Analysis of decentralized potential field based multi-agent navigation via primal-dual Lyapunov theory

Transcription

1 Analyss of dcnralzd ponal fld basd mul-agn navgaon va prmal-dual Lyapunov hory Th MIT Faculy has mad hs arcl opnly avalabl. Plas shar how hs accss bnfs you. Your sory mars. Caon As Publshd Publshr Dmarogonas, Dmos V., and Emlo Frazzol. Analyss of dcnralzd ponal fld basd mul-agn navgaon va prmal-dual Lyapunov hory. 49h IEEE Confrnc on Dcson and Conrol, Dc. 5-7, 00, Hlon Alana Hol, Alana, GA, IEEE IEEE. hp://d.do.org/0.09/cdc Insu of Elcrcal and Elcroncs Engnrs Vrson Fnal publshd vrson Accssd Sa Mar 6 ::46 EDT 09 Cabl Ln Trms of Us Dald Trms hp://hdl.handl.n/7./65637 Arcl s mad avalabl n accordanc wh h publshr's polcy and may b subjc o US copyrgh law. Plas rfr o h publshr's s for rms of us.

2 49h IEEE Confrnc on Dcson and Conrol Dcmbr 5-7, 00 Hlon Alana Hol, Alana, GA, USA Analyss of Dcnralzd Ponal Fld Basd Mul-Agn Navgaon va Prmal-Dual Lyapunov Thory Dmos V. Dmarogonas and Emlo Frazzol Absrac W us a combnaon of prmal and dual Lyapunov hory for almos global asympoc sablzaon and collson avodanc n mul-agn sysms. Prvous wor provdd local analyss around h crcal pons wh h us of h dual Lyapunov chnqu. Ths papr provds analyss of h whol worspac wh h us of h rcnly nroducd combnd us of prmal and dual Lyapunov funcons. I. INTRODUCTION Th dvlopmn of advancd dcnralzd collson avodanc algorhms for larg scal sysms s an ssu of crcal mporanc n fuur Ar Traffc Conrol ATC archcurs, whr h ncrasng numbr of arcraf wll rndr cnralzd approachs nffcn. In rcn yars h applcaon of robocs collson avodanc ponal fld basd mhods o ATC has bn plord [6],[5] as a promsng alrnav for such algorhms. A common problm wh ponal fld basd pah plannng algorhms n mul-agn sysms s h snc of local mnma [7],[9]. Th smnal wor of Kosch and Rmon [8] nvolvd navgaon of a sngl robo n an nvronmn of sphrcal obsacls wh guarand convrgnc. In prvous wor, h closd loop sngl robo navgaon mhodology of[8] was ndd o mul-agn sysms. In [0],[3] hs mhod was ndd o a no accoun h volum of ach robo n a cnralzd mul-agn schm, whl a dcnralzd vrson has bn prsnd n [4], [6]. Formaon conrol for pon agns usng dcnralzd navgaon funcons was dal wh n [9], [3]. Dcnralzd navgaon funcons wr also usd for mulpl UAV gudanc n []. In [4],[6] h navgaon funcons wr dsgnd n such a way o allow agns ha had alrady rachd hr dsrd dsnaon o coopra wh h rs of h am n h cas of a possbl collson. In hs papr, a consrucon smlar o h nal navgaon funcon consrucon n [8] s usd. Hnc ach agn no longr parcpas n h collson avodanc procdur f s nal conon concds wh s dsrd goal. In ssnc, h agns mgh convrg o crcal pons whch ar no longr guarand no o concd wh local mnma. In hs papr w amn h convrgnc of h sysm usng a combnaon of prmal and dual [6] Lyapunov chnqus. In parcular, prmal Lyapunov analyss s usd o show ha h sysm convrgs o an arbrarly small nghborhood of h crcal pons and h dsrd goal confguraon. Dnsy funcons ar hn usd o yld Th auhors ar wh h Laboraory for Informaon and Dcson Sysms, Massachuss Insu of Tchnology, Cambrdg, MA, U.S.A. dmar,frazzol@m.du}. Ths wor was suppord by NASA undr h IDEAS gran NNX08AY5A. a suffcn conon for h aracors of h undsrabl crcal pons o b ss of masur zro. Ths combnaon has bn usd n [],[],[]. In parcular n [],[] a dnsy funcon s provdd for a sngl robo drvn by a navgaon funcon n a sac obsacl worspac. Prmal analyss s usd o show convrgnc o a nghborhood of h crcal pons and dnsy funcons o prov h nsably of h undsrabl crcal pons usng h proprs of h navgaon funcons. Th ffrnc n our cas s ha w consdr a sysm of mulpl movng agns drvn by dcnralzd ponal funcons and h ponal funcons ar no consdrd a pror navgaon funcons. On h conrary, h dsgnd ponals ar und proprly o sasfy appropra conons o guaran asympoc sably from almos all nal conons. Th combnaon of prmal and dual Lyapunov mhods as dscrbd abov was frs usd n [] and mor rcnly n [0] for h sablzaon of a nonlnar aud obsrvr. Ths papr s a connuaon of [5], whr w provdd a conncon bwn dnsy funcons and dcnralzd navgaon funcons, by applyng h dual Lyapunov horm of [6] o h crcal pons of hs parcular navgaon funcons, and no h whol confguraon spac, as n h currn papr. Morovr n hs papr a lss consrvav suffcn conon wh a ffrn dfnon of dcnralzd ponal funcon s drvd. Th rs of h papr s organzd as follows: Scon II prsns h sysm and dcnralzd mul-agn navgaon problm rad n hs papr. Th ncssary mahmacal prlmnars ar provdd n Scon III, whl Scon IV provds h dcnralzd conrol dsgn. Scon V ncluds h convrgnc analyss and a smulad ampl s found n Scon VI. Scon VII summarzs h rsuls of h papr and ncas furhr rsarch rcons. II. DEFINITIONS AND PROBLEM STATEMENT Consdr a group of N agns oprang n h sam planar worspac W R. L q R dno h poson of agn, and l q = [q T,...,q T N ]T b h sac vcor of all agns posons. W also dno u = [u T,...,u T N ]T. Agn moon s dscrbd by h sngl ngraor: q = u, N =,...,N} whr u dnos h vlocy conrol npu for ach agn. W consdr cyclc agns of spcfc raus r a 0, whch s common for ach agn. Th rsuls can rvally b ndd o h cas of agns wh no ncssarly common ra. Collson avodanc bwn h agns s man n h sns /0/$ IEEE 5

3 ha no nrscons occur bwn h agns scs. Thus w wan o assur ha q q j > r a,,j N, j for ach m nsan. For h collson avodanc objcv ach agn has nowldg of h poson of agns locad n a cyclc nghborhood of spcfc raus d a ach m nsan, whr d > r a. Ths s T = q : q q d} s calld h snsng zon of agn. Th proposd framwor may covr varous mul-agn objcvs. Ths nclud navgaon of h agn o dsrd dsnaon and formaon conrol. Th funcon s agn s goal funcon whch s mnmzd onc h dsrd objcv wh rspc o hs parcular agn s fulflld. In h frs cas, l q W dno h dsrd dsnaon pon of agn. W hn dfn = q q as h squard sanc of h agn s confguraon from s dsrd dsnaon q. In h formaon conrol cas, h objcv of agn s o b sablzd n a dsrd rlav poson c j wh rspc o ach mmbr j of N, whr N N s a subs of h rs of h am. Th dfnon of h ss N spcfs h dsrd formaon. W assum ha j N N j n hs papr. In hs cas, w hav = q q j c j. Whl w consdr solly h j N frs cas hr, h ovrall framwor can b ndd o h formaon conrol problm as wll as ohr dfnons of. In ordr o ncod nr-agn collson scnaros, w dfn a funcon j, for j =,...,N,j, gvn by j β j = β j, 0 β j c φβ j, c β j d 4r a, d 4r a β j 3 whr β j = q q j 4ra. W also dfn h funcon 0 whch rfrs o h worspac boundary ndd by 0 and s usd o manan h agns whn h worspac. W hav β 0 = R W r a q. Th funcon 0 s dfnd n h sam way as j,j > 0. Th posv scalar paramr c and h funcon φ ar chosn n such a way so ha j s vrywhr wc connuously ffrnabl. For ampl, w can chos φ o b a ffh dgr polynomal funcon of h form φ = a a a a + a + a 0. Th coffcns a, = 0,...,5 ar calculad so ha j s vrywhr wc connuously ffrnabl, and n parcular, a h pons β j = c,β j = d 4ra. Ths provds a sysm of s lnar algbrac quaons, h soluon of whch provds h coffcns n rms of c and d 4ra. In h squl, w wll also us h noaon = q for brvy. Movad by applcaons n ATC, and n parcular from h nd o provd congson mrcs n larg scal Ar Traffc Conrol sysms [8], w no ha n h analyss ha follows w consdr pon agns. Ths s no ncssary for h analyss of h frs par prmal Lyapunov analyss, howvr, faclas h calculaons for h dual Lyapunov analyss of h scond par. Pon agns ar acually also consdrd n [9],[3]. A scusson on h nson of hs assumpon s provdd afr h convrgnc analyss. III. MATHEMATICAL PRELIMINARIES A. Dual Lyapunov Thory For funcons V : R n R and f : R n R n h noaon V = [ V... V n f = f f n n s usd. Th dual Lyapunov rsul of [6] s sad as follows: Thorm : Gvn h quaon ẋ = f, whr f C R n, R n and f0 = 0, suppos hr ss a nonngav dnsy funcon ρ C R n \ 0}, R such ha ρf / s ngrabl on R n : } and [ fρ] > 0 for almos all 4 Thn, for almos all nal sas 0 h rajcory ss for [0, and nds o zro as. Morovr, f h qulbrum = 0 s sabl, hn h concluson rmans vald vn f ρ as ngav valus. In hs papr w us a combnaon of prmal and dual Lyapunov chnqus for almos global sably. In parcular, w us prmal Lyapunov analyss o show ha h sysm convrgs o a s ha ncluds a nghborhood of h crcal pons and h dsrd goal confguraon. Dnsy funcons ar hn usd o yld a suffcn conon for h aracors of h undsrabl crcal pons o b ss of masur zro. Ths s guarand by h sasfacon of conon 4 n a nghborhood of h crcal pons. No ha whl Thorm appls o h whol R n, w apply hr for h worspac W. Th applcaon of dnsy funcons o navgaon funcon basd sysms was also usd n []. A local vrson of Thorm was usd n [7]. Rlad conons for convrgnc o an qulbrum pon n subss of R n wr provdd n [4]. ] T IV. DECENTRALIZED CONTROL In [4],[6] h conrol law allowd agns ha had alrady rachd hr dsrd dsnaon o coopra wh h rs of h am n h cas of a possbl collson. In hs papr, w us a consrucon smlar o h nal navgaon funcon consrucon n [8]. Hnc ach agn no longr parcpas n h collson avodanc procdur f s nal conon concds wh s dsrd dsnaon. As a rsul, h drvd dcnralzd ponal funcons ar no guarand o hav h Mors propry. A local analyss of h proposd dcnralzd ponal around h crcal pons was hld n [5] usng dual Lyapunov hory [6]. In hs papr h global sably proprs of h closd-loop sysm ar amnd. Spcfcally, w qup ach agn wh a dcnralzd ponal funcon ϕ : R N [0,] dfnd as ϕ = / 5 + G Th ponn s a scalar posv paramr. Th funcon G s consrucd n such a way n ordr o rndr h moon 6

4 producd by h ngad gran of ϕ wh rspc o q rpulsv wh rspc o h ohr agns. Th conrol law s of h form u = K ϕ 6 whr K > 0 s a posv scalar gan. A. Consrucon of h G funcon In h conrol law, ach agn has a ffrn G whch rprsns s rlav poson wh all h ohr agns. In h squl w rvw h consrucon of G for ach agn, whch was nroducd n [4], [6]. In hs papr, h funcon G s consrucd o a no accoun h local snsng capabls of ach agn. To ncod all possbl nr-agn promy suaons, h mul-agn am s assocad wh an unrcd graph whos vrcs ar ndd by h am mmbrs. W us h followng noons: Dfnon : A bnary rlaon wh rspc o an agn s an dg bwn agn and anohr agn. Dfnon : A rlaon wh rspc o agn s dfnd as a s of bnary rlaons wh rspc o agn. Dfnon 3: Th rlaon lvl s h numbr of bnary rlaons n a rlaon wh rspc o agn. Th complmnary s R,C j l of rlaon j wh rspc o agn s h s ha conans all h rlaons of h sam lvl apar from h spcfc rlaon j. Th funcon j dfnd abov s calld h Promy Funcon bwn agns and j. A Rlaon Promy Funcon RPF provds a masur of h sanc bwn agn and h ohr agns nvolvd n h rlaon. Each rlaon has s own RPF. L R dno h h rlaon of lvl l wh rspc o. Th RPF of hs rlaon s gvn by b R l = j R j whr l j R l dnos h agns ha parcpa n h rlaon. Whn s no ncssary o spcfy h lvl and h spcfc rlaon, w also us h smplfd noaon b r = j P r j for h RPF for smplcy, whr r dnos a rlaon and P r dnos h s of agns parcpang n h spcfc rlaon wh rspc o agn. A Rlaon Vrfcaon Funcon RVF s dfnd by: g R l = b R l + λb R l b R l + B R,C /h l whr λ,h > 0 and B R,C l = m R C b m l l whr as prvously dfnd, R,C l s h complmnary s of rlaons of lvl-l,.. all h ohr rlaons wh rspc o agn ha hav h sam numbr of bnary rlaons wh h rlaon R. Agan for smplcy w also us h noaon B R,C l b r = b s for h rm B R,C l s S r s r whr S r dnos h s of rlaons n h sam lvl wh rlaon r. Th RVF can b now also b wrn as gr = b λb r + r. I s obvous ha for h hghs lvl b r + b r /h l = n only on rlaon s possbl so ha R,C n = and g R l = b R l for l = n. Th basc propry ha w dmand from RVF s ha assums h valu 7 of zro f a rlaon holds, whl no ohr rlaons of h sam or ohr hghr lvls hold. In ohr words should nca whch of all possbl rlaons holds. W hav h followng lms of RVF usng h smplfd noaon: a lm lm g b r 0 r b r 0 b r, b r = λ b lm b r 0 g r b r 0 b r, b r = 0. Ths lms guaran ha RVF wll bhav as an ncaor of a spcfc rlaon. Th funcon G s now dfnd as G = n L l= n R l j= g Rj l whr n L h numbr of lvls and n R l h numbr of rlaons n lvl-l wh rspc o agn. Hnc G s h produc of h RVF s of all rlaons wr. Usng h smplfd noaon, w hav G = N gr r= whr N s h numbr of all rlaon wh rspc o. W hn hav ϕ = +G/ +G/ so ha + G +G /, ϕ = / + G G G 8 W can also compu ϕ j = dj / + G j dj G j 9 A crcal pon of ϕ s dfnd by ϕ = 0. Th followng Proposon wll b usful n h followng analyss: Proposon : For vry ǫ > 0 hr ss a posv scalar Pǫ > 0 such ha f Pǫ hn hr ar no crcal pons } of ϕ n h s F = q W g r ǫ, r =,...,N \ }. Proof: A a crcal pon, w hav ϕ = 0, or G = G, whch mpls G = G. snc =. A suffcn conon for hs qualy no o hold n F s gvn by > G G, q F. An uppr bound for h rgh hand sd s gvn by G N r= g r g r ε ma W } N G g r } = ma r= W ǫ, r =,...,N. No ha h rms W W P, snc g r} ma, ma gr } ar boundd du o h bounddnss of h worspac. Ths proposon also mpls ha all h crcal pons ar rsrcd o h s B rǫ = q : 0 g r < ǫ} for som r =,...,N. No ha snc G s a produc of boundd funcons g r, s sraghforward ha for all ǫ > 0, hr ss ǫ ǫ > 0 such ha 0 G < ǫ ǫ mpls 0 g r < ǫ for a las on rlaon r =,...,N. V. CONVERGENCE ANALYSIS Th convrgnc analyss of h ovrall sysm consss of wo pars. Th frs par uss prmal classc Lyapunov analyss o show ha h sysm convrgs o an arbrarly small nghborhood of h crcal pons. W hn us dual Lyapunov analyss o show ha h s of nal conons ha drvs h sysm o pons ohr han h goal confguraons s of zro masur. 7

5 A. Prmal Lyapunov Analyss No ha h closd loop nmacs of sysm undr h conrol law 6 ar gvn by q = fq = K d + G / G d d } G. K dn + G N / GN N dn dn N G N } Dfn ϕ = ϕ. Th drvav of ϕ can b compud by ϕ = ϕ T q = K N = ϕ T ϕ = K N N = j= ϕ T ϕ j whr ϕ s dfnd n 5. Consdr ε > 0. Thn w can furhr compu N ϕ = K ϕ = + ϕ T ϕ j j = K ϕ T ϕ j K ϕ : ϕ >ε + j ϕ T ϕ j ϕ : ϕ ε + j K ϕ T ϕ j K : ϕ >εε + j ϕ T ϕ j : ϕ ε j Th rms n h frs sum, whr ϕ > ε, ar lowr boundd as follows: ε + ϕ T ϕ j ε ε j j ϕ j Usng 9 w hav ϕ j = dj / dj + G j G j For dj > mn, >, h rm dj + G j /+ n h abov quaon s mnmzd by mn so ha ε + ϕ j T ϕ j ε ε dj G j mn j W wan o achv a bound of h form ε + j ϕ T ϕ j ρ > 0, whr 0 < ρ N mn < ε. A suffcn conon for hs o hold s ma j dj G j } ε ρ ε, or quvalnly ε N ε ρ mn ma dj G j } 0 j W n compu a lowr bound on h rms n h scond sum, whr ϕ ε. No frs ha ϕ T ϕ j = G = T G dj G j /+ /+ + G dj + G j = djg T G j + dj G T G j /+ /+ + G dj + G j 8 so ha ϕ T ϕ j djg G j dj G G j 4 mn W wan o achv a bound of h form j ϕ T ϕ j ρ, whr ρ > 0. A suffcn conon for hs o hold s ha mn 4 boh and dj G mn 4 G j ρ and dj G G j ρ or quvalnly, ha ma j dj G G j } ρ 4 mn ma j dj G G j } ρ 4 mn hold. Provdd ha sasfs 0,, w hav ϕ Kρ + KN ρ assumng hr ss a las on agn such ha ϕ > ε. Th lar s srcly ngav for 0 < N ρ < ρ < ε. In ssnc, ϕ can b rndrd srcly ngav as long as hr ss a las on agn wh ϕ > ε. Thus h sysm convrgs o an arbrarly small rgon of h crcal pons, provdd ha 0 < N ρ < ρ < ε and h conons on hold. W hav: Proposon : Consdr h sysm wh h conrol law 6. Assum ha mn > 0. Pc ε > 0,ρ,ρ > 0 sasfyng 0 < N ρ < ρ < ε and assum ha 0,, hold. Thn h sysm convrgs o h s ϕ ε for all n fn m. B. Dual Lyapunov Analyss Havng sablshd convrgnc o an arbrarly small nghborhood of h crcal pons, dnsy funcons ar now usd o pos suffcn conons ha h aracors of undsrabl crcal pons ar ss of masur zro. For ϕ = ϕ, dfn ρ = ϕ α,α > 0 whch s dfnd for all pons n W ohr han h dsrd qulbrum = 0, for all N. No also ha ach ϕ s C and as valus n [0,] and hus boh h funcon ϕ and s grans ar boundd funcons n W. Hnc, ρ fulfls h ngrably conon of Thorm and s a suabl dnsy funcon for h qulbrum pon = 0, N. No ha h us of a navgaon funcon as a canda dnsy funcon n sphr worlds was also usd n [], nvolvng a sngl agn navgang n a sac obsacl nvronmn. W hav ρ = αϕ α ϕ and fρ = ρ f +ρ f == αϕ α ϕ f + ϕ α f. Whnvr ϕ = 0 for all N, w hav f = 0 and fρ = ϕ α f = ϕ α ϕ K + ϕ y

6 A suffcn conon for h rgh hand sd of h las quaon o b srcly posv s for all N. W hav ϕ + ϕ y G 4G snc ϕ ϕ y ϕ < 0 + y G = y + ϕ y + G y =, and G < 0 ϕ = 0 = + G /+ = + G /+ < 0 3 G G y + G < 0 y G G y,. Thrfor, n ordr o hav [ fρ] > 0, suffcs ha 4G G + G y < 0 4 In prvous wor w provdd a suffcn conon for 4 o hold for a smplr consrucon of h G funcon. I urns ou ha a lss consrvav conon can b drvd wh h dfnon of G usd n h currn papr. N r= W can compu ḡ r g r + ḡr g r G = N r= } ḡr g r and G = Smlar rlaons hold for h y - paral drvavs. For G = 0 w hav gr = 0 for on and only on rlaon of agn,.., gr 0 for all r r. In hs cas, s asy o chc gr λ for all r r, so ha G λ N g N r ḡ + r g } r r= Rmmbrng ha P r dnos h numbr of bnary rlaons n a rlaon, w g g r = P r + λ b r b r + b r /h No ha for gr = 0 w hav b r = 0 and b r > 0. Assumng r a = 0 w hn also hav b r = 0, and g r = 0. In hs cas w can show ha h rgh rm s srcly posv. In parcular, consdrng b r = 0, afr som calculaons w hav b r = b b r + b r /h r /h b r /h P r b r /h b r /h b r b r. Th frs rm of h rgh hand sd s srcly posv whl h scond and hrd ons ar canclld du o h snc of b r. Thrfor w hav g r > P r and G > λ N P r + N ḡ r g r }. W can also show ha n h cas whr r= r a = 0, h scond rm s qual o zro. Jus no ha ach summaon rm s a produc conanng hr g r = 0 or gr = 0. Thrfor G > λ N P r, so ha G > 0 s srcly posv for G = 0. No also ha for all rlaons, P r. By connuy, w hn now ha for small nough ǫ ǫ, w hav G λ N > 0 for 0 G < ǫ ǫ whch mpls ha G λ N > 0 for q B rǫ for som r =,...,N. Smlarly, w can show ha G λ N > 0. y Snc w can rsrc h crcal pons of ϕ o occur a h rgon q Brǫ Proposon for som r =,...,N, w hav ha G + G λ N for y ϕ = 0. For h analyss ha follows, w us h noaons ma N } ma gr } = M, W W r= ma j dj G j } = M, ma djg G j } = M 3, j ma dj G G j } = M 4. j W can now choos accorng o M = ma = ma ǫ, εn M } M 3 M 4 ε ρ mn, ρ mn 4, ρ mn whch sasfs Proposon and h conons 0,,. A suffcn conon for 4 o hold a h crcal pons s hn gvn by ǫ ǫ ma λ > mn N 5 W now us h argumn of [] mnonng ha snc 4 s sasfd acly a h crcal pons, s sasfd also n an arbrary small nghborhood around hm. From h prmal Lyapunov analyss, w now ha ndd h sysm convrgs o an arbrarly small nghborhood of h crcal pons. Th dual Lyapunov analyss guarans ha h aracors of h undsrabl crcal pons ar ss of masur zro. Th followng hn holds: Proposon 3: Consdr h sysm wh h conrol law 6. Assum ha h assumpons of Proposons, hold and ha 5 also holds. Thn h closd-loop sysm s asympocally sabl for almos all nal conons. Rmar : Th prvous analyss holds for h cas of pon agns,.., r a = 0. Th nson o h cas of arbrary r a holds for h prmal analyss cas, howvr s no sraghforward n h dual analyss, snc n hs cas h rms b r b r + b r /h and N r= ḡ r g r } ar no guarand o b posv and zro rspcvly. For h cas of nonpon agns, a consrvav conon was drvd n [5]. Th nson of h lss consrvav rsul of hs papr o nonpon agns s currnly undr nvsgaon. No also ha h agns ar only guarand o convrg o a nghborhood of h dsnaon pons, snc mn > 0 should b boundd away from zro o prvn and λ from plong o nfny. 9

7 y Fg.. Svn agns navga undr h conrol law 6. Th blac crcls rprsn h nal posons of h agns whl hr fnal locaons ar dnod by a gry crcl. Dsancs Tm Fg.. Dsancs of a parcular agn from ach of h rmanng ons as h closd-loop sysm volvs. Ths furhr jusfs h nd of an adonal lmn o h funcons so ha agn coopras wh h rs of h am n h cas of a possbl collson vn onc s alrady arbrarly clos o s dsrd dsnaon. Th radr s rfrrd o h rlvan dsgn n [4],[6]. VI. SIMULATIONS In ordr o llusra h ffcvnss of h dsgnd conrollr w hav s up a smulaon nvolvng svn agns, wh conrollr paramrs gvn by = 4,λ =,h =,R w =. Th voluon of h sysm s dpcd n Fgur, whr h blac crcls rprsn h nal posons of h agns and hr fnal dsnaons ar dnod by a gry crcl. Fgur dpcs h sancs of a parcular agn from ach of h rmanng ons. On can s ha nragn sancs sasfy h collson avodanc propry. Afr som m, nr-agn sancs bcom consan snc agns convrg o hr corrsponng dsrd dsnaons. VII. CONCLUSIONS W usd a combnaon of prmal and dual Lyapunov Thory o drv suffcn conons for asympoc sablzaon from almos all nal conons n mul-agn sysms drvn by dcnralzd navgaon-l funcons. Th prmal Lyapunov analyss guarand convrgnc of h sysm o an arbrary small nghborhood of h crcal pons. Th dual analyss hn provdd suffcn conons for h undsrabl crcal pons o hav aracors of masur zro. Fuur rsarch nvolvs nng h dual analyss rsul o h cas of non-pon agns, as wll as applyng h dcnralzd navgaon funcons framwor o h dsgn of [5]. REFERENCES [] D. Angl. An almos global noon of npu-o-sa sably. IEEE Transacons on Auomac Conrol, 496: , 004. [] J. Chn, D.M. Dawson, M. Salah, and T. Burg. Mulpl UAV navgaon wh fn snsng zon. 006 Amrcan Conrol Confrnc, pags [3] M.C. DGnnaro and A. Jadbaba. Formaon conrol for a cooprav mulagn sysm wh a dcnralzd navgaon funcon. 006 Amrcan Conrol Confrnc, pags [4] D. V. Dmarogonas, S. G. Lozou, K.J. Kyraopoulos, and M. M. Zavlanos. A fdbac sablzaon and collson avodanc schm for mulpl ndpndn non-pon agns. Auomaca, 4:9 43, 006. [5] D.V. Dmarogonas and K.J. Kyraopoulos. An applcaon of Ranzr s dual Lyapunov horm o dcnralzd navgaon. 5h IEEE Mrranan Confrnc on Conrol and Auomaon, 007. [6] D.V. Dmarogonas and K.J. Kyraopoulos. Dcnralzd navgaon funcons for mulpl agns wh lmd snsng capabls. Journal of Inllgn and Roboc Sysms, 483:4 433, 007. [7] O. Khab. Ral-m obsacl avodanc for manpulaors and mobl robos. Inrnaonal Journal of Robocs Rsarch, 5:90 98, 986. [8] D. E. Kosch and E. Rmon. Robo navgaon funcons on manfolds wh boundary. Advancs Appl. Mah., :4 44, 990. [9] S. LaVall. Plannng Algorhms. Cambrdg Unvrsy Prss, 007. [0] S. G. Lozou and K. J. Kyraopoulos. Closd loop navgaon for mulpl holonomc vhcls. Proc. of IEEE/RSJ In. Conf. on Inllgn Robos and Sysms, pags , 00. [] S.G. Lozou and A. Jadbaba. Dnsy funcons for navgaon funcon basd sysms. IEEE Transacons on Auomac Conrol, 53:6 67, 008. [] S.G. Lozou and V. Kumar. Wa npu-o-sa sably proprs for navgaon funcon basd conrollrs. 45h IEEE Conf. Dcson and Conrol, pags , 006. [3] S.G. Lozou and K.J Kyraopoulos. Navgaon of mulpl nmacally consrand robos. IEEE Transacons on Robocs, 008. o appar. [4] I. Masubuch. Analyss of posv nvaranc and almos rgonal aracon va dnsy funcons wh convrs rsuls. IEEE Transacons on Auomac Conrol, 57:39 333, 007. [5] L. Pallono, V.G. Scoro, A. Bcch, and E. Frazzol. Dcnralzd cooprav polcy for conflc rsoluon n mulvhcl sysms. IEEE Transacons on Robocs, 36:70 83, 007. [6] A. Ranzr. A dual o Lyapunov s sably horm. Sysms and Conrol Lrs, 4:6 68, 00. [7] A. Ranzr and S. Prajna. On analyss and synhss of saf conrol laws. 4nd Allron Confrnc on Communcaon, Conrol, and Compung, 004. [8] K. Spsr, D.V. Dmarogonas, and E. Frazzol. On h ransfr m comply of cooprav vhcl roung. Amrcan Conrol Confrnc, 00. o appar. [9] H.G. Tannr and A. Kumar. Formaon sablzaon of mulpl agns usng dcnralzd navgaon funcons. Robocs: Scnc and Sysms, 005. [0] J.F. Vasconclos, A. Ranzr, C. Slvsr, and P. Olvra. Combnaon of lyapunov funcons and dnsy funcons for sably of roaonal moon. 48h IEEE Conf. Dcson and Conrol, pags ,