Role Based Operations 28 9 Re f e r e n ces B elbin, M. ( 1993 ) . T eam Roles at Wor k , Butterworth-Heinemann. Fong, T., Kaber, D., Lewis, M., Scholtz , J., Schultz, A., Steinfeld, A. ( 2004 ) . C ommon M etrics for Human-Robot Interaction . Unpublished. Lewis, M., Sycara, K., Payne, T. (2003). Age n t roles in human teams. Proceedings of the AAMAS-03 Workshop on Humans and Multi-Agent Systems. Partsakoulakis, I., Vouros. G. ( 2002 ) . R oles in Collaborative Activit y . In I. Vlahavas and C. Sp y ropoulos, editors, Methods and Applications of Artificial Intelli g ence, Second Hellenic C on f erence on AI, LNAI 2308, pa g es 449-460. Partsakoulakis, I., Vouros, G. ( 2003 ) . Roles in MAS: Mana g in g the complexit y of tasks and environments. Multi-Agent Systems: An Application Science. Payne. T., Lenox, T., Hahn, S., Sycara, K., Lewis, M. (2000). Agent-based support fo r h uman/agent teams. Proceedings of the Software Demonstration; ACM Computer Human Interaction Conference; The Hague, Netherlands. Tambe, M. ( 1997 ) . Towa r ds F lexible T eamwo r k . Journal of Artificial Intelligence Research, 7: 83 - 124 . ERGODIC DYNAMICS BY DESIGN: A ROUTE TO PREDICTABLE MULTI-ROBOT SYSTEMS Dy l an A. S h e ll ,C h r i s V. Jones, Ma j a J. Matar i ´ c Computer Science Department, University of Sout h ern Ca l iforni a L os Angeles, California 90089-0781, USA d shell @ cs.usc.edu, cv j ones @ cs.usc.edu, mataric @ cs.usc.ed u Abs tr act W e define and discuss a class of multi-robot systems possessing ergodic dy - namics and show that they are realizable on physical hardware and useful for a variety of tasks while being amenable to analysis. We describe robot controller s synthesized to possess these dynamics and also physics-based methodologie s that allow macroscopic structures to be uncovered and exploited for task execu - tion in systems with large numbers of robots . Keywords: M ulti-robot systems, Ergodicity, Formal methods . 1. Introduction Mu l t i -ro b ot systems can b ot h en h ance an d expan d t h e capa bili t i es o f s i ng le r o b ots, b ut ro b ots must act i n a coor di nate d manner. So f ar, examp l es o f co - o r di nate d ro b ot systems h ave compr i se d o fl arge l y d oma i n-spec ifi cso l ut i ons , w i t hf ew nota bl e except i ons. In t hi s paper we d escr ib e our ongo i ng wor k o n t h e d eve l opment o ff orma l met h o d o l og i es f or synt h es i so f mu l t i -ro b ot system s t h at a dd ress t h ese i ssues i napr i nc i p l e df as hi on. W e f ocus h ere on i nter-ro b ot d ynam i cs, t h ero l es p l aye db yt h ose d ynam - i cs towar d tas k ac hi evement, an d t h e i r i mp li cat i ons i n f eas ibl e f orma l met h o ds f or synt h es i san d ana l ys i s. We d escr ib e automate d synt h es i so f contro ll er s t h at cap i ta li ze on so-ca ll e d e r g o d i c d ynam i cs, w hi c h ena bl e mat h emat i ca l ar - g uments a b out system b e h av i or to b es i mp lifi e d cons id era bl y. Sensor- b ase d s i mu l at i ons an d p hy s i ca l ro b ot i mp l ementat i ons s h ow t h at t h ese contro ll ers t o b e f eas ibl e f or rea l s y stems. We f urt h er su gg est t h at t hi s approac h w ill sca l eto s y stems w i t hl ar g e num b ers o f ro b ots. 2 91 L .E. Parker et al. ( eds.) , Multi-Robot S y stems. From Swarms to Intelli g ent Automata. Volume III , 2 9 1–2 9 7.  c 2005 S prin g er. Printed in the Netherlands . 2 9 2 Shell , et al . 2. Related formal methodologies Forma l met h o d o l og i es f or synt h es i san d ana l ys i so f mu l t i -ro b ot systems dif- f er b ase d on t h e type o f systems t h ey a i mtoa dd ress. One success f u l met h o d f or ana l ys i so f swarm systems i s b ase d on t h et h eory o f stoc h ast i c processes: f or examp l e, i nt h ep h enomeno l og i ca l mo d e li ng an d ana l ys i so f mu l t i -ro b o t c ooperat i ve st i c k -pu lli ng, a macroscop i c diff erence equat i on f or t h e rates o f ch ange o f eac h type o f ro b ot state i s d er i ve df rom t h e stoc h ast i c master equa - t i on an d sensor- b ase d s i mu l at i ons are use d to est i mate parameter va l ues (Mar - t i no li et a l ., 2004). An extens i on to t h e same t h eory ( b ut us i ng cont i nuou s diff erent i a l equat i ons i nstea d )a ll ows a d apt i ve systems to b emo d e l e d (Lerma n a n d Ga l styan, 2003). W h en app li e d to f orag i ng, t h e ana l ys i s ena bl e d syste m d es i gn i mprovements. Exp li c i t l y coor di nate d systems are typ i ca ll ya dd resse d a tt h ea lg or i t h m i c l eve l , suc h as i nt h e Computat i on an d Contro l Lan g ua ge ( K l av i ns, 2003 ) an df orma l stu di es o f mu l t i -ro b ot tas k a ll ocat i on met h o d o l o - gi es (Ger k e y an d Matar i c, 2004). Also related is Donald’s (1995) Information ´ Invar i ants T h eor y ,an d Er d mann’s (1989) stu di es o f t h ea d vanta g es o f pro b a - bilistic controllers . 3 . Behavioral configuration space and ergodicity T h ep h ys i ca l con fi gurat i on space, common i nro b ot i cs f or represent i ng p h ys - i ca l arrangements, can b e augmente d to i nc l u d ea ddi t i ona ldi mens i ons f or eac h of t h ero b ot’s i nterna l contro lv ar i a bl es t h at o b ser v a bl e b e h a vi or. We ca ll t his t h e b e h aviora l con fi guration spac e (BCS). It i s a use f u l menta l representat i o n f or a mu l t i -ro b ot system an df or reason i ng a b out t h e overa ll system d ynam i cs. For pract i ca l app li cat i ons, we w ill on l y cons id er part i cu l ar su b spaces, neve r t h e f u ll con fi gurat i on space . T h e BCS o f as i ng l ero b ot cons i sts o fdi mens i ons f or t h ep h ys i ca l con fi g - u rat i on (e.g,. t h e pose var i a bl es, an d ve l oc i t i es if necessary) an ddi mens i on s f or i nterna l state var i a bl es (cont i nuous or di screte va l ues w i t hi n memory). T he r ange o f eac hdi mens i on i s d eterm i ne db y constra i nts on state var i a bl es. T he BCS o f an ensem bl eo f ro b ots i s constructe df rom essent i a lly a Cartes i an pro d- u ct o fi n di v id ua l s spaces an d t h e spaces o f mova bl eo b stac l es, etc. T h e con - stra i nts (e. g ., two ro b ots s i mu l taneous ly occup yi n g t h e same l ocat i on) su b trac t c omponents f rom t hi s pro d uct. Coup li n g s b etween t h ero b ots v i a commun i ca - tion channels, mutual observation, etc., further restrict this space . The g lobal state of the multi-robot s y stem at an y specific time can be rep - r esented b y a point in BCS and likewise, the time-evolution of the s y stem , a s a tra j ector y .As y stem that exhibit s e rgodic dynamic s c ompletel y visits al l p arts of the confi g uration space with probabilit y that is dependent onl y on th e v olume of that part of the space. Lon g term histor y is unimportant in predict - in g the d y namical behavior because the s y stem “for g ets” previous tra j ectories . E r g odic D y namics b y Desi gn 29 3 Ti me averages o f some system property (over a d urat i on l onger t h an t h eun d er - l y i ng d ynam i cs t i mesca l e), are equa l to (con fi gurat i on) spat i a l averages. Fe w u se f u l ro b ot i cs systems are ent i re l yergo di c, b ut var i ous su b -parts o f t h eBC S may b eergo di c. T h e next sect i on d escr ib es one suc h system . 4. Automated s y nthesis for se q uential tasks J ones an d Matar i c (2004a, 2004b) have developed a framework for auto- ´ mat i can d systemat i c synt h es i so f m i n i ma li st mu l t i -ro b ot contro ll ers f or se - quent i a l tas k s. T h e f ramewor k cons i sts o f asu i te o f a l gor i t h ms t h at ta k eas i n - put a f orma l spec ifi cat i on o f t h eenv i ronmenta l e ff ects, t h e tas k requ i rements , a n d t h e capa bili t i es o f t h ero b ots. T h ea l gor i t h ms pro d uce e i t h er prova bl y cor - rect ro b ot contro ll ers, or po i nt to t h e exact scenar i os an d tas k segments w hi c h ma k e(a l gor i t h m i ca ll y) guarantee d tas k comp l et i on i mposs ibl e. T h e type o f contro ll er an d prospect o f success f u l tas k execut i on d epen d on t h e capa bili- ti es o f t h e i n di v id ua l ro b ots. Current opt i ons i nc l u d et h e poss ibl yo fb roa d cas t i nter-ro b ot commun i cat i ons ( Jones an d Matar i c, 2004a), and a local memory ´ on eac h o f t h ero b ots perm i tt i ng non-react i ve contro ll ers (Jones an d Matar i ´ c , 2004 b ). Two comp l ementary ana l ys i s tec h n i ques a ll ow var i ous stat i st i ca l per - f ormance c l a i ms to b ema d ew i t h out t h e cost o f a f u ll i mp l ementat i on an d ex h aust i ve exper i mentat i on . W e d o not prov id e f u ll d eta il so f t h e f ramewor kh ere, b ut i nstea df ocus o n t he (non-obvious) role of er g odic d y namics in the work. The framework uses a set o f states S t o denote the possible states that the (assumed to be Markovian ) world can be in. The se t A c ontains actions which act upon the world state , p roducin g state transitions defined in some probabilistic manner (see Fi g ure 1) . A particular sequence of states, sa y T , ( T ⊂ S ) makes up the task. In actualit y t he robots are onl y interested in producin g the sin g le task sequence, and thu s onl y those transitions need to be stored. Thus , S is never stored or calculated , onl y T need be ke p t, and | T |  | S | . Robots then mo v e around the en v ironmen t makin g observations, perhaps consultin g internal memor y or listenin g to th e b roadcast communication channel if suitabl y equipped. If a robot has sufficien t information to ensure that the p erformance of a p articular action (fro m A ) ca n onl y result a world transition that is part of the task (i.e., result in a state in T ) t hen it ma y perform that action . F i g ure 1 2 94 Shell , et al . Return i ng to t h e not i on o fb e h av i ora l con fi gurat i on space, eac h o f t h ewor ld states i n S r epresents ent i re su b spaces o f t h e overa ll system’s space. F i gure 2 s h ows t h at t h e ent i re b e h av i ora l con fi gurat i on space as i t fi ts i nto t hi s f orma l- i sm. A h ypot h et i ca l pro j ect i on o f t hi s ent i re ( h uge) con fi gurat i on space sep - a rates t h e con fi gurat i ons i nto su b spaces, eac h su b space represent i ng a s i ng le state i n T . Act i ons (f rom A ) evo l ve t h ewor ld state an dh ence trans i t i on t he system f rom one su b space to anot h er. We d esi g nt h es y stem so t h at wit h in eac h su b space t h e dy namics are er g o d ic . Wor k t h at h as use d t hi s f orma lf ramewor k e nsure d t hi s property b y h av i ng t h ero b ots per f orm ran d om i ze d exp l orat i o n p o li c i es. T h e ran d om i ze d strategy nee d sto h ave su ffi c i ent e ff ect to overpowe r o t h er s y stemat i c bi ases i nt h es y stem t h at cou ld pro d uce l ar g e sca l ee ff ects an d ig nore some part o f t h e con fig urat i on space . Bot h contro ll ers w i t h memor y (Jones an d Matar i c, 2004b) and ones en- ´ d owe d w i t h commun i cat i on capa bili t i es (Jones an d Matar i c, 2004a) were demon- ´ strate di ns i mu l at i on an d on p hy s i ca lh ar d ware i namu l t i -ro b ot construct i o n domain. The task involves the sequential placement of colored cubes into a p lanar arran g ement. The sequence T contains simpl y the required evolution o f the structure, actions A b ein g the placement of an individual cube. Referrin g back to Fi g ure 2; in the construction domain the motions within each subspac e a re random walks b y the robots, and the transitions between spaces are cub e p lacement actions . A nal y tical techniques developed in order to predict task execution are aide d b y the er g odic components of the robots behavior in this domain. One exampl e is in the macrosco p ic model (Jones and Matar i c, 2004b, pp. 4–5) applied to ´ t his formal framework. This model calculates the probabilit y of successful tas k completion b y calculatin g a lar g e multiplication of all possible memor y state s t hat g et set, in each possible world state, after each possible observation, calcu - l atin g the probabilit y that onl y the correct action will result and includes term s for when actions may result in other, or null, world transitions. A fundamenta l a ssum p tion for that calculation is that no “structure” in the world results in th e o bservation and action sequences that correlate. When endowed with naviga - t ional controllers that have ergodic dynamics, we know that this is true becaus e Fi g ure 2 E r g odic D y namics b y Desi gn 295 th eo b servat i on o f an ergo di c system a t N ar bi trary i nstants i nt i me i s stat i st i - ca ll yt h e same as N a r bi trary po i nts w i t hi nt h e b e h av i ora l space (McQuarr i e , 1976, pp. 55 4) . T hi s sect i on h as d emonstrate d t h at d ynam i cs w i t h a hi g hd egree o f ergo di c - i ty are ac hi eva bl eonp h ys i ca l ro b ot systems. T h ey can p l ay a ro l e i n system s f or w hi c h ana l yt i ca l met h o d sex i st, an d as a very s i mp l e f orm o fd ynam i cs t h e y can a id i ns i mp lif y i ng part i cu l ar aspects o f system d es i gn . 5 . Large-scale multi-robot systems We cons id er l arge-sca l emu l t i -ro b ot systems t h ose w i t h ro b ots on t h eor d e r o f t h ousan d s. In sp i te o f t h e f act t h at manu f actur i ng an d tracta bl es i mu l at i o n rema i n open c h a ll enges, a var i ety o f tas k s h ave b een propose df or system s o f t hi s type. Increas i ng t h e num b er o f ro b ots i ncreases t h e tota l num b er o f d egrees-o f - f ree d om i n a system, an d resu l ts i na hi g hl y di mens i ona l BCS. Co - or di nat i on approac h es t h at coup l ero b ot i nteract i ons as l oose l y as poss ibl ear e most lik e l ytosca l eto l arge s i zes . M at h emat i ca l tec h n i ques emp l oye di n stat i st i ca l mec h an i cs are use f u lf o r esta bli s hi ng t h ere l at i ons hi p b etween m i croscop i c b e h av i or an d macroscop ic s tructures (McQuarrie, 197 6 ). Typical system sizes for classical work are sig- n ifi cant l y l arger ( ∼ 10 23 )t h an t h e num b ers current l y conce i va bl e f or ro b ots. I n th e case o fl arge (or i n fi n i te) systems, i nterest i ng macroscop i c structures ca n resu l teven f rom ergo di c l oca ld ynam i cs. g l o b a l structures lik e equ ilib r i u m p h ases, p h ase trans i t i ons, coex i stence li nes, an d cr i t i ca l po i nts are w id e l y stu d - i e di nt h ermo d ynam i cs. Recent wor k attempts to re f ormu l ate many o f t h es e c l ass i ca l not i ons f or systems w i t hf ewer ent i t i es (Gross, 2001). We are pursu i ng a met h o d o l ogy f or coor di nat i on o fl arge-sca l e system s th roug h t h e stu d yo f a sma ll set o f mec h an i sms f or pro d uc i ng genera l macro - s cop i cp h enomena. One can did ate mec h an i sm i s a protoco lf or ac hi ev i ng con - s ensus. The Potts (19 5 2) model is illustrative; it is an archetypal magnetic spi n s ystem t h at mo d e l s i nteract i ons b etween part i c l es at a num b er o ffi xe dl oca - ti ons w i t hi na g rap h or l att i ce. T h eIs i n g mo d e l (a spec ifi c Potts mo d e l ) h a s al so b een use d to mo d e lg as fl ow. Ne i t h er mo d e li s a per f ect fi t f or ro b ots, b u t ill ustrates macroscop i c structure f rom s i mp l e i nteract i ons. M app i n g t h esp i n i nteract i ons at sp i ns i tes to ro b ots a ll ows f or t h e d eve l op - ment of a communication al g orithm that possesses er g odic d y namics (and a n ener gy conservation constraint) that permits the definition of a partition func - t i o n Z t hat can be solved usin g a numerical method for pseudo-d y namics sim - u lation (or in trivial cases anal y ticall y ). This admits a prediction of g lobal be - h avior because exhaustive p arameter variations enable construction of a p has e d ia g ram. In the case of the Potts and Isin g models this phase dia g ram is wel l known. Particular re g ions of the phase space in the Isin g model represent re - 2 96 Shell , et al . gi ons o f max i ma l or d er. For ro b ots t hi s means unan i m i ty; consensus i s reac h e d t h roug h a secon d -or d er p h ase trans i t i on. T h ea bili ty to prescr ib eergo di c d ynam i cs f or l arge-sca l ero b ot systems ma k e s t h ose ana l yt i ca l approac h es t h at f ocus on l y on constra i nt space topo l ogy f ea - s ibl e f or pre di ct i ons o f g l o b a l structure. T hi s means t h at tas kdi recte d act i on s c an b e tac kl e ddi rect l y f rom a macroscop i c perspect i ve . 6 . S ummar y and Conclusion W e h ave ta k en a d ynam i cs-centr i c approac h to d escr ibi ng mu l t i -ro b ot b e - h av i or. T hi sv i ew h as suggeste d t h at t h e not i on o f ergo di c i ty may b e use f u l w i t hi naro b ot i cs context, somet hi ng t h at we h ave d emonstrate d t h roug h ou t t h e paper. A f ter d e fi n i ng a b e h av i ora l con fi gurat i on space, we d emonstrate d t h at su b spaces i nw hi c h t h ero b ot d ynam i cs are essent i a ll yergo di c can b e use d to pro d uce mean i ng f u lb e h av i or, an d a ll ow automate d synt h es i s tec h n i ques t o f ocus on a sma ll set o f tas k -or i ente d states, rat h er t h en t h e ent i re ensem ble c on fi gurat i on space. A l so, i nat l east one case, ergo di c i ty s i mp lifi es ana l ys i so f system b e h av i or. Imp l ementat i ons on p h ys i ca l an d s i mu l ate d ro b ots s h ow t h a t e rgo di c i ty i s i n d ee d ac hi eva bl e i nt h e rea l wor ld . Future prom i se o f t hi s genera l a pproac hi ssu gg este di na di scuss i on o fl ar g e-sca l emu l t i -ro b ot s y stems. A cknowledgment s T hi s researc h was con d ucte d at t h e Interact i on La b , p art o f t h eRo b ot i c s Researc h La b at USC an d o f t h e Center f or Ro b ot i cs an d Em b e dd e d Systems . It was supporte db yt h eO ffi ce o f Nava l Researc h MURI Grant N00014-01-1 - 0 8 9 0 . Reference s D onald, B. R. (1995). On information invariants in robotics . AI ,7 2(1 - 2) : 21 7– 304 . E rdmann, M. A. (1989). O n Probabilistic Strategies for Robot Task s . PhD thesis, M.I.T . Gerkey, B. P. and Matar i c, M. J. (2004). A formal analysis and taxonomy of task allocation in ´ multi-robot systems. I nternational Journal of Robotics Researc h , 23(9):939–954. Gross, D. H. E. (2001). M icrocanonical Thermodynamics: Phase Transitions in “Small” Sys - t em s . World Scientific Lecture Notes in Physics - Volume 66. World Scientific, Singapore . Jones , C. V. and Matari c, M. J. (2004a). Automatic Synthesis of Communication-Based Coor- ´ dinated Multi-Robot Systems. I n P roc. of the IEEE/RSJ International Conference on Intel - ligent Robots and Systems (IROS-04 ) , Sendai, Japan . Jones , C. V. and Matari c, M. J. (2004b). Synthesis and Analysis of Non-Reactive Controllers ´ for Multi-Robot Sequential Task Domains. I n Proc. of the International Symposium on Ex - p erimental Robotic s , Singapore . Klavins, E. (2003). Communication Complexity of Multi-Robot Systems. In Boissonnat, J D. , B urdick, J., Goldber g , K., and Hutchinson, S., editors , Algorithmic Foundations o f Robotic s V , volume 7 of VV S prin g er Tracts in Advanced Robotic s ,pa g es 275–292. Sprin g er . E r g odic D y namics b y Desi gn 29 7 L erman, K. an d Ga l styan, A. (2003). Macroscop i c Ana l ys i so f A d apt i ve Tas k A ll ocat i on i nRo - bots . In Proc. of the IEEE/RSJ International Conference on Intelligent Robots and System s ( IROS-03), pages 1951–1956, Las Vegas, NV., USA . Martinoli, A., Easton, K., and Agassounon, W. (2004). Modeling Swarm Robotic Systems: A Case Study in Collaborative Distributed Manipulation . IJ RR, 23(4):415–436 . McQuarrie, D. A. (1976) . Statistical Mechanic s . Harper and Row. reprinted by University Sci - ence Books , Sausalito , CA. , USA in 2000 . Potts, R. B. (1952). Some generalized order-disorder transformations. I n Proc. of the Cambridge Philosophical Society, volume 48, pages 106–109 . A uthor Inde x A dams , Julie A. , 1 5 A ghazarian„ 235 A llen , James , 25 7 A m i gon i , Francesco, 133 A miranashvili , Vazha , 25 1 B ata li n , Max i mA. , 2 7 B ruce , James , 15 9 C h a i mow i cz , L. , 22 3 Chambers , Nathanael , 257 C h an d ra, Maureen, 119 C h erry, Co li n, 79 Choset, Howie, 14 5 C h ox i , Heeten, 283 C l ar k , Just i n, 171 Commur i ,Ses h , 171 Cosenzo, Keryl, 18 5 Cow l ey, A., 22 3 D ellaert, Frank, 10 7 D erenick, Jason, 263 Fierro, Rafael, 17 1 G alescu, Lucian, 2 57 G as p arini, Simone, 13 3 G erke y , Brian P., 6 5 G ini, Maria, 13 3 G omez-Ibanez, D., 22 3 G oodrich, Michael A., 18 5 G ordon, Geoff, 6 5 G rocholsk y , B., 22 3 Hou g en, Dean, 171 Housten, Drew, 283 Hsieh , M. A. , 22 3 Hsu , H. , 223 Hull , Richard A. , 4 1 Huntsber g er, Terr y ,23 5 Jain , Sonal ,3 Jones , Chris V. , 29 1 Jun g ,H y uckchul, 257 K eller , J. F. , 22 3 K umar , V. , 223 La g ou d a ki s, M i c h a il G., 3 Lakeme y er, Gerhard, 25 1 Lin , Kuo-Chi , 2 6 9 M ata r i ´ c, Maja J., 291 ´ McMillen , Colin P. , 5 3 New, Ai Peng, 14 5 O ’Hara , Ke i t h J. , 27 7 O kon , Avi , 23 5 P ar k er, Lynne E., 119 P owers, Matt h ew, 10 7 Q u, Z hih ua, 41 Q uigley, Morgan, 18 5 R av i c h an d ran, Ramprasa d , 107 R ekleitis, Ioannis, 14 5 R oth,Maayan, 9 3 R ybski, Paul E., 53 S atterfield, Brian, 28 3 S ellner, Brennan, 197 S hell, D y lan A., 29 1 S immons, Reid, 93, 197 S in g h, San j iv, 197 Sp ears, Diana, 21 1 Sp letzer, John, 263 S troupe, Ashle y ,23 5 S ukhatme , Gaurav S. , 2 7 S ven, Koeni g , 3 S waminathan , R. , 22 3 T an g ,Fan g , 119 T a yl or, C. J., 22 3 T horne, Christopher, 2 6 3 T hrun , Sebastian , 6 5 T ovey, Cra i g, 3 V ig, Lovekesh, 15 W a lk er , Dan i e l B. , 27 7 W ang,J i ng, 4 1 Zarz hi ts k y, D i m i tr i ,21 1 Z h ang, Hong, 79 B a l c h, T uck TT er R . , 10 7 , 2 77 V eloso, VV M anue la M., 53, 93, 15 9 299 . t hi s approac h w ill sca l eto s y stems w i t hl ar g e num b ers o f ro b ots. 2 91 L .E. Parker et al. ( eds.) , Multi-Robot S y stems. From Swarms to Intelli g ent Automata. Volume III , 2 9 1–2 9 7.  c 2005 S prin g er a partition func - t i o n Z t hat can be solved usin g a numerical method for pseudo-d y namics sim - u lation (or in trivial cases anal y ticall y ). This admits a prediction of g lobal be - h avior. Printed in the Netherlands . 2 9 2 Shell , et al . 2. Related formal methodologies Forma l met h o d o l og i es f or synt h es i san d ana l ys i so f mu l t i -ro b ot systems dif- f er b ase d on