Proceeding of he 7 h Inernaional Symposium on Arificial Inelligence, Roboics and Auomaion in Space: i-sairas 2003, NARA, Japan, May 19-23, 2003 Dynamic Neworks for Moion Planning in Muli-Robo Space Sysems Chrisopher M. Clark & Sephen M. Rock Aerospace Roboics Lab Deparmen of Aeronauics & Asronauics Sanford Universiy {chrisc, rock}@sun-valley.sanford.edu Jean-Claude Laombe Deparmen of Compuer Science Sanford Universy laombe@cs.sanford.edu KeyWords: Moion Planning, Roboics, Neworks ABSTRACT A new moion planning framework is presened ha enables muliple mobile robos wih limied ranges of sensing and communicaion o maneuver and achieve goals safely in dynamic environmens. The framework is applicable o boh planeary rover and free-floaing space robo applicaions. To combine he respecive advanages of cenralized and decenralized planning, his framework is based on he concep of cenralized planning wihin dynamic robo neworks. As he robos move in heir environmen, localized robo groups form neworks, wihin which world models and robo goals can be shared. Whenever a nework is formed, new informaion becomes available o all robos in his nework. Wih his new informaion, each robo uses a fas, cenralized planner o compue new coordinaed rajecories on he fly. Planning over muliple robo neworks is decenralized and disribued. The applicabiliy of he framework o planeary rovers is demonsraed in boh simulaions and real robo experimens. Also, he framework s applicabiliy o free-floaing robos in a 3D space environmen is demonsraed in simulaion. 1. INTRODUCTION Conceps for fuure space roboic sysems involve many robos under he direcion of a few human operaors, (e.g. assembly of large space srucures [1], humanrobo colonies [24]). To enable his ype of humanrobo operaion, robos mus be given a high degree of auonomy for compleing asks. Many challenges mus be overcome o achieve his level of auonomy. This research focuses on one of hese challenges: muli-robo moion planning. When many robos operae in he same environmen, high-level moion planning is required for he robos o accomplish asks auonomously. They mus be able o reach heir goals while avoiding collisions among hemselves and wih saic and moving obsacles. In unknown or parially known environmens, i is unlikely ha a sysem of sensors can provide global knowledge. In addiion, coninuous iner-robo communicaion is usually no feasible. Insead, only robos ha are sufficienly close o each oher can exchange informaion, e.g., share heir goals and local world models. This paper inroduces a new planning framework ha explois he changing communicaion links beween robos, as he robos move, o combine he respecive advanages of cenralized and decenralized planning. Figure 1: Moion Planning for 4 free-floaing robos in a 3D space environmen. Yellow lines denoe robo rajecories ha end a goal locaions denoed by red cube laices. The large gray cubes denoe obsacles. More precisely, our approach is based on dynamic robo neworks ha are capable of: 1) forming dynamically whenever communicaion and sensing capabiliies permi; 2) sharing world models and robo goals wihin each nework; and 3) consrucing on he fly coordinaed rajecories for all robos in each nework using a fas cenralized moion planner. A brief overview of his approach is presened in Secion 2. Then, a background review (Secion 3) jusifies he choices made in our approach. We hen describe some aspecs of our framework in more deail, namely he represenaion of parial world models (Secion 4) and he planning echnique used (Secion 5). Secion 6 presens he Micro-Auonomous RoverS (MARS) es-plaform and discusses experimens involving rovers in a 2D workspace. In Secion 7, resuls from free-floaing space robo simulaions are provided. 2. PLANNING IN DYNAMIC NETWORKS 2.1 Nework Formaion When any wo robos are wihin communicaion range of each oher, hey esablish a communicaion link. Define G o be he graph whose nodes are he robos and edges are he communicaion links. A nework of robos is any group of k 1 robos forming a maximal conneced componen of G. So, any wo robos in a nework can communicae hrough one or several communicaion links, bu wo robos from differen neworks can no.
Figure 2a shows an environmen wih 5 robos, where 2 neworks have formed. In Ne1, he op and boom robos can exchange informaion via heir communicaion links wih he middle robo. Because robos are moving o achieve heir goal locaions, he neworks are dynamic. Robos may leave neworks and/or form new neworks (see Figure 2b). An applicaion level proocol ensures ha a any ime robos in each nework can access he local sensing informaion of all oher robos in he same nework, and hence share a common world model. Ne 0 Ne 1 Ne 2 a) All hree robos (grey circles) are a heir iniial locaions. The wo lef robos are in communicaion range and form a nework. Their cenralized planners creae coordinaed collision-free rajecories for hem oward he goals (crosshairs). The righ robo forms a nework by iself, and is rajecory is planned independenly from he oher wo. The robos sar moving along hese rajecories. a) b) Figure 2: Example wih 5 robos. Dashed lines beween robos depic communicaion links. In a) he robos form wo disinc neworks Ne0 and Ne1. In b), wo robos have moved, and he wo neworks in a) have merged ino Ne2. b) As he robos move along heir rajecories, he middle robo and he righ robo ener communicaion range wih each oher, and all hree robos now form a larger nework. 2.2 Planning Process Moion planning in a nework N is riggered by any one of he following evens: N jus go formed, i.e., wo robos from differen neworks enered one anoher s communicaion range. A significan change in he world model occurs, e.g., a robo in N senses a new obsacle. c) A new plan is made for all hree robos in he nework. This plan consiss of collision-free rajecories for all hree robos. A new goal locaion is requesed for one or several robos in N. When such a riggering even occurs, daa is exchanged beween he robos in N, so ha each one ges an updaed world model ha combines he local world model and goal of every robo. Once robos have shared his informaion, each robo runs is own copy of a cenralized moion planner o consruc coordinaed rajecories for all robos in he nework. When he planner erminaes, each robo broadcass is plan o all oher robos in he nework. Each robo selecs he same bes plan and immediaely sars execuing is rajecory in his plan. The planner is a single-query probabilisicroadmap (PRM) planner similar o he one presened in [14] (see Secion 5). This process is illusraed in Figure 3 on a simple example involving 3 robos, wih no obsacles. A riggering even auomaically occurs a he sar of he process, as he firs neworks ge formed. Since robos also have limied sensing, he world model shared hrough a nework is parial. Planning is done using his model. As robos move, heir sensors may deec previously unknown obsacles or a change in he rajecory followed by a known obsacle. Such an even riggers a re-plan operaion wihin he nework where he new obsacle or change of rajecory was deeced. d) As robos move along heir new rajecories, hey leave communicaion range of each oher and some nework links are broken. They keep following he planned rajecories. Figure 3: Top-down view of a planning example wih hree robos. In each of he fours snapshos, he illusraion on he lef shows he robos on heir rajecories o heir respecive goals (cross-hairs). The diagram on he righ depics he communicaion range of each robo and he exising communicaion links. 3. BACKGROUND REVIEW Mos previous work on muli-robo moion planning can be grouped ino cenralized and decenralized planning [3,27]. While cenralized planning considers all robos ogeher as if hey were forming a single muli-body robo [5,8,26,19,30,31], a decenralized planner plans for each robo separaely before coordinaing he individual plans by uning he robo velociies along heir respecive pahs [2,4,11,16,17,22,25,29]. A varian of decenralized, called prioriizing planning, plans for one robo a a ime, in some sequence, considering he robos whose rajecories have already been planned as moving obsacles [6,12].
Cenralized planners can be advanageous because hey allow he possibiliy of compleeness and global opimizaion. For example, i was shown in [27] ha a cenralized planner based on PRM echniques can reliably solve problems requiring he igh coordinaion of muliple ariculaed arms, while decenralized planners based on similar PRM echniques fail ofen. On he oher hand, cenralized planning may ake more ime due o he high dimensionaliy of he configuraion spaces ha are searched. A worse drawback is ha hey require all informaion (parial world models and robo goals) o be cenralized in one single place, which is only possible if he robos have unlimied communicaion abiliies. This is no he case in many pracical seings. A major advanage of decenralized planning is ha i allows for disribued planning. Each robo can hen plan is own rajecory using is own parial model of he environmen. If wo robos evenually ge close o one anoher and risk colliding, simple velociy-uning echniques or oher reacive echniques can be used o locally coordinae heir moions. However, such a fully disribued approach fails o exploi he fac ha localized groups of robos can exchange informaion o improve planning By searching several configuraion spaces of smaller dimensionaliy, decenralized planning is poenially less compuaionally inensive. Bu i canno offer any compleeness or opimaliy guaranee. Various aemps have been made o improve he oucome of decenralized planners (e.g., [4,6,13]). In paricular, a negoiaion scheme beween localized groups of robos is used in [4] o assign prioriy orders o robos, which allow he decenralized planner o compue rajecories of reduced lenghs. This negoiaion scheme demonsraes he benefis of localized iner-robo communicaion, and is he echnique mos closely relaed o he robo nework planning framework presened in his paper. However de-cenralized planning remains inrinsically incomplee. The planning approach presened in his paper explois he respecive advanages of cenralized and decenralized planning. In each robo nework, i uses a cenralized single-query PRM planner o increase compleeness and sill provide fas on-he-fly planning. However, planning is disribued over he various neworks hence, planning over muliple neworks is decenralized o accommodae he fac ha robos from differen neworks canno share informaion. The riggering even caused by he merging of wo previously disinc neworks ino a single nework leads he robos in his new nework o ake advanage of he informaion hey now share by cenrally re-planning heir coordinaed rajecories. Planning wih incomplee world models and on-he-fly re-planning when a sensor deecs he presence of a sill unknown obsacle or a change in an obsacle s rajecory have previously been described in [14, 18] for a single robo. We use similar echniques, bu exend hem o muliple robo neworks. 4. WORLD MODEL Describing he world model in a concise bu useful form is necessary o allow for informaion sharing beween robos in he same nework. In he experimenal sysem ha we have buil, world models simply consis of a lis of robos and heir descripions, and a lis of obsacles and heir descripions. The following able oulines he informaion sored in each lis: World Model Descripion 1) Lis of Robo Descripions - Sae (posiion and velociy) - Size (Radius) - Mos Recen Updae Time - Informaion Source - Goal posiion - Curren Trajecory 2) Lis of Obsacle Descripions - Sae (posiion and velociy) - Size (Radius) - Mos Recen Updae Time - Informaion Source Robos repor heir own size and sae, while obsacle sizes and saes are esimaed by robo sensors. The mos recen updae ime is useful when updaing world models wih informaion received from oher robos. The informaion source is a robo idenificaion number ha indicaes which robo sensed (or communicaed wih) he objec. I is used o keep rack of which robos are currenly in he nework. Several assumpions were made o allow such a concise world model: Each robo has access o is own sae relaive o a global coordinae sysem (e.g., GPS). Each objec is approximaed as a circular objec o allow is geomery o be described by a single parameer, is radius. Each obsacle has consan linear velociy esimaed by a robo s sensor. As in [11], if a any laer ime is rajecory is found o diverge by more han some hreshold from he prediced rajecory (eiher because he obsacle did no move a consan velociy, or because he error in he velociy esimae was oo high), hen he robo ha deecs his divergence calls for he consrucion of a new plan wihin is nework. The planner grows he obsacles (and he robos) o allow for some errors in prediced rajecories of he objecs. All objecs in he environmen are easily idenifiable by robo sensors, which can also precisely esimae heir posiions and velociies. Any discrepancy beween wo local world models can be easily resolved.
The second assumpion is raher easy o eliminae, as i has been shown before ha PRM planners can efficienly deal wih geomerically complex robos and obsacles (e.g., [26]). In [14], he hird assumpion has been shown o be quie reasonable, even when obsacle velociies change frequenly, provided ha (re-) planning is fas enough. The las assumpion is more crucial. In our experimenal sysem, i is enforced by engineering he vision sysem appropriaely (see Secion 6.2). In he fuure, i will be imporan o relax his assumpion by using more general sensing sysems and daa fusion echniques [23]. 5. MOTION PLANNING ALGORITHM As indicaed earlier, moion planning wihin a robo nework is done using a cenralized single-query PRM planner (more precisely, several copies of his planner running in parallel). This planner searches he join sae ime space C of he k robos in his nework. The sae of each robo is defined by he wo coordinaes of is cener and wo velociy parameers, so C has 4k+1 dimensions. This represenaion can easily be exended o oher robos. For insance, we have implemened a version of he planner for robos in hree-dimensional space [10]. The planner searches C for a collision-free rajecory from he iniial sae of he robos o heir goal sae. The resuling rajecory defines he coordinaed moions of he robos o heir respecive goals. Our planner searches C by incremenally building a ree of milesones (he roadmap), as described in [14,15,19]. A each ieraion, i selecs a milesone m in he curren roadmap, generaes a collision-free sae m a random in a neighborhood of m in C and, if he pah from m o m ess collision-free, insalls m as a new milesone in he roadmap. The search erminaes when m falls ino an endgame region around he goal. See [14] for deails. As in [14,28], our planner saisfies kinodynamic consrains as follows: o generae each new milesone m, i picks a conrol inpu a random and inegraes he equaions of moion of he robos over a shor duraion. We name our planner Kinodynamic Randomized Moion Planning (KRMP). As shown in [14], under reasonable assumpions on he free space, he probabiliy of no finding a plan when one exiss decreases exponenially o 0 as he number of milesones increases. This is a major advanage over our previous work in [9,11], which used a decenralized prioriized planning approach. Noe, however, ha he fac ha he planner is probabilisically complee does no imply ha he enire sysem is also probabilisically complee. The robos use parial world models and hus need o re-plan heir rajecories when hey encouner discrepancies in heir model, (e.g. new obsacles). Since here is no guaranee ha a series of complee plans is iself a complee plan, he robos are no guaraneed o find a global plan if one exiss. While i is unclear o wha exen he noion of compleeness applies when planning for global goals wih only parial knowledge of he environmen, i is sill desirable o achieve compleeness in he sysem s componens whenever his is possible. The work in [14] also demonsraed empirically ha he above echniques successfully compue rajecories for a single robo wih kinodynamic moion consrains, in real-ime, (i.e. fas enough o be run on he fly). To enable moion planning wihin robo neworks, KRMP exends his previous work o accommodae several robos. Modified echniques are needed o 1) selec milesones for expansion, 2) generae new milesones, and 3) define he endgame region. Below we presen only he echnique we use o generae a new milesone m. No all modificaions are presened in his paper. When planning for muliple robos, one may generae m using he following parallel approach: firs, pick he conrol inpus for all he robos a random; nex, inegrae he moions of all he robos concurrenly; if no collision is deeced, hen record he endpoin as a new milesone, oherwise pick anoher se of conrol inpus. We found ha his echnique yields a high rejecion rae, especially in igh space. This led us o develop he following sequenial approach: consider he robos in some order, pick he conrol inpu for each robo and inegrae is moion (while considering he previous robos as moving obsacles); if he moion collides, pick anoher conrol or change he moion of a previous robo. Our experimens show ha his sequenial approach makes i possible o ge each new milesone much faser, wihou affecing he probabilisic compleeness of he overall planner. Finally, we ake advanage of he various processors available in a robo nework by concurrenly running a separae copy of KRMP on each robo of he nework. Each copy uses a differen seed of he random number generaor, hence consrucs differen roadmaps. We se he same imeou consrain (ypically, a small fracion of a second) on every robo. Each robo hen reurns a plan or is failure o generae one. The same bes plan is seleced by he robos and each robo immediaely swiches o execuing is new rajecory. This is made possible because we use a PRM planning approach. 6. ROVER PLANNING To validae our planning approach, i was implemened on he MARS es-plaform. This secion describes he hardware used for rover experimens, followed by a brief summary of experimenal resuls. For deails abou he implemenaion of he planner on planeary rovers, refer o [11]. 6.1 Micro-Auonomous RoverS Tes-Plaform Locaed in he Aerospace Roboics Lab a Sanford Universiy, he Micro-Auonomous RoverS (MARS) es-plaform is used o model mobile robos in a wodimensional workspace. The plaform consiss of a large 12 x 9 fla, granie able wih six auonomous robos ha move abou he able s surface.
The robos are cylindrical in shape and use wo independenly driven wheels ha allow hem o roae on he spo, bu inhibi laeral movemen (nonholonomic consrain). Each robo is equipped wih is own planner (copy of KRMP) and conroller ha are locaed off-board. An overhead vision sysem is used o rack he saes of all objecs on he able. The vision sysem processor calculaes hese saes and publishes hem o all applicaions ha subscribe (see Figure 4). This makes global sae informaion available o all robos. To simulae he limied sensing range ha would occur when sensors are mouned on robos, he objec saes are filered such ha robos only receive sae informaion regarding objecs wihin some predeermined range of he robo. Figure 4 shows he compuer/nework archiecure of he MARS es-plaform. All he processing is done offboard. Two processors are assigned o each robo, respecively for planning and conrol. These compuers are conneced hrough a LAN. All communicaion wihin he LAN is accomplished wih Real Time Innovaion's Nework Daa Delivery Service (NDDS) sofware. Because a LAN is used for iner-robo communicaion insead of a wireless medium, here are no physical barriers o limi he range of communicaion. Hence he communicaion barrier is simulaed. experimen. The boom phoo shows he physical hardware, and was aken a he same ime as he GUI screensho. In he GUI, robos and objecs are depiced as small and large circles, respecively. Robo goal locaions are indicaed by cross-hairs, and lines leading o he goal locaions depic he rajecories. When robos form a nework as described in Secion 2, i is indicaed by a color change. Hence robos wihin a nework have a common color, and his color will differ beween neworks. In he experimen presened, all five robos are iniially locaed a he near end of he able (i.e. boom of he GUI screen). Communicaion and sensing ranges were limied o 0.75 m. Robo colors indicae ha 2 neworks have formed, one wih he 2 robos in he boom lef and one wih he 2 robos in he boom righ. As he experimen progresses, he robos follow heir rajecories o reach heir goal locaions a he far end of he able. Throughou he experimen, robos planned an average of 3.4 imes, and planning imes were an average of 9 ms. NDDS is based on a publish/subscribe archiecure. To broadcas messages by flooding a robo nework, he sender will publish a message o which all robos subscribe. Before robos can receive heir subscripions, he messages are filered so ha only robos wihin some predeermined range of he sender will receive he message. This effecively simulaes a discree physical communicaion range. GUI Planners Conrollers Visualizaion ( ( ( ) ) ) Vision Figure 5: Example experimen on he MARS esplaform involving 5 robos and 3 obsacles. Successful planning was demonsraed in more complex simulaions involving up o 12 planeary rovers in a bounded workspace ha conained 12 saic and moving obsacles, (see [10]). 7. FREE-FLOATING ROBOT PLANNING Figure 4: Nework archiecure of MARS es-plaform 6.2 Planeary Rover Experimens To illusrae he applicabiliy of he planner o a physical sysem, real robo experimens wih up o 5 robos have been carried ou. One example of such an experimen is illusraed in Figure 5. The op phoo is a screensho of he GUI aken a one poin in he This secion deails planner implemenaion issues, provides a brief noe on he 3D visualizaion, and gives resuls from simulaions of free-floaing robos in a 3D environmen. 7.1 Free-Floaing Robos Planner Implemenaion Free-Floaing Robo Model - A simple cube-shaped robo equipped wih 6 independen on/off hrusers was used o model a space robo in simulaion. Noe his
does no allow for any change in orienaion. Fuure work will include addiional hrusers o allow roll, pich and yaw variaion. The sae of he robos can be described by X = (x 1, x 2, x 3 ) R 3 represening he posiion wih respec o he inerial frame. Milesones are specified by he sae of he k robos a a paricular ime, (X 0, X 1,..., X k, ). To implemen his in pracice, one mus creae a lis of milesones o ge from m e o m g, he milesone defining he goal saes of each robo. Each milesone in his lis corresponds wih he change in acuaion necessary for obaining he bang-off-bang conrol sequence. u 1 x 3 u 2 x 2 u 3 x 1 Figure 6: Sae-space model of he free-floaing robo Milesone Generaion - To generae a new milesone for he road map, hruser conrol inpus are randomly seleced ha will propagae robos o new saes. Firs, he ime for which he hrusers will be acuaed, ( ac ), is randomly seleced where: [ ] min, max x 1 x 2 x 3 m e m 0 m 1 m 2 m 3 m 4 m 5 m g ac Figure 7: Example of acuaion required o move one Nex, he conrol inpus (ON/OFF) are randomly seleced for each hruser. We assume ha only one of wo opposie-facing hrusers should be enabled a he same ime. This can reduce he number of random variables. Tha is, for each pair of opposie-facing hrusers, a conrol inpu variable u ac is seleced where: u { 1, 1,0} ac u ac 1-1 0 Thruser 1 ON OFF OFF Thruser 2 OFF ON OFF Table 1: Mapping he random variable u ac o hruser acuaion. Wih he random variables seleced, a candidae milesone m can be generaed. Given any paren milesone m p, and using 1/s 2 dynamics, robo saes in m can be easily calculaed: robo from (0, 0, 0) o a goal sae. The series of milesones required is {m p, m 0, m 1, mm 3, m 4, m 5, m g } 7.2 Free-Floaing Robo Simulaions To simulae moion planning experimens ha involve free-floaing robos maneuvering in a 3D space environmen, he es plaform described in Secion 6 was augmened o incorporae a 3D visualizaion. The applicaion was coded in C++ and OpenGl. A screensho from he applicaion can be seen in Figures 1 and 9. As depiced in Figure 4, he applicaion acs only as a lisener o receive sae informaion. The applicabiliy of he planner o a 3D environmen was validaed wih simulaions ha include up o 8 robos and 8 obsacles. A es scenario is provided in which robos mus cross pahs several imes. A GUI screensho of he scenario is provided in Figure 8, (Noe ha he hird dimension is no displayed here.) u = + x 2M uaci i = ac + x i M + x aci 2 x i ac i, p ac i, p x, p Endgame Region - The endgame region E in his implemenaion is defined as he subspace ha includes all milesones m e, such ha all robos can be propagaed wihou collisions from saes defined by m e o heir respecive goal locaion via a bang-off-bang conrol sequence. An advanage his sequence has is ha i allows us o limi he velociy of he robo, making i easier o re-plan in he fuure. Figure 8: A es scenario involving 8 robos and 8 obsacles. On he lef, four robos denoed by ligh gray circles have formed a nework. On he righ, wo neworks have formed, each wih wo robos, (denoed by blue and yellow circles). Lines indicae rajecories. Obsacles are depiced as dark circles.
The es scenario was simulaed 25 imes o produce he resuls in Table 2. From hese resuls i is clear ha he planner was capable of planning on he fly wih average planning imes of 67 ms. An average of 12.2 neworks were formed hroughou each simulaion. Avg. number of plans made by each robo 4.77 Avg. number of robos in each plan 1.84 Avg. planning ime (ms) 67.0 Avg. number of neworks formed 12.2 Table 2: Daa from 25 simulaions of he es scenario depiced in Figure 8. Provided in Figure 9 is a visualizaion of a simulaion of 4 free-floaing robos planning in a bounded 3D environmen ha conains 4 obsacles. The robos have formed a nework and carried ou cenralized planning o consruc rajecories o heir respecive goals. The view has been roaed beween screenshos in a clockwise direcion o provide differen poins of view of he simulaion. 8. CONCLUSIONS a) b) The moion planning framework presened has demonsraed is effeciveness in planning for muliple mobile robos wihin a bounded workspace. I plans wih a high probabiliy of success in environmens involving robos, saionary obsacles and moving obsacles. Planning imes of less han 100 ms allowed he robos o re-plan on he fly and reac in real-ime o changes in he environmen. Fuure work includes incorporaing more sophisicaed mehods of modeling he environmen ino he communicaion sysem. Anoher fuure direcion will be o invesigae he effecs of varying he raio beween sensor range and communicaion range. For he applicaion o he hree-dimensional workspaces, planning for all degrees of freedom should be incorporaed. c) d) e) Figure 9: Visualizing a simulaion involving 4 robos and 4 obsacles. Large gray cubes denoe he obsacles. Trajecories are denoed by yellow lines ha end a robo goal locaions, (denoed by red cube laices).
BIBLIOGRAPHY [1] D. L. Akin, M. L. Bowden. EVA, Roboic, and Cooperaive Assembly of Large Sapce Srucures, Proc. IEEE Aerospace Conference, 2002. [2]R. Alami, F. Rober, F. Ingrand, & S.Suzuki. Muli-Robo Cooperaion Through Incremenal Plan-Merging, Proc. IEEE In. Conf. on Roboics and Auomaion, p. 2573-2678, 1995. [3] T. Arai & J. Oa. Moion Planning of muliple mobile robos. Proc. IEEE/RSJ In. Conf. on Inelligen Robos and Sys., p. 1761-1768, 1992. [4] K. Azarm & G. Schmid. Conflic-Free Moion of Muliple Mobile Robos Based on Decenralized Moion Planning and Negoiaion, Proc. IEEE In. Conf. on Roboics and Auomaion, p. 3526-3533, 1997. [5] J. Barraquand, B. Langlois, & J.C. Laombe. Numerical Poenial Field echniques for Robo Pah Planning, IEEE Tr. On Sys., Man, and Cyb., 22(2):224-241, 1992. [6] M. Bennewiz, W. Burgard & S. Thrun. Opimizing Schedules for Prioriized Pah Planning of Muli-Robo Sysems, Proc. In. Conf. on Roboics and Auomaion, 2001. [7] Z. Bien & Jihong Lee. A Minimum-Time Trajecory Planning Mehod for Two Robos, IEEE Tr. on Roboics and Auomaion, pg 443-450, 1992. [8] S.J. Buckley. Fas Moion Planning for Muliple Moving Robos. Proc. IEEE In. Conf. on Roboics and Auom., p. 1419-1424, 1989. [9] C. Clark & S. Rock. Randomized Moion Planning for Groups of Nonholonomic Robos, Proc. In. Symp. of Arificial Inelligence, Roboics and Auomaion in Space, 2001. [10] C. Clark, S. Rock & J. C. Laombe. Dynamic Neworks for Moion Planning in Muli-Robo Space Sysems, Proc. In. Symp. of Arificial Inelligence, Roboics and Auomaion in Space, 2003. [11] C. Clark, T. Brel, & S. Rock. Kinodynamic Randomized Moion Planning for Muli-Robo Space Sysems, Proc. of IEEE Aerospace Conf., 2002. [12] M. Erdmann & T. Lozano-Perez. On Muliple Moving Objecs, Proc. IEEE In. Conf. on Roboics and Auomaion, p. 1419-1424, 1986. [13] Y. Guo & L. E. Parker. A Disribued and Opimal Moion Planning Approach for Muliple Mobile Robos, Proc. IEEE In. Conf. on Roboics and Auomaion, p. 2612-2619, 2002. [14] D. Hsu, R. Kindel, J.C. Laombe, & S. Rock. Randomized Kinodynamic Moion Planning wih Moving Obsacles, In. J. of Roboics Research, 21(3):233-255, March 2002. [15] D. Hsu, J.C. Laombe, & R. Mowani. Pah planning in expansive configuraion spaces, Proc. IEEE In. Conf. on Roboics and Auomaion, p. 2719-2726, 1997. [16] K. Kan & S. Zucker. Toward efficien Trajecory Planning: The pah-velociy decomposiion, In. J. of Roboics Research, 5(3):72-89,1986. [17] S. Kao, S. Nishiyama, & J. Takeno. Coordinaing mobile robos by applying raffic rules, Proc. IEEE/RSH In. Conf. on Inelligen Robos and Sysems, p. 1535-1541, 1992. [18] J.J. Kuffner. Auonomous Agens for Real-Time Animaion. PhD Thesis, Compuer Science Dep., Sanford U., 1999. [19] S.M. LaValle & S.A. Huchinson. Opimal Moion Planning for Muliple Robos Having Independen Goals, IEEE Tr. on Roboics and Auomaion, 14:912-925, 1998. [20] S.M. LaValle & J.J. Kufner. Randomized Kinodynamic Planning, In. J. of Roboics Research, 20(5):278-300, 2001. [21] Lee, Lee, & Park. Trajecory Generaion and Moion Tracking for he Robo Soccer Game, Proc. IEEE In. Conf. on Inelligen Robos and Sysems, p. 1149-1154, 1999. [22] V.J. Lumelsky & K.R. Harinarayan. Decenralized Moion Planning for Muliple Mobile Robos: The Cockail Pary Model, Auonomous Robos J., 4:121-135, 1997. [23] P. Mouarlier & R. Chaila. Sochasic Mulisensory Daa Fusion for Mobile Robo Locaion and Environmen Modelling. Proc. In. Symp. on Roboics Research, Tokyo, 1989. [24] P. S. Schenker, T. L. Hunsberger, P. Pirjanian & E. T. Baumgarner. Planeary Rover Developmens Supporing Mars Exploraion, Sample Reurn and Fuure Human-Roboic Colonizaion, Proc. 10 h Conf. on Advanced Roboics, p. 31-47, 2001. [25] D. Parsons & J. Canny. A Moion Planner for Muliple Mobile Robos, Proc. IEEE In. Conf. on Roboics and Auom., p. 8-13, 1992 [26] G. Sánchez & J.C. Laombe. On Delaying Collision Checking in PRM Planning : Applicaion o Muli-Robo Coordinaion, In. J. of Roboics Research, 21(1):5-26, Jan. 2002. [27] G. Sánchez-Ane & J.C. Laombe. Using a PRM Planner o Compare Cenralized and Decoupled Planning for Muli-Robo Sysems, Proc. IEEE In. Conf. on Roboics and Auom.., 2002. [28] S. Sekhava, P. Sveska, J.P. Laumond, & M.H. Overmars. Mulilevel Pah Planning for Nonholonomic Robos Using Semiholonomic Subsysems. In. J. of Roboics Res., 17:840-857, 1998. [29] T. Siméon, S. Leroy, & J.P. Laumond. Pah Coordinaion for Muliple Mobile Robos: a Geomeric Algorihm, Proc. In. Join Conf. on Arificial Inelligence, 1999. [30] P. Sveska, & M.H. Overmars. Coordinaed Moion Planning for Muliple Car-Like Robos Using Probabilisic Roadmaps, Proc. IEEE In. Conf. on Roboics and Auom., p. 1631-1636, 1995. [31] C.W. Warren. Muliple Pah Coordinaion using Arificial Poenial Fields, Proc. IEEE In. Conf. on Roboics and Auom., p. 500-505, 1990.