MOTION PLANNING FOR MULTI-ROBOT ASSEMBLY SYSTEMS. M. Bonert L.H. Shu B. Benhabib*
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1 Poceengs of the 999 ASME Desgn Engneeng Techncal Confeences Septembe -, 999, Las Vegas, Nevaa DETC99/DAC-869 MOTION PLANNING FOR MULTI-ROBOT ASSEMBLY SYSTEMS M. Bonet L.H. Shu B. Benhabb* Compute Integate Manufactung Laboatoy, Depatment of Mechancal an Inustal Engneeng, Unvesty of Toonto, Kng s College Roa, Toonto, ON, Canaa, MS G8 *emal: beno@me.utoonto.ca Keywos: assembly, mult-obot, genetc algothms, moton annng, electonc component acement ABSTRACT Ths pape aesses a mult-obot optmal assembly annng poblem whch n essence s an augmente Tavellng Salespeson Poblem (TSP+). In ou TSP+, both the salespeson (a obot wth a tool) as well as the ctes (anothe obot wth a wopece) move. Namely, n aton to the sequencng of tass, futhe annng s eque to choose whee the salespeson shoul enezvous wth each cty. The use of a genetc algothm (GA) s chosen as the seach engne fo the soluton of the mult-obot TSP+ optmzaton. As an exame nustal apcaton the optmzaton of the electonc-component acement pocess s aesse. In the most genealze component-acement system confguaton, the acement obots meet the component elvey systems (CDSs) at optmal enezvous locatons fo the pc-up of components an subsequently meet the pnte ccut boa (on the moble XY-table) at optmal enezvous locatons fo the acement. In aton to the soluton of the componentacement sequencng poblem an the enezvous-pont annng poblem, the collson-avoance ssue s also aesse.. INTRODUCTION Autonomous obotc systems ae nceasngly utlze n nustal envonments, equng the evelopment of moellng methoologes an tools to optmze the opeatonal effcency. In ths context, the classcal assembly-annng optmzaton poblem has been extensvely stue, wth numeous ecent attempts at apyng the eale eseach esults to obotc-base assembly, [e.g., ]. Ths pape pesents a novel pont-to-pont (PTP) motonannng technque fo multe coonate assembly obots, whch can be moelle as a TSP, usng Genetc Algothms (GAs) []. As an exame nustal apcaton, the optmzaton of an electonc component acement pocess s utlze [e.g.,, ]. The Tavellng Salespeson Poblem: Many vaatons of ths combnatoal poblem have been aesse n the lteatue, nclung the Asymmetc, Symmetc, Euclean, Chebyshev, Pze Collectng an Tme-epenent TSP vaatons [e.g.,, 6, 7, 8, 9, an ]. Leu et al. [] solve the electonc-component acement optmzaton poblem usng genetc algothms. Thee types of assembly machnes wee moelle: () A sngleobot pc-an-ace poblem wth fxe feees an a fxe Pnte Ccut Boa (PCB), () Poblem extene to have feee component assgnment optmzaton, an () A multhea fxe acement tuet wth a movng XY-table an feee component optmzaton. The Augmente Tavellng Salespeson Poblem (TSP+): In contast to the sngle-obot TSPs, whee the pmay objectve s to fn the best sequence fo N tass, fo mult-coonateobot poblems, one must also solve the enezvous-pont annng poblem. In such augmente TSP, (TSP+), the salespeson (one obot) as well as the ctes (PCB acement locatons that ae move aoun by anothe obot) have moton capablty. Namely, futhe annng s eque to choose whee the salespeson shoul enezvous wth the cty. Ths mnmum-tme optmzaton poblem s futhe comcate when two acement obots ae use concuently n a tas-shang moe. As one of the few eseach pojects on pont-to-pont (PTP) TSP+, Cao et al. [,] aess the ssue of nspecton-tas-sequence annng fo two coonate obots usng the smulate-annealng technque. A sees of locatons on a sphee ae nspecte by the obot pa. As expecte, they showe that the coopeatve obot confguaton, Copyght 999 by ASME
2 whee both obots move, was faste than when only one obot move an the othe acte as a fxtue. Augmente Tavellng Salespeson Poblem wth Multe Robots: If an atonal obot s ntouce nto the TSP+, whch contnuously shaes ts wospace wth the othe acement obot, t woul be necessay to ectly aess the collson avoance poblem. Such multe-obot collson avoance poblems have been aesse n the lteatue n two pmay ways: () Collson avoance though path annng, an () Collson avoance though scheulng (o tme elays), [,, 6, 7]. Fo exame, Lee an Lee [8] popose solvng the collson-avoance poblem by spee changes, whch ntouce a tme elay n one of the two obots. Each obot s epesente by a sngle sphee at the wst, an staght-lne moton s assume. Fo paths that ae close togethe, collson maps of tme vesus stance tavelle along the obot path ae geneate.. PROBLEM DEFINITION & SOLUTION APPROACH In electonc assembly, components must be ace onto the PCB n a tme-effcent manne. The fst tas s the confguaton of the PCB, whee component locatons ae etemne subject to constants an objectves. In ths eseach, t s assume that ths tas has aleay been cae out. It s also assume that the component-acement machne pcs an aces one component at a tme. Although vaous othe acement stateges exst, an ae futhe etale n the lteatue, [e.g., 7], the objectve of ths pape s the nvestgaton of the funamental TSP+ poblem as t apes to electonc component acement... Poblem Defnton () Set-Up Geomety Fgue shows the most genealze physcal set up of the two-obot-acement machne moelle. The system compses fve man sub-systems: two X-Y ganty obots fo component pc-an-ace opeatons; a numecally contolle X-Y table, on whch the PCB s locate; an, two sngle-of multecomponent elvey systems, wth contollable motons n the Y ecton. The two ganty obots shae a common wospace, whch nclues the wospace of the espectve component-elvey systems an the wospace of the X-Y table. Each CDS s accesse by only the obot assgne to t. The two obots ae not allowe to cossove each othe. Component Delvey System Ganty Robot X-Y table Component Delvey System y x Placement Hea Ganty Robot Fgue : The Two-Robot Electonc-Component Placement Machne Confguaton. () Poblem Stuctue Fo the two-obot confguaton n Fgue, thee exst fve sub-poblems: Assgnng components to obots; Detemnng the acement sequence; Fnng the enezvous locatons; Coonatng obot movements; an, Plannng the evce paths. Although components ae ace one at a tme, accong to an oveall sequence, each component s ace by only one of the two obots. Ths eques each obot to be assgne the components t s esponsble fo acng. Snce components ae assume to be ace sequentally, the acement sequence s entfe as a vaable. The enezvous-annng poblem s the pocess of etemnng the meetng postons of the acement obot an the CDSs, an the meetng postons of the acement obot an the moble X-Y table. When the combne of of the two movng sub-systems s above the mnmum neee, an nfnte numbe of possble enezvous-locaton solutons exst fo evey potental pc o acement exchange between two movng evces. Theefoe, fo a gven sequence of pcs an acements, a coesponng set of optmal enezvous locatons must be etemne. Once the enezvous postons have been etemne, snce thee ae two ganty obots shang the wospace, t s necessay to coonate the actons of the two obots to pevent collsons. The fnal poblem s the obot-path-annng poblem. Fo a gven pont-to-pont (PTP) (enezvous) moton, the fastest obot path s nomally etemne usng the obot ynamcs. In ou case, fo a gven mult-component acement sequence, the optmalty of a potental set of enezvous locatons can be etemne by measung the oveall moton tme. To acheve optmal esults, nvual obot paths between these enezvous locatons must also be optmze. Ths obot path sub-optmzaton poblem s not aesse n ths pape snce t has been extensvely aesse by the obotcs eseach communty [e.g., 9]. Heen, t s smy assume that Copyght 999 by ASME
3 mnmum obot-moton tme can be acheve by mnmzng the Catesan stance tavelle by the nvual evces. () The Assembly Cycle In oe to calculate acement cycle tmes, the oveall boa-populaton assembly poblem s ve nto nvual cycles. Fgue shows the pocess of a genec pc-an-ace cycle. The obot stats ths cycle at ts pevous acement locaton an the CDS stats at ts pevous pc locaton. The CDS moves to the cuent pc locaton, path # (), whle the obot smultaneously moves thee as well to enezvous wth the CDS, path # (). The obot then pcs the component fom the CDS. Whle () an () ae happenng, the X-Y table moves to the cuent acement locaton, path # (), whee t meets wth the obot avng fom the cuent pc locaton, path # (). Whle () an () ae happenng, the CDS s allowe to move, wthout watng, to the next pc locaton.. The obot then aces the component an the cycle epeats tself fo the next component n the acement sequence (wth ethe the same obot o the secon obot). Last Pc Locaton Cuent Pc Locaton then, geneate new spng, whch ae evaluate, an the ente pocess s epeate untl the best genome s etemne. The soluton stategy must conse collson avoance between the two acement obots. Snce the paametes of sequencng an evce postons ung pc-an-ace opeatons affect both the collson avoance poblem an the pefomance of a gven acement stategy, t s avantageous to chec both smultaneously when seachng fo an optmum soluton, hence the ntegaton nto a sngle objectve functon n ths eseach. To encoe the two-obot paametes, a compoun genome consstng of a sees of sub-genomes s popose: a sequencng sub-genome, two obot-assgnment sub-genomes, an a enezvous-pont annng sub-genome.. THE TWO-PLACEMENT-ROBOT PROBLEM The goal of the two-acement-obot TSP+ poblem s to mnmze the total assembly tme. The cycle tme s use ectly as the bass fo the ftness scoe of a gven GA genome. Ths soluton s an extenson of a smla sngle-obot poblem etale n []. Fo claty puposes, the collson avoance ssue wll be aesse, only afte the sequencng an enezvous-pont-annng poblems have been scusse. Last Placement Locaton Cuent Placement Locaton CDS Robot X-Y Table Fgue : Illustaton of a Sngle Cycle... Soluton wthout Collson Avoance Conseatons... The Objectve Functon The objectve functon evaluates an assgns a ftness value to a genome. Assembly tme s use ectly as the bass fo the ftness scoe. A genome fo the two-obot case, wth no collson-avoance conseaton conssts of fou pats: The sequencng sub-genome, two obot-assgnment sub-genomes, an the enezvous-pont sub-genome... Popose Soluton Appoach The goal of ou eseach was the evelopment of a methoology to optmze the assembly pocess outlne above. Rathe than optmzng one paamete at a tme, a metho that allows us to optmze all paametes smultaneously s chosen. Ths s avantageous snce many of the paametes ae nteepenent an the mofcaton of one poblem paamete affects anothe. Use of a GA was chosen as the soluton appoach fo the poblem at han. The paametes fo the poblems ae encoe nto genomes (GA ata stngs). The ftness (objectve functon value) s use to etemne whch genomes ae chosen fo epoucton. The genetc opeatos,... Calculatng Cycle Tme Fo a potental acement sequence, the cycle tme pe component can be calculate base on the selecte enezvous locatons between the evces an the ntal locatons. In oe to convet ths collecton of postons nto tme, one nees to now the tajectoy of each evce fom one pont to the next. Fo smcty, n ths pape, t s assume that all evces move n staght lnes between two ponts at evces maxmum allowable spees. (Due to the moulaty of the popose soluton, one may utlze one of the many obot-tajectoy optmzaton technques popose n the lteatue). Copyght 999 by ASME
4 Fom Fgue, one can see that thee ae fou moton tmes that nee to be calculate: () The obot moton tme-to-pclocaton, t, () The CDS-moton-tme to pc locaton, t, () The obot moton tme-to-acement-locaton, t, an, (v) The X-Y table moton tme-to-acement locaton, t t. PK - t t PK t () The obot tme befoe the pc opeaton The cometon of the fst pat of the obot cycle tme,, epens on both the CDS moton an the obot s moton t st to the pc locaton. Theefoe, the fst pat of the obot cycle tme s the maxmum of the tme the obot taes to each the pc locaton an the tme the CDS taes to each t. The fome s the obot moton tme to pc locaton, t, mnus the obot tme, t. The latte s the CDS moton tme to pc locaton, t, mnus the CDS tme, t. Theefoe, st t = max[( t t ),( t t )]. () PL - tt PL The secon tem n Equaton (), t, s the tme the cuent obot has been snce ts last acement opeaton. If the th component s ace by the same obot as the (-) th component, then; CDS Robot X-Y Table Fgue : Illustaton of Cyclc Devce Moton Tmes. The oveall assembly tme fo a comete populaton of a PCB, t, s calculate heen as follows: N C = t =, () whee C s the tme t taes to comete cycle an N s the numbe of components on the PCB. One can note that the cycle C can be ve nto two pats: () The tme befoe the en of the component pc opeaton, an () the tme afte the component pc opeaton.... Calculatng Moton Tmes The cycle tme C fo the two-obot system s efne by: t C = max[ t, t ] + c, () whee the obot tme, t, s the tme the obot une conseaton executes all tass eque an moves fom the last acement locaton to be eay to ace the next component at the cuent acement locaton. The X-Y table tme, t t, s the tme the table taes to move fom the last acement locaton to the cuent acement locaton. The obot cycle tme, t, can be ve nto: () the tme befoe the pc opeaton, an () the tme afte an nclung the pc opeaton. =. () t Othewse, t s ace by the othe obot an; j j= + t = C, () whee the nex epesents the last cycle n whch the cuent obot move. The fouth tem n Equaton (), t, s the tme peo that the cuent CDS has not been nvolve n a pc opeaton an has ha tme to move towa ts next pc locaton. In oe to calculate t, the total tme of the last cycle n whch the CDS was pce fom, C, s taen, an the tme the CDS was busy s subtacte fom t. The CDS s busy fo the fst pat of st the obot cycle tme, t, an the component pc tme, c. If the th component s pce fom the same CDS as the (-) th component, then, =- an: t t t t c = C max[( ),( t )]. (6) Othewse, t s pce fom the othe CDS, whee t s also necessay to a all the othe cycles that the CDS has been : j= t = C max[( t t ),( t t )] c, (7) j whee the nex s the last cycle n whch a component was pce fom the CDS une conseaton. Copyght 999 by ASME
5 () The obot tme afte an nclung the pc opeaton n The secon pat of the obot cycle tme, t, epens on the pc opeaton tme, c, an the obot moton tme to acement locaton, t, whch occu sequentally, n t = c + t. (8) Ang the fst an secon pats of the obot cycle tme, the oveall obot moton tme eque n Equaton () s obtane as follows: st t t = + n t t t = max[( t ),( t t )] + c +.(9).. Two-Robot Poblem wth Collson-Avoance Conseatons As scusse eale, the two basc collson-avoance appoaches note n the lteatue ae () the path-annngbase appoach, an () the tme-elay-base appoach. The collson-avoance appoach popose heen as a safety elay to one of the two obots, elayng ts moton untl the othe obot s no longe on ts path. Ths metho may appea to be an appoach base on tme elays, howeve, snce collson avoance s ntegate nto the GA (as escbe below) the en esult s a combnaton of path changes an tme elays. Fo a gven GA genome, potental collsons ae avete by ang a safety elay to one of the two obots. Thus, the tme t taes to comete the assembly fo that patcula confguaton becomes longe. (Namely, the ftness of the genome that s base on the assembly tme becomes lowe). Ths s whee the coecton fo collson avoance as pat of the GAoptmzaton evaluaton becomes valuable. Snce the vaables that affect the optmzaton of the system also etemne whethe collsons occu, a sepaate collson-avoance coecton cae out afte the path-annng optmzaton coul ntefee wth the esult. Heen, wth the collson avoance ntegate nto the optmzaton, as s possble wth the GA, t s stll possble to ensue that the soluton s not only collson fee but also optmze.... The Objectve Functon The GA objectve functon evaluates a genome an assgns a ftness value to t. In ths secton, the objectve functon s entcal to the one pesente eale wth an atonal tem elate to collson avoance.... Calculatng Moton Tmes The cycle tme C s calculate usng Equaton (). Howeve, the expesson fo the obot cycle tme s mofe by s one atonal safety-elay tem, t : t = max[( t t ),( t t )] + c + t + t. () Befoe the use of the safety-elay tem n Equaton (), one must etemne f a collson s possble at all. It s assume that the obots o not stop anywhee except at the pc an acement locatons. Namely, mmeately afte a acement opeaton, the cuent obot stats to move bac to ts pc locaton. To cay out a collson chec, the obot state must be moelle n space an tme. In oe to smfy the collson etecton pocess, t was ece to moel the obots as walls movng acoss the wospace, as shown n Fgue. Ths moellng smfcaton euces the state of the obot (fo collson avoance calculatons) to two vaables, the obot X- coonate an a tme at whch the obot occupes the coesponng poston. In Fgue, fo the acement of a component, Robot moves fom CDS to the acement locaton. Its maxmum tavel towa the othe obot s the X-coonate of the acement locaton of component, w. The cuent obot (Robot, n the exame) can colle wth the othe obot (Robot ) that may stll be n the common wospace. Theefoe, n etemnng whethe a safety elay s eque, one must chec all pevous cycles, up to the last cycle, n whch the cuent obot move. Checng these pevous cycles s necessay snce the cycle sequence etemnes only when the components ae ace, an not when the assocate acement obot stats movng. Wth a lage enough obot- tme, a obot n cycle can stat movng ung cycle +, an each the acement locaton (potentally be n the othe obot s path) to wat fo the X-Y table. Fo a collson to occu, the poston an tme fo the two potentally collng obots have to match. Theefoe, the collson chec fo the cuent obot (n cycle ) checs all pevous cycles untl the cuent obot last move (cycle ). To chec fo an ovelap, cycles (+) to (-) ae examne to ensue that the cuent obot acement locaton n the X axs ( w ) oes not cossove wth any of the othe obot s pevous acement locatons ( w (+) to w (-) ). If an ovelap occus n one o moe of the pevous cycles, futhe testng to chec the tme vaables s necessay to confm o spove a collson. s Copyght 999 by ASME
6 Robot XY-Table the components n the bns o have the GA optmze the locatons. y CDS w w - Robot CDS.. Smulaton Exames The followng two confguatons wee consee fo a 6- component boa: () All evces movng wth use-efne-cds allocaton; Collson-avoance outne s not emoye. () All evces movng, wth use-efne-cds allocaton; Collson-avoance outne s emoye. x Fgue : The Robots Moelle as Walls fo Collson Avoance Detecton. Fo Case (), the smulaton yele a mnmum total tme of.6 s wth a acement sequence of (,,,,, ). The two obot paths ae shown n Fgue. In ths pape, collson s efne as a stuaton n whch the cuent obot entes the ovelap egon befoe the othe obot has left t. In the case whee both poston an tme vaables ncate that the obots ae n the ovelap egon at the same tme, a peventve stategy must be evelope. The appoach hee s to ntouce a elay to the cuent (cycle s) obot s moton to allow the othe obot to leave the ovelap egon befoe the cuent obot entes t. Y (mm), Afte all of the cycles, j= (+) to (-), have been teste fo collson, t s possble to calculate the exact amount of tme elay eque to avo a collson fo a patcula cycle j. The elay ntouce s the tme eque by the obstuctng obot to clea the ovelap aea befoe the cuent obot entes the ovelap aea, []... Seach Methoology Wth the above efnton of the objectve functon an ts vaous nput paametes, t s possble to moel any mult-obot coonate system. The methoology aapte n ths pape s the smultaneous soluton of the mult-level optmzaton poblem usng a GA as the seach engne. Namely, the optmal genome compses the best acement sequence of the components as well as the best enezvous locatons between the obotc evces. Snce one of the goals of ou wo was to exoe the effect of the ntoucton of moe of to an assembly system, ou softwae s econfguable to un ffeent set-ups: optmzng the postons of all-fxe evces; optmzng the fxe locaton of the X-Y table wth a moble CDS confguaton; optmzng the fxe locatons of the CDSs wth a moble X-Y table confguaton; an, an all-evces-movng confguaton. The softwae also allows uses to ethe efne the locatons of all - X (mm) Fgue : The Robot Paths; Case (), No Collson Avoance Routne. Fgue 6 shows the ot of the two obots postons (Xcoonate) vesus tme allowng fo a quc vsual collson entfcaton. On such ots, any locaton whee the lne epesentng Robot s movement cosses Robot s lne ncates a collson event. Thee ae two collson events n the exame consee. The fst occus when Robot colles wth Robot whle tyng to ace Component. The secon collson occus when Robot tes to ace Component an colles wth Robot acng Component. 6 Copyght 999 by ASME
7 ..8 Robot Robot..8 Robot Robot Tme (sec).6. Tme (sec) X (mm) Fgue 6: Robot X-Coonate vs. Tme; No Collson- Avoance Routne. - X (mm) Fgue 8: Robot X-Coonate vs. Tme; Wth Collson- Avoance Routne. Fo Case (), whee collson-avoance s consee, the smulaton yele a longe mnmum total tme of. s wth a ffeent acement sequence of (,,,,, ). Fgue 7 shows the obot paths fo Case (). Fgue 8 shows the obots postons vesus tme. In ths case, as expecte, the obot tme lnes o not coss. Namely, no collsons occu. Y (mm) - X (mm) Fgue 7: The Robot Paths; Case (), Wth Collson Avoance Routne.. CONCLUSIONS Ths pape pesente a genealze pont-to-pont (PTP) moton-annng technque fo multe coonate assembly obots. The popose appoach mnmzes assembly tmes usng Genetc Algothms (GAs). Specfcally, as an exame nustal apcaton, the optmzaton of the electonc component acement pocess was aesse. The collson avoance stategy popose s a ule-base appoach an s ectly ncopoate nto the objectve functon of the GA. Snce the paametes of sequencng an evce postons ung pc-an-ace opeatons affect both the collson avoance poblem an the pefomance of a gven acement stategy, both ae smultaneously optmze. REFERENCES. Dezne, Z. an Nof, S. Y., On Optmzng Bn Pcng an Inseton Plans fo Assembly Robots, IIE Tans., 6() pp. 6-7, 98.. Dav E. Golbeg. Genetc Algothms, n Seach, Optmzaton an Machne Leanng. New Yo: Ason Wesley Publshng Company, 9.. Chang, C., Smth, M. L. an Bla, E. L., Analyss of Assembly Plannng Poblem fo a Robotze Pnte Ccut Boa Assembly Cente, Poc. of the 9th Intenatonal Confeence on Poucton Reseach, Cncnnat, Oho, pp. 7-8, Fancs, R. L., Hamache, H. W. an Lee, C. Y., Fnng Placement Sequences an Bn Locatons fo Catesan Robots, IIE Tans., 6(), pp. 7-9, 99.. Boze, Y. A., Schon, E. C., an Shap, G. P., Geometc 7 Copyght 999 by ASME
8 Appoaches to Solve the Chebyshev Tavelng Salesman Poblem, IIE Tans., (), pp. 8-, J, Z., Leu, M. C. an Wong, H., Apcaton of Lnea Assgnment Moel fo Plannng of Robotc Pnte Ccut Boa Assembly, Tans. of the ASME; Jounal of Electonc Pacagng, (), pp. -6, Dubowsy, S. an Blubaugh, T. D., Plannng Tme- Optmal Robotc Manpulato Motons an Wo Places fo Pont-to-Pont Tass, IEEE Tans. on Robotcs an Automaton, (), pp. 77-8, Nelson, K.M. an Wlle, L.T., Compaatve Stuy Of Heustcs Fo Optmal Pnte Ccut Boa Assembly, Southcon Confeence Reco, pp. -7, Naft, J. NEUROPT: Neuocomputng fo Multobjectve Desgn Optmzaton fo Pnte Ccut Boa Component Placement, IJCNN: Intenatonal Jont Confeence on Neual Netwos, New Yo, NY, Vol., pp. -6, Huang, Y., Sha, K., Aance, J. an Westby, G., A Soluton Methoology fo the Multe Batch Suface Mount PCB Placement Sequence Poblem, Tansactons of the ASME; Jounal of Electonc Pacagng, 6(), pp. 8-89, 99.. Leu, M. C., Wong, H. an J, Z., Plannng of Component Placement/Inseton Sequence an Feee Setup n PCB Assembly Usng Genetc Algothm, Tans. of the ASME; Jounal of Electonc Pacagng, (), pp. -, 99.. Cao, B., Dos, G. I. an Iwn, G. W., Tme-Suboptmal Inspecton Tas Sequence Plannng fo Two Coopeatve Robot Ams Usng Mxe Optmzaton Algothms, Poc. of the IEEE Intenatonal Conf. on Robotcs an Automaton, Albuqueque, New Mexco, pp. -8, Cao, B., Dos, G. I. an Iwn, G. W., A Pactcal Appoach to Nea Tme-Optmal Inspecton-Tas- Sequence Plannng fo Two Coopeatve Inustal Robot Ams, Intenatonal Jounal of Robotcs Reseach, 7(8), pp , Baba, N. an Kubota, N., Collson Avoance Plannng of a Robot Manpulato by Usng Genetc Algothm. A Conseaton fo the Poblem n Whch Movng Obstacles an/o Seveal Robots ae Inclue n the Wospace, Poc. of the Fst IEEE Confeence on Evolutonay Computaton, New Yo, NY, Vol., pp. 7-79, 99.. Zuaws, R. an Phang, S., Path Plannng fo Robot Ams Opeatng n a Common Wospace, Poc. of the 99 Intenatonal Confeence on Inustal Electoncs, Contol, Instumentaton, an Automaton; Powe Electoncs an Moton Contol, New Yo, NY, Vol., pp. 68-, Chang, C., Chung, M. J. an Lee, B. H., Collson Avoance of Two Geneal Robot Manpulatos by Mnmum Delay Tme, IEEE Tans. on Systems, Man an Cybenetcs, (), pp. 7-, Baptste, P., Legea, B. an Vane, C., Host Scheulng Poblem: An Appoach Base on Constant Logc Pogammng, Poc. of the IEEE Intenatonal Confeence on Robotcs An Automaton, Los Alamtos, CA, Vol., pp. 9-, Lee, B. H. an Lee, C. S. G., Collson-Fee Moton Plannng of Two Robots, IEEE Tansactons on Systems, Man an Cybenetcs, 7(), pp. -, Cao, B. an Dos, G. I. Imementaton of Nea-Tme- Optmal Inspecton Tas Sequence Plannng fo Inustal Robot Ams, Poceengs of the th Intenatonal Woshop on Avance Moton Contol, Me, Japan, pp , Bonet, M., Tullet, F., Shu, L. H, Sela, M.N, Benhabb, B. Optmal Moton Plannng of Robotc Systems Pefomng Pont-to-Pont Electonc-Assembly Opeatons, n Wol Automaton Congess, ISORA, Alasa, pp..-.6, Bonet, M., Moton Plannng fo Mult-Robot Assembly Systems, M.A.Sc. Thess, Depatment of Mechancal an Inustal Engneeng, Unvesty of Toonto, Januay Copyght 999 by ASME
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