a gula hxagonal ostacl. In th appndix, w psnt a polynomial-tim algoithm to comput th U st fo an nionmnt with polygonal ostacls. Th oall timcomplxity of this algoithm is O(n 2 log n) wh n is th num of dgs in th nionmnt. Figu 8 shows th ad in a polygonal nionmnt. Th gion nclosd y th dashd lins is th U st gnatd y th nionmnt fo th initial position of th ad. Th U st fo any nionmnt haing polygonal ostacls is a conx polygon with at most n sids[6]. Figu 9 shows th U st fo an nionmnt fo Fig. 7. B st and U st fo an nionmnt containing of a gula hxagonal ostacl and = 0.5. Th polygon oundd y thick lins is th B st and th polygon oundd y thin lins is th U st Fig. 8. Bounday of U st U st fo a gnal nionmnt aious atio of th maximum spd of th ad to that of th pusu. In Figu 9, it can sn that as th spd atio twn th ad and th pusu incass, th siz of th U st dcass. Th siz of th U st diminishs to zo at a citical spd atio. At spd atios high than th citical atio, th ad has a winning statgy fo any initial position of th pusu. Hnc th polm coms dcidal [5] whn th atio of th maximum spds is high than a citical limit. Th nxt thom poids a sufficint condition fo scap of th ad in an nionmnt containing ostacls using th U st. Thom 4 If th U st dos not contain th initial position of ith th pusu o th ad, th ad wins th gam. Poof: To po th thom w nd th following lmma. Lmma 2: Fo 1, th ad lis insid th U st. Poof: Fo 1, p. If th pusu lis at th sam position as th ad, its statgy to win is to maintain th sam locity as that of th ad. Hnc if th pusu and th ad ha th sam initial position, th pusu can tack th ad succssfully. Sinc all th initial positions fom

6 T y d o a θ t d d x C h t Fig. 9. U st fo a aious spd atios of th ad to that of th pusu Fig. 11. A cicula ostacl in f spac d a K KMKMKMKMKMK KNKNKNKNKNK KNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKM KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKN KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNKNK KMKMKMKMKMKMKMKMKMKMKMKMKMKMK KNKNKNKNKNKNKNKNKNKNKNKNKNKNK a KOK KPKK K K i K K d K a K Fig. 10. A polygon in f spac. Th gion shadd in d is otaind y using Lmma 1. Th gion shadd in gn gts addd y using a tt appoximation schm. which th pusu can win th gam must containd insid th U st, th ad position must also insid th U st. Rfing ack to th poof of Thom 4, y dfinition of th U st, if th pusu lis outsid th U st, it loss. If th ad lis outsid th U st, Lmma 2 implis > 1. If > 1, > p. If > p, th ad wins th gam in any nionmnt containing ostacls. Its winning statgy is to mo on th conx hull of any ostacl. B. Discussion In th pious sctions, w ha poidd a simpl appoximation schm fo computing th st of initial pusu positions fom which th ad can scap asd on th intsction of a family of half-spacs. A slight modification to th poposd schm lads to a tt appoximation. In th poof of Lmma 1, w psntd an algoithm to find a half-spac fo y dg of th polygon such that if th initial position of th pusu lis in th half-spac, th ad wins th gam. All th points in th half-spac a at a distanc gat than da fom l i. By imposing th condition that th minimum distanc of th dsid st of points fom l i in th f wokspac should gat than da, w can includ d a l i h i mo points in th dcidal gions as shown in Figu 10. Th figu shows an ostacl in f spac. Fom th poof of Lmma 1, w gt th half-spac shadd in d. By adding th nw condition, th gion shadd in gn gts includd. Whn w pat this fo y dg, th st of initial positions fom which th pusu might win th gam gts ducd and lads to a tt appoximation of th dcidal gions. Th ounday of th shadd gion consists of staight lins and ac of cicls. Th ounday of th dsid st is otaind y computing th intsctions among a unch of ays and acs of cicls gnatd y ach dg. In this cas a tt appoximation coms at th cost of xpnsi computation. W li that tt appoximation schms xist and on of ou ongoing ffots is in th diction of otaining computationally fficint appoximation schms. Non of th appoximation schms w ha suggstd so fa stict th initial position of th pusu to in th ads isiility gion. This condition can imposd y taking an intsction of th output of th appoximation algoithm with th isiility polygon at th ads initial position. Efficint algoithms xist fo computing th isiility polygon of a static point in an nionmnt[10]. In th nxt sction w xtnd th ida of U st to nionmnts containing non-polygonal ostacls. V. APPROXIMATION METHODS FOR NON-POLYGONAL OBSTACLES In this sction w xtnd th appoximation schms psntd in th pious sction to non-polygonal ostacls. In od to illustat th tchniqus quid to handl nonpolygonal ostacls, w comput an appoximation fo th initial positions of th pusu fom which th ad wins th gam fo th simpl cas of an ad in an nionmnt containing a cicula ostacl. Thn w psnt th algoithm fo any nionmnt containing conx ostacls with smooth oundais. Figu 11 shows an ad,, in an nionmnt containing a cicula ostacl of adius a in f spac. Th ounday of th ostacl is dnotd y C. Lt t a point on C such that Ot = θ and t = d. T dnots th tangnt to C at t. Lt

7 h t a lin at a distanc of d fom T on th sam sid of T as th ostacl. By Lmma 1, th ad wins th gam if th pusu lis in th half-spac h + t, shown y th shadd gion. Th quation of lin h t is y + x cot θ (a d ) csc θ = 0. Fo y point t on C, th xists a lin h t and th cosponding half-spac h + t such that if th initial position of th pusu lis in h + t, th ad wins th gam. Hnc if th initial pusu position lis in t C h + t, th ad wins th gam= if th initial pusu position lis outsid t C h t, th ad wins th gam. Lt us call t C h t as th U st. Now w comput th ounday of th U st. Lt l(x, y, θ) dnot th family of lins h t gnatd y all points t lying on C. Du to symmty of th nionmnt aout th x- axis, th U st is symmtic aout th x-axis. W psnt th constuction of th ounday of th U st gnatd as θ incass fom 0 to π. Lt U dnot th ounday of th U st. Thom 6- U is th nlop of th family of lins l(x, y, θ). Poof: Consid any point q on U. Sinc q longs to th ounday of th U st, it longs to som lin, h q, in th family l(x, y, θ). Eith h q is tangnt to U o ls it intscts U. In cas it intscts U, th is a nighohood aound q in which U lis in oth th half-spacs gnatd y h q. This is not possil sinc on of th half-spacs gnatd y h q has to ntily outsid th U st. Hnc h q is tangnt to U. Sinc q is any point on B, it implis that fo all points q on U, th tangnt to U at q longs to th family l(x, y, θ). A cu satisfying this popty is th nlop to th family of lins l(x, y, θ). Hnc th poposition follows. Using th Enlop thoms [20], th nlop of a family of lins l(x, y, θ) can otaind y soling th following quations simultanously l(x, y, θ) = y + x cot θ (a d ) csc θ = 0 (1) l θ = 0 (2) d as a function of θ is gin y { d a2 + d (θ) = 2 d2 2ad cos θ if θ θ 0 a 2 + a(θ θ 0 ) if θ θ 0 wh θ 0 = cos 1 a d. Th solution is A. Cas 1 (θ θ 0 ) a2 + d x = (a 2 2ad cos θ ad sin 2 θ ) cos θ+ a 2 + d 2 2ad cos θ a2 + d y = (a 2 2ad cos θ ad sin θ cos θ ) sin θ a 2 + d 2 2ad cos θ (a) (c) () (d) Fig. 12. (a) shows a cicula ostacl with th initial position of th ad. Th small cicl is th ad. In (),(c) and (d), d = 5, 7and 9 units spctily. In ach of th figus (), (c) and (d), th lack ounday is fo = 0.5, th gn ounday is = 1 and th d ounday is fo = 10 B. Cas 2 (π θ θ 0 ) x = (a y = (a d2 a 2 + a(θ θ 0 ) d2 a 2 + a(θ θ 0 ) ) cos θ + sin θ ) sin θ cos θ Sinc U is symmtical aout th x axis, th oth half of U is otaind y flcting th ao cus aout th x axis. Figu 12(a) shows an ad in an nionmnt consisting of a disc-lik ostacl. Figus 12(),(c) and (d) show th ounday of th U st fo aying distanc twn th ad and th ostacl. In ach of ths figus, th ounday of th U st is shown fo th diffnt alus of. W can s that fo 1, th ad lis insid th U st as gin y Lmma 2. Th ao pocdu can usd to constuct th U st fo any conx ostacl with smooth ounday. Gin th initial position of th ad, w psnt th pocdu to constuct th ounday of th U st fo a ostacl with smooth ounday. Consid an ostacl with smooth ounday gin y th quation f(x, y) = 0. Th pocdu to gnat th ounday of th U st is as follows 1) Gin any point t on th ounday, comput th minimum distanc of th point fom th ad. Lt it d t. 2) Find th quation of th lin h t at a distanc of dt fom th tangnt to th ostacl at t. 3) Find th family l(x, y, θ) of lins gnatd y h t as t ais along th ounday of th ostacl. θ is a paamt that dfins t.

8 4) Comput th nlop of th family l(x, y, θ). This is th ounday of th U st. This is tu sinc th poof of Thom 5 dos not dpnd on th shap of th ostacl. VI. CONCLUSION AND FUTURE RESEARCH In this wok w addss th polm of tagt-tacking in gnal nionmnts. W po that in a gnal nionmnt containing ostacls, gin th initial position of th ad, th st of initial positions fom which th pusu might al to tack th ad is oundd. Moo w poid an appoximation algoithm to constuct a conx polygonal gion to ound that gion. W poid a sufficint condition fo scap of th ad in a gnal polygonal nionmnt that dpnds on th gomty of th ostacls, th initial position of th ad and th atio of th maximum spd of th ad to that of th pusu. W xtnd th appoximation schms to ostacls with smooth oundais. Gin th complt map of th nionmnt, ou sults dpnd only on th initial position and th maximum spds of th pusu and ad. Hnc ou sults hold fo aious sttings of th polm such as an unpdictal o pdictal ad [14] o localization unctaintis in th futu positions of th plays [8] o dlay in pusus snsing ailitis [17]. In th futu, w would lik to poid an algoithm to appoximat th initial positions of th pusu fom which it can tack th ad and also th statgis usd y th pusu to tack succssfully. W a using gam-thoy as a famwok to poid fdack statgis fo th pusu to tack succssfully. W a also instigating th polm of tagt-tacking with multipl pusus. An intsting diction of futu sach would to xtnd ou sults to th tagt-tacking in R 3. Rsachs ha addssd th polm of tagt-tacking in R 3 [1]. W li that som of ou sults can usd in 3-d y considing polyhdons as ounding sts instad of polygons. Anoth diction of futu sach would to incopoat dynamics in th plays motion modl. REFERENCES [1] T. Bandyopadhyay, M.H. Ang J, and D. Hsu. Motion planning fo 3-D tagt tacking among ostacls. Intnational Symposium on Rootics Rsach, [2] T. Bandyopadhyay, Y. Li, M.H. Ang J, and D. Hsu. Stalth Tacking of an Unpdictal Tagt among Ostacls. Pocdings of th Intnational Wokshop on th Algoithmic Foundations of Rootics, [3] T. Bandyopadhyay, Y. Li, M.H. Ang J., and D Hsu. A Gdy Statgy fo Tacking a locally Pdicatal Tagt among Ostacls. Rootics and Automation, Pocdings. ICRA02. IEEE Intnational Confnc on, pags , [4] C. Bck, H. Gonzalz-Banos, J.C. Latom, and C. Tomasi. An intllignt os. Pocdings of Intnational Symposium on Expimntal Rootics, pags 94 99, [5] Souah Bhattachaya, Salato Candido, and Sth Hutchinson. Motion statgis fo suillanc. In Rootics: Scinc and Systms - III, [6] M. d Bg, M. an Kld, M. Omas, and O. Schwazkopf. Computational Gomty- Algoithms and Applicaions. Sping-Vlag, Blin Hidlg, [7] A. Efat, HH Gonzalz-Banos, SG Koouo, and L. Palaniappan. Optimal statgis to tack and captu a pdictal tagt. Rootics and Automation, Pocdings. ICRA03. IEEE Intnational Confnc on, 3, [8] P. Faiani and J.C. Latom. Tacking a patially pdictal ojct with unctainty and isiility constaints: a gam-thotic appoach. Tchnical pot, Tchnical pot, Uniisty of Stanfod, Dcm stanfod. du/.(citd on pag 76). [9] HH Gonzalz-Banos, C.Y. L, and J.C. Latom. Ral-tim cominatoial tacking of a tagt moing unpdictaly among ostacls. Rootics and Automation, Pocdings. ICRA02. IEEE Intnational Confnc on, 2, [10] J. E. Goodman and J. O. Rouk. Handook of Disct and Computational Gomty. CRC Pss, Nw Yok, [11] B. Jung and G.S. Sukhatm. Tacking Tagts Using Multipl Roots: Th Effct of Enionmnt Occlusion. Autonomous Roots, 13(3): , [12] Pak L. Algoithms fo Multi-Root Osation of Multipl Tagts. Jounal on Autonomous Roots, 12: , [13] J. P. Laumond, S. Skhaat, and F. Lamiaux. Guidlins in Nonholonomic motion planning fo Moil Roots. Sping, [14] S. M. LaVall, H. H. Gonzalz-Banos, C. Bck, and J. C. Latom. Motion statgis fo maintaining isiility of a moing tagt. In Rootics and Automation, Pocdings., 1997 IEEE Intnational Confnc on, olum 1, pags , Aluququ, NM, USA, Apil [15] S. M. LaVall and J. Hinichsn. Visiility-asd pusuit-asion: Th cas of cud nionmnts. IEEE Tansactions on Rootics and Automation, 17(2): , Apil [16] T.Y. Li, J.M. Lin, S.Y. Chiu, and T.H. Yu. Automatically gnating itual guidd tous. Comput Animation Confnc, pags , [17] R. Muita, A. Saminto, and S. Hutchinson. On th xistnc of a statgy to maintain a moing tagt within th snsing ang of an os acting with dlay. Intllignt Roots and Systms, 2003.(IROS 2003). Pocdings IEEE/RSJ Intnational Confnc on, 2, [18] R. Muita-Cid, R. Monoy, S. Hutchinson, and J. P. Laumond. A complxity sult fo th pusuit-asion gam of maintaining isiility of a moing ad. Accptd in IEEE Intnational Confnc on Rootics and Automation, [19] R. Muita-Cid, T. Muppiala, A. Saminto, S. Bhattachaya, and S. Hutchinson. Suillanc statgis fo a pusu with finit snso ang. Intnational Jounal of Rootics Rsach, pags , [20] E. Silg. Th Vin-Wong Enlop Thom. Jounal of Economic Education, 30(1):75 79, VII. APPENDIX A. Algoithm fo gnating th U-st Algoithm CONSTRUCTUSET(S,,(x, y )) Input: A st S of disjoint polygonal ostacls, th ad position = (x, y ), atio of maximum ad spd to maximum pusu spd Output: Th coodinats of th tics of th U st 1) Fo y dg i in th nionmnt with nd-points a i, i 2) l 1 =DIJKSTRA(VG(S),, a i ) 3) l 2 =DIJKSTRA(VG(S),, i ) 4) d i = min(l1,l2) 5) Find th quation of h i using Lmma 1. 6) INTERSECTHALFPLANES(h 1,...h n ) Th suoutin VG(S), computs th isiility gaph of th nionmnt S. Th suoutin DIJKSTRA(G,I,F) computs th last distanc twn nods I and F in gaph G. Th suoutin INTERSECTHALFPLANES(h 1,..., h n ) computs th intsction of th half plans h 1,..., h n [6]. Th tim complxity of th ao algoithm is O(n 2 log n), wh n is th num of dgs in th nionmnt.