Coverage Assessment and Target Tracking in 3D Domains

Transcription

1 Snsos 211, 11, ; do:1.339/s OPEN ACCESS snsos ISSN Atcl Covag Assssmnt and Tagt Tackng n 3D Domans Nouddn Boudga 1,, Mohamd Hamd 1 and Sthaama Iynga 2 1 Communcaton Ntwoks and Scuty Rsach Lab, Unvsty of Cathag, Aana, 283, Tunsa; E-Mal: [email protected] 2 Robotcs Rsach Laboatoy, Lousana Stat Unvsty, Baton-Roug, LA 782, USA; E-Mal: [email protected] Autho to whom cospondnc should b addssd; E-Mal: [email protected]; Tl.: xt Rcvd: 2 August 211; n vsd fom: 15 Sptmb 211 / Accptd: 15 Sptmb 211 / Publshd: 2 Octob 211 Abstact: Rcnt advancs n ntgatd lctonc dvcs motvatd th us of Wlss Snso Ntwoks (WSNs) n many applcatons ncludng doman suvllanc and mobl tagt tackng, wh a numb of snsos a scattd wthn a snstv gon to dtct th psnc of ntuds and fowad latd vnts to som analyss cnt(s). Obvously, snso dploymnt should guaant an optmal vnt dtcton at and should duc covag hols. Most of th covag contol appoachs poposd n th ltatu dal wth two-dmnsonal zons and do not dvlop statgs to handl covag n th-dmnsonal domans, whch s bcomng a qumnt fo many applcatons ncludng wat montong, ndoo suvllanc, and pojctl tackng. Ths pap poposs ffcnt tchnqus to dtct covag hols n a 3D doman usng a fnt st of snsos, pa th hols, and tack hostl tagts. To ths nd, w us th concpts of Voono tssllaton, Vtos complx, and tact by dfomaton. W show n patcula that, though a st of tatv tansfomatons of th Vtos complx cospondng to th dployd snsos, th numb of covag hols can b computd wth a low complxty. Moblty statgs a also poposd to pa hols by movng appopatly snsos towads th uncovd zons. Th tackng objctv s to st a non-unfom WSN covag wthn th montod doman to allow dtctng th tagt(s) by th st of snsos. W show, n patcula, how th poposd algothms adapt to cop wth obstacls. Smulaton xpmnts a cad out to analyz th ffcncy of th poposd modls. To ou knowldg, pang and tackng s addssd fo th fst tm n 3D spacs wth dffnt snso covag schms.

2 Snsos 211, Kywods: wlss snso ntwoks; covag hols; 3D Voono dagams; Vtos-Rps complx 1. Intoducton On among th man WSN ssus that should b addssd whl dalng wth tagt tackng and montong applcatons, n 3D nvonmnts wth obstacls, s aa covag. Ths s bcaus a snso can dtct th occunc of vnts o th psnc of hostl tagts only f thy a wthn ts snsng ang. Covag flcts how wll a zon s montod o a systm s tackd by snsos. Thfo, th WSN dtcton pfomanc dpnds on how wll th wlss snsos obsv th physcal spac und contol. Sval mtcs hav bn povdd n th ltatu to masu th qualty of covag. Among ths mtcs, on can mnton th followng: (a) th numb of covag hols; (b) th popoton of uncovd aa wth spct to th aa und montong; and (c) th so calld Avag Lna Uncovd Lngth (ALUL), whch has bn dvlopd n 2D zons to stmat th avag dstanc a mobl tagt can tavs bfo bng dtctd by on snso [1]. Th ALUL can b usd to assss th dtcton ffcncy of th WSN n mo gnal spacs. Howv, th majo shotcomng of ths appoach s ts havy computatonal load makng t non-confomng wth th sv pocssng and ngy lmtatons chaactzng WSNs. Obstacls n montod 3D domans may complcat sously th ol of th montong snsos, ncas th pow consumpton, and lmt th covag ffcncy of th pocss povdng covag contol [2,3]. Pocdus st up to mplmnt covag contol and tagt tackng ffcncy should b optmal. Thy should tak nto consdaton th gogaphc natu of th montod aa and cop wth th numb and th shap of obstacls. Ths pap poposs a covag assssmnt appoach amnabl to mplmnt advancd tagt tackng functonalts. Fst, t povds a tchnqu basd on th concpt of tacton by dfomaton appld to a spcal spac, calld th Rps complx, assocatd wth th dploymnt of a st of snsos to dvlop a low complxty algothm fo locatng covag hols. Scond, t constucts a collaboatv mchansm to pa covag hols, assumng that th snsos hav moblty capablts. Thd, th pap bulds on hgh-od Voono dagams to dfn an ffcnt schm to coodnat tackng actvts of sngl and multpl tagts. To th bst of ou knowldg, ths s th fst tm wh tacton by dfomaton and hgh-od Voono tssllatons a usd fo hol assssmnt and tagt tackng n 3D domans wth obstacls usng snsos. Th majo contbutons of ths pap a as follows: Th dfnton poposd to dstbutvly duc th Rps complx assocatd to th snsos s gnal, n th sns that t appls to a lag vaty of snso, dtcton tchnqus, montod domans, and obstacls. Th poposd coopatv covag pang appoach consdably ducs th uncovd aas and povds ffcnt handlng of obstacls wth spct to xstng mthods. Th dtcton and localzaton of hols s don wth low complxty.

3 Snsos 211, W show that th hgh-od Voono tssllatons w utlz a usful fo pfomng multpl tasks ncludng actvty schdulng and coodnaton. In addton, w show that local covag nfomaton, whn gathd usng th Voono dagam, can b usd to mplmnt covag psvng moblty modls. Th manng pat of ths pap s oganzd as follows: Scton 2 dscbs th stat of th at of covag contol n vaous aas n gnal and n 3D spacs n patcula. Scton 3 suvys th dfnton of th mathmatcal objcts ndd fo covag and tackng contol, th Vtos complx and th Voono dagam and dscusss th tacton by dfomaton. Scton 4 dscusss dffnt schms basd on th Vtos complx to dtct and count th covag hols n 3D domans, locat ths hols, and pa thm. It also dfns a spcal pocdu to duc th complxty of th Vtos complxs wthout modfyng th topologcal popts. Scton 5 sts up modls fo covag assssmnt, snso moblty, and tagt tackng. Scton 6 analyzs th complxty of th algothms constuctd n ths pap and sts som xtnsons of ou sults to mo gnal typs of snsos. Scton 7 dvlops smulaton xpmnts to valuat th pfomanc of a montong systm mplmntng ou tchnqus. Scton 8 concluds ths pap. 2. Rlatd Wok Studs on covag, hols, and bounday dtcton hav bn addssd usng th man catgos of tchnqus: gomtc mthods, statstcal/pobablstc mthods, and topologcal mthods. Studs usng pobablstc appoachs usually mak assumptons on th pobablty dstbuton of th snso dploymnt. Fkt t al. [4] assum unfomly andomly dstbutd snsos nsd a gomtc gon fo th bounday dtcton algothm. Th appoach hngs on th da that th bounday nods would hav low avag dgs than that of th nto nods and statstcally povd a dg thshold to dffntat nto and bounday nods. Kuo t al. [5] popos an o modl fo locaton stmaton usng pobablstc covag, whl Rn t al. [6] psnts an analytcal modl basd on pobablstc covag to tack movng objcts n a dnsly covd snso fld. Most of pobablstc appoachs hav focusd on th dtcton and tackng of objcts n a snso fld. Thy dd not addss oth latd ssus such as locaton of th hols, numb of such hols and pang. A numb of ltatu has addssd th statc o blankt covag. Dynamc o swpng covag [7] has bn also a common and challngng task wth applcatons angng fom scuty to houskpng. Two pmay appoachs to statc covag poblms n th ltatu. Th fst uss computatonal gomty tools appld to xact nod coodnats. Such appoachs a vy gd wth gads to nputs: on, fo xampl, must know xact nod coodnats and must know th gomty of th doman to dtmn th Dlaunay complx. To allvat th fom qumnt, many authos hav tund to pobablstc tools. Fo xampl, n [8], th autho assums a andomly and unfomly dstbutd collcton of nods n a doman wth a fxd gomty and povs xpctd aa covag. Oth appoachs gv pobablstc o pcolaton sults about covag fo andomly dstbutd nods. Th dawback of ths mthods s th fact that unfom dstbuton of nods may not b always alstc. Mo cntly, th obotcs communty has xplod how ntwokd snsos and obots can ntact and augmnt ach oth: (s.g., [9] fo mo dtals). Th a sval nw appoachs to ntwoks

4 Snsos 211, wthout localzaton that com fom sach woks n ad hoc wlss ntwoks that a not unlatd to covag qustons. On xampl s th outng algothm of [1], whch gnally woks n pactc but s a hustc mthod nvolvng hat-flow laxaton. Ths wok nvstgats th ssus of mantanng covag and connctvty by kpng mnmum numb of snso nods to opat n th actv mod. Th authos show that f th ado ang s at last twc th snsng ang, thn complt covag mpls connctvty. A dcntalzd and localzd dnsty contol algothm, calld OGDC, s dvsd to contol and mantan covag and connctvty. Howv, th appoach qus knowldg of nod locaton. Th authos clam that ths qumnt can b laxd to that ach nod knows ts latv locaton to ts nghbos.on th oth hand, Hsn and Lu [11] gv mthods fo localzng an nt ntwok f localzaton of a ctan poton s known. Thy addss th poblm of tagt tackng n fac of patal snsng covag by consdng th ffct of dffnt andom and coodnatd schdulng schms. In th coodnatd-covag algothm, a snso mght dcd to slp fo som tm aft acknowldgmnts fom ts nghbo(s) that must b actv. Ths dcsons a not synchonzd as ndvdual snsos could ngotat wth sponsos ndpndntly. Snc covag vfcaton s nhntly a gomtc poblm, many sach don n ths aa a basd on computatonal gomty, and mo pcsly on th Voono Tssllaton (and ts dual, Dlauny Tangulaton). Motvatd fom th aly succss of th applcaton of gomtc tchnqus to cop wth covag poblms (At Gally Poblm), sachs hav appld ths tchnqus to ad-hoc dstbutd snso ntwoks ([12 15]). Th most mpotant dawback of ths appoachs s that thy a too computatonally xpnsv to b mplmntd n al-tm contxts. Anoth sv lmtaton s th mpact of localzaton unctanty on th pfomanc of ths appoachs. Ths clams a wll-documntd n ([16]). In fact, to dtct covag hols, th locatons of th snsos must b xactly known. Obvously, ths cannot b always povdd, spcally whn th snsng nods a mobl. Moov, quppng snsos wth localzaton dvcs may consdably ncas th dploymnt cost of th WSN and duc ts soucs. In th followng paagaphs, w summaz th mthodologs, poblms addssd and sults of som of th cnt, notabl studs n th aa of dtcton and covag n wlss snso ntwoks. Mgudchan t al. [13] study th poblm of computng a path along whch a tagt s last o most lkly to b dtctd. Thy povd an optmal polynomal tm algothm that uss gaph thotc and computatonal gomtc (Voono dagam) mthods. Thy addss th ssus of maxmal bach path, maxmal suppot path and povd bst and wost cas covag usng computatonal gomty. Dlaunay tangulaton was usd to fnd th bst-covag path. In addton, dploymnt hustcs a povdd to mpov covag. Snc computatonal gomtc mthods qu locaton nfomaton, th authos mplmnt a locaton pocdu po to th covag schm. Ths pocdu qus that a fw of th dployd nods (calld bacons) must know th locatons n advanc (th fom GPS o p-dploymnt). L t al. [14] uss local Dlauny tangulaton, latv nghbohood gaph and th Gabl gaph to fnd th path wth th bst-cas covag. Huang t al. [15] study th poblm of k-covag. Thy popos solutons to th k-uc and k-nc (Unt Dsks and Non-Unt Dsks) covag poblms whch a modld as dcson poblms whos goal s to dtmn f ach locaton of a tagt snsng aa s suffcntly covd. Thy psnt a polynomal-tm algothm wth a gomtc appoach that uns n O(nd log d) tm.

5 Snsos 211, Ghst t al. [17] us topologcal mthods to dtct nsuffcnt snso covag and hols. In th smnal wok on usng homologcal concpts fo addssng hol dtcton and covag, th algothm dtcts hols wth no knowldg of th locaton. Although th appoachs by Ghst t al. hav many dsabl popts, th assumpton of a statc ntwok and th cntalzd schm a not sutabl fo dynamc ntwoks. 3. Mathmatcs fo Covag and Tackng Th objctv of ths scton s to povd a mathmatcal modl fo accuatly gaugng th covag dg of a montod doman n th 3D spac R 3 and pang th covag hols. Ths modl uss th Vtos Complx [6,8]. Th followng assumptons wll b usd n th nxt subsctons: Lt M b a boundd doman (o manfold) n R 3 wth non-mpty bounday M. Th bounday s assumd to b an ontabl topologcal sufac (.., a closd sufac homomophc to som numb of sphs and som numb of connctd sum of g to, fo g 1, [18]). Lt δ : R 3 R 3 R + dnotng th Eucldan dstanc. W dnot by S a st of snsos dployd n R 3 to monto M, and by S th numb of ths snsos. W wll dsgnat ndffntly by p S th snso n S and ts locaton (x p, y p, z p ) n R 3. Lt us notc, fnally, that th snsos can b dployd nsd M o outsd t Voono Dagams fo Sphcal-Dtcton Snsos Lt us assum that th snsos n S hav dntcal covd aa psntd by a ball wth adus ρ. Fo vy pa p, q S, w dnot by B(p, q) th plan, n R 3 ppndcula to sgmnt [p, q] and passng by ts mddl pont and by H(p, q) th half spac of R 3 contanng th p and dlmtd by B(p, q). Thus, B(p, q) and H(p, q) a xpssd as follows: B(p, q) = { x R 3 /δ(p, x) = δ(q, x) }. (1) H(p, q) = { x R 3 /δ(p, x) δ(q, x) }. (2) W also dnot by H M (p, q) and B M (p, q) th ntscton of H(p, q) and B(p, q) wth M, spctvly. Th Voono cll gnatd by p S s nothng but th common aa to th ( S 1) closd half spacs contanng p nvolvng th oth snsos. Thfo, th Voono cll gnatd by p s xpssd by: V S (p) = q S\p H(p, q) (3) Th Voono cll of a snso s convx and contactbl. Th common bounday of two Voono clls V S (p) V S (q) s ncludd n H(p, q). It can b a plan, a half plan, an dg, a pont, o an mpty st. Th Voono dagam assocatd to th st S of snsos dployd to monto M s th unqu subdvson dfnd n R 3 by th Voono clls assocatd to all snsos. Thus, vy cll of th subdvson contans

6 Snsos 211, th nast nghbos dfnd n S fo a snso p. Th Voono dagam of S s th st of pont blongng to th all th Voono cll. Hnc w hav: V D (S) = p S V S (p). (4) In patcula, th Voono dagam V D (S) has no vtcs and no dgs whn th snsos a locatd at collna ponts. In that cas, th facs of th Voono dagam a paalll plans. In addton, on can notc that whn p S ls on th bounday of th convx hull of S, thn th Voono cll of p s unboundd n R 3. Snc n ths pap, w a ath ntstd n pattonng a doman M nto clls accodng to k-nast nghbos n S, fo a gvn ntg 1 k n 1, w tun now to th dfnton of th Hgh-od Voono dagams, as thy a usful concpts to dfn ths sts and suppot tagt tackng. An od k Voono dagam s dfnd as follows: Lt T S contanng k snsos, th T -gnatd cll s dfnd by V (T ) = {x R 3 p T, q S T, δ(x, p) δ(x, q)}. (5) Th od k Voono dagam s gvn by: V D k (S) = T S, T =k V (T ). (6) On can asly s that th od 1 Voono dagam V D 1 (S) s just V D S, that V (T ) can b mpty, and that V D (S) nducs a patton on th doman M nto boundd componnts Vtos-Rps Complxs W consd a st of ponts S = {v 1,.., v n } cospondng to th locatons of a st of snso nods n a 3D spac. Fo bvty, (v ) 1 n wll b smply fng ndffntly to as snso nods and ponts. W suppos that ach snso s capabl of covng a dsk of adus c and communcat wth th oth snsos wthn a dstanc b 3 c. Th total gon covd by th snso ntwok can b psntd by: Γ(S) = Γ v, c (7) v S wh Γ v, c = {x R 3 : x v c }. A k-smplx (o a smplx of dmnson k) σ s an unodd st σ = {v, v 1,.., v k } S, wh v v j and δ(vı, vj), fo all j. A fac of th k-smplx σ s a (k 1)-smplx fomd by k lmnts (o vtcs) of σ. Claly, any k-smplx has xactly k + 1 facs. Th collcton of all k-smplcs of S s calld th abstact assocatd wth Γ(S). In fact, an abstact smplcal complx X s a fnt collcton of smplcs whch s closd wth spct to th ncluson of facs; manng that, f σ X, thn all facs of σ a also n X. It s notwothy that a smplcal complx s a gnalzaton of a gaph; that s, th connctvty gaph s nothng but th st of 1-smplcs of th smplcal complx assocatd to a st V of ponts n th 3D spac. Now lt us dscuss th dfnton of th Vtos-Rps complx. Ths complx captus th fatus latd to connctvty and covag of WSNs.

7 Snsos 211, Dfnton 3.1. (Vtos-ps complx) Lt S b a st of ponts n a 3D spac and a gvn adus ɛ. Th Vtos-ps complx of S, dnotd by R ɛ (S), s th smplcal complx whos k-smplcs cospond to unodd (k + 1)-tupls of ponts n S whch a paws wthn Eucldan dstanc ɛ of ach oth. A subst of k + 1 ponts n S dtmn a k-smplx of fo th Vtos-ps complx f, and only f, ach of ths ponts ls wthn th ntscton of th balls of adus ɛ cntd at th oth k ponts. Th ad, howv, may wond whth such topologcal stuctu can b computd n pactc by tny mots quppd wth ado dvcs and lmtd stoag capablts. To answ ths quston, w popos a smpl mchansm allowng a fully dstbutd constucton of th Vtos-ps complx. Though a 3-stp boadcast of connctvty nfomaton, ach snso nod can b awa of what smplcs t blongs to, and what oth smplcs ts nghbos blong to. To ths nd, w assum that vy snso nod has a unqu dntf (typcally a lay-2 addss) and has nough spac to mantan a tabl of dntfs. Th potocol pfoms as follows: 1. Intalzaton: Evy snso v boadcasts ts dntty to ts nghbos. Upon cpt of th mssag, ach snsos bulds th lst, dnotd by Σ, of -smplcs fomd by ts nghbos. 2. Edg constucton: Snso v appnds ts dntty to th vtcs n Σ to constuct th lst, say Σ 1, of all 1-smplcs t blongs to. It also dtmns th numb n of ts nghbos. Thn t nfoms ts nghbos about th 1-smplcs t bult. 3. Smplcal taton: On cvng th nfomaton fom ts nghbos, snso v stats buldng th lsts Σ j, 2 j n, by smply addng appopatly th stuctus t has cvd to th ons t has alady constuctd. An nfomal xplanaton of th constucton algothm s as follows. Smplcs of hgh dmnson a constuctd tatvly. In th fst taton, th 2-smplcs a constuctd by applyng th followng ul: < v, v j > Σ 1, < v, v k > Σ 1, < v j, v k > Σ j 1 < v, v j, v k > Σ 2 fo vy, j, and k, povdd that j, k, j k. Th uls usd fo th followng tatons a smla Homotopy and Rtacton Lt X and Y b two topologcal spacs and f, g : X Y b two maps (o contnuous functons). W say that f and g a homotopc f th s a map F : [, 1] [, 1] X such that F (x, ) = f(x), F (x, 1) = g(x), x, y X Lt x X b a gvn baspont of X. A loop basd on x s a map α : [, 1] X, such that x = α() = α(1). An quvalnc laton on th st of all loops basd at x can b dfnd by statng that loops α 1 and α 2 a quvalnt f thy a homotopc wth spct to x ; manng that th xsts a homotopy F btwn α 1 and α 2 such that F (, t) = F (1, t) = x, t [, 1].

8 Snsos 211, W dnot th quvalnc class of a loop α : [, 1] X basd at x by [α] and call t th basd homotopy class of th loop α. Th st of quvalnc classs of loops basd at x s dnotd by π 1 (X, x ) and s calld th fundamntal goup. It can b quppd wth a multplcaton dfnd by [α 1 ] [α 2 ] = [α 1.α 2 ], fo all loops [α 1 ] and [α 2 ] basd at x, wh α 1.α 2 s th loop obtand by attachng α 1 to α 2. A scond goup of homotopy, dnotd by π 2 (X, x ) can b dfnd as th st of homotopy quvalnc classs of applcatons β : [, 1] 2 X, basd at x. It s an Ablan goup, [19]. On th oth hand, a map f : X Y s calld a homotopy quvalnc f th s a map g : Y X such that f g s homotopc to th dntty functon n X and g f s homotopy to th dntty functon Y. Thus, on can say that two spacs a homotopy quvalnt f thy hav th sam shap. A dfomaton tacton of a spac X onto a subspac A X s a map f : X [, 1] X such that: f(x, ) = x, f(x, 1) A, f(a, t) = a, x X, a A, t 1. In oth wods, Th subst A s a tacton by dfomaton of th spac X f, statng fom th ognal spac X at tm, w can contnuously dfom X untl t bcoms th subspac A at tm 1 and dfomaton s pfomd wthout v movng th subspac A n th pocss. It s obvous that, f A s a tacton by dfomaton of X, thn X and A a homotopcally quvalnt. Fnally, lt K b complx, a tacton fltaton of K s a nstd fnt squnc of subcomplxs K, K K 1... K n = K. such that, fo all k, K k s a tact by dfomaton of K k+1. Thus, t can b shown asly that, K and K n hav th sam typ of homotopy and th sam homotopy goup. Lt T = {p 1,.., p k } b a smplx and A T 1 = {p 2,.., p k } b on of ts facs. Thn A = T (T 1 T 1 ) b th pat of th bounday of T that s not ntnal to T 1. Thn A s a dfomaton tact of T. Lt R(S) b th Rps complx assocatd wth S, patng th pocss of tacton of smplxs that a on th bounday of R(S), wth facs xtnal to R(S), would lad to a fltaton of R(S), say K k, k n, such that, fo all k, K k s a tact by dfomaton of K k+1 and K k+1 s obtand fom K k by addng on smplx, xtnal to K k and blongng to R(S). Th objct K has no smplx wth xtnal fac that s tactbl. 4. Covag Hol Managmnt of Sphcal Snsos In ths scton, w popos a novl dstbutd tchnqu to count th covag hols of WSN usng th tacton thoy of spacs. In patcula, w show that th Vtos-ps complx assocatd wth th WSN can b ducd to a smpl spac that s tghtly latd to th numb of hols. In th followng, lt D R 3 b a compact doman n th 3D spac R 3 and D b ts bounday. W consd that D contans no obstacls. W also consd that a collcton S = {v 1,.., v n } s dployd ov doman D and that th snsos a quppd wth local communcaton and snsng capablts. In fact, ach snso s capabl of communcatng dctly wth oth snsos n ts poxmty (wthn a gvn dstanc b ) and has a lmtd snsng ang ɛ.

9 Snsos 211, Rducng th Vtos-Rps Complx W assum, n ths subscton, a complt absnc of localzaton capablts and mtc nfomaton, n th sns that th snsos n th ntwok can dtmn nth dstanc no dcton. Und ths assumptons, w a ntstd n dsgnng dstbutd algothms fo covag assssmnt and hol dtcton. To ths nd, w nd fst to ntoduc a spcal pocdu, calld Rtact, that ducs th sz of th Vtos-ps complx whl kpng ts typ of homotopy. Rpatng ths pocdu sval tms wll lmnat all th 3-clls of th Vtos-ps complx. Lt R ɛ (S) b th Vtos-ps complx. Lt {v,..., v 3 } b a 3-smplx n R ɛ (S) such that on of th 2-cll {v, v 1, v 2 } dos not blong to anoth 3-smplx n R ɛ (S). If such a stuaton dos not xst, thn on can asly dduc that R ɛ (S) has no 3-clls. Lt X 1 and A 1 b th st of ponts x R ɛ (S) blongng to smplx {v, v 1, v 2 } and th subst of X 1 gnatd by th oth two facs, spctvly. Thn ts s asy to constuct a map h 1 : X 1 [, 1] X 1 such that: h 1 (x, ) = x, h 1 (x, 1) A 1, h 1 (a, t) = a, x X, a A 1, t 1. Map h 1 can b asly xtndd to a map such that: Rtact : R ɛ (S) [, 1] R ɛ (S) Rtact(x, ) = x, Rtact(x, 1) A, Rtact(a, t) = a, x X, a A, t 1. wh A s R ɛ (S) X 1 ) A 1. Rpatng th map Rtact sval tms wll lad to lmnatng all th 3-smplcs n R ɛ (S). Th map Rtact can also b appld sval tms to dlt all 2-smplcs and 1-smplcs that a f fac. Th sultng spac, say Rɛ d (S). Poposton 4.1. Lt S b a st of snsos. followng popts: If R ɛ (S) s path-connctd, thn Rɛ d (S) satsfs th 1. Rɛ d (S) s homotopy quvalnt to R ɛ (S) 2. th numb of hols dlmtd by Rɛ d (S) s qual to th numb of hols of th vtos spac R ɛ (S) Poof. Applyng th map Rtact sval tms hlps catng a tacton fltaton of R ɛ (S) such that: Rɛ d (S) = K K 1... K n = R ɛ (S). wh n s numb of 3-smplcs n R ɛ (S). Snc, fo vy, K s homotopy quvalnt to K +1, w can dduc that Rɛ d (S) s homotopy quvalnt to R ɛ (S). Th scond statmnt of th thom can b dducd fom th followng fatus:

10 Snsos 211, a hols s a path connctd componnt that s suoundd by th dlmtng spac (Rɛ d (S) and R ɛ (S)). Rtactng a 3-smplx n R ɛ (S) may nlag a hol but dos not lmnat t. Th tacton pocss dos not cat hols snc t opats on th smplcs that hav f facs Countng and Locatng Covag Hols To count and locat hols, w st up a 3-stp algothm. In th fst stp, w constuct th xtnal bounday of R ɛ (S). Ths s th subst of S contanng all th nods occung on f facs and facng th bounday D of th doman. In th scond stp, w dfn an algothm that dtcts hols by pogssvly tansfomng th xtnal bounday by tactng all ts xtnal smplcs. In th thd stp, th followng pocss s patd: on xtnal 2 smplx s dflatd, th Rtact map s appld sval tms to duc appang smplcs wth f facs, and th xtnal bounday s updatd. Th numb of tatons of ths pocss gvs th numb of covag hols Constuctng th Bounday of R ɛ (S) Lt us assum that th bounday D of th doman D und montong can b sn (o dtctd) by th snsos n S and that th nods n S boadcast podcally th unqu ID numbs. Th constucton s basd on th th followng actons: Evy snso nod dtctng a bounday componnt of D o fndng tslf on an xtnal fact snds ths nfomaton to ts nghbos. Th nfomaton latd to bounday dtcton, whn cvd by snsos should b put togth to fom th xtnal bounday of R ɛ (S), by smply allowng vy snso nod to know whch nghbo s on th xtnal bounday. Th nods boadcast nfomaton latd th xtnal bounday of R ɛ (S) so that vy nod on th bounday can hav a pcs pctu of th bounday Countng Covag Hols Countng th covag hols can b st up by an algothm that pats tatvly th followng majo pocdus: Bounday tacton: Lt C n b a n-smplx on th bounday of R ɛ (S) and C n 1 b on of ts xtnal facs, thn C n can b tactd usng th pocdu Rtact and th bounday s updatd by addng a nw nod (th on n C n C n 1 ), f n 2, o by dltng th nod occung n C n 1, f n = 1. Bounday dflaton: Whn all th smplcs on th bounday of hav bn tactd, a p-slctd nod n S (n chag of th count) slcts on of th nods of th nw xtnal bounday, wthdaws t fom th bounday, and ncmnts th count.

11 Snsos 211, Locatng Covag Hols It s woth notcng that, whn a dflaton of a 2-smplx on th bounday R ɛ (S) s appld aft tacton s complt, a hol s ducd fom th covag zon. Ths bcaus th slctd nod, fo dflaton, s obsvng th hol, snc t s on of th nast nods suoundng th ducd hol. Thus, ths nod can stat th constucton of th bounday of th ducd hol by dtmnng th lst of th nods suoundng mmdatly th hol. On can conclud, thfo, that any tm a dflaton s opatd, a hol can b locatd by smply constuctng ts bounday usng th nast nods to that hol Rpang Covag Hols Lt us h assum that th 3D doman D und montong has no obstacls and lt us dnot by χ (χ = 4πɛ 3 /3) th volum of th aa covd by a snso and by V ol(d) th volum of D. On can stat that th numb S of snsos n S should b hgh than th numb N = V ol(d)/χ to b abl to guaant full covag of D, at last aft hol dtcton and covag optmzaton. Thfo, w wll assum n th squl that ths condton s satsfd. Fnally, w assum that th snsos a abl to mov and dtct th xtnal bounday of D, whn thy a clos to t, lk n th abov subscton. Rpang hols ams at xtndng th covag by lmnatng th hols, o at last by shnkng consdably th sz. An algothm can b dfnd to ths pupos. It can b bult basd on th followng gnal uls: A nod dtctng th xtnal bounday M should kp sng th bounday whn t movs. A nod on th xtnal bounday of R ɛ (S) should mov towads th uncovd aa, whn t dos not s th bounday. Whn two nghbo nods on th xtnal bounday of R ɛ (S) a spaatd by a dstanc hgh than a pdfnd thshold, say θ 1, and on of thm s not sng th bounday of D, thn th snso unabl to s th bounday asks ts succsso (.., a nghbo nvolvd n th tacton of th smplx contanng ths snso) to mov towads th xtnal bounday. A nod sng th bounday should nfom ts nghbos so that thy can mov accodngly. Whn th dstanc btwn a snso s and ts nghbos on th bounday of a hol s low than a pdfnd valu, say θ 2, thn s should mov n th oppost dcton of th hol, whl th oth snsos should mov towads th hol so that whn thy s ach oth, s can wthdaw tslf fom th mnmal sufac aft nfomng ts nghbos. A nod on th xtnal bounday, fndng tslf unabl to mov nfoms, ts succsso to mov towads ts dcton. 5. Tagt Tackng n 3D Domans In ths scton, w us 3D Voono dagams to optmz snso covag and tagt tackng pfomanc. W fst popos a statgy to masu th uncovd zons of th montod gon.

12 Snsos 211, Thn, w dvlop two moblty modls that povd tagt tackng usng od k Voono dagams and optmz th covag ato of a zon usng Voono clls. Fnally, w xtnd ths modls to multpl tagt tackng. W assum n ths scton that th snsos hav sphcal covag. Th vcto-gudd cas can b addssd usng smla tchnqus Masung Uncovd Aas Assum that a locaton x wthn th suvllanc aa s not covd by any snso. Lt L(x, θ) dfn th Lna Uncovd Lngth (LUL) at locaton x wth dcton θ. Ths s th undtctd path lngth of a tagt tavlng fom locaton x wth dcton θ = (θ 1, θ 2 ), fo θ 1 2π, π/2 θ 2 π/2). Th Avag Lna Uncovd Lngth (ALUL), dnotd by ALU L(x), ntoducd n [2,21], fo th 2D spac, gvs an appoxmaton of th avag dstanc that can b mad by a tagt, movng n 3D spac, bfo bng dtctd by th snso ntwok. Th Avag Lna Uncovd Lngth (ALUL) functon can b dfnd by th followng fomula:, f x s covd. ALUL(x) = 1 π/2 2π L(x, θ (2π) 2 π/2 1, θ 2 )dθ 1 dθ 2, othws. Mo gnally, whn A s a subgon of th 3D doman und supvson, th Avag Lna Uncovd Lngth latd to A, ALUL(A), that a tagt can tavl wthn A wthout bn dtctd by a snso s gvn by th xpsson: ALUL(A) ALUL(x)dx x A, (8) A wh A s th volum of A. th ALUL mtc was dvlopd to dal wth a statc dploymnt, whch s not th cas of ou study. Whn a moblty modl s mplmntd, th topology of th WSN s no long statc. To ovcom ths, w xtnd ths noton so as to suppot snso nod moblty. Th ALUL should also vay accodng to tm and should us a functon, dnotd by th L(x, θ, t), that dfns th Lna Uncovd Lngth at locaton x wth dcton θ, at tm t. Basd on ths asonng, w dfn th mtc ALUL m (x, t) psntng th ALUL n a locaton x at tm t and gvn by: Du to snso nod moblty, th ALUL, ov tm, n a pont x wll b xpssd by: ALUL m (x) = ALUL m (x, t)dt. (9) Fnally, ALUL m (A) can b computd by Equaton (8) by placng ALUL(x) by ALUL m (x). Fom th pfomanc valuaton pspctv, two mpotant ponts should b hghlghtd: ALUL m (A, t) gvs nfomaton about th covag-psvng capablts of th moblty modl. It can b usd to stat whth th stady stat s apdly achd, and whth th moblty modl affct th dtcton pfomanc of th snso ntwok. ALUL m (A) povds nfomaton about th long-tm bhavo of th moblty modl. It can b usd to valuat th mpact of moblty on th possblty fo a tagt to b undtctd wthn th montod gon.

13 Snsos 211, Moblty Modls fo Tagt Tackng In ths scton, w show how th Voono clls can b usd to mplmnt tagt tackng usng a snso moblty modl. In fact, w dfn two moblty modls: Th fst modl s calld k-moblty modl. Snso nods n ths modl mov towad th gons wh th hostl tagt s supposd to b and collaboat to kp th tagt contolld by k snsos all th tm, To ths nd, th od k Voono dagams a usd and mantand all th tm. Th scond modl s calld smplfd modl. It ls on stmatng th uncovd zons wthn th Voono clls, usng th ALUL mtcs and movng snso nods towad th uncovd zons. Whl th fst modl s tggd by th occunc of tagts, th scond modl ams at adaptng th covd aa so that th tagts can b dtctd wth hgh pobablts. Obvously, th k-moblty modl s mo ngy-consumng than th scond snc t ncompasss th pdcton of th tagt poston and qus tackng usng k snso nods. Thfo, w suppos that th scond modl can b usd whn ngy soucs bcom scac. Th pfomanc of both modls wll b assssd n Scton 7. Moov, on can notc that th pdcton functon w a usng s tghtly latd to th covag of th zons wh th tagts a xpctd and that th moblty modls assum that nast snso nods can mov to ths zons whl ducng th covag of oth zons wh tagts a not xpctd. In fact, th gat s th numb of tagt dtcton sgnals, th btt s th pdcton pcson to command snso movmnts Th k-moblty Modl In th followng, w dstngush two cass: (a) a tagt cossng a k covd aa and (b) a tagt cossng non k covd zon Fo a Tagt Cossng a k covd Zon Th moblty algothm s tggd upon th dtcton of a tagt psnc. Evy dtctng snso snds ts dtcton sgnal to th lvant ntmdat snso (calld IS). Th latt collcts all dtcton sgnals, vfs th ntgty, dducs th cunt zon wh th tagt mght b, stmats th postons of th tagt n th nxt of tm slot, and commands k snsos to mov to monto th nw zon to nsu tackng contnuty. Typcally, th slctd zon of tagt psnc s takn among oth zons (whn mo thn k snsos dtct th tagt psnc). Ths zons a odd accodng to th pobablty of psnc of th tagt. Th zon slctd s th on psntng th hghst pobablty among thos whch a k covd. Th moblty algothm s dfnd though fv stps: 1. Assum that k snsos dtct th tagt (k > k). Th k snsos s, 1 k, snd th dtcton data d to an ntmdat nod und th fom: d = ( t,, θ t,, τ t,, s )

14 Snsos 211, wh t, = δ(x, z t, ) s th Eucldan dstanc spaatng s fom th poston z t, of th tagt as sn by s, θ t, = (α t,, β t, ) s th dcton of th vcto z t, x, and τ t, s th dtcton nstant. 2. In th cas wh dtcton sgnals a snt to dffnt ntmdat nods; th ntmdat coodnat to gath all sgnals (o at last k of thm) at a unqu nod IS, whch vfs fst th authntcaton of th mssags. 3. IS constucts: Th zon of tagt psnc Zt, tau fo ach snso basd th os mad fo th valus potd. Ths zon s dlmtd by th followng ght ponts: as dfnd by th stmatd dtcton os. ( t, ±, (α t, ± α, β t, ± β) Th most lkly tagt psnc zon Z τ (t). Sval statgs can b usd fo ths ncludng slctng th lagst ntscton of k zons of th fom Z tau t,. It can also b th lagst unon of k zons. Lt T b th st of k snsos nvolvd n th dfnton of Z τ (t). Thn, IS computs th od k Voono cll V S (T ). Obvously, t contans Z τ (t). 4. IS stmats th zon Z τ,+ (t), wh tagt z t s lkly to b n th nxt tm slot. Sval statgs can b usd fo ths stmaton ncludng xtapolaton of old postons o som nfomaton latd to tagt dcton and spd. It also stmats th most lkly nw poston of z t. 5. IS slcts k snsos basd on a spcfc cta and od thm to mov towads Z τ,+ (t) to ncas ts covag. If no cta s usd, thn th od gos to th snsos n T. A cta can smply to duc snso movmnt. Whn a cta s appld fo th slcton of k snsos to cov th nw poston, som of th slctd snsos (say k snsos) may blong to T and th oth (say k k ) hav to b addd among th nghbos of T. Ths stuaton s addssd n th followng subscton Fo a Tagt Cossng a Non k covd Zon In ths cas, only k (k k) dtcton sgnals a cvd by th ntmdat snso IS, whch should pocds at th constucton of th pobabl cunt zon of psnc of th tagt th way th pcdng algothms dos. Thn t stats th slcton of th manng (k k ) qud sgnals. Thn, t ods th movmnt of th k snso povd k montong to th tagt. Fo ths pupos, IS xcuts th followng stps: 1. IS computs th most lkly zon of tagt psnc lt z t usng th k pots fom k snsos dnotd by s 1,..., s k. 2. Fo ach k, IS slcts th nast k snsos to s. It computs th latd k Voono cll V (k) and dducs th ntscton z t V (k)

15 Snsos 211, Fo ach k, IS gts th numb of snsos k, k < k, that hav snt dtcton sgnals to IS. 4. IS classfs th k Voono clls accodng to th valu of k. Th gat k s, th most mpotant s th pobablty of psnc of th tagt n V (k). A small valu of k j nducs that th tagt s gong n o out th cll V (k). 5. IS slcts th nast k snsos nvolvd n V (k), wh k = max j k k j, and guds th (k k ) addd snsos (among th nast snsos to s ) to mov towads V (k). Fo that, t snds thm a moblty nstucton ncludng th pobablty of psnc of th tagt. A moblty nstucton s dfnd by th 3-tupl. (, α, π ) wh δ(s, p) such that q V (k), δ(s, p) δ(s, q). and α = agmax xs y wh x, y v and v s th st of th vtcs of th bounday V, π = k /k s th pobablty of psnc of th tagt n δv (k). To nhanc covag whl kpng mo moblty fdom, w mplmnt a goup moblty modl n whch gound snsos mov n goups n od to psv th k covag. To ths pupos, fo ach moblty stp, th snsos dfn andomly goups of k mmbs fo ach, th latt a not qud to b th nast nghbos. Each goup chooss andomly a had whch chooss th fst moblty stp. Th manng mmbs of th goup tak nto account ths choc to dtmn th nxt moblty stp. By ths way, vy snso s moblty wll dpnd on th ntgatng goup. Futhmo, a snso may mov fom on goup to anoth n ach moblty stp. Ths modl nabls th dfnton of ovlappng k Voono goups whch ncass th guaant of havng a k covag Smplfd Moblty Modl W popos haft a moblty modl whch s basd on th us of smpl Voono dagam to dntfy and duc covag hols. Ths modl can sv to mplmnt a moblty statgy wh a snso nod looks fo on o mo nghbos that a at last 2ρ-dstant fom t. If such nods xst, th snso nod movs towad th most dstant nghbo, dnotd by n f, wth a dstanc δ(s,n f ) 2ρ 2. Th followng sult xtnds ths statgy to th cas wh th montod gon s qud to b k-covd usng th smplfd algothm. It uss a st, dnotd by X(s, V (S)), whch dfnd th st of ntscton ponts xpssd as follows: X(s, V D (S)) = D ( V D (S \ {s }) ) Γ(s, R s ), (1) wh D, fo a gon R R 3, dnots th bounday of R. Fo th sak of pasmony, w do not povd poofs fo ths coollas n ths pap. Lmma 5.1. Fo s n S, f N(s, V D (S)) < k, wh. dnots st cadnalty, thn V D (s ) s not k-covd. Fo s n S, f X(s, V D (S)) < k, thn V D (s ) s not k-covd.

16 Snsos 211, Ths lmma shows how smpl Voono dagams can b usd to dtct th covag hols basd on th dstanc btwn th snso nod and th dgs of ts Voono cll. It s basd on th concpt that th Voono tssllaton s a patton of th ponts blongng to th montod aa accodng to th poxmty to th snso nods. In oth tms, f a pont s not dtctd by th snso nod locatd at th gnato of th Voono cll t blongs to, t cannot b dtctd by any oth snso nod. If a snso dtcts that th dstanc to on among th dgs of ts Voono dgs s mo than ts covag ang, t has to mov towads ths dg to cov th cospondng hol. Th uncovd can thfo b gadually ducd usng ths dstbutd statgy. Howv, a snso nod can dtct that mo than on of ts Voono nghbos do not fulfll th condton of th lmma, t wll thfo mov towads th most dstant nghbo. Th majo advantag of ths statgy, wth spct to th advancd statgy, s that t ls on smpl Voono dagams to dal wth k-covag whl th advancd modl poposd n th pvous subscton s basd on od k Voono tssllatons whch a mo complx to buld. A mo accuat compason btwn th two modls wll b cad out n th smulaton scton Mult-Tagt Tackng Th two tackng modls psntd n th abov can b xtndd to th tackng of multpl tagts. To dscb th xtnson lt us assum, fo th sak of claty, only two tagts a dtctd by snsos n S. Lt z t and z t b th potd postons. Th xtnson of th smplfd modl consds two cass: Only on nod has dtctd th psnc of th two tagts: In that cas, th snso kps montong on of th tagts and nvts th nast nghbo to th scond tagt to monto th scond and povds t wth lvant nfomaton t collcts. Mo than on nod hav dtctd th tagts: In that cas,two snso among thos that hav dtctd th tagts a slctd to kp montong th tagts ndpndntly. On th oth hand, th k-moblty modl xtnds n followng way: f d nods dtct th tagts, ths snsos a dvdd nto two substs, ach n chag of montong on tagt, thn th substs a xtndd so that any of thm contans k snsos. 6. Complxty Analyss of Covag Managmnt and Tackng 6.1. Complxty In ths scton, w analyz th complxty of th dffnt algothms w hav dvlopd n th pvous sctons fo th dtct and locat hols o to pa covag hols. Ou appoach to stmat th complxty can b basd on th followng mtcs: Th numb of mssags xchangd btwn th snsos dung th xcuton of th algothm. Th numb of addtons and dltons of smplcs to th Vtos complx. Th numb of snso movmnts mad dung th xcuton of an algothm.

17 Snsos 211, Som oth opatons can b addd fo a mo accuat stmaton of complxty. Ths mtcs may nclud, fo xampl, th numb of stong opatons mad at th nod lvl to updat th latd data stuctus. Th mssags xchangd dung th xcuton of an algothm can b of dffnt typs. In patcula, thy can b snt to a nghbo to tll t to chang ts status fom ntnal (to th Vtos complx) to xtnal (.., on th bounday of Vtos complx). Thy also can b usd to constuct th ntal bounday of Rps complx, o usd to duc th xtnal bounday. Thy also can b snt aft th tacton o th dflaton of a smplx, o thy a snt by a lad nod th command a coodnatd movmnt of snsos. Fo th sak of claty, w wll focus on th complxty on th dtcton and countng of covag hol. In ths cas, lt n b th numb of snsos n S, and b th numb of 1-smplcs, f th numb of 2-smplcs, and t th numb 3-smplcs n RIPS complx of S). Lt also p th numb of vtcs at th ntal bounday of th Rps complx. Th numb of mssags snt dung th xcuton of th algothm should b low o qual to th numb of mssags xchangd f all th polyhda (xtnal and ntnal) hav bn tactd fst and that aft dflaton all th facts hav bn tactd. In that cas, on can stat that th numb N 1 of mssags snt s gvn by: N = p + (f p) + ( (f p)) = p + S +, wh p s th numb of xtnal vtcs. Ths sult can b dducd fom th pcdng and th fact that p S. Now lt us assum, wthout loss of gnalty, that th dploymnt of snsos (ntal and cunt) guaants that vy nod n th Rps complx of S has at most v nghbos (v s a fxd valu copng wth th volum of th aa to monto and th adus of covag). Thn, on can conclud that s small than v S, w dduc that N n + v S (v + 1)n. Lt us notc that th assumpton s not mandatoy and a dct poof can b gvn. Ths shows that th algothm to dtct and count th covag hols has a lna complxty Extndng Rsults Th sults psntd n th pvous sctons can b xtndd n two dmnsons: th typ of th snsos and th occunc of obstacls n th doman und montong. Copng wth obstacls. Th algothms dvlopd n th pcdng sctons can b adaptd to th occunc of obstacls. Obstacls n montod 3D aas may complcat sously th ol of montong snsos, ncas th pow consumpton, and lmt th covag ffcncy. Two patcula objcts hav to b modfd n ou algothms. Fst, th covag hols that hav to b countd should not contan obstacls. on can assum fo ths that th snsos a abl to cognz an obstacl). Scond, th moblty modl usd to ncas covag o to povd tackng should consd movng th snso vtcally as an altnatv. Copng wth sm sphcal. Th algothms dvlopd n th pcdng sctons can b xtndd to sm-sphcal snsos (snso havng a sm sphcal covd aa). It s woth notcng, at ths

18 Snsos 211, pont, that ths typ s suffcntly gnal to psnt vaous snsos-bas applcatons. In patcula, th modl can b usd to psnt f and smokng-basd snsos o cama-basd snsos. To cop wth sm-sphcal snsos, on can notc that th concpts of Vtos-ps and Voono dagam can b xtndd, so that covag hols can b handls n a smla way. Howv, whn pang a hol, th moblty modl of th snso should nclud otatng a snso to ncas th covag of a spcfc aa by th snso. It s woth notcng that th addtons mad to th dvlopd algothms do not modfy sgnfcantly th complxty of th algothms. In patcula, th complxty of th hol count mans lna (as shown n th smulaton dscussd n th followng scton). 7. Expmntal Rsults In ths scton, w cay out a st of xpmnts to pov th ffcncy of th poposd tchnqus. W fst addss th covag hol poblm by valuatng th pfomanc of th hgh-od Voono-basd statgy fo covag optmzaton. To ths pupos, w dfn a mtc psntng th ato of uncovd aa wth spct to th total aa of th montod gon. Scond, W assss th tagt tackng appoach by stmatng th maxmum lna dstanc that can b mad by a hostl tagt wthout bng dtctd. Fnally, w valuat th complxty of ou covag contol and moblty tchnqus. W us th numb of tansmttd mssags as a man cton to stmat ths complxty snc data tansmsson consums much mo pow than computatonal stps n WSNs Covag Contol and Hol Rducton Th fst xpmnt ams at valuatng th hol ducton statgy basd on th-dmnsonal Voono tssllatons wth sphcal covag. W dfn th followng mtc to valuat th pfomanc of hol ducton. µ = Sum of hol volums Total volum of th montod aa. (11) Fgu 1 shows th voluton of µ accodng to th numb of tatons of th covag hol ducton algothm. W compad th Voono-basd hol ducton statgy wth th Homotopy-basd statgy poposd n Scton 4. It can b notcd that th ncas n tms of nomalzd uncovd popoton s about 15% whn th numb of taton s low. In addton, whn th numb of taton xcds 3, both appoachs pfom wll snc th nomalzd sum of uncovd aas bcoms hgh than 9%. A smla xpmnt s conductd fo vcto gudd snsos, assumng that at vy stp of th taton, th moblty s povdd along wth an ontaton of th vcto to achv btt covag. Fgu 1 shows th voluton of µ accodng to th numb of tatons of th covag hol ducton algothm and compas t to th Voono-basd hol ducton statgy wth th Homotopy-basd statgy. On can conclud that whl th homotopy-basd appoach s lss complx, snc lna fo dtcton and localzaton, th Voono-basd mthod achs btt sults. In addton, a compason btwn th sults obtand fo sphcal snsos and sm sphcal snsos shows th followng:

19 Snsos 211, Fgu 1. Evoluton of th uncovd aa popoton accodng to tm. (b) Sm-sphcal covag 1 1 No mal zdsum ofuncov da as No ma l z ds u mo fu n c o v da a s (a) Sphcal covag Cov ag duct on t at on Co v a g d u c t o n t a t o n Th appoach pfoms btt wth sphcal snsos fo th fst tatons. Indd, th nomalzd uncovd popoton achs 7%, wth sphcal snsos, aft 1 tatons, whl t stays und 1% fo sm sphcal snsos. Th appoach pfoms th sam fo both typs of snsos aft 3 tatons. Ths can b xpland by th fact that th dnsty of snsos s th sam fo both typs and, thfo, t taks mo moblty fo sm sphcal snsos th covag hols Moblty Modlng Th Avag Lna Uncovd Lngth (ALUL), dnotd by L(x, θ), gvs an appoxmaton of th avag dstanc that can b mad by a tagt, movng n 3D spac, bfo bng dtctd by th snso ntwok. Th mtc ALU Lm (x, t) psntng th ALUL n a locaton x at tm t s gvn by: ALU Lm (x, t) =, 1 (2π)2 f x s covd by a snso. R π/2 R 2π π/2 L(x, θ1, θ2, t)dθ1 dθ2, othws. Fom th pfomanc valuaton pspctv, ALU Lm (A) povds nfomaton on th covag-psvng capablts of th moblty modl and th long-tm bhavo of th moblty modl. In od to vsually llustat th pfomanc of covag ducton modls, w us th local nod dnsty dstbuton that gvs th numb of snsos that cov vy pont of th montod gon. Fgu 2 shows that, n th smpl contxt wh on tagt s movng wthn a 1 m2 -sz montod zon, th covag dg consdably vas accodng poxmty to th mobl tagt. In fact, aft 5 moblty stps, th local snso dnsty s lss than 1 n gons that a fa fom th tagt locaton (whch s (7,3)) and achs 2.7 n ponts that a clos to ths tagt. Mo ntstngly, Fgu 3 addsss th cas wh two tagts a psnt wthn th gon of ntst. W notc that th snsos a ntally unfomly dstbutd. Th dnsty thn ncass fo th th followng tatons n th gons wh th tagts a. In fact, ths povs that ou tackng schm s pcs nough to dstngush btwn th two dffnt tagts.

20 Snsos 211, Fgu 2. Local nod dnsty dstbuton aft 5 moblty tatons. Co v a g d g xpos 5 t on on t pos y- 8 1 To confm ths sults, w usd th ALULm mtc to valuat th voluton of th uncovd aa wth spct to tm. In fact, ths allows to know whth uncovd gons a catd du to th dnsty ncas n th zons that a clos to th tagt. W compad ou schm to fou known moblty modls, whch a: th andom walk modl, th andom waypont modl, th andom dcton modl, and th Gauss-Makov modl. Th sults of ths compason a dpctd n Fgu 4. W notc that th poposd moblty modls, dnotd by Advancd Voono-basd Moblty Modl (AVBMM) and Dstbutd Voono-basd Moblty Modl (DVBMM), claly outpfom th xstng modls. Thy also tun a btt pfomanc than th Dnsty-Psvng Moblty Modl. Ths s bcaus th latt modl, dspt ts ablty to guaant a naly unfom nod dnsty wthn th montod aa, dos not tak nto account th psnc of hostl tagts n th zon of ntst Complxty Evaluaton In ths subscton, w valuat th communcaton ovhad sultng fom th poposd tact-basd covag contol appoach. To ths nd, w only consd th complxty of th dtcton and localzaton stps n ou algothm and do not addss th complxty of th pa stp, snc th pa stp complxty s manly dpndnt on th fst dploymnt. Howv, on can asly dduc that f th dploymnt guaants that hols sz do not xcd a thshold, thn th lna complxty can b vfd. W consdd that th dmnsons of th montod zon a 1 m 1 m 3 m. W vad th numb of nods dployd wthn ths zon and w masud th numb of mssags qud to stup ou covag contol potocol. W fst supposd that all snso nods hav a sphcal covag of ang.5 m. Fgu 5 dpcts th numb of mssags fo dnsts angng fom.5 snsos p m2 to 5 snsos p m2. Th majo mak s that ths numb s naly lna wth spct to th numb of snsos p aa unt. Moov, w consdd th cas wh snsos hav sm-sphcal covag (wth th sam ang). Fgu 5 shows that th communcaton ovhad s also lna n ths stuaton but wth a smooth slop.

21 Snsos 211, Fgu 3. Illustaton of 3 moblty tatons fo a contxt wh two tagts a consdd. (a) Itaton 2 Co v a g d g (b) Itaton 3 1 (c) Itaton Co v a g d g Co v a g d g On can dduc th followng statmnts fom th afomntond fgus: Th numb of xchangd mssags s ndpndnt of th dnsty. It s clos to 4 fo th sphcal snsos and 1 fo sm sphcal snsos. Ths fact may appa stang; howv, on can notc that whn a dploymnt s pfomd, th dtcton and locaton wll only sach fo hols suoundd by th Vtos spac. Th latt s ducd whn th dnsty s low. Th numb of mssags xchangd by th sm sphcal snsos fo dtcton and localzaton s 2.5 tms hgh than th numb obsvd fo sphcal. Two asons can b mntond fo ths. Fst, th aa covd by a sm sphcal snsos s half th aa covd by sphcal snsos. Scond, th gudng vctos s andomly ontd.

22 Snsos 211, Fgu 4. Evoluton of th ALULm mtc accodng to tm. 12 DPRMM Ra ndomwa l k Ra ndomd c t on Ra ndomwa ypo nt Ga us s Ma kov BVBMM AVBMM 1 ALUL Numbofsnso s Fgu 5. Illustaton of complxty. (a) Itaton 1 (b) Itaton Nu mb o fm s s a g s Nu mb o fm s s a g s S ns od ns t y S ns od ns t y 4 Concluson Ths pap dvlopd a low complxty appoach to dtct and localz snsng hols n 3D spacs. It also constuctd ffcnt algothms to pa hols and tack (multpl tagts). Ou appoach has bult on two concpts, th Vtos complx and th Voono dagam, and dmonstatd that th tchnqu calld tacton by dfomaton achvs low complxty algothms fo th dtcton of covag hols n WSNs. 5

23 Snsos 211, Ou appoach can b asly xtndd to mo gnal snsos, fo whch th Vtos complx and th Voono dagam can b dfnd. Such snsos can b calld concal snsos o vcto gudd snsos and can psnt cama snsos. Rfncs 1. Gu, C.; Mohapata, P. Pow consvaton and qualty of suvllanc n tagt tackng snso ntwoks. In Pocdngs of th 1th Annual Intnatonal Confnc on Mobl Computng and Ntwokng (Mobcom 4), Phladlpha, PA, USA, 26 Sptmb 1 Octob 24; pp Lu, B.; Towsly, D. A study of th covag of lag-scal snso ntwoks. In Pocdngs of 24 IEEE Intnatonal Confnc on Mobl Ad-hoc and Snso Systms, Fot Lauddal, FL, USA, Octob 24; pp Zou, Y.; Chakabaty, K. Snso dploymnt and tagt localzaton n dstbutd snso ntwoks. ACM Tans. Embd. Comput. Syst. 24, 3, Fkt, S.P.; Koll, A.; Pfst, D.; Fsch, S.; Buschmann C. Nghbohood-basd topology cognton n snso ntwoks. In Pocdngs of ALGOSENSORS, Tuku, Fnland, July 24; pp Kuo, S.-P.; Tsng, Y.-C.; Wu, F.-J.; Ln, C.-Y. A pobablstc sgnal-stngth-basd valuaton mthodology fo snso ntwok dploymnt. In Pocdngs of 19th Intnatonal Confnc on Advancd Infomaton Ntwokng and Applcatons, AINA 25, Tap, Tawan, 28 3 Mach 25; Volum 1, pp Rn, S.; L, Q.; Wang, H.; Chn, X.; Zhang, X. A study on objct tackng qualty und pobablstc covag n snso ntwoks. SIGMOBILE Mob. Comput. Commun. Rv. 25, 9, Muhammad, A.; Jadbaba, A. Dynamc covag vfcaton n mobl snso ntwoks va swtchd hgh od Laplacans. In Pocdngs of Robotcs: Scnc and Systms, Atlanta, GA, USA, Jun Tahbaz-Salh, A.; Jadbaba, A. Dstbutd covag vfcaton n snso ntwoks wthout locaton nfomaton. In Pocdngs of 47th IEEE Confnc on Dcson and Contol, CDC 28, Cancun, Mxco, 9 11 Dcmb 28; pp Cok, P.; Ptson, R.; Rus, D. Localzaton and navgaton assstd by ntwokd coopatng snsos and obots. Int. J. Rob. Rs. 25, 24, Zhang, H.; Hou, J. Mantanng snsng covag and connctvty n lag snso ntwoks. Wl. Ad Hoc Snso Ntw. 25, 1, Hsn, C.-F.; Lu, M. Ntwok covag usng low duty-cycld snsos: Random & coodnatd slp algothms. In Pocdngs of Th 3d Intnatonal Symposum on Infomaton Pocssng n Snso Ntwoks, IPSN 4, Nw Yok, NY, USA, Apl 24; pp Abdlkad, M.; Hamd, M.; Boudga, N. Usng hgh-od Voono tssllatons fo WSN-basd tagt tackng. In Pocdngs of th IASTED Intnatonal Symposum, Dstbutd Snso Ntwoks, DSN 8, Calgay, AB, Canada, Sptmb 28; pp

24 Snsos 211, Mgudchan, S.; Koushanfa, F.; Potkonjak, M.; Svastava, M. Covag poblms n wlss ad-hoc snso ntwoks. In Pocdngs of INFOCOM 21 Twntth Annual Jont Confnc of th IEEE Comput and Communcatons Socts, Anchoag, AK, USA, Apl 21; Volum 3, pp L, X.-Y.; Wan, P.-J.; Fd, O. Covag n wlss ad hoc snso ntwoks. Comput. IEEE Tans. 23, 52, Huang, C.-F.; Tsng, Y.-C. Th covag poblm n a wlss snso ntwok. In Pocdngs of Th 2nd ACM Intnatonal Confnc on Wlss Snso Ntwoks and Applcatons, WSNA 3, San Dgo, CA, USA, 19 Sptmb 23; pp Kosknn, H. On th covag of a andom snso ntwok n a boundd doman. In Pocdngs of th 16th ITC Spcalst Smna on Pfomanc Evaluaton of Wlss and Mobl Systms, Antwp, Blgum, 31 August 2 Sptmb 24; pp Ghst, R.; Muhammad, A. Covag and hol-dtcton n snso ntwoks va homology. In Pocdngs of th 4th Intnatonal Symposum on Infomaton Pocssng n Snso Ntwoks, IPSN 5, Los Angls, CA, USA, Apl 25; pp Gaman, A. Topology of Sufacs; BCS Assocats: Moscow, ID, USA, Hatch, A. Algbac Topology; Cambdg Unvsty Pss: Cambdg, UK, D Slva, V.; Ghst, R. Covag n snso ntwoks va psstnt homology. Alg. Gom. Topology 27, 7, D Slva, V.; Ghst, R. Coodnat-f covag n snso ntwoks wth contolld boundas va homology. Int. J. Rob. Rs , c 211 by th authos; lcns MDPI, Basl, Swtzland. Ths atcl s an opn accss atcl dstbutd und th tms and condtons of th Catv Commons Attbuton lcns (