A Parallel Motion Planner for Systems with Many Degrees of Freedom Pekka Isto

Transcription

1 A Parallel Moion Planner for Sysems wih Many Degrees of Freedom Pekka Iso Laboraory of Informaion Processing Science Helsinki Universiy of Technology P.O. Box 9700, FIN HUT Absrac During he several decades of research, a number of algorihms inended o solve pracical moion planning problems have been presened. However, he inracabiliy of he problem makes i difficul o design algorihms capable of solving hard problems, especially when he number of degrees-of-freedom is large. I is necessary o use all available means o exend he domain of pracically solvable problem insances. This paper repors resuls for a parallel implemenaion of a moion planner based on wo-level search algorihm. The planner can solve difficul problems wih many degrees-of-freedom wihin pracicable ime limis. Furhermore, easier problems can be solved wih unprecedened search resoluion. 1. Inroducion The compuaion of collision-free moion for an objec among obsacles is an imporan problem wih applicaions in roboics and virual realiy [1]. However, he problem is inracable. The curren undersanding is ha he planning of a collision-free moion for an objec among saic obsacles is PSPACE-hard problem wih an upper bound ha is exponenial in he number of degrees-of-freedom () [2]. The analysis of he problem has produced several heoreical algorihms for moion planning, bu none of he heoreical algorihms have been implemened for pracical use [3]. Insead, a large number of approximae and heurisic algorihms have been developed o solve various insances of pracical moion planning problems [4]. Due o he complexiy of he problem even he bes heurisic planners ake impracical amoun of ime o compue a soluion, if he number of is sufficienly high and he ask is geomerically difficul. In order o solve realworld moion planning problems as efficienly as possible and o exend he scope of pracically solvable moion planning asks, all he available echniques from compuer science mus be used, including parallel processing. The purpose of his paper is o demonsrae ha a wo-level search algorihm [5] can be used o solve geomerically difficul moion planning problems for sysems wih many degrees-of-freedom (here up o 18). Furhermore, he algorihm can solve less difficul asks wih a search resoluion ha is an order of magniude higher han oher mehods [6]. High-resoluion planning is required in workspaces ha conain igh clearances or hin obsacles like in assembly planning and spo welding. The srucure of he algorihm makes i easy o parallelize i efficienly. A parallel implemenaion of he algorihm running on a PC-cluser can solve he Hwang and Ahuja benchmark ask [2] in average ime comparable o he cycle ime enabling near real-ime operaion. The Alpha Puzzle 1.2 benchmark ask [7] can be solved in minues demonsraing significan improvemen over previous resuls. The nex secion will presen a brief overview of he previous relaed research in moion planning. Secion 3 provides an overview of he implemened algorihm and he seleced parallelizaion sraegy. Secion 4 presens he experimenal resuls. Finally, secion 5 presens conclusions. 2. Previous Relaed Research A considerable amoun of lieraure has been accumulaed during he pas hree decades of research in moion planning. Several good reviews and books have been published on he subjec [1,2,3,4,8]. Therefore, his secion is limied o he conex of he resuls presened here. Like mos conemporary algorihms for moion planning, he presened algorihm performs search in discreized represenaion of he robo s join space or configuraion space (C-space) [9]. In order o deal wih he exponenial cos of global planners and he suscepibiliy o local minima of local planners, Faverjon and Tournassoud proposed o combine boh mehods ino a single wo-level planner [10]. Glavina presened randomized planner using similar approach ha builds a subgoal graph by connecing random subgoals or landmarks in C-space wih a sliding local planner [11]. The SANDROS search sraegy performs selecive and non-uniform subdivision of he C-space and uses a sliding local planner o connec subgoals

2 from differen porions of he C-space [12]. As heir local planners are relaively weak, hese moion planners are quie dependable on he qualiy of he generaed subgoals. Recen developmens along his line of research involve he heurisic placemen of he subgoals [13,14,7]. Many of he moion planning approaches and algorihms originally inroduced in serial form have been laer formulaed ino parallel form. Only a limied number is covered here; for a more exensive discussion, see he review by Henrich [15]. Glavina conjecures ha a significan reducion in ime for he firs soluion can be aained by running his algorihm in parallel on muliple worksaions [16]. Challou e al. [6] and laer Caselli and Reggiani [17] use he same approach o consruc a parallel implemenaion of he influenial Randomized Pah Planner (RPP) [18]. Insances of he RPP solving he original problem are run on muliple processors and he firs soluion is kep. Experimenal resuls demonsrae good expeced speedup and reducion in variance of he planning ime when he problems are sufficienly difficul. Baginski presens a parallel version of Glavina s algorihm for ime-varying workspaces [19]. Raher han running muliple insances of he algorihm on he processors, he implemenaion disribues he local planners ino he available slave processors and builds he subgoal graph in he maser processor. Lile experimenal daa is provided, bu a nework of 30 worksaions provides only a speed-up of four. A similar approach is used by Qui and Henrich bu wih a beer speed-up [20]. Their local planner is rule-based. Mazer e al. presen a moion planner based on parallel geneic algorihms [21]. In addiion o running he componens of he planner in parallel, hey implemen parallel collision deecion. The planner should be quie scalable due o he perfecly parallel naure of geneic algorihms. 3. A Parallel Moion Planner The moion planner presened in his paper is a parallel and improved version of he adapive wo-level heurisic search algorihm presened earlier [5]. The upper level is a subgoal graph buil wih random subgoals. A unique feaure of he algorihm is ha raher han using a single local planner or a combinaion of local planners o aemp he pah segmens beween he subgoals, i uses a local planner wih coninuously adapable capabiliy. As more subgoals are needed and generaed for solving he problem, he global planner increases he capabiliy of he local planner. The subgoal graph is essenially a ask decomposiion mehod and he global planner aemps o consruc a soluion from soluions o subproblems wih a cos measure below a cerain increasing limi. The local planner originaes from he free-space enumeraion algorihm presened by Kondo [22]. I uses a bi-direcional A * algorihm guided by four differen heurisics o search a grid represenaion of he C-space. The heurisics are execued in round-robin fashion and he efficiency of each heurisics = 1,...,4 a configuraion node C is esimaed by he formula: p g( C) = F where g(c) is he disance from he sar configuraion o he curren configuraion C in grid seps (Manhaan disance) and F (C) is he oal number of configuraions examined by he heurisics unil he examinaion of C. The overall efficiency P (j) of heurisics a round j is esimaed by aking an average of p (C) over he las 20 configuraions. A he firs round of execuion each heurisics is allocaed 25 node expansions in he A * algorihm. Subsequenly, he number of examinaions is deermined by he relaive efficiency of he heurisics wih he following formula: P ( j) E ( + 1) = max 25 j, 1 max{ P ( ),..., ( )} 1 j P T j where E sands for he number of examinaions allocaed for heurisics. A second evaluaion value is calculaed for each examined configuraion: F O = gc ( ) If he evaluaion value O (C) for he heurisics increases above a hreshold value O h, he execuion of he heurisics is disconinued for ha round. If all heurisics are disconinued, he local planner fails. The heurisic evaluaion of a configuraion uses he sandard form of A * guiding funcion f(c) = g(c) + h(c), where g(c) is as above and h(c) is an esimae for he cos from he evaluaed configuraion o he goal configuraion. The following four weigh ses are used in he esimae: Manipulaor heurisics: + 1 i a i = 9, i = 1,...,. Posiion heurisics: i d ai = 9, = 1,..., 1, i = d + 1,..., Roaion heurisics: i d ai = 1, = 1,..., 9, i = d + 1,...,

3 Even heurisics: a i = 5, i = 1,...,. d= ( + 0.5)/ 2. An addiional greediness muliplier A=3 and a ie-breaking erm ρ are added o he full expression of h(c): hc ( ) = A ad i i( CG, ) ρaj, i= 1 D i (C,G) is he disance beween he curren configuraion and he goal configuraion G in grid seps along axis i. The ie-breaking erm has a value of 0.5 if he evaluaed node was expanded in he same direcion as he paren node along he axis j or 0 oherwise. All of he heurisics share he search space represenaion, and herefore progress made by any of he heurisics will immediaely and coninuously benefi all of hem. The evaluaion values conrol he search so ha relaively more efficien heurisics are used more and he individual heurisics and evenually he local planner are disconinued if sufficien progress is no made. The memory consumpion of he local planner grows exponenially as he number of degrees-of-freedom is increased. The hreshold parameer O h can be used o conrol he memory consumpion, bu an addiional limi on he maximum number of configuraion nodes for each subask is needed. This is a machine specific parameer ha depends on he amoun of main memory in he machine. The global planner generaes random subgoals and builds wo rees of subgoals and successful subpahs, one rooed a he sar configuraion and he oher a he goal configuraion. Pohl s cardinaliy principle [23] is used o selec he ree for expansion. As more subgoals are generaed, he hreshold parameer is updaed according o he formula O h = O h R S, where S is he curren number of subgoals, O h and R are consans wih values of 3 and This means ha he local planner becomes gradually more powerful and he balance of compuaion shifs increasingly from global planning o local planning. The main moivaion for he parallel implemenaion of he moion planner is o reduce he wall-clock ime spen in solving he planning asks. Alhough no esablished limi exiss o denoe wha is pracicable planning ime, Gupa considers run imes in few minues and few ens of minues reasonable [8] and imes in hours impracicable [24]. A parallel implemenaion should be able o reduce he running ime from hours o minues or ens of minues wih reasonable compuing resources. Several alernaive parallelizaion sraegies could be used. The whole planner can be run on muliple machines for he firs soluion like parallel RPP. Or he Figure 1: A version of he benchmark problem proposed by Hwang and Ahuja. Figure 2: Alpha Puzzle 1.2 benchmark ask. Two inerwined L shaped ubes mus be separaed. Figure 3: The sar and goal configuraions for he 12 ask dogs wih bones. Figure 4: The sar and goal configuraions for he 18 ask. The robos mus avoid he gaes and each ohers while performing a righ hand roaion. local planners can be disribued o he slave processors while he global planner is run a he maser processor. A hybrid sraegy would combine hese approaches by running muliple copies of he whole planner on subasks a he slave processors and running an addiional global planner a he maser. The resuls presened here are for he second sraegy as i has

4 5 HR5 6 HR CPUs Min. 25% 50% 75% Max. Ave. Sd Table 1: Run imes in wall clock seconds for he various asks on a single 500 MHz CPU and on he whole cluser of 11 CPUs. Sample size is 100 runs. Perceniles are rounded up. (HR6). These are significan improvemens, since he smalles granulariy of compuing, and hus is he he earlier resuls are in hours on a single processor [7]. wors case for scalabiliy. The planner has also demonsraed capabiliy o plan 4. Experimenal Resuls The parallel moion planner was esed on a Linux PC cluser comprised of 11 processors wih clock speeds beween 450 MHz and 550 MHz and memory sizes of 128 MB or 512 MB. The compuing nodes are conneced wih 100 Mbi Eherne. For he scalabiliy experimens he compuing nodes are grouped so ha each group has an average clock speed of 500 MHz. The implemenaion uses RAPID collision deecion library [25] and MPICH message passing library [26]. As seen in able 1, he planner can solve he Hwang and Ahuja benchmark ask (figure 1) in seconds wih a grid represenaion of he C- space (label: 5). The search resoluion is he same as ha of he similar AdepOne ask [12]. A high-resoluion version of he ask (HR5) can be solved in few minues using he whole cluser. The benchmark ask is designed o force a large backracking moion o he soluion and many oher planners would need considerable effor for producing he backracking moion a such a high search resoluion. The Alpha Puzzle 1.2 ask (figure 2) is inended o represen disassembly problems wih a narrow passage. The parallel planner can solve i in minues wih a resoluion of 128 posiions for each (6). A high-resoluion version wih 1280 posiions can be solved in ens of minues on he whole cluser moions hrough narrow space in he previous experimens [5]. The main moivaion of his paper is o complemen earlier experimens wih many degrees-of-freedom problems and demonsrae ha he planner can be used for such problems despie he poenially excessive memory consumpion of he local planner [5]. Various noions of wha is considered a large number of degrees-of-freedom exis. Gupa saes ha from he pracical poin of view, sysems wih more ha four mus be included in he caegory [24,8]. Faverjon and Tournassoud would only include sysems wih 8 or more [10]. Kavraki and Laombe inroduced he PRM approach especially for sysems wih many degrees-of-freedom [27]. Their es cases have 8 and 12. The minimum number of degrees-of-freedom required o place an objec ino an arbirary posiion and orienaion in 3D space is six. Therefore, moion planners inended for general use should be able o generae moions for 6 degrees-of-freedom sysems. Indusrial robo cells usually have auxiliary degrees-offreedom for placemen of he work piece. Mulirobo cells can easily have wo dozens degrees-of-freedom. Recenly inroduced new applicaions, such as planning for flexible objecs, call for several dozens or hundreds of degrees-of-freedom [28,29,30]. In his paper, sysems wih 10 or more degrees-of-freedom are included in he caegory of problems wih many degrees-of-freedom. The Hwang and Ahuja benchmark ask can be exended ino a 10 version by planning he moion

5 Number of Processors ask 10 ask Speed up Figure 5: Speed-up for asks 5 and 10. The speed-up is calculaed from he average run ime for 5 runs of he planner. simulaneously for wo independen SCARA robos (10). Similarly, 12 and 18 asks are consruced from wo and hree Puma ype robos as shown in figures 3 and 4. The 12 and 18 asks are planned wih 100 discree posiions for each degreeof-freedom. The daa in able 1 shows ha also hese problems can be solved in minues wih he cluser. Finally, he scalabiliy of he planner is shown in figure 5. Even he wors-case parallelizaion sraegy provides an accepable speed-up. Addiional reducion in planning ime could be aained by adding more processors o he cluser. However, he graph shows diminishing reurns and evenually a larger granulariy sraegy mus be used. The superlinear speed-up for small number of processors is probably caused by cache effecs. 5. Conclusions This paper presened experimens wih a parallel moion planner based on wo-level heurisic search in he configuraion space. The experimens demonsraed a moderaely good scalabiliy of he algorihm on an inexpensive parallel compuer buil form commodiy hardware and free sofware. While he algorihm shows diminishing speed-up as he number of processors is increased, alernaive parallelizaion sraegies may be used o provide improvemen in he behavior of he planner. Even in he curren form, he parallel implemenaion can be used o solve very difficul moion planning problems wihin near real-ime and pracicable off-line ime limis. The es problems included cases wih many degrees-of-freedom demonsraing ha A * search-based approach can be used o solve such cases. Addiionally, easier bu nonrivial problems can be solved wih excepionally highresoluion represenaion of he C-space. The resuls on Alpha Puzzle 1.2 presened here are a significan improvemen over previous resuls. Acknowledgmens This research is funded by he Helsinki Graduae School in Compuer Science and Engineering. The parallel implemenaion is based on sofware developed by Johannes Lehinen and he IMPACT group of Tik Sofware Projec course. Mr. Lehinen also provided he geomeric model for he 5 es problem. Alpha Puzzle 1.2 was designed by Boris Yamrom, GE Corporae Research & Developmen Cener. The model was provided by he DSMFT research group a Texas A&M Universiy. Janne Ravani provided access o he cluser compuer a he Bamford Laboraory, Universiy of Helsinki. The auhor hanks Dr. Juha Tuominen for helpful commens and suggesions. References [1] J. C. Laombe, Robo Moion Planning, Kluwer Academic Publishers, Norwell, Mass [2] Y. K. Hwang, N. Ahuja, Gross Moion Planning - A Survey, ACM Compuing Surveys, vol. 24, no. 3, Sep. 1992, [3] J. C. Laombe, Moion Planning: A Journey of Robos, Molecules, Digial Acors, and Oher Arifacs, Inernaional Journal of Roboics Research, vol. 18, no. 11, November 1999, [4] K. Gupa, A. P. del Pobil (eds.), Pracical Moion Planning in Roboics: Curren Approaches and Fuure Direcions, John Wiley & Sons, Wes Sussex, [5] P. Iso, A Two-level Search Algorihm for Moion Planning, Proceedings of he 1997 IEEE Inernaional Conference on Roboics and Auomaion, IEEE Press, [6] D. J. Challou, D. Boley, M. Gini, V. Kumar, C. Olson, Parallel Search Algorihms for Robo Moion Planning, In [4], [7] D. Vallejo, C. Jones, N. M. Amao, An Adapive Framework for `Single Sho' Moion Planning, Proceedings of he IEEE/RSJ Inernaional Conference on Inelligen Robos and Sysems (IROS), [8] K. Gupa, Moion Planning for Flexible' Shapes (Sysems wih Many Degrees of Freedom): A Survey, The Visual Compuer, vol. 14, no. 5/6, [9] T. Lozano-Pérez, M. Wesley, An Algorihm for Planning Collision-free Pahs among Polyhedral

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