CELLULAR AUTOMATA BASED PATH-PLANNING ALGORITHM FOR AUTONOMOUS MOBILE ROBOTS. Rami Al-Hmouz, Tauseef Gulrez & Adel Al-Jumaily

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1 CELLULAR AUTOMATA BASED PATH-PLANNING ALGORITHM FOR AUTONOMOUS MOBILE ROBOTS Ram Al-Hmouz, Tauseef Gulrez & Adel Al-Jumaly Informaon and Communcaons Group ARC Cenre of Ecellence n Auonomous Sysems Unversy of Technology Sydney, Ausrala ralhmouz gulrez Absrac: Ths paper presens he applcaon of Cellular Auomaa (CA) model n solvng he problem of pah plannng. I s shown ha a CA allows he effcen compuaon of a pah from an nal o a goal confguraon on a physcal space cluered wh obsacles. The cellular space represens a dscree verson of he workspace. The mehod was epermenally esed on an auonomous moble robo on real-me sofware player/sage, n all cases very good pahs were obaned wh neglgble compuer effor. These smulaon resuls ndcae ha he Cellular Auomaa approach s a very promsng mehod for real me pah plannng. Copyrgh 005 IFAC. Keywords: Cellular Auomaa, pah plannng.. INTRODUCTION Pah plannng s essenal problem need o be solved n auonomous moble robo. Auomac moon plannng has applcaon n many areas such as robocs, vrual realy sysems, and compuer-aded desgn. Alhough many dfferen moon plannng mehods have been proposed, mos are no used n pracce snce hey are compuaonally nfeasble ecep for some resrced cases lke he case of moble robos Many researchers have proposed dfferen soluons durng he las 0 years. Snce (979, Lozano-Pérez e.al ) frs proposed a pahplannng algorhm among polyhedral obsacles based on he vsbly graph, hen eended o he Confguraon Space approach (T. Lozano-Pérez e.al 983). These ypes of approach requre a geomercal descrpon of he envronmen. The reconsrucon of geomercal prmves from sensoral daa s usually dffcul and me-consumng when n he real lfe he envronmen s changng. Therefore he updang phase of he world model becomes a relevan par of he navgaon algorhm. The pah-planners workng on hese models generae very precse opmal rajecores and can solve really dffcul problems, especally n cluered worlds, also akng no accoun non-holonomc consrans, bu hey are very me consumng, oo. In he leraure dverse algorhms have been proposed o ackle hs problem: Some of hem, such as he randomzed poenal feld mehods (H. Chang e.al 995) represen he robo as a parcle movng under he nfluence of an arfcal poenal feld produced by he sum of a repulsve poenal, generaed by he obsacles, and an aracve poenal, generaed by he goal confguraon. The pah s obaned by a descen along he negave graden of he oal poenal. Oher researcher s work s based on

2 roadmaps (D. Hsu, 000) and (L. Kavrak, e.al 995) n whch a nework of one-dmensonal curves, called he roadmap, lyng n he free space of he workspace s consruced. The fnal pah resuls from he concaenaon of hree sub-pahs, one connecng he nal confguraon o he roadmap, anoher belongng o he roadmap and a fnal one from he roadmap o he goal confguraon. Anoher general approach s based on a dvson of he free space no a se of eac or appromae cells (E. Anshelevch, e.al 000) and (H. Guowz, e.al 990). Ths paper descrbes a very fas, safe and complee roung mehod for a cluered envronmen based on he very smple rules of cellular auomaa. The ask s o buld a robo navgaon sysem ha works n realme, whle neracng wh he envronmen and reacng as fas as possble o s dynamcal evens. The organzaon of he paper s as follows: In he ne wo secons nroducons o he confguraon space formalsm and Cellular Auomaa are presened. In he fourh secon he proposed pah plannng algorhm s descrbed. In he ffh secon he compley of he algorhm s eamned and n he las secon some epermenal resuls are dscussed, wh he concluded remarks and he fuure work.. CONFIGURATION SPACE Pah Plannng problem can generally be consdered as a search n a confguraon space: Le A be a sngle rgd objec, movng n a Eucldean space W= R N, N = or 3, Le B...Bn be fed rgd bodes dsrbued n W. The B ' s are called as obsacles n workspace W, and he obsacles B...Bn are he closed subses of W, a confguraon of A s a specfcaon of he poson of every pon n A wh respec o F W, where F W s a Caresan coordnae sysem. The confguraon space of A s he space denoed by C, wh all possble confguraons of A. The subse of W occuped by A a confguraon q s denoed by A(q). A pah from an nal confguraon qn o a goal confguraon q goal s a connuous map τ :[0,] C wh τ (0) = q n and τ () = q goal. The workspace conans a fne number of obsacles denoed by B wh = N. Each obsacle B maps n C o a regon { q C A( q) B 0} C ( B ) = whch s called as C obsacle. The Unon of all Cobsacle s he C, U n obsacle regon = C( B ) and he se C free = C U n = C( B ). A free collson pah beween wo confguraons s any connuous pah. The confguraon space s a τ :[0,] C free powerful concepual ool because seems o be he naural space where he pah-plannng problem les. Ths s manly because any ransformaon of a rgd or arculaed body becomes a pon n he confguraon space. 3. CELLULAR AUTOMATA The proposed pah plannng mehod, consss on he successve applcaon of wo smple local Cellular Auomaa (CA) ranson rules (H. Guowz, e.al 990). CA s a decenralzed eended sysem conssng of a large number of dencal enes wh local connecvy arranged on a regular array. I consss of he followng componens: () A cellular space: a regular lace of cells each wh dencal fne sae machnes and he same local paern of connecvy along wh defne boundary condons. Here a square D bmap lace. () A se of saes Σ wh cardnaly sae k = Σ over whch he fne-sae machnes akes values. Each cell of he lace s denoed by an nde and s sae a me s represened by S. The neghborhood η, + ranson rule S = φ ( η ) whch esablshes he way n whch each cell of he auomaa s o be updaed. The ranson rule s appled synchronously o each cell n he CA, defnng an nrnsc parallel dynamc. + fs = 0 η S = S = () S oherwse wh S as he sae of he auomaa a me and η Moore neghborhood of cell. The objecve of hs nal dynamcs s he growh of each obsacle a number of cells o accoun for he physcal sze of he pah pon. The number of eraons wll depend on he sze of he pah gap (safes pah s condons) hs number s up o four. The schemac dagram n fgure shows hs process. Fg..Cellular Auomaa Process. (a) The absrac of envronmen showng free space and bounded space shows obsacle regon, (b) Afer applyng

3 he ranson rules on he envronmen he obsacles grow. In he hrd phase, he fnal confguraon resulng from phase s used as nal sae for a second CA dynamcs ha compues he shores dsance beween he nal and goal posons. In hs second cellular auomaa he possble saes for he cells are he followng: (0) free, () obsacle, () nal poson, (3) goal poson, (4) dsance l o he goal... (3+l) dsance l o he goal. The ranson rule appled o evolve hs cellular auomaa s: + = + S fs 0 η S 3 () S = S oherwse Ths process ha resembles a flood from he goal o he nal poson s shown n fgure. The flood dynamcs s sopped eher when he cell wh he nal poson s reached or when all cells n he cellular space are dfferen from 0 n whch case no pah s possble. In he former case a pah s calculaed by gong backwards from he goal o he sar n a descendng manner. Due he esence of saddle pons n he navgaon funcon, he shores pah beween an nal and goal poson s no well defned. Fg.. Flood Dynamcs hrough CA. (a) Frs CA eraon (epanson sars), (b) The sdes collde shown n black, (c) The resul from hrd eraon and more collsons shown n black, (d) The fourh and las eraon and more collsons recorded. Some mes a cell has more han one neghbor wh he same shores dsance o he goal. To follow always he seepes descend of he funcon may no work because here could be cases where he graden may have he same value n several drecons. In order o selec a reasonable good pah wo heurscs are appled among he neghbors of a cell: () The cells ha fulfll he condon of beng n drecon of he seepes descend and allows he conservaon of he drecon of movemen for he roue are seleced n he pah wh he hghes prory or () he ne prory s for hose cells n drecon Norh, Souh, Wes and Eas and have he smalles angle o he movng drecon of he robo are seleced. In general s found epermenally ha hese wo heurscs effcenly produces reasonable good pahs wh few mnmal changes of he drecon of movemen (commands for he roue). The knowledge of he sequence of cells ha consue a pah ogeher wh he nal drecon of movemen of he roue allows n a sraghforward manner he producon of a ls of commands whch gudes he roue along he pah o s goal. 4. THE ALGORITHM The workspace space s an Eucldean space n D, and s decomposed n a fne collecon of conve polygons, squares n hs case, called cells such ha her nerors do no nercep. These consue he cellular space of he CA. The confguraon space s hence represened by a se of hese dscree square cells, wh he obsacles occupyng a ceran number of cells. Assumng a grd of gven sze, he dscree confguraon space can be defned by: C = {(, y) { 0,..., }, y { 0,..., y } (3) Every confguraon q corresponds o a pon (,y), each obsacle s a se : B = q U j= (, y ) j j If each of he n obsacles of sze r, s decomposed no a se of rb of sze one each, hen C obsacle = nr U( j, y j ) j= The free space s defned by C free = C-C obsacle. The compuaonal cos for cellular auomaa, wh a smple ranson rule, s proporonal o he number of updaes eecued on he cells. Hence for a wo dmensonal auomaa wh a cellular space of ( y ) where s he number of updae eraons. In he applcaon of he ranson rule each cell eecues wo operaons: readng and wrng, of he sae of he cell self and of hose cells n s neghborhood. In consequence, n a complee evoluon, he number accessed cell s: ( y ) (+ η ) (6) where η s he sze of neghborhood. The frs CA dynamcs n he proposed algorhm produces a growh of he congesons (obsacles) n ω cells, herefore each cell wll be vsed: ( y (7) ) (+ η ) ω mes. The upper bound for he order of hs CA process s : O ( y ). (4) (5) Sar. Preprocessng of he Envronmen

4 . Compue he confguraon space Free, obsacle,nal-poson & Goal-poson 3. Updae he compued confguraon space wh CA No of updae eraons ( y ) Transon rule each (Blob) cell eecues wo operaons: readng and wrng ( y ) (+ η ), where η s he sze of neghborhood. 4. Compue he Shores dsance The epanson ll he edges mee wh each oher. Read and wre If No Goo 3 Else f he map fully cellularly auomaed Compue he shores pah from nal o goal Fnsh The second CA dynamcs compues he dsance and he basc number of cell accessed s: ( y ) (+ η ) l, where l s he Shores dsance from nal o goal poson. The wors case occurs when he pah conans nearly half of he cells, so he number of cell accesses s gven by ( y ) (+ η ) l y ( ). The order of hs process s O( y ).In he bes case he nal and goal posons are neghbors, wh a number of accesses equal o ( y ) (+ η ). The fnal phase of he algorhm deals wh he calculaon of he opmal roue pah: each neghbor of a cell on he pah has o be vsed n order o selec he bes heursc pah. Then (+ η ) l vss have o be performed and dependng on one of he followng y cases: wors case (+ η ) l ( ), and bes case (+ η ). Fnally when he algorhm deermnes he pah, commands for each cell of he pah has o be eamned along he enre pah lengh, herefore he upper bound s of order O(l). 5. SIMULATION AND RESULTS Fg. 3. The fnal envronmenal eracon wh CA, and many pahs has been creaed whch are mananng he safe dsance from he obsacles. Once he lnes generaed are joned, hs wll creae many pahs whn he same map, he selecon of pah sars when he sar and goal desnaon are added o he envronmen as shown n fgure 4. These pons wll deermne he real safes and shores pah selecon for he robo. Based upon he sar and goal pons fnally a real pah s obaned as n fgure 4. Fg. 4. The shores pah has been seleced from he predefned sar and goal poson, resulng n he dsance l. The algorhm was esed usng real-me sofware player/sage see (R. Vaughan e.al 004) on auonomous robo. The resuls were frs esed on he real-me smulaon envronmen, where he map was creaed wh arbrary obsacles, compuer generaes a very fne pah for he robo and robo raveled very safely as n fgure 5,6. Afer runnng he program, he occuped space obsacles mgh epand and collde wh each oher; he collson lnes produced by he epanson of obsacles could be joned o make a pah. Ths pah s so far he safes pah as far as he presen scenaro s concerned. Aferward, he collson lnes are conneced o each oher as shown n fgure 3.

5 Fg. 5. The map sen o he sofware for robo o move, rgh hand sde robo (red) s movng and on he lef hand sde s movemen can be seen on he wndow n grey movemen. compuer effor depends on he sze of he cellular space and he lengh of he resulng pah. Ths reduced me compley ogeher wh he smplcy of he cellular auomaa smulaon allows he algorhm o perform que well on seral machnes. Some of he key praccal advanages of he mehod are: does no requre parameer unng, real me performance for any knd of workspace scene, conssen behavor over repeaed epermens; can handle pah-plannng n dynamc envronmen. In concluson he praccal performance s very sasfacory however furher sudy nvolvng many more benchmarkng eamples are necessary. 7. FUTURE WORK The ne sep s o use CA n a cluered envronmen, whle usng probablsc roadmaps mehods as a shores pah mehod, whle ryng o mplemen he same heory on he dsrbued wreless neworks projec as well, where many unforeseen problems arse. REFERENCES Fg. 6. The robo successfully reaches s desnaon. The resuls show he possbly n a real me pah palnnng scenaro usng he proposed CA based algorhm. In epermenal rals wh a Penum III.6 GHz. machne, and usng an npu mage wh 60*0 pels resoluon, an average response of 399 ms per pah was obaned. When usng mages of 80*60 pels resoluon, a ms average response per pah was measured. The resuls are a rue pcure of real me Poneer DX auonomous robo (fg 7), whle usng CA based planner; hence robo can be moved fas among he cluered obsacle envronmen. Fg. 7. Poneer DX Robo. 6. CONCLUSIONS A cellular auomaa approach for solvng he pah plannng problem yelds very effcen epermenal performance n real me suaons. The cellular auomaa algorhms were esed wh dfferen workspace confguraons and cellular space szes over real me mages from a dgal camera. The Anshelevch. E, S. Owens, F. Lamrau, and L. Kavrak, Deformablevolumes n pah plannng applcaons, n Proc. IEEE In. Conf. Robocs and Auomaon, 000, pp Chang. H and T. Y. L, Assembly mananably sudy wh moonplannng, n Proc. IEEE In. Conf. Robocs and Auomaon, 995, pp Donald. B, K. Lynch, and D. Rus, New Drecons n Algorhmc and Compuaonal Robocs. Nack, MA: A. K. Peers, 000. Fnn P.W and L. E. Kavrak, Compuaonal approaches o drug desgn, Algorhmca, vol. 5, pp , 999. Guowz H, Cellular Auomaa. The M. Press,. Cambrdge, 990. Hsu D, Randomzed sngle-query moon plannng n epansve spaces, Ph.D. dsseraon, Dep. Compu. Sc., Sanford Unv., Sanford, CA, 000. Kavrak L, J. Laombe, R. Mowan, and P. Raghavan, Randomzed query processng n robo moon plannng, n Proc. ACMSymp. Theory of Compung, 995, pp Lozano-Pérez T, M.A. Wesley, An algorhm for plannng collson-free pahs among polyhedral obsacles, Commun. ACM (0) (979) Lozano-Pérez T, Spaal plannng: a confguraon space approach, IEEE Trans. Compu. C 3 () (983) Vaughan R and B. Gerkey, Player Verson.4rcManual, hp://playersage.sourceforge. ne/doc/player-manual-.4-hml/node.hml. (004).