Wall Following to Escape Local Minima for Swarms of Agents Using Internal States and Emergent Behaviour

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1 Wall ollowng to Escap Local Mnma for Swarms of Agnts Usng Intrnal Stats and Emrgnt Bhaour Mohamd H. Mabrouk and Coln R. McInns Abstract Natural xampls of mrgnt bhaour, n groups du to ntractons among th group s ndduals, ar numrous. Our am, n ths papr, s to us complx mrgnt bhaour among agnts that ntract a parws attract and rpuls potntals, to sol th local mnma problm n th artfcal potntal basd nagaton mthod. W prsnt a modfd potntal fld basd path plannng algorthm, whch uss agnt ntrnal stats and swarm mrgnt bhaour to nhanc group prformanc. Th algorthm s usd succssfully to sol a ract pathplannng problm that cannot b sold usng conntonal statc potntal flds du to local mnma formaton. Smulaton rsults dmonstrat th ablty of a swarm of agnts to prform problm solng usng th dynamc ntrnal stats of th agnts along wth mrgnt bhaour of th ntr group. Indx Trms Agnt Intrnal Stats, Local Mnma Escap, Swarm Emrgnt Bhaour, Wall ollowng. I. INTRODUCTION Common mrgnt pattrns n natural systms such as cohrnt flock, sngl-mll stats, and doubl-mll pattrn ha bn obsrd and rportd for arous spcs whos mmbrs ha hgh rats of nformaton xchang [] [3]. As th rsarchrs bcom mor concrnd n nstgatng such phnomnon, trms lk complxty, mrgnc, and stgmrgy ha bn dfnd [4] and modls of natural or artfcal ndduals, whch ntract through par-ws long-rang attracton and short-rang rpulson wthn a swarm, ha bn ntroducd [5] [8]. Such bhaour may offr nw approachs to many classs of nformaton procssng problms, whch currntly pro nfasbl, and to dsgn systms that can accomplsh thr tasks mor rlably, fastr and chapr than could b achd by conntonal systms [9]. II. SWARM MODEL Instgatng th moton of swarms usng artfcal potntal flds shows that swarms of ntractng partcls can rlax nto ortx-lk stats [8]. Th modl conssts of N p agnts wth mass m, poston r, locty and rlat dstanc r j btwn th th and j th agnts. Th agnts ntract by mans of a cohs two-body gnralzd Mors potntal V ntracton (r ) wth wak long rang attracton and strong short rang rpulson. Manuscrpt rcd bruary 27, M. H. Mabrouk s wth th Dpartmnt of Mchancal Engnrng, Unrsty of Strathclyd, Glasgow, UK. (phon:+44 (0) ; fax: +44 (0) ; -mal: mohamd.mabrouk@ strath.ac.uk). C. R. McInns s wth th Dpartmnt of Mchancal Engnrng, Unrsty of Strathclyd, Glasgow, UK. or smplcty, w wll consdr dntcal agnts of unt mass. To control th spd of th th agnt a lnar dsspat trm wth a post coffcnt β s addd [0]. Th total potntal fld, whch affcts th th agnt, s thn charactrzd by othr agnt s attract and rpuls potntal flds of strngth C a and C r wth rangs l a and l r rspctly along wth obstacl potntals V obstacls (r ) of strngth C o wth rang l o (C a, C r, l a, l r, C o, l o 0). or th goal w us a hyprbolc attract wll of strngth w g to nsur conrgnc of th agnts to th goal []. To mak th swarm of agnts dsspat nrgy whl th total angular momntum s consrd as th swarm rlaxs, th agnts ncountr orntaton forcs orntaton (r, ), whch act drctly on th agnts locts to ornt th ndduals locts wth rspct to on anothr [8]. Th constant C A s th magntud of th orntaton forc and l A s th rang or whch th orntaton ntracton occurs (C A, l A 0). In gnral, th quatons of moton for N p agnts mong n a workspac that contans N o pont obstacls at locatons r o and on goal G at poston r g ar thn dfnd by: m = & () r & = r, ) (2) total ( whr, total (r, ) s th sum of all forcs xrtd on th th agnt. To calculat th forc n Eq. (2), th global potntal s now dfnd as: V = V + V + V (3) total ntracton obstacls goal whr, th ntracton potntal V ntracton (r ) s dfnd as th sum of th rpulson potntal and th attracton potntal among th agnts. W us gnralzd Mors potntal, of th xponntally dcayng natur, to obtan ntractons that ar clos to ral bologcal systms as follows: V N p N p r rj lr j r rj la j ntracton ( r ) Cr C j a (4) = j j j Th obstacls and goal potntals ar dfnd as: V V No r roz obstacls ( r ) = Co z l (5) oz z= goal ( + r ) 2 r ( r ) w (6) = g g ISBN: WCE 2008

2 total (r, ) n Eq. (2) conssts of th followng: total ( r, ) = dsspaton ( ) + ntracton ( r ) + orntaton ( r, ) + r ) + ( r ) (7) whr, dsspaton ntracto n goal goal( obstacls ( ) = β (8) ( r ) = V ( r ) = V obstacls goal ( r ) = V ntracto n ( r ) = obstacls ( r ) = ( r ) N p j w ( r r g + r r No = z= C l G G C l a j a j ) 2 oz oz r r j la j r ro z lo z rˆ C l oz rj r j r r j lr j Th orntaton forc s dfnd n (McInns,2007) as: orntaton ( r, rˆ j (9) (0) () N p rj / l A ) = C A( j.ˆ rj ) rˆ (2) j j Substtutng from Eq. (3), Eq. (7-2) n Eq. (2), t can b sn that: us th agnts ntrnal stats to mploy ths mrgnt bhaour to mak th swarm of agnts scap local mnma by followng th boundars of obstacls. Th ract problm of a swarm of agnts attractd to a goal pont at poston G s shown n g., whr w can s th group mos towards th goal as a flock wth ndduals locts ncrasng untl rachng th goal. Thn thy ar trappd n th local mnmum, whch s a barrr that conssts of a numbr of dntcal obstacl ponts locatd n th path of th swarm to th goal such that th goal s sbl from th swarm ndduals ntal postons but thy cannot pass through th barrr. Consdrng ths cas, th whol swarm wll b trappd at th barrr bcaus th agnts trappd nsd th barrr wll suffr two oppost forcs; th frst forc s th rpulson from th barrr whl th othr on wll b th attracton to th goal. g. 2 shows th narly snusodal chang n th group angular momntum for th swarm n g. ndcatng th frqunt attmpts of th group to go to th goal through th obstacls and that th angular momntum of th swarm dcrass as th swarm s rpulsd. Thn, th swarm group angular momntum almost dcays wth tm. In that cas, th swarm rotats around ts cntr wth a dcayng angular momntum, whch ndcats that th swarm wll nr scap th local mnmum. m & = ) + ( r, ) V ( r ) (3) dsspaton ( orntaton total or a complx potntal fld such as that rprsntd by Eq. (3-6), th potntal can posss multpl local mnma. A ky ssu s to dntfy how th agnts wll ralz that thy ar trappd n a local mnmum so that thy can thn attmpt to scap. To sol th problm n ths cas, th agnts must dscount thr mmdat snsory nformaton (attracton of th goal) by ndowng thm wth hghr-ll prcpton concrnng th nronmnt. III. PROBLEM DEINITION Th local mnma problm has bn an ssu of concrn for potntal fld mthods [2]. Sral attmpts, whch can b catgorzd nto local mnma aodanc (LMA) tchnqus and local mnma scap (LME) tchnqus, ha bn mad to orcom t [3]. In our prous work w ntroducd th approach of usng dynamc ntrnal stats for a swarm of robots to scap local mnma by manpulatng th global potntal of th nronmnt. Th prformanc of th swarm was nhancd by usng som aspcts of swarmng bhaours such that swarm ladr concpt [4], and th collct bhaour [0], whch can b found n ral bologcal systms. In ths papr w ntroduc a soluton to th problm prformd by a swarm of robots, whch ncountr mutual ntracton. By choosng th propr ntracton paramtrs, a ortx-lk pattrn wll mrg [8]. W wll g..a. Th swarm starts from poston S and thn bcoms stuck n th local mnmum, t = 95 ISBN: WCE 2008

3 g..b. Th swarm fals to scap th local mnmum, t = 250 g.. Bhaour of a swarm of agnts that us fxd ntrnal stats. g. 2. Group angular momntum wth tm for th swarm n g.. IV. INTERNAL STATE MODEL W us on of th most ntrstng aspcts n swarmng bhaours, whch s th mrgnc of ortx pattrn among agnts that ntract a par-ws attract and rpuls potntals [5], as a nw tchnqu to scap th local mnmum poston. Th soluton dpnds manly on ncrasng th group prcpton about th swarm stat by lnkng th goal gradnt potntal n th quaton of moton to on of th swarmng paramtrs, th swarm cntr locty c, n a way that whn th locty of th swarm dcrass th goal ffct dmnshs. Ths hlps n lmnatng th local mnmum from th global potntal, whch n turn nabls th formd ortx pattrn amongst th group to sol th problm. Th attracton strngth w g n Eq. (3) s now dfnd as: c c w = ( λ λ. k (4) g g ) whr k s a post coffcnt. Ths ffct wll not sol th problm by t slf bcaus th ortx pattrn mrgs and th local mnmum dsappars but th agnts may rotat around thr cntr bhnd th wall, as shown n g., whch ndcats that th swarm wll not follow th obstacl boundars. At ths pont coms th rol of manpulatng on of th agnts ntrnal stats, th dsspaton coffcnt β, to ach a pur rollng n a way that maks th group follow obstacl boundars. Th dsspaton coffcnt wll b dfnd as: R λ β. c β = βo + (5) and, R N o = Mn( r ) (6) z = oz whr β o s th mnmum dsspaton coffcnt ncssary to prnt th agnts from scapng th group [0], R s th mnmum dstanc btwn th agnt and th obstacls, λ c s a post coffcnt that controls th ffct of th swarm cntr locty on th goal attracton potntal strngth n Eq. (4), λ g s a post coffcnt that guarants that w g s always post and λ β s a post coffcnt that controls th ffct of th swarm cntr locty on th dsspaton coffcnt n Eq. (5). Th ffct of manpulatng th dsspaton coffcnt β guarants that whn th swarm s trappd, th swarm s ndduals closr to th boundary of th obstacls wll gan hghr alus of dsspaton coffcnt (.. lowr locts). Manwhl, th ndduals who ar far from th obstacls wll gan lowr alus of dsspaton coffcnt and consquntly hghr locts to form a pur rollng acton. V. STABILIT ANALSIS W wll follow Moglnr s approach [6] to dscuss th stablty of th systm. rom Eq. (3) t can b dducd that: N P rj / l A m (.ˆ ) ˆ & = β C A j rj rj Vtotal ( r ) (7) j whr V total (r ) s from Eq. (3). Now, th total nrgy of th systm s dfnd as: NP φ = m 2 + Vtotal ( r ) (8) = 2 so that N = = ( m & + V ( r )) P & φ. (9) total Substtutng from Eq. (8), Eq. (0) and Eq. (7) n Eq. (9): N P N P 2 = = N P & φ = β C ˆ (20) j Thn, t can b concludd that: rj / la 2 A (. j ) rj ISBN: WCE 2008

4 & N P N P N P 2 2 rj / l A 2 φ = β ˆ C A j r (2) j = 2 = j Knowng that β > 0, w g 0, λ g 0, C A 0, k >0, thn & φ < 0, thrfor th systm s Lyapuno stabl, so that th group wll slowly lak nrgy and rlax to a mnmum-nrgy stat. th swarm wll follow th obstacl wall n a pur rollng moton n n th absnc of th goal ffct and consquntly th absnc of th drct touch of th swarm to th wall. VI. NUMERICAL RESULTS A. Problm Solng Smulaton rsults, dmonstratd n g. 3, show a swarm of agnts scapng local mnmum whl g. 4 shows th group angular momntum durng th problmsolng phas. W can s n g. 3.a) that th swarm mos towards th goal as almost algnd flock untl t ntrs th local mnmum. Ths s shown rgm I of g. 4, whr th group angular momntum s low and of almost constant alu. Th ffct of th trm n that ncrass th prcpton of th group about th nronmnt s obous n g. 3.b) to g. 3.), whch show that whn th swarm s stuck th goal ffct on th group s dcrasd. Ths maks th local mnmum dsappars, as shown n g. 3.b) to g. 3.), and th ortx pattrn mrg wth hghr angular momntum amongst th group. g. 3.b) to g. 3.) also show th ffct of usng Eq. (5) and Eq. (6) whch manpulat th alus of th dsspaton coffcnt makng th ndduals closr to th obstacl to gan hghr dsspaton coffcnt that maks thm of lowr locty than thos who ar locatd far from th obstacl walls. Ths guarants pur rollng moton, makng th group to follow th wall boundary n f thr s no drct contact to th obstacl wall. Th pur rollng-wall followng acton s ry clar n g. 3.b) to g. 3.). g. 4 shows that th swarm mantans almost constant group angular momntum to follow boundars of th obstacls. Zons II, III, IV and V of g. 4 rspctly show th group angular momntum corrspondng to th boundars followng for th lowr horzontal nnr wall, rtcal nnr wall, th hghr horzontal nnr wall, and th outr boundars of th hghr horzontal wall untl th swarm scaps from th local mnmum. Agan th ffct of th trm that ncrass th prcpton of th group about th nronmnt s obous n g. 3.f) and g. 3.g) n a way that as th agnts scap from th local mnmum, th swarm cntr locty ncrass and consquntly th goal ffct on th group ncrass whch maks th group mos towards th goal wth rlatly hghr locty as an almost algnd flock. Ths s shown n rgon VI of g. 4 whr th swarm group angular momntum dcrass as th swarm mos toward th goal. Usng th modl nsurs snusodal chang n th agnt s dsspaton coffcnt, spcally th prphral ons, wth tm as shown n g. 5. Ths ffct guarants that g. 3.a. Th swarm at th ntal poston, t=7 g. 3.b. t= 39 ISBN: WCE 2008

5 g. 3.c. t= 55 g. 3.. t= 220 g. 3.d. t= 200 g. 3.f. t= 260 ISBN: WCE 2008

6 B. Solng a Maz Applcaton W now consdr two groups of agnts attmptng to rach a sngl goal n a maz whos potntal fld has multpl local mnma. Th groups nagat from a startng pont S and attmpt to rach a goal poston G through a 4-ll maz. On of ths two groups, swarm A, s usng th ntrnal stat modl supportd by th wall followng tchnqu to sol th maz whl th othr group, swarm B, s usng a conntonal statc potntal fld. Th smulaton rsults, shown n g. 6, dmonstrat th capablty of th swarm usng th ntrnal stat modl to sol th problm and rach th goal, whl th othr conntonal swarm s trappd n th frst ll of th maz. g. 7 shows th path of th cntr-of-mass of swarm A through th maz to th goal. g. 3.g. Th swarm scap th local mnmum, t=270 g.3. Bhaour of a swarm usng th ntrnal stat modl wth agnts abo zoom wndow. I II III IV V VI g.4. Group angular momntum wth tm for swarm n g.3. g.6.a. Th two swarms as th startng poston, t=0 g.5. A prphral agnt s dsspaton coffcnt wth tm ISBN: WCE 2008

7 g. 6.b. t=7 g. 6.d. t=346 g. 6.c. t=25 g.6.. t=550 ISBN: WCE 2008

8 g. 6.f. Th swarm that us ntrnal stat modl sols th maz whl th swarm wth fxd ntrnal stats fals. t=80 g.6. Two swarms n a maz applcaton g.7. Th path of th swarm cntr nsd th maz VII. CONCLUSION Ths papr prsnts a dlopmnt of our work n orcomng th local mnma problm by usng th agnt s ntrnal stats along wth th mrgnt bhaour of th agnts. Th modl uss th swarm cntr locty to sol th problm n two ways. Th frst way lnks th goal attracton potntal strngth n th quaton of moton to th swarm cntr locty n a way that as th swarm cntr dcrass, th goal ffct dcrass and th local mnmum dsappars. At th sam tm whn th goal ffct dcrass, a swarm ortx pattrn mrgs. Ths actats th pur rollng moton of th swarm through whch th agnts nar to th wall of obstacls acqur hghr dsspaton coffcnt, consquntly hang lowr locts than thos ndduals that ar far from th wall. Ths wll nabl th swarm of agnts to ach pur rollng moton n whch th swarm follows th wall boundars n n cas of ndrct contact. Th smulaton rsults show that, rathr than mong n a statc potntal fld, th agnts ar abl to manpulat th potntal accordng to thr stmaton of whthr thy ar mong towards th goal or stuck n a local mnmum and th mthod allows a swarm of agnts to scap from and to manour around a local mnmum n th potntal fld to rach a goal. Ths nw mthodology succssfully sols ract path plannng problm, such as a complx maz wth multpl local mnma, whch cannot b sold usng conntonal statc potntal flds. REERENCES [] Czrok A., Bn-Jacob E., Cohn I., and Vcsk T., ormaton of complx bactral colons a slf-gnratd ortcs, Phys. R. E 54, 79, 996. [2] Parrsh J. K., and Edlstn-Ksht L., Complxty, pattrn, and olutonary trad-offs n anmal aggrgaton, Scnc (284), 99-0,999. [3] Camazn S., Dnubourg J. L., ranks N. R., Snyd J., Thraulaz G., and Bonabau E., Slf Organzaton n Bologcal Systms, Prncton Unrsty Prss, Prncton, NJ, [4] Holland O., and Mlhush C., Stgmrgy, Slf-Organzaton, and Sortng n Collct Robotcs. Artfcal Lf, 5: , 999. [5] Ln H., Rappl W. J., and Cohn I., Slf organzaton n systms of slf-proplld agnts, Phys. R. E, Vol. 63,, 070, [6] Moglnr A., Edlstn-Ksht L., Bnt L., and Spros A., Mutual ntractons, potntals, and nddual dstanc n a socal aggrgaton, Journal of Thortcal Bology 47, , [7] D Orsogna M. R., Chuang. L., Brtozz A. L., and Chays L., Slf-proplld agnts wth soft-cor ntractons: pattrns, stablty, and collaps, Phys. R. Ltt., Vol. 96, 04302, [8] McInns C., Vortx formaton n swarm of ntractng partcls, Phy. R. E 75, , [9] Taylor T., Ottry P. and Hallam J., Pattrn formaton for multrobot applcatons: Robust, slf-rparng systms nsprd by gntc rgulatory ntworks and cllular slf-organsaton, Tchncal Rport EDI-INRR- 097, School of Informatcs, Unrsty of Ednburgh, [0] Mabrouk M., and McInns C., Swarm Potntal lds wth Intrnal Agnt Stats and Collct Bhaour, In Procdng of Towards Autonomous Robotc Systms, TAROS, pp , 2007a. [] Badawy A. and McInns C., "Robot Moton Plannng usng Hyprbolod Potntal unctons", World Congrss on Engnrng, London, 2-4 July 2007, papr ICME 5. [2] Khatb O., Ral-tm obstacl aodanc for manpulators and mobl robots, Int. J. for Robotcs Rsarch, ol. 5, no., pp , 986. [3] Mabrouk M., and McInns C., Solng th potntal fld local mnmum problm usng ntrnal agnt stats, Submttd to th Journal of Robotcs and Autonomous systms, 2007b. [4] Mabrouk M., and McInns C., Swarm robot socal potntal flds wth ntrnal agnt dynamcs, In Procdng of th 2 th Intrnatonal Confrnc on Arospac Scncs & Aaton Tchnology (ASAT2), Caro, Egypt, ROB-02:-4, 2007c. [5] Mabrouk M., and McInns C., Socal potntal modl to smulat mrgnt bhaour for swarm robots, Accptd by th 3 th Intrnatonal Confrnc on Appld Mchancs and Mchancal Engnrng, AMME-3, ISBN: WCE 2008