Formation Control and Collision Avoidance for Multi-Agent Systems and a Connection between Formation Infeasibility and Flocking Behavior

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1 Formaton Contro and Coson Avodance for Mut-Agent Systems and a Connecton between Formaton Infeasbty and Focng Behavor Dmos V Dmarogonas and Kostas J Kyraopouos Abstract A feedbac contro strategy that acheves convergence of a mut-agent system to a desred formaton confguraton avodng at the same tme cosons s proposed The coson avodance obectve s handed by a decentrazed navgaton functon that vanshes when the desred formaton tends to be reazed When nter-agent obectves that specfy the desred formaton cannot occur smutaneousy n the state space the desred formaton s nfeasbe It s shown that under certan assumptons, formaton nfeasbty forces the agents veocty vectors to a common vaue at steady state Ths provdes a connecton between formaton nfeasbty and focng behavor for the mut-agent system I ITRODUCTIO Mut-agent avgaton s a fed that has recenty ganed ncreasng attenton both n the robotcs and the contro communtes, due to the need for autonomous contro of more than one mobe robotc agents n the same worspace Whe most efforts n the past had focused on centrazed pannng, specfc rea-word appcatons have ead researchers throughout the gobe to turn ther attenton to decentrazed concepts The motvaton for ths wor comes from many appcaton domans one of the most mportant of whch s the fed of mcro robotcs, ([1],[7]), where a team of a potentay arge number of autonomous mcro robots must cooperate n the sub mcron eve Among the varous specfcatons that the contro desgn ams to mpose on the mut-agent team, formaton convergence and achevement of focng behavor are two obectves that have been pursued extensvey n the ast few years The man feature of formaton contro s the cooperatve nature of the equbra of the system Agents must converge to a desred confguraton encoded by the nteragent reatve postons Many feedbac contro schemes that acheve formaton stabzaton to a desre formaton n a dstrbuted manner have been proposed n terature, see for exampe [],[1],[11],[3] for some recent efforts Of partcuar nterest s aso the so-caed agreement probem, n whch agents must converge to the same pont n the state space ([15], [18],[],[9]) On the other hand, focng behavor nvoves convergence of the veocty vectors and orentatons of the agents to a common vaue at steady state; contrbutons ncude [8], [19],[17] The authors are wth the Contro Systems Lab, Department of Mechanca Engneerng, atona Technca Unversty of Athens, 9 Heroon Poytechnou Street, Zografou 1578, Greece ddmar,yra@mantuagr In ths paper, the probem of formaton contro s consdered The man feature of formaton contro s the cooperatve nature of the equbra of the system Agents must converge to a desred confguraton encoded by the nter-agent reatve postons Inspred by our prevous wor ([3],[5]) nvovng decentrazed navgaton and coson avodance of mutagent systems to non-cooperatve equbra (e each agent had a specfc goa confguraton not reated to the postons of the others) n ths paper we propose a methodoogy that handes the probem of formaton contro satsfyng at the same tme, the coson avodance obectve for the sphere word case We must note that the same probem has been consdered n [] for the pont word case In ths paper, we expot our prevous resuts to treat the sphere word case In most cases, formaton convergence nvoves nematc modes of the agents moton, whe focng behavor dynamc ones Hence the probem of focng moton has rarey been examned n the context of nematc modes of moton In ths paper, a connecton between formaton nfeasbty and focng behavor for mutpe nematc agents s estabshed Formaton nfeasbty s equvaent to the case when nter-agent obectves cannot occur smutaneousy n the state space By decoupng the two obectves (coson avodance and formaton convergence) t s shown that under certan assumptons formaton nfeasbty forces the agents veocty vectors to a common vaue at steady state The rest of the paper s organzed as foows: secton II presents the system defnton and probem statement Secton III presents the proposed contro scheme The stabty anayss s provded n secton IV Secton V contans an nterestng resut reatng formaton nfeasbty and focng behavor In secton VI computer smuaton resuts are presented whe secton VII summarzes the concusons and ndcates our current research II SYSTEM AD PROBLEM DEFIITIO Consder a system of spherca agents operatng n the same worspace W R Each agent occupes a dsc: R = {q R : q q r } n the worspace where q R s the center of the dsc and r s the radus of the agent The confguraton space s spanned by q = [q 1,, q ] T The moton of each agent s descrbed by the snge ntegrator: q = u, = [1,, ] (1) where u denotes the veocty (contro nput) for each agent Each agents obectve s to converge to a desred reatve

2 confguraton wth respect to a certan subset of the rest of the team, n a manner that w ead the whoe team to a desred formaton Specfcay, each agent s assgned wth a specfc subset of the rest of the team, caed agent s neghborng set wth whch t can communcate n order to acheve the desred formaton Foowng the terature on formaton contro [16],[19], the desred formaton can be encoded n terms of a formaton graph: Defnton 1: The formaton graph G = {V, E, C} s an undrected graph that conssts of () a set of vertces V = {1,, } ndexed by the team members, () a set of edges, E = {(, ) V V } contanng pars of nodes that represent nter-agent formaton specfcatons and () a set of abes C = {c }, where (, ) E, that specfy the desred nter-agent reatve postons n the formaton confguraton The obectve of each agent s to be stabzed n a desred reatve poston c wth respect to each member of, avodng at the same tme cosons Coson avodance s meant n the sense that no ntersectons occur between the agents dscs Thus we want to assure that q (t) q (t) > r +r,,, for each tme nstant t Ths s a ey dfference of coson avodance of non-pont agents wth respect to pont agents In the atter case, cosons (n the two dmensona word) occur ony when q (t) q (t) = for some, Ths s not the case for the non-pont word as can be seen by the prevous equaton Thus a dfferent machnery s used n the non-pont case In prevous wor [5],[13],[3] we used the navgaton functons approach, estabshed by Kodtsche and Rmon n the semna paper [1], to acheve coson avodance and destnaton convergence for mutpe spherca agents In ths paper the destnaton convergence obectve s repaced by formaton convergence Hence, the probem treated n ths paper can be stated as foows: derve a set of contro aws (one for each agent) that drves the team of agents from any nta confguraton to the desred formaton confguraton avodng, at the same tme, cosons The foowng assumptons hghght the eve of decentrazaton of the approach: 1) Each agent has ony nowedge of the poston of agents ocated n a cycc neghborhood of specfc radus d C at each tme nstant, where d C > max, (r + r ) Ths set S = {q : q q d C } s caed the sensng zone of agent Hence apart from nowedge of agent ocated n, has aso nowedge of the postons of agents n S ) Each agent nows the exact number of agents n the worspace 3) The worspace s bounded and spherca Specfcay W = {q : q R w }, where R w denotes the worspace radus ) The formaton graph s undrected, n the sense that,,, It s obvous that (, ) E ff 5) There are no confctng nter-agent obectves, n the sense that c = c,,, 6) The formaton confguraton s feasbe, n the sense that q : q q c =, (, ) E The set E q = {q : q q c = (, ) E} s caed the equbrum set of the formaton The next fgure shows two exampes of feasbe formaton confguratons n a team of four and seven agents respectvey, as we as the correspondng neghborng sets for each agent n the second case The ne formaton confguraton of the second fgure s mpemented n the smuaton secton It s obvous that the rad of the agents do not have to be equa {1,,5,7} 5 {1,6} {3,7} 6 {5,7} 3 {,} 7 {1,,6} {1,3} (a) (b) 3 Rectanguar Formaton 1 3 Lne Formaton Fg 1 Feasbe formaton confguraton exampe of (a) a rectanguar and (b)a ne formaton III COTROL STRATEGY The proposed feedbac contro strategy for agent s defned as ϕ γ u = K D () q q where K, D are postve gans The functon γ : W R + represents the contro obectve for agent : convergng to a desred reatve confguraton wth respect to each A sutabe choce s: γ = 1 q q c (3) Functon ϕ s a navgaton functon that ensures coson avodance between agents n the team Inspred by our prevous wor on decentrazed navgaton functons ([5],[3]), ϕ s constructed to assure coson avodance between the agents n a decentrazed manner: ϕ = f (G ) ((f (G )) + G ) 1/ () The functon G serves as an encoder of a possbe coson schemes between agent and the rest of the team It s desgned n such a way to ensure that the boundary of the free space of each agent s repusve wth respect to the produced gradent moton The free space for each agent s defned as the subset of W whch s free of cosons wth the other agents Coson avodance s reassured n a bounded worspace and for approprate tunng of the controer gans Under the assumptons of the prevous secton, G s defned

3 so that each agent taes nto account the postons of agents that are wthn ts sensng zone at each tme nstant However, decentrazaton s restrcted by the fact that the constructon of G requres nowedge of the exact number of agents n the state space The parameter s a postve constant whch as sha be shown n the seque must be suffcenty arge to guarantee system stabty The constructve procedure to defne G and more detas can be found n [3](see aso [] for the goba sensng case) Functon f s defned n such a way to ensure that the repusve potenta vanshes when nter-agent dstances are suffcenty arge Ths functon has aso been used n our prevous wor ([5],[3]) and was ntroduced n [1] We defne the functon f by: a + 3 a G f (G ) =, G X (5), G > X where X, Y = f () > are postve scaar constants The parameters a are evauated so that f s maxmzed when G and mnmzed when G = X We aso requre that f s contnuousy dfferentabe at X Therefore we have: a = Y, a 1 =, a = 3Y X, a 3 = Y X The parameter X 3 serves as a sensng parameter that actvates the f functon whenever possbe cosons are bound to occur The ony requrement we have for X s that t must be sma enough to guarantee that f vanshes whenever the system has reached ts equbrum set, e when q S In mathematca terms: X < G (q), q E q, (6) Ths constrant ensures that the repusve potenta vanshes at the formaton confguratons IV STABILITY AALYSIS A Toos from Agebrac Graph Theory In ths subsecton we revew some toos from agebrac graph theory that we sha use n the stabty anayss the next sectons The foowng can be found n any standard textboo on agebrac graph theory(eg [1]) For an undrected graph G wth n vertces the adacency matrx A = A(G) = (a ) s the n n matrx gven by a = 1 f (, ) E and a = otherwse The degree d of vertex s defned as the number of ts neghborng vertces, e d = {# : (, ) E} Let be the n n dagona matrx of d s The (combnatora) Lapacan of G s the symmetrc postve semdefnte matrx L = A The Lapacan captures many nterestng topoogca propertes of the graph Of partcuar nterest n our case s the fact that for a connected graph, the Lapacan has a snge zero egenvaue and the correspondng egenvector s the vector of ones, 1 The ast property has ead to the nterestng resut regardng the connecton between formaton non-feasbty and focng behavor dscussed n secton V The next subsecton contans the stabty anayss of the formaton scheme B Stabty of a feasbe formaton Convergence of the agents to the desred formaton confguraton s guaranteed by the foowng theorem: Theorem 1: Assume that the foowng hod: The equbrum set n nonempty, e E q X s sma enough to guarantee that f : G < X then δ > : (L I ) q + c δ where L s the Lapacan of the formaton graph, the vector c s defned by c = [c 11,, c ] T, wth c = c and where denotes Kronecer product Then, under the feedbac contro strategy (), the state of the system converges to E q, provded that s bounded from beow by a fnte ower bound Proof : Functon V = (ϕ + γ ) s used as a canddate Lyapunov functon for the whoe system Tang ts dervatve we have V = { } (ϕ + γ ) V = ( ϕ + γ ) T q ϕ Rememberng that u = K γ q D q and that ϕ = the cosed oop dynamcs of the system f (G ) ((f (G )) +G ) 1/ are gven by: q = K 1 A (1+1/) 1 σ 1 G 1 q 1 D 1 γ 1 q 1 K A (1+1/) = A K ΣQq D (Lq + c ) σ G q D γ q = where σ = G σ(g ) f, A = f + G, σ(g ) = 3 a G 1 and the matrces Σ = ( ) K A K = dag 1 A (1+1/) 1, K 1 A (1+1/) 1,, K A (1+1/), K A (1+1/) }{{} D = dag (D 1, D 1,, D, D ) }{{} Σ }{{} 1,, Σ, Σ }{{} = dag,, σ, σ }{{}, 1,, Wthout oss of generaty, we assume that D = D for a We w use nterchangeaby the notaton D both for the matrx D as we as for ts equa eements The matrx Q s defned by the foowng reaton: G = G 1 G = Q 1 Q q = Qq

4 Anaytc expressons for the eements of the matrces Q can be found n [] Each Q s symmetrc, e Q = Q T We aso have Q = Q = Q and Q whenever The matrx L corresponds to the desred nter-agent reatve postons and can be shown to be reated wth the Lapacan of the formaton graph by the reaton L = L I That s a cruca resut regardng the behavor of the system as we sha we n the next secton The gradent of γ can be cacuated by ( γ ) T = = ( q T L + c T ) Smary, the gradent of ϕ s cacuated by ( ϕ ) T = = q T Q T A Σ, where A Σ = A Σ1 A Σ, A Σ = dag ( A (1+1/) A (1+1/) σ σ,, ) } {{ } By usng the notaton z = Lq + c, the dervatve of the canddate Lyapunov functon s now cacuated as { } V = ( ϕ + γ ) T q = = D z [ z T q ] [ ] (7) z T M q [ ] where M = D A K ΣQ Q T A Σ D Q T Hence A Σ A K ΣQ V D z + λ mn (H(M)) ( z + q ) ( where H(M) = ) 1 M + M T the Hermtan part of the matrx M and λ mn (H(M)) ts argest negatve egenvaue The postve defnteness of the matrx M cannot be guaranteed due to the fact that each agent has to tae nto account n the confct resouton procedure agents that do not beong n ts neghborng [ set The Hermtan ] part of the matrx M s H(M) = 1 H1 H, where H H 3 H 1 = D, H = A K ΣQ + DA T Σ Q, H 3 = Q T A Σ D + Q T Σ T A K, H = Q T A Σ A K ΣQ+Q T Σ T A K A T ΣQ After some cacuaton, we have (H ) = (H 3 ) = K σ A (1+1/) Q + DA (1+1/) σ Q, = = DA (1+1/) σ Q, (H ) = (A A ) (1+1/) σ σ K Q Q and + (A A ) (1+1/) σ σ K Q Q Specfc bounds on each of the terms n the ast reatons are provded n [] In partcuar n [] t s shown that the term σ s aways negatve and bounded and ts bounds are gven by σ Θ where Θ s a scaar postve parameter Ths reaton, aong wth the fact that the σ terms are present n the anaytc expressons of (H ), =, 3, heps us n dervng bounds on that ensure asymptotc stabty In the foowng anayss we mae use of the foowng theorems from matrx anayss ([6]) to provde an estmate of λ mn (H(M)): Theorem 1: Gven a matrx A R n n then a ts egenvaues e n the unon of n dscs: n =1 z : z a n a = n R (A) = R(A) =1 Each of these dscs s caed a Gersgorn dsc of A Coroary : Gven a matrx A R n n and n postve rea numbers p 1,, p n then a the egenvaues of A e n the unon of n dscs: n =1 z : z a 1 p n p a By usng p 1 = = p = p for the frst rows of H(M), Coroary provdes the foowng estmates for the egenvaues correspondng to these rows: z D p + p (H ) p max (H p + ) (H The form of (H ) guarantees that max p + ), where a postve fnte bound that corresponds to the upper bounds of the terms A ( ), Q,K The fact that these bounds are fnte s dscussed n [] can be cacuated expcty after a seres of maxmzatons on the terms nvoved n (H ) The fact that the exponent appears n the denomnator s a drect consequence of the fact that a term of the form σ s present n every byproduct of (H ) The correspondng egenvaues of the matrx H(M) can be rendered strcty postve by tunng arge enough as shown n the foowng: z D p z > D p Then z > s guaranteed by > p D otce that there s no restrcton on how to choose the parameters p + of the ast rows, provded that they are fnte Repeatng the procedure for the submatrces H 3, H does not guarantee postve defnteness because a carefu examnaton of the anaytc forms of (H 3 ), (H ) reveas that some eements of the man dagona of H coud be zero whe some correspondng eements of the -th row correspondng to H 3 can be nonzero Hence the Gersgorn dscs may ntersect wth the eft haf pane of the magnary axs However, the foowng procedure shows that the argest negatve egenvaue can be rendered suffcenty sma to guarantee negatve defnteness of V

5 In the worst case, the egenvaues of rows +1,, of the matrx H(M) e n the dsc p z < max (H 3 ) + (H ) p + Assumng wthout oss of generaty that p + = p = 1,, and usng the same ogc as above t s straghtforward to see that there exsts a fnte Θ 3 > such that max p p + (H 3 ) + (H ) p p Hence λ mn (H(M)) s bounded by λ mn (H(M)) p p Θ 3 ote that the term p p can be chosen arbtrary sma, whe Θ 3 s aways bounded n a bounded worspace Hence V D z + p p Θ 3 ( z + q ) The atter s guaranteed to be negatve f s chosen arge enough: V < D z > p p Θ 3 ( z + q ) ( ) > p p Θ 3 D 1 + R w δ By vrtue of the ast reaton we have that V < f G < X for some In the set G X we have Θ 3 V = D z = D Lq + c (8) Appcaton of LaSae s nvarance prncpe ensures the convergence of the system to the argest nvarant subset of the set S = {q : Lq + c = } whch corresponds to the desred formaton confguraton ote that the sets S = {q : Lq + c = }, q : X < G (q) aways ntersect due to the constrant (6), provded that the equbrum set s non-empty It s obvous that the second assumpton of the theorem statement guarantees the fnteness of the bound on that eads to asymptotc stabty However t s not as restrctve as t seems snce X can be chosen sma enough to ensure the vadty of the assumpton V FORMATIO O-FEASIBILITY RESULTS I FLOCKIG BEHAVIOR The ey assumpton behnd the stabty anayss of the prevous secton s formaton feasbty, namey that there exsts a confguraton q W such that Lq+c = But what happens when there does not exst such a confguraton n the state space? The answer s contaned n the next theorem: Theorem : Assume that the foowng hod: 1) mn q W Lq + c > ) X < G (q ), q : Lq + c = mn q W Lq + c > 3) The formaton graph s connected Under these assumptons, the system reaches a confguraton n whch a agents have the same veoctes and orentatons Proof : Equaton (8) guarantees that the system converges to a confguraton than mnmzes Lq + c Snce Lq + c mn > we have Lq+c = c at steady state, where c s a constant nonzero vector Hence L q = at steady state Assumpton and eq(7) guarantee that at steady state the dynamcs of the system are gven by q = D (Lq + c ) Usng the notaton v x, v y for the n-dmensona stac vectors of the components of the agents veoctes n the x, y drectons at steady state, we have L q = q T (L I ) q = v T x Lv x + v T y Lv y = The ast reaton and the fact that q guarantees that at east one of the vectors v x, v y s nonzero Hence at east one of the vectors v x, v y s an egenvector of L correspondng to the zero egenvaue For a connected graph, the egenvector assocated wth the snge zero egenvaue of the Lapacan s the vector of ones, 1 Hence at steady state, at east one of the vectors v x, v y beongs n span{ 1 }, whch ensures that a agent veocty vectors w have the same components at steady state Ths smpe resut shows that formaton non-feasbty s drecty reated to a phenomenon wth many smartes to what s nown as focng behavor n mut-agent systems The nteragent reatve postons at steady state as we as the vaue of the veocty norms are captured by mn Lq + c q W It s obvous that the form of c,l s drecty reated to the form of the resutng foc The exact hdden reaton s qute nterestng and s a topc of current research VI SIMULATIOS To verfy the resuts of the prevous paragraphs we provde two nontrva computer smuatons The frst smuaton nvoves convergence to a feasbe formaton confguraton Specfcay, we mpement the ne formaton of fgure 1(b) The neghborng sets of each agent are defned on that pcture Screenshots I-VI show the evouton n tme of the mut agent team The vaues of the parameters n ths smuaton are: = 9, D = 1K = 1,X = Y = 1 3 and d C = max,={1,,7} (r + r ) = 9 The coson avodance as we as formaton confguraton propertes are both verfed ote that X has been chosen sma enough to guarantee that at the equbrum confguraton the condton G > X hods even f the agents are very cose to each other as wtnessed n screenshot VI The second smuaton nvoves four agents and a nonfeasbe formaton confguraton The vaues of the parameters n ths smuaton are the same as prevousy The neghborng sets and desred nter-agents reatve poston vectors are 1 = {, 3, }, = {1, }, 3 = {1}, = {1, }, c 1 = c 1 = c = [ 1 ] T, c13 = [ 1 ] T It can easy be seen that ths s not a feasbe formaton confguraton and that the formaton graph s connected Screenshots I-V show the evouton n tme of the mut agent team As can be seen n ths fgure, the nteragent veoctes vectors are stabzed at steady state to a common vaue Ths s shown aso n the veocty dagram of the ast screenshot

6 nter-agent desred postons vector c wth the resutng foc n the case of formaton nfeasbty as we as budng a smar contro scheme for formaton convergence of nonhoonomc agents I II VIII ACKOWLEDGEMETS The authors want to acnowedge the contrbuton of the European Commsson through contracts MICRO(IST ) and I-SWARM (IST--576) Fg III V IV VI Seven agents convergng to a ne formaton Veoctes x Tme Fg 3 Focng behavor for agents Agents veoctes converge to a common vaue VII COCLUSIOS A feedbac contro strategy that acheves convergence of a mut-agent system to a desred formaton confguraton avodng at the same tme cosons was proposed The coson avodance obectve s handed by a decentrazed navgaton functon that vanshes when the desred formaton tends to be reazed When nter-agent obectves that specfy the desred formaton cannot occur smutaneousy n the state space the desred formaton s nfeasbe It was shown that under certan assumptons, formaton nfeasbty forces the agents veocty vectors to a common vaue at steady state Ths provdes a connecton between formaton nfeasbty and focng behavor for the mut-agent system Current research nvoves further studyng the connecton between the Lapacan of the formaton graph and of the REFERECES [1] B Boobás Modern Graph Theory Sprnger Graduate Texts n Mathematcs # 18, 1998 [] J Cortes, S Martnez, and F Buo Robust rendezvous for mobe autonomous agents va proxmty graphs n d dmansons IEEE Transactons on Automatc Contro, submtted for pubcaton, [3] D V Dmarogonas and K J Kyraopouos Decentrazed stabzaton and coson avodance of mutpe ar vehces wth mted sensng capabtes 5 Amercan Contro Conference, pages [] D V Dmarogonas, S G Lozou, KJ Kyraopouos, and M M Zavanos A feedbac stabzaton and coson avodance scheme for mutpe ndependent non-pont agents Automatca, to appear, 5 [5] DV Dmarogonas, MM Zavanos, SG Lozou, and KJ Kyraopouos Decentrazed moton contro of mutpe hoonomc agents under nput constrants nd IEEE Conference on Decson and Contro, pages , 3 [6] R A Horn and C R Johnson Matrx Anayss Cambrdge Unversty Press, 1996 [7] Proect ISWARM [8] A Jadbabae, J Ln, and AS Morse Coordnaton of groups of mobe autonomous agents usng nearest neghbor rues IEEE Transactons on Automatc Contro, 8(6):988 11, 3 [9] M J and M Egerstedt Connectedness preservng dstbuted coordnaton contro over dynamc graphs 5 Amercan Contro Conference, pages [1] D E Kodtsche and E Rmon Robot navgaton functons on manfods wth boundary Advances App Math, 11:1, 199 [11] G Lafferrere, A Wams, J Caughman, and JJP Veerman Decentrazed contro of vehce formatons Systems and Contro Letters, 5(9):899 91, 5 [1] Z Ln, B Francs, and M Maggore ecessary and suffcent graphca condtons for formaton contro of uncyces IEEE Transactons on Automatc Contro, 5(1):11 17, 5 [13] S G Lozou and K J Kyraopouos Cosed oop navgaton for mutpe hoonomc vehces Proc of IEEE/RSJ Int Conf on Integent Robots and Systems, pages , [1] Proect MICRO mcron/ [15] L Moreau Stabty of contnuous-tme dstrbuted consensus agorthms 3rd IEEE Conf Decson and Contro, pages , [16] A Muhammad and M Egerstedt Connectvty graphs as modes of oca nteractons 3rd IEEE Conf Decson and Contro, pages 1 19, [17] R Ofat-Saber and RM Murray Focng wth obstace avodance: Cooperaton wth mted communcaton n mobe networs st IEEE Conf Decson and Contro, pages 8, 3 [18] R Ofat-Saber and RM Murray Consensus probems n networs of agents wth swtchng topoogy and tme-deays IEEE Transactons on Automatc Contro, 9(9): , [19] HG Tanner, A Jadbabae, and GJ Pappas Focng n fxed and swtchng networs IEEE Transactons on Automatc Contro, submtted for pubcaton, 5 [] HG Tanner and A Kumar Formaton stabzaton of mutpe agents usng decentrazed navgaton functons Robotcs: Scence and Systems, 5 [1] MM Zavanos and KJ Kyraopouos Decentrazed moton contro of mutpe mobe agents 11th Medterranean Conference on Contro and Automaton, 3