Real Time Auction Based Allocation of Tasks for Multi-Robot Exploration Problem in Dynamic Environments

Transcription

1 Real Tme Auton Based Alloaton of Tasks for Mult-Robot Exploraton Problem n Dynam Envronments Sanem Sarel 1, Tuker Balh 2 1 Istanbul Tehnal Unversty, Department of Computer Engneerng, Maslak, Istanbul TURKEY TR Georga Insttute of Tehnology, College of Computng, Atlanta, Georga, USA sarel@s.tu.edu.tr, 2 tuker.balh@.gateh.edu Abstrat Sngle auton-based methods are known to be effent for mult-robot problem solvng. In ths work, we nvestgate performane of our general mult robot oordnaton framework for mult robot mult target exploraton problem under unertantes n dynam envronments. Our framework offers a real tme sngle tem alloaton method featurng dfferent mehansms for falure reovery. In mult robot exploraton problem, a dfferent verson of well known NP-hard MTSP Multple Travelng Salesman Problem), eah target s vsted by at least one robot n ts open tour. Overall objetve funton for ost optmzaton whle vstng targets vares by dfferent exploraton domans. In ths work, we present performane results for total ost mnmzaton objetve. There are many effent entralzed heurst methods for generatng lose to optmal solutons. These heursts may be used to alloate targets to robots. However, when the envronment s dynam and/or unknown, ntally assgned targets may need to be realloated durng run tme. In our framework, redundant alulatons are elmnated by means of nremental assgnments based on up-to-date stuatons of the envronment. Offered preautons n the framework mantan the qualty of solutons as lose to optmal as possble. Experments are onduted on smulatons. The omparson of the proposed method s made wth Prm Alloaton method. Introduton Searh and resue operatons, spae exploraton, reonnassane/survellane applatons requre effetve mult robot exploraton. Overall objetve ost funton of the robot team may be dfferent n these domans. Whle tme s a hard onstrant n searh and resue operatons, n spae operatons, the objetve may be mnmzng the overall ost value whh s proportonal to the total energy onsumed by robots. We propose a framework for heterogeneous teams of robots oordnatng to omplete omplex mssons that requre dverse apabltes and olletve work wthout a entral planner/deson maker. Robots are always n vald plans satsfyng onstrants on nterdependenes among tasks. Dfferent objetve funtons an be optmzed by ths framework when effetve ost funtons and onstrants are suppled. The framework also onssts of preauton routnes for falure reovery. Detals of the proposed framework and mehansms an be found n Sarel 2005). In ths work, we nvestgate performane of some heurst funtons ombned wth our framework for mult robot exploraton problem as a ase study. Ths applaton doman s seleted to analyze the devatons of results generated by the framework from optma. Optmal results an be easly obtaned by effetve nteger programmng formulaton. For ths partular ase problem, robot team s homogeneous; tasks do not requre dfferent robot apabltes and an be exeuted ndependently. Sngle robot mult target exploraton problem as a dfferent verson of TSP) s an NP-hard problem. For mult robot ase, alloatons of targets to robots are also very affetve on the qualty of the soluton. After alloatons are made, the problem turns nto a TSP problem for eah robot. When the knowledge of the robot s not ertan or the envronment s dynam, ntally alloated targets to robots are subjet to hane. Therefore alulatons for alloatng all targets may beome redundant. Our framework elmnates these redundant alulatons by means of nremental assgnments based on up-to-date stuatons of the envronment. Moreover, robot systems are open to falures ether n ommunaton or robots. Espeally robot falures may ause some of the ntally alloated targets beome unalloated, and these stuatons should be onsdered. Our framework an handle suh stuatons by preauton routnes. Communaton falures may prevent ntal alloatons from beng optmal. Our framework an also detet these stuatons and mantans hgh soluton qualty by dynam target seleton and task exhange shemes. Related Work Both ombnatoral and sngle auton methods are studed for mult robot exploraton problem n lterature. In GRAMMPS Brummt 1998), one of the earlest works on MTSP Mult Travelng Salesman Problem) problem, msson planner works entrally ether on one of the robots or as an operator. Msson planner selets the orrespondng robot for eah target. The system an regenerate plans when the envronment s hanged. Authors lam that for the ntal state of the system, the

2 alloatons may be sub-optmal. But n later steps when the number of open mssons dereases, the system an fnd lose to optmal solutons. By usng the smulated annealng algorthm, a randomzed searh over all possble alloatons s made. In ther latest work, Das and Stentz propose a market based sheme ntrodung leader approah for ombnatoral task exhanges Das 2002). These leaders are responsble for mult-party mult-task optmzatons for obtanng optmal results. In ombnatoral autons, dfferent ombnatons of tasks are offered and bdders bd by onsderng all tasks n these ombnatons. However ths method may beome ntratable for large nstanes or dynam stuatons n whh alulatons should be made frequently. Espeally when the envronment s dynam, alloatons may beome sub-optmal. Then a ombnatoral exhange mehansm s neessary to mantan optmalty. Computatonal requrements for ombnatoral autons nrease drastally for dynam envronments. Prm Alloaton method s a sngle auton based alloaton method. All the targets are assgned to the robots ntally. Whenever world knowledge s hanged, targets are reassgned aordngly wth the same algorthm Lagoudaks 2004). MSF Mnmum Spannng Forest) of all the targets and robots are onstruted by ths algorthm. Then routes of robots are onstruted for eah robot by depth frst traversal of the MST s Mnmum Spannng Tree) and addng shortuts. Sne ths algorthm s based on Prm Algorthm, ts worst ase performane s bounded by 2*OPT. The dstanes are onsdered from any of the nodes n robot routes whle onsderng a new target to assgn. Addtonal ost of vstng a target s not onsdered. Therefore some alloatons may be suboptmal. TSP Heursts Although TSP problem s NP-hard there are many effent heurst methods n lterature generatng k-opt solutons. Some of the heurst methods use trangle nequalty prnple gven n Eq. 1. The trangle nequalty prnple for tes, j and k assumes that the shortest dstane ) between two tes s a straght lne dret route). k j + Eq.1 jk In the followng heurst funton defntons, T s a partal tour and k s a ty not on T, and {, j} s one of the edges of T. Mnmum Spannng Tree Heurst MST) In ths method, ether Prm algorthm or Kruskal algorthm may be used to generate MST for the gven set of tes. Then a depth frst searh of T s onstruted. After ntrodung shortuts nto the depth frst searh, t s ensured that tes are vsted only one. Nearest Merger Heurst NMH) Ths method orresponds to the Kruskal s mnmum spannng tree algorthm. The algorthm starts wth n partal tours, eah onsstng of a sngle ty. Then suessvely merges the tours untl a sngle tour ontanng all the tes s obtaned. Eah tme trees to be merged are hosen so that mn {j: T and j T } s as small as possble. Nearest Neghborhood Heurst NNH) The algorthm starts wth an empty tour T. Ctes k and j, for whh j,k) mnmzed are found. {, j} s replaed by {, k} and {k, j} to obtan a new tour nludng k. The algorthm ontnues untl all the tes are added to T. Nearest Inserton Heurst NIH) The algorthm starts wth an empty tour T. Ctes k and j, for whh j,k) mnmzed are found. {, j} s the edge of T whh mnmzes, k) + k, j) -, j), and t s replaed by {, k} and {k, j} to obtan a new tour nludng k. The algorthm ontnues untl all the tes are added to T. Cheapest Inserton Heurst CIH) The algorthm starts wth an empty tour T. If T does not nlude all tes, for eah k, the edge {, j} of T whh mnmzes T, k) =, k) + k, j) -, j) s found. Then ty k mnmzng T, k) s found. If {, j} s the edge of T for whh T, k) s mnmzed, t s replaed by {, k} and {k, j} to obtan a new tour nludng k. The algorthm ontnues untl all the tes are added to the T. Farthest Inserton Heurst FIH) The algorthm starts wth an empty tour T. Cty k, whh s the farthest of all the tes out of T, s nserted to T. The algorthm ontnues untl all the tes are added to the T. Chrstofdes Algorthm CA) MST of the gven set of tes s onstruted. Mnmum mathng M* for the set of all odd-degree vertes n T s onstruted. An Euleran tour for the Euleran Graph that s the unon of the T and M* s found, and t s onverted nto a tour usng shortuts. Exept from FIH and NNH, all the algorthms presented above has a worst ase soluton bounded by 2*OPT. The worst ase soluton of the NNH s bounded by log n)*opt. CA soluton s bounded by 1.5*OPT. Mult Robot Exploraton Problem Mult robot exploraton problem s a dfferent verson of MTSP Multple Travelng Salesman Problem). In ths problem, eah target.e. nodes or tes of TSP) should be vsted by at least one robot n ts open tour. Even n sngle robot TSP problem, there s a trade-off between the

3 soluton qualty and the omputatonal requrements for generatng optmal solutons. If the envronment s dynam and/or knowledge of the robots s not ertan, alloatons made ntally may result n sub-optmal solutons n later steps beause of the desons based on unertan nformaton and estmates. A very smple llustratve example s gven n Fg. 1. In a) the estmated dstanes among targets and robots are seen. Wth ths knowledge, r 1 : t 3 and r 2 : t 1, t 2 assgnment gves a total ost of 13. However when the obstale s deteted, all the alloatons should be hanged as r 1 : t 1, t 2 and r 2 : t 3. R R1 7 3 a) R R1 7 3 b) Fgure 1. A smple llustratve example for realloatons a) Robots do not have omplete nformaton on the loatons of obstales. t 3 s assgned to r 1 ; t 1 and t 2 are assgned to r 2 b) Obstale s deteted; alloatons should be hanged. Falures are also ommon n robot systems. Therefore n addton to the dffultes of dynam envronments, robots may also beome natve durng runtme. That requres realloaton of all targets assgned to the faled robots when the world knowledge or the stuaton s hanged. In ths ase, for mantanng optmalty, nformaton on the loatons of the all atve robots and unvsted targets should be reonsdered. As the frequeny of updates on world nformaton nreases the alulatons for realloatons may beome ntratable. Therefore applyng an algorthm searhng all over the target set and assgnng all targets to robots may result n redundant alulatons wthout beneft of soluton qualty. Heurst funtons/algorthms used for fndng nearoptmal tours are onvenent for envronments wth unertantes Lawler 1985). In ths ase, omputaton s mplemented for a loal nstane of the overall problem. It s more logal to make loal searh for stuatons wth unertantes. However t should be noted that there s not a polynomal tme algorthm guaranteeng optmal soluton. All sngle TSP heurst algorthms presented n the prevous seton are desgned to generate a omplete sngle TSP tour. For mult robot exploraton problem, more than one TSP tour should be generated for eah robot. Moreover assgnments of the targets to robots affet the soluton qualty to a great extent. Clusterng methods generate lasses of the targets for the gven number of lusters. For ths partular problem, ntal luster number an be seleted as the number of robots or t may also allowed to be smaller than the number of robots related to the seleted objetve funton. One problem wth these algorthms s that they only take nto onsderaton of the dstanes between target pars. However osts of addng targets to routes of robots should be onsdered whle generatng lusters. Ths onsderaton may omplate the lusterng approah. Ths approah may be used for ntal seleton of targets to be onsdered. Vehle Routng Problem VRP) Ghan 2003) from transportaton and logsts researh s smlar to mult robot exploraton problem. Espeally dynam and stohast mult-depot VRP problem solutons an be used. In mult-depot VRP, vehles may be n dfferent depots, as n our ase robots may be n dfferent loatons ntally. Prm Alloaton Method Prm Alloaton method Lagoudaks 2004) nspred from Prm Algorthm generates a MSF Mnmum Spannng Forest) of the targets and robots. MSF onssts of separate robot trees. These trees are onstruted by addng eah unalloated target to the losest robot path ontanng the node wth the mnmum dstane to the target, untl all targets are alloated. In other words, addton of a new target s mplemented by onsderng the dstanes between the target and nodes of the robot routes nstead of onsderng the last poston of the robot. The target s autoned and the robot offerng the smallest bd value s seleted to alloate that target. Before robots run and vst the targets, all the alloatons are made. Whenever the world knowledge s hanged, alloatons for the remanng unvsted targets are made wth the same algorthm. Real Tme Sngle Auton Based Dynam Alloaton Sheme Our general framework Sarel 2005) s for a heterogeneous robot team workng on a omplex msson onsstng of tasks requrng dfferent apabltes and ooperatve work and havng dependenes on eah other ayl preedene relaton). In ths framework, optmal solutons are sought whle handlng wth ommon falures n robot systems. Ths framework s based on dstrbuted market based approah. However to mantan optmalty addtonal routnes suh as dynam target seleton and task exhange shemes alled as oalton mantenane n the general defnton) are added to the framework. In ths work, we nvestgate the performane of our framework for mult robot exploraton problem wth total ost mnmzaton objetve. In our approah, the targets are alloated n an nremental way. Robots vst targets and then offer autons for the next best unvsted target for them. Loatons of robots at the tme of the auton are onsdered whle alulatng dstanes to targets for all

4 expermented heurst ost funtons wth our framework. Therefore routes are onstruted onlne based on the world knowledge at hand. In the problem defnton, the robots are not supposed to ome bak to ther ntal loatons, or the mnmzaton of the dstane between last target of the route and the ntal loaton s not requred;.e. routes are not losed. Sne the targets are nrementally assgned to robots n our framework, onsderng robots up-to-date loaton nformaton s a better approah than onsderng all targets from routes of robot n the ost alulaton for our framework. Target Seleton Targets and related tasks are onsdered n eah step when the world knowledge s hanged f the robot s already assgned to a target or n eah step when the robot s dle. Stuatons n whh world knowledge s updated are arrval of new onlne targets, hanges n knowledge about envronment and dstane estmatons aordngly, new ost value nformaton on tasks under exeuton or deteton of robot falures. Ths proedure s alled as selettarget n our framework for the mult robot exploraton problem. selettarget method s the modfed verson of the seletaton method Sarel 2005) of the general framework and s gven n Fg. 2. selettarget for rj For eah known and unvsted target t f the target t s not n one of the followng types, skp The targets for whh the robot s aepted for exhangng wth hgher ost value than the robot has) Awarded targets by the autoneers Free targets whh are not under onsderaton else add the target t n a prorty queue ordered by ost wth the prortes n gven order Selet the mnmum ost target n the prorty queue f rj has already been alloated to a dfferent target anel vstng the target at hand f the seleted target s a free target begn AutonNegottatonProess else set the new target as the urrent target and begn exeutng f t s an awarded target send aept beome a member message to the autoneer Fgure 2. Target Seleton method The targets/tasks are prortzed based on the stuatons. If there are two targets at the same ost, the one wth the hghest prorty, otherwse the mnmum ost target s seleted. The targets sutable to exhange have the hghest, awarded targets by autoneers have the seond and the targets not beng onsdered free) have the lowest prorty. The targets are pushed nto a prorty queue n whh they are ordered based on ther estmated osts and prortes. The mnmum ost target top element of the queue) s seleted. If the robot s already assgned to a dfferent target than the seleted one, urrent task s aneled. If t s dle and the seleted target s a free target, an auton negotaton proess s ntated. Otherwse the target s set as the new assgned target and the robot begns exeutng t. If the target s an awarded target, a onfrmaton message s sent to the autoneer. Robots may onsder targets at the same tme n the general framework. However for obtanng optmal results for the gven objetve funton, we just allowed only one robot to vst a target or offer an auton. If there s more than one robot offerng an auton, the one havng the smallest ost value ontnues auton negotaton proess; other robots anel ther auton negotaton proesses. Auton Negotaton Proess In the auton negotaton proess, the orrespondng autoned target s assgned to one of the robots bdders) havng the mnmum ost value. Intally the autoneer offers the auton. If the auton s nvald, a warnng message s sent to the autoneer. Invaldty may our f the autoneer has nomplete knowledge about the task status. Possble stuatons may be that the task related to the target s ompleted or t has already been exeutng. If the auton passes the valdty hek, the anddate robot beomes a bdder of the task, alulates the ost and sends the ost value as a bd. The other anddate robots behave as so. The autoneer robot s also a bdder and generates a bd for the task at hand. It wats untl the end of the deadlne. If the autoneer does not get bds from other robots untl the deadlne, t selets tself as the awarded bdder. Otherwse t ranks all the bds and selets the robot wth the mnmum ost/bd value. A bdder robot may get onfrmaton from dfferent autoneers. However n the aton seleton step, t selets the mnmum ost target for t. Therefore t sends a message to only one of these autoneers. It s allowed that more than one robot offer an auton for the same task. In ths ase, when the robots detet ths onflt, the robot havng the mnmum ost value s the wnner of ths rae. If the ost values are the same, the robot havng the smaller dentfaton number s the wnner. Ths deson only provdes to reover from the onflt. There may be other solutons but we avod addng extra omplexty to the negotaton proess. Besdes that, sne these robots are n relable ommunaton range they an detet that both them autoned for the same task), the bd of eah of them s onsdered n ether ase not affetng the optmalty. Task Exhange Mehansm To keep onssteny, eah robot exeutng a task s responsble for broadastng an exeuton message to nform others that the task s under exeuton, the robot s alve and about the ost value for that moment. Sne n the deson stage, eah robot takng these knds of messages onsders the ost value from eah robot, f a robot detets that ts ost s lower than the broadasted ost of a task, t sends a task exhange message to the robot exeutng the task. The orrespondng robot anels ts exeuton after

5 reevng task exhange message and the sender of the message begns exeutng the task. Preautons For dealng unertantes beause of the message losses, eah robot keeps trak of the models of known tasks targets) and other robots n ther world knowledge. Up-todate knowledge of the known tasks s mantaned by representng eah known stuaton n a FSM. When robots get nformaton from others they update ther world knowledge aordngly. Whenever ommunaton beomes unrelable, world knowledge of eah robot may be nonsstent. The proposed framework ensures an update mehansm when onflts are deteted to redue nonssteny. When robots reeve nonsstent messages, they ether warn others or orret themselves. These nonsstenes our when robots are not nformed about the tasks that are ompleted, under exeuton or under auton. It s assumed that robots are trusted and benevolent. Exeuton onflts: If a parallel exeuton of the same task s deteted, all robots exeutng the same task for the same target) anel ther exeuton. Ths s for ensurng optmalty. A new auton s generated to onsder all the robots osts. The robots are allowed to generate dfferent autons for the same task. However one of them ontnues the auton negotaton proess. The robot gettng exeuton messages s nformed about the task and assumes that the orrespondng task s under exeuton for a perod of tme. If a robot annot get any message related to a task for a perod of tme, the task s assumed to be free. The exeuton message s a lue that the exeuter robot s stll alve and the task s beng exeuted. The same type of update s performed when robots offer autons for tasks. In ths ase the world knowledge s updated as the robot s not exeutng a task and the task s under onsderaton. Auton onflts: Whenever a robot gets an auton message, t frst heks the valdty of the auton by onsderng the task nformaton. If t s a vald auton, t always sends bd nformaton to the autoneer. Therefore a more sutable task for the robot may be seleted n future, whether t s urrently exeutng a task or not. Updates are mplemented when exeuton, anellaton messages are reeved. Robot Falures: Sne world knowledge of a robot s updated based on nomng nformaton from others, f a robot exeutng a task fals, after a perod of tme other robots not reevng messages from the faled robot mark the related task as left/free to onsder t n the target seleton proess. Cost Funton and Heursts: In the proposed framework, the general ost funton j for robot r j and target t s alulated as summaton of two separate omponents namely base dstane ost j1 ) and addtonal ost j2 ) as n Eq. 2. j = j1 + Eq.2 j2 These funtons hange based on the heurst methods HM). The base dstane ost funton for robot r j and target t s alulated as gven n Table 1 for dfferent stuatons. It an be seen that when a robot s n target seleton step, t always alulates the dstane to target from the urrent loaton of tself. When another robot offers an auton for that target, t alulates the dstane ether from the urrent loaton when dle) or from the assgned target t k when not dle) and sends the ost nformaton to the autoneer. Table 1. Base dstane ost j1 Robot s dle Robot onsderng the target Target s autoned by another robot Robot s already assgned to target t k dst r j, t ) dst r j, t ) dst r j, t ) dst t k, t ) Addtonal ost funton j2 ) for robot r j and target t depends on the seleton of the heurst method. We performed experments on three dfferent heurst methods gven n Table 2. These methods are Closest Cost CC) Heurst, Nearest Addton Cost NAC) Heurst and Farthest Addton Cost FAC) Heurst. We nspred from TSP heursts whle desgnng these ost funtons. Table 2. Cost funton omponents for dfferent heurst methods Base HM j2 ost 0 CC j1 NAC j1 FAC α * j1 dst t, ) dst rj, ) f dst t, ) > dst rj, ) 0 otherwse 1- α) * dst t, t ) max dst r, t ), dst r, t ))) f 1 f 2 j f 1 j f 2 T U s the unalloated target set, T Uj T U s the set of targets losest to robot r j. t k T U s the target wth the mnmum dstane dstt, t k ) to t. In CC heurst method, j2 s 0. Therefore robots onsder only dstanes to targets. NAC heurst method provdes one further look ahead for an addtonal ost of vstng target t. If the dstane between the next losest target t k to t s greater than the dstane from the robot loaton to t that means seleton of vstng target t k may ause addtonal ost gven n the Table 2. FAC heurst method onsder t f1 and t f2 n T Uj whh are the targets havng the maxmum dstt f1, t f2 ) value. The bas dea behnd ths heurst funton s that f a rle s drawn ontanng all the

6 targets n T Uj, t f1 and t f2 are the targets determnng the dameter of the rle and they should also be vsted. Ths method forwards robots to these farthest targets to some degree. By ntrodung a onstant α) ths addtonal ost onsderaton an be adjusted. We seleted α as 2/3 n the experments. Expermental Results The experments are onduted on a 100*100 grd envronment wth 20 targets and 1-4 robots. Target and robot loatons are assgned randomly for eah run. The results are presented as average of 50 runs for eah set. In the experments, the performane of the heurst methods are measured on a stat envronment and ompared wth Prm Alloaton method. Addtonally result of a sample run n whh robots do not have omplete nformaton on a dynam envronment s gven as an llustratve example. The dstanes are taken as symmetr n our problem meanng that the dstane between any two ponts are the same n both ways. The optmal results are obtaned by CPLEX program. The nteger programmng formulaton for obtanng optmal results an be found n Lagoudaks 2004). % Devaton from Optmal PRIM CC NAC FAC Number of Robots Fgure 3. Performane results of the heurst approahes ombned wth the framework and the Prm Alloaton, averaged over 50 runs gven wth standard devaton Results are presented as average perentage devaton from optmal values wth standard devaton n Fgure 3. As an be seen from the graphs, all heurst methods ombned wth our framework outperform Prm Alloaton method. The reason for even losest ost heurst CC) generates better results s takng nto aount of the up-to-date loaton of the robot. Although ntally some targets are lose to a robot, they may beome farther when the robot moves or ve versa. FAC heurst s the most suessful of all, t generates lose to optmal results even for experment wth one robot t should be noted that n ths ase, the problem s a sngle TSP problem). NAC heurst performane approahes to the other two suessful heursts CC and FAC) for the experment wth 4 robots. Devaton from optmum dereases when the number of robots nreases beause of the nrease n densty and the derease n dstanes for all approahes. Reonfguratons n a dynam envronment mplemented by the proposed framework an be seen n Fgure 4. In ths senaro, there are 6 targets and 3 robots n the envronment. Robots are shown as small squares. Targets are marked as rosses. World knowledge of robots s not ertan. In a) the envronment and n b) world knowledge of robots s gven. Robots begn exeutng the tasks. In ) robot r 1 detets the obstale and marks the target as unreahable. In d) robot r 3 detets the obstale n whh ase the target s reahable. However robot r 2 has the smallest ost for ths target. Then robot r 3 anels the exeuton. Robot r 2 onsders all the targets and dedes on the losest target e). Robot r 2 fals n f). Robots r 1 and r 2 detet ths falure. Robot r 1 ompletes the msson. Conluson In ths work we nvestgate performane of our general framework for a mult-robot team n an exploraton problem. Overall msson s explorng a set of targets wth mnmzaton of total ost objetve. The system an handle real tme ommunaton and robot falures. Experments on dfferent heurst ost funtons ombned wth the framework valdate that the results are lose to optmal. Desons on eah new stuaton and alloatons of targets n an nremental way make the results loser to optmal. Reovery solutons provded by preauton routnes for dfferent knds of falures ensure the approah s omplete. The approah s also sutable for dynam verson of the problem n whh targets are generated on-lne. Some stohast approahes may be used for forwardng to robots or for ther deson makng. Robots may selet hgh ost alulatons guaranteeng optmalty by usng Lnear Programmng methods f t s ertan that the envronment wll not hange. However the stuatons are subjet to hange for dynam envronments, and t s more reasonable to perform a loal searh by usng a heurst funton as n presented n ths work. Currently ombned tests wth ommunaton and robot falures for dynam envronments are beng expermented. In ths work, we analyzed performane on an objetve funton for mnmzng the total ost. As a future work, we are gong to perform experments for dfferent objetve funtons on mult robot exploraton problem n dfferent domans. Aknowledgments Ths work s supported by Semens TURKEY, Tnel Kultur Vakf TURKEY and NSF. Authors would lke to thank to Mhal G. Lagoudaks for provdng optmal CPLEX results.

7 Referenes Brummt, B. and Stentz, A., GRAMMPS: A Generalzed Msson Planner for Multple Robots n Unstrutured Envronments. In Proeedngs of the IEEE Internatonal Conferene on Robots and Automaton. Das, M.B. and Stentz, A., Opportunst Optmzaton for Market-Based Multrobot Control. In Proeedngs of the IEEE/JRS Internatonal Conferene on Intellgent Robots and Systems IROS). Ghan G. et al Real-Tme Vehle Routng: Soluton Conepts, Algorthms and Parallel Computng Strateges. European Journal of Operatonal Researh 151: Lagoudaks, M. G. et al., Smple Autons wth Performane Guarantees for Mult-Robot Task Alloaton. In Proeedngs of the IEEE/JRS Internatonal Conferene on Intellgent Robots and Systems IROS). Lawler, E.L., and Lenstra, J.K., eds The Travelng Salesman Problem. John Wley and Sons Ltd. Sarel S. and Balh T., A Robust Mult-Robot Coordnaton n Nosy and Dangerous Envronments. GVU Tehnal Report: GIT-GVU-05-17, Georga Insttute of Tehnology. R1 R1 R2 R3 a) Intal State of the Sene: 6 targets and 3 robots b) World Knowledge of the robots: unertantes are rled ) Robot r1 detets that the target s unreahable d) Robot r3 detets the obstale and anels the task e) Robot r2 begns to vst targets on ts way f) Robot r2 fals g) Robots r1 and r2 detet the falure. Robot r1 ompletes the msson Fgure 4. A sample senaro n a dynam envronment