Planning with Reachable Distances: Fast Enforcement of Closure Constraints

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1 Plnning with Rechle Distnces: Fst Enforcement of Closure Constrints Xinyu Tng, Shwn Thoms, nd Nncy M. Amto Astrct Motion plnning for closed-chin systems is prticulrly difficult due to dditionl closure constrints plced on the system. In fct, the proility of rndomly selecting set of joint ngles tht stisfy the closure constrints is zero. We propose Plnning with Rechle Distnce (PRD) to overcome this chllenge y first precomputing the suspce stisfying the closure constrints, then directly smpling in it. To do so, we represent the chin s hierrchy of suchins. Then we clculte the closure su-spce s pproprite rechle distnce rnges of su-chins stisfying the closure constrints. This provides two distinct dvntges over trditionl pproches: (1) configurtions re quickly smpled nd converted to joint ngles using sic trigonometry functions insted of more expensive inverse kinemtics solvers, nd (2) configurtions re gurnteed to e closed. In this pper, we descrie this hierrchicl chin representtion nd give smpling lgorithm with complexity liner in the numer of links. We provide the necessry motion plnning primitives for most smpling-sed motion plnners. Our experimentl results show our method is fst, mking smpling closed configurtions comprle to smpling open chin configurtions tht ignore closure constrints. Our method is generl, esy to implement, nd lso extends to other distnce-relted constrints esides the ones demonstrted here. I. INTRODUCTION Closed-chin systems, s in Figure 1, re involved in mny pplictions in rootics nd eyond, such s prllel roots [12], closed moleculr chins [16], nimtion [4], reconfigurle roots [7], [14], nd grsping [6]. However, motion plnning for closed-chin systems is prticulrly chllenging due to dditionl constrints, clled closure constrints, plced on the system. Using only the trditionl joint ngle representtion, the proility tht rndom set of joint ngles lies on the constrint surfce is zero [9]. Insted of rndomly smpling in the joint ngle spce to find closed configurtions, we propose Plnning with Rechle Distnce (PRD). PRD overcomes this chllenge y precomputing the suspce where closed constrints re stisfied nd then directly smpling in this suspce. We represent the chin s hierrchy of su-chins y recursively reking down the prolem into smller suprolems. Ech su-chin in the hierrchy my e prtitioned into other, smller su-chins forming closed loop. For ny su-chin, we cn compute the ttinle distnce (rechle distnce or length) etween its two endpoints recursively. With this informtion, we simply smple distnces in these rnges, nd then use sic geometry reltionships to clculte the Reserch supported in prt y NSF Grnts EIA-13742, ACR-8151, ACR , CCR , ACI-32635, y the DOE, nd y Hewlett- Pckrd. Thoms supported in prt y n NSF Grdute Reserch Fellowship, PEO Scholrship, nd DOE GAANN Fellowship. W l4 θ5 l5 θ4 l3 l θ3 θ1 θ2 l2 l1 (c) Fig. 1. Exmples of different closed-chin systems. Ech must stisfy certin closure constrints. Single loop: loop must remin closed. W is the world coordinte frme. Multiple loops: ll loops must remin closed. (c) Multiple root grsping: ll root end-effectors must remin in contct with the grsped oject. joint ngles. Thus, we cn directly smple in the rechle spce, which is expensive to compute explicitly. While linkge s rechle distnce hs een used in other computtions, to our knowledge it hs not een used to guide smpling in smpling-sed plnners. PRD hs two distinct dvntges over trditionl pproches: configurtions re quickly smpled nd converted to joint ngles using sic trigonometry functions insted of more expensive inverse kinemtics solvers. configurtions re gurnteed to e closed. In this pper, we formlly descrie this new chin representtion nd give recursive smpling lgorithm with complexity liner in the numer of links in the chin. This lgorithm explores the closure constrint surfce y recursively smpling in the fesile rechle distnce rnge for ech su-chin. It cn quickly determine whether or not it is possile to stisfy the closure constrints of su-chin s children. Our method does not gurntee collision-free configurtions; like mny methods, they must e checked fterwrds. PRD cn significntly improve the performnce of smpling-sed plnners for closed-chin systems, such s Proilistic Rodmp Methods (PRMs) [5] nd Rpidlyexploring Rndomized Trees (RRTs) [1], which hve een widely successful in solving other high degree of freedom (dof) prolems. However, the trditionl joint ngle representtion mkes it difficult for these methods to smple closed configurtions. While severl strtegies use heuristics to improve the proility of smpling closed configurtions [9], [19], [2], [18], [1], [17], smpling is still difficult nd expensive for lrge systems. Here, we provide two necessry motion plnning primitives (smpling nd locl plnning) to implement most of these smpling-sed plnners. Our experimentl results show tht our method is fst nd efficient in prctice mking smpling configurtions with closure constrints comprle to smpling open chin

2 configurtions tht ignore closure constrints entirely. Our method is esy to implement nd generl it cn e pplied to other rticulted systems. It is lso extendile to more distnce-relted constrints esides the ones here. For exmple, it cn e used to smple configurtions with the end effector in specific regions. II. PRELIMINARIES This pper focuses on closed-chin systems. A closedchin system differs from other systems in tht it must lso stisfy certin closure constrints. Figure 1 gives exmples of different types of closed-chin systems. Ech system hs different set of closure constrints to stisfy. Closed-chin systems re trditionlly represented y the position nd orienttion of the se or se link nd one or more ngles for ech joint (corresponding to the joint s dof). For exmple, the single loop in Figure 1 is represented y 11 vlues: 6 for the position nd orienttion of link l nd 5 for the ngles θ1... θ5. While this representtion sufficiently descries configurtion, it is extremely difficult to smple these prmeters rndomly while stisfying the closure constrints. This is ecuse this joint ngle representtion does not lso encode the closure constrints they re hndled seprtely. In fct, it hs een shown tht the proility of rndomly smpling set of joint ngles tht stisfy the closure constrints is zero [9]. III. RELATED WORK In theory, exct motion plnning lgorithms [15], [8] cn hndle systems with closure constrints. However, since exct lgorithms re exponentil in the dimension of C-spce (the set of ll possile root configurtions, vlid or not), they re generlly imprcticl for lrge closed-chin systems. Rndomized lgorithms such s Proilistic Rodmp Methods (PRMs) [5] nd Rpidly-exploring Rndomized Trees (RRTs) [11] re widely used tody for mny pplictions. They first construct rodmp (grph or tree) tht represents the connectivity of the root s free C-spce, nd then query the rodmp to find vlid pth for given motion plnning tsk. Initilly, they were minly limited to rigid odies nd rticulted ojects without closure constrints. Kvrki et l. [9], [19] rndomly smples configurtions nd then pplies n itertive rndom grdient descent method to push the configurtion to the constrint surfce. They solved plnr chins with up to 8 links nd 2 loops in severl hours with PRM nd severl minutes with RRT. Hn et l. [2] uses kinemtics-sed PRM pproch y first uilding rodmp ignoring ll ostcles, nd then populting the environment with copies of this kinemtics rodmp, removing invlid portions. This method cn solve chin prolems with 7 9 links in under minute. This work is extended to increse the numer of links the method cn hndle in [1], [18]. Trinkle nd Milgrm proposed pth plnning lgorithm [17] sed on configurtion spce nlysis [13]. Their method does not consider self-collision ut still my e pplied s locl plnner. Recently, Hn et l. [3] proposed set of geometric prmeters for closedchin systems such tht the prolem cn e reformulted s system of liner inequlities. Then, liner progrmming nd other lgorithms cn e used for smpling nd locl plnning. However, they do not discuss the lgorithm s complexity or consider collisions in plnning. Our method is very fst nd hs complexity liner in the numer of links. It gurntees tht closed configurtion will e generted or reports tht one cnnot e otined. Moreover, our method is generl nd esy to implement for smpling-sed plnners. It is extendile to more distncerelted constrints thn the ones considered here. IV. REACHABLE DISTANCE REPRESENTATION Here we descrie our hierrchicl representtion sed on rechle distnces. The min dvntge of this representtion over the trditionl joint ngle representtion is tht it lso encodes the closure constrints. This new representtion llows us to esily rndomly smple closed configurtions. Intuitively, in closed-chin system, the length of ech link hs to e in n pproprite rnge to stisfy the closure constrints. For exmple, Figure 2 shows simple tringulr closed-chin with 3 links, nd c of vrile length. To smple closed configurtion, we hve to mke sure the length of ech link is in n pproprite (fesile) rnge. In other words, the length of ech link needs to stisfy the tringle inequlity: c >= + nd c <=. To smple closed configurtion, we first cn clculte the link s remining ville rnge sed on the ville rnges of the other links. In this wy, we cn eventully smple vlid length for ech link to stisfy the closure constrints or report tht it is impossile to find closed configurtion. For the tringulr cse in Figure 2, once we get vlid length for ech link, we get tringulr configurtion tht is gurnteed to e closed. Below we show how we extend this smpling strtegy to hndle generl closedchin system. This scheme only ensures tht the closure constrints re stisfied. Collision checking must still e performed to determine the configurtion s vlidity. Note tht this representtion cn lso support joint ngle limits y restricting the fesile rnge of the links. For simplicity, we ignore joint ngle limits in this pper. c c c (c) Fig. 2. Configurtions of n rticulted system with 3 links., nd c ech hs vrile length. Ech link hs n pproprite length to close the configurtion. c is too long. (c) c is too short. A. Closed-Chin s Hierrchy of Su-Chins For simplicity, here we consider single loop closedchin. However, this representtion is generl nd my e pplied to other closed-chin systems. The chin (clled the prent) my e prtitioned into severl su-chins (clled

3 the children) where ech su-chin is set of consecutive links nd the union of the children represents ll the links in the prent. For exmple, the virtul link 16, see Figure 3, connects the two ctul links nd 1. In this wy, the prent is closed if nd only if the virtul links nd its children form closed-chin. We recursively prtition su-chins until ll the children hve only 1 link. This defines hierrchy of su-chins nd virtul links, s displyed in Figure 3. B. Rechle Rnge of Su-Chin The rechle distnce of su-chin is the distnce etween the endpoints of its virtul link. The set of ll possile rechle distnces for su-chin is clled its rechle rnge. For exmple, the rechle rnge of single link is its length. A prent link s rechle rnge depends on the rechle rnges of its children. If we uild the su-chin hierrchy such tht ech su-chin hs only or 2 children, then the rechle rnges re computed s follows. Consider su-chin with no children. Let l min nd l mx e the minimum nd mximum link length, respectively. (For non-prismtic link, l min = l mx.) The rechle rnge is then [l min, l mx ]. Now consider the su-chin with 2 children in Figure 4. The su-chin nd its two children form tringle. Let [ min, mx ] nd [ min, mx ] e the rechle rnge of the first child nd the second child respectively. The rechle rnge of the prent is then [l min, l mx ] where mx(, min mx ), min < min l min =, min = min (1) mx(, min mx ), min > min nd l mx = mx + mx. Note tht Eqution 1 is generl. Given the rechle rnges of ny two links in the sme tringulr su-chin, it clcultes the rechle rnge of the third one to stisfy the tringle inequlity. lmin x l min (c) min lmin mx x min lmx (d) mx min Fig. 4. A su-chin with 2 children nd nd its virtul link l. A configurtion where l is minimum when min < min nd x = min( mx min, min min). (c) A configurtion where l is minimum when min > min nd x = min( mx min, min min). (d) A configurtion where l is mximum l mx. A closed configurtion is then set of distnces for ech virtul link in the hierrchy such tht ech distnce is within the virtul link s rechle rnge. We cn then compute its joint ngles using only sic trigonometry reltionships insted of more expensive inverse kinemtics solvers. We discuss smpling in more detil in Section V-A. C. Aville Rnge of Su-Chin The ville rechle rnge (ville rnge) of virtul link is set of distnces/lengths which llow it to close with the other links in the sme su-chin (i.e., stisfy the tringle inequlity). The ville rnge is suset of the rechle rnge tht is function of the ville rnges (or lengths) of other links in the sme su-chin loop. For exmple, in the eginning, for ech su-chin with 2 children, the rechle rnges of ll 3 links cn stisfy the tringle inequlity nd thus the ville rnge is the sme s the rechle rnge. However, once we fix the length of link, portions of the rechle rnges of other links in the sme su-chin my no longer e vlid. When this hppens, we need to reclculte their ville rnges using Eqution 1. Note tht Eqution 1 is used in more generl wy here., nd l cn e ny link in the sme tringulr su-chin. V. APPLICATION TO SAMPLING-BASED PLANNING Here we descrie two primitive opertions for most rndomized smpling-sed motion plnners such s PRMs nd RRTs: smpling nd locl plnning (i.e., finding vlid pth etween two smples). We show tht these opertions re fst nd efficient, therey llowing smple-sed motion plnners to e directly pplied to lrge closed-chin prolems. A. Smpling To smple closed configurtion nd determine its joint ngles, we first recursively smple the lengths of ech virtul link. We then smple the orienttion of ech su-chin (e.g., concve or convex for chins in the plne, dihedrl ngles for non-plnr chins). Finlly, we use the virtul link lengths nd orienttions to compute the pproprite joint ngles. Note tht this smpling only ensures tht the configurtion is closed it will still need to e checked for collision. We descrie ech step in more detil elow. 1) Recursively smple link lengths: Becuse we know the rechle rnge for ech su-chin in the hierrchy, smpling closed configurtion simply ecomes smpling distnces in the ville rechle rnges of ech su-chin. Once we fix the length or rechle distnce of prent su-chin, portions of its children s rechle rnges my ecome invlid. We define the remining vlid portions of their rechle rnge s their ville rechle rnge. Note tht n ville rechle rnge my never ecome empty, t the very lest its minimum nd mximum my ecome equl. Thus, we smple rechle distnces nd updte ville rechle rnges strting t the the root of the hierrchy until ll su-chin rechle distnces re smpled. Algorithm V.1 descries this recursive smpling strtegy. Recll tht the su-chin hierrchy is inry tree nd tht there is one internl node for ech virtul link. The smpling lgorithm is clled once on ech virtul link. Becuse there re O(n) internl nodes in the inry tree (n is the numer of ctul links), the smpling lgorithm runs in O(n) time.

4 Fig. 3. A chin (indicted y solid links in the first imge) my e prtitioned into set of su-chins represented y virtul links (indicted y dshed links). This prtitioning repets to form hierrchy where the virtul links in one level ecome the ctul links of the next level. The tree represents the entire su-chin hierrchy where nodes correspond to virtul links. Algorithm V.1 Smple Input. A su-chin c. Let c.rr e c s ville rechle rnge, c.left nd c.right e c s children, nd c.len e the length of c s virtul link. Let p e c s prent nd s e c s siling. 1: Updte c.rr from p.rr nd s.rr. 2: Rndomly smple c.len from c.rr. 3: Set c.rr to [c.len, c.len]. 4: if c hs children then 5: Smple(c.lef t). 6: Smple(c.right). 7: end if 2) Smple dihedrl or concve/convex orienttion: Ech su-chin nd virtul link forms tringle. In 2D, two different configurtions hve the sme virtul link length: concve orienttion nd convex orienttion (see Figure 5). In 3D, virtul link length cn represent mny configurtions depending on the dihedrl ngle etween its tringle nd its prent s tringle (see Figure 5). Thus, we lso smple the orienttion of the virtul link. configurtions to trnsform q s into q g t some user-defined resolution. Here we descrie simple locl plnner tht uses stright-line interpoltion in the rechle spce to determine the sequence of configurtions. While this is certinly not the only locl plnning scheme possile, we find it performs well in prctice. For ech virtul link, this locl plnner interpoltes etween its length in q s nd its length in q g to get sequence of lengths. We then must determine the virtul link s orienttion sequence (i.e., concve/convex in 2D nd the dihedrl ngle in 3D) to fully descrie the sequence of configurtions. In 2D, if the orienttion is the sme in q s nd q g, we keep it constnt in the sequence. If it is not the sme, we must flip the links s descried elow. In 3D, we simply interpolte etween the dihedrl ngle in q s nd the dihedrl ngle in q g. We determine the vlidity of the sequence y checking tht ech configurtion is closed (i.e., ech virtul link s length is in its ville rechle rnge) nd collision-free. 1) Concve/convex flipping for 2D chins: Consider the first nd lst configurtions of su-chins in Figure 6. When the orienttion is different, s in this exmple, we need to find trnsformtion from one to the other. ρ Fig. 6. Flip the links to chnge its orienttion. Fig. 5. In 2D, the sme virtul link represents two configurtions: concve tringle nd convex tringle. In 3D, the sme virtul link represents mny configurtions with different dihedrl ngles ρ etween the su-chin s plne nd its prent s plne. 3) Clculte joint ngles: Consider the joint ngle θ etween links nd. Links nd re connected to virtul link c to form tringle. Let l, l, nd l c e the lengths of links,, nd c, respectively. The joint ngle cn e computed using the lw of cosines: B. Locl Plnning θ = cos( l2 + l 2 l2 c) 2l l ). (2) Given two configurtions, q s nd q g, locl plnner ttempts to find sequence {q s, q 1, q 2,..., q g } of vlid To flip the virtul link, we need to open the prent s virtul link enough so its children cn chnge orienttion while remining in their ville rechle rnge. In other words, t some point the prent s rechle rnge must e lrge enough to ccommodte the summtion of its children s rechle distnce (i.e., llow the children to e flt ). Such constrint on rechle-distnce cn e esily hndled y our method. There re other more powerful options. In our current implementtion, we first clculte the ville rnge for this minimum flt constrint. Then we smple n intermedite configurtion where the virtul links re flt nd try connecting it to oth the strt nd gol configurtions. VI. RESULTS AND DISCUSSION In this section we show how our PRD method performs in prctice. We give performnce study in Section VI-A nd

5 present the method in complete motion plnning frmework in Section VI-B. All experiments re performed on 3GHz desktop computer nd implementtions re compiled using gcc4 under linux. Our current implementtion supports plnr joints nd sphericl joints. A. Performnce Study Here we study nd compre the performnce of our smpling lgorithm oth with nd without collision detection. In the first experimentl set, we ignore collision detection. We mesure the running time to generte 1 configurtions in 1 different environments for chins of size 1, 2, 5, 1, 2, 5, 1, 5, 1, nd 1 links. In ech environment, the rticulted system is composed of different sized links rnging from.1 to 1.. We compre the performnce of generting oth open nd closed configurtions using rechle-distnce. As shown in Tle I nd in Figure 7, the running time hs two prts: the time to smple nd the time to convert the representtion into trditionl joint ngles. Note tht it tkes much more time for conversion thn for ctul smpling. However, the performnce of oth scles well even for lrge numers of links. These results show tht smpling closed configurtions is comprle to smpling open chin configurtions tht ignore closure constrints entirely. This demonstrtes the dvntge of using rechle distnces to represent configurtions over the trditionl joint ngle representtion. Method CD PRD N Open Y PRD N Closed Y KBPRM N N/A N/A N/A N/A Closed Y N/A N/A N/A N/A TABLE II RUNNING TIMES OF DIFFERENT SAMPLING METHODS BOTH WITH AND WITHOUT COLLISION DETECTION. Time (sec) Open Chin using PRD Closed Chin using PRD Closed Chin using KBPRM Numer of Links Fig. 8. Running time of open configurtion smpling using rechle distnces nd closed configurtion smpling using rechle distnces ll with collision detection s function of the numer of links. The performnce for oth methods is comprle since they re ech dominted y collision detection. Time (sec) Running Time for Open Chin Smpling Smple Convert Smple+Convert Numer of Links x 1 4 Time (sec) Running Time for Closed Chin Smpling Numer of Links x 1 4 Fig. 7. Running time to generte 1 open or closed configurtions in the rechle-distnce spce without considering collision. Ech plot shows the totl running time, the pure smpling time in rechle distnce spce, nd the time to convert configurtions into joint-ngle spce. Note tht the running time for closed nd open configurtions re similr. In the second set of experiments, we mesure the performnce when self-collision is considered. Agin we mesure the time to smple nd convert 1 self-collision-free configurtions. We studied 6 different environments with chins of size 1, 2, 5, 1, 2 nd 5. In ech environment, we mesured the running time to generte open-chin nd closed-chin configurtions in rechle-distnce spce. We lso mesure the performnce using nother closed-chin plnner, KBPRM [2]. We stop n experiment fter 1 hours. Tle II nd Figure 8 show the performnce results. Note tht when considering collision detection, the performnce of using PRD to smple n open-chin is comprle to the performnce to smple closed-chin, even though closed configurtions involve more self-collision. This gin shows tht y directly smpling in the rechle spce, PRD cn smple closed configurtions efficiently. This mens tht our method mkes it prcticl to solve motion plnning prolems for lrge rticulted systems with distnce-relted constrints. Also note tht the PRD method out-performs the KBPRM method s expected. B. Appliction in Motion Plnning Frmeworks In this section, we show how we pply the smpling nd locl plnning primitives to solve closed-chin prolems. 1) Multiple root grsping: Here, two rticulted rootic rms with moile ses grsp n oject nd work together to move it from one end of the environment to the other (see Figure 9). Ech rm is composed of 3 links. Considering the grsping oject nd the root s se, there re 8 dof in totl. The root must mneuver the oject underneth the ostcle in the ceiling to solve the query. We use n incrementl PRM method to construct the rodmp nd solve the query. It egins with smll rodmp nd incrementlly genertes more nodes until the query is solved. Our method required 4 nodes to solve the query. It took 58.4 seconds to uild the mp nd found vlid pth with 31 intermedite configurtions (see Figure 9). 2) Single-loop closed-chin: A single-loop chin with 1 links of vrile length must pss through series of nrrow pssges to trverse the entire environment (see Figure 1). Agin, we used incrementl PRM mp genertion to uild rodmp until it solves the query. Our method successfully

6 Numer of Links , 5, 1, 1, Smple Open Chin (sec) Smple Open Chin nd Convert (sec) Smple Closed Chin (sec) Smple Closed Chin nd Convert (sec) TABLE I TIME TO GENERATE 1 OPEN/CLOSED CONFIGURATIONS IN THE REACHABLE-DISTANCE SPACE WITHOUT COLLISION DETECTION. Fig. 9. Grsping experiment where 2 rms collorte to trnsport n oject cross the environment. Pth s swept volume. found vlid pth with only 2 nodes in 96.2 seconds of computtion time. Figure 1 shows the swept volume of the pth with 168 intermedite configurtions. Fig. 1. Single-loop chin tht must pss through series of nrrow pssges to trverse the entire environment. VII. CONCLUSION We proposed new method to pln the motion of closedchin prolems sed on hierrchicl representtion of the chin. It cn e used to quickly generte closed configurtions or determine tht it is impossile to stisfy the closure constrints. We descried this new representtion nd gve smpling lgorithm to generte closed configurtions with complexity liner in the numer of links. It cn e used to significntly improve smpling-sed plnners for closedchin systems y overcoming the difficulty of smpling closed configurtions. We provide the necessry motion plnning primitives (nmely smpling nd locl plnning) to implement smpling-sed motion plnners. Our experimentl results show tht our method is fst nd efficient in prctice, mking the cost of generting closed nd open configurtions comprle. In Proc. IEEE Int. Conf. Root. Autom. (ICRA), pges , 22. [2] L. Hn nd N. M. Amto. A kinemtics-sed proilistic rodmp method for closed chin systems. In Rootics:New Directions, pges , Ntick, MA, 2. A K Peters. Book contints the proceedings of the Interntionl Workshop on the Algorithmic Foundtions of Rootics (WAFR), Drtmouth, Mrch 2. [3] L. Hn, L. Rudolph, J. Blumenthl, nd I. Vlodzin. Strtified deformtion spce nd pth plnning for plnr closed chin with revolute joints. In Proc. Int. Workshop on Algorithmic Foundtions of Rootics (WAFR), July 26. [4] M. Kllmnn, A. Auel, T. Aci, nd D. Thlmnn. Plnning collision-free reching motion for interctive oject mnipultion nd grsping. In Eurogrphics, 23. [5] L. E. Kvrki, P. Svestk, J. C. Ltome, nd M. H. Overmrs. Proilistic rodmps for pth plnning in high-dimensionl configurtion spces. IEEE Trns. Root. Automt., 12(4):566 58, August [6] O. Khti, K. Yokoi, K. Chng, D. Ruspini, R. Holmerg, nd A. Csl. Vehicle/rm coordintion nd multiple moile mnipultor decentrlized coopertion. In Proc. IEEE Int. Conf. Root. Autom. (ICRA), pges , [7] K. Koty, D. Rus, M. Von, nd C. McGry. The self-reconfiguring rootic molecule: Design nd control lgorithms. In Proc. Int. Workshop on Algorithmic Foundtions of Rootics (WAFR), pges , [8] J.-C. Ltome. Root Motion Plnning. Kluwer Acdemic Pulishers, Boston, MA, [9] S. LVlle, J. Ykey, nd L. Kvrki. A proilistic rodmp pproch for systems with closed kinemtic chins. In Proc. IEEE Int. Conf. Root. Autom. (ICRA), pges , [1] S. M. LVlle nd J. J. Kuffner. Rndomized kinodynmic plnning. In Proc. IEEE Int. Conf. Root. Autom. (ICRA), pges , [11] S. M. LVlle nd J. J. Kuffner. Rpidly-Exploring Rndom Trees: Progress nd Prospects. In Proc. Int. Workshop on Algorithmic Foundtions of Rootics (WAFR), pges SA45 SA59, 2. [12] J. P. Merlet. Still long wy to go on the rod for prllel mechnisms. In Proc. ASME Int. Mech. Eng. Congress nd Exhiition, 22. [13] R. J. Milgrm nd J. Trinkle. The geometry of configurtion spces for closed chins in two nd three dimensions. Homology Homotopy Appl., 6(1): , 24. [14] A. Nguyen, L. J. Guis, nd M. Yim. Controlled module density helps reconfigurtion plnning. In Proc. Int. Workshop on Algorithmic Foundtions of Rootics (WAFR), 2. [15] J. H. Reif. Complexity of the mover s prolem nd generliztions. In Proc. 2th Annu. IEEE Sympos. Found. Comput. Sci., pges , [16] A. Singh, J. Ltome, nd D. Brutlg. A motion plnning pproch to flexile lignd inding. In 7th Int. Conf. on Intelligent Systems for Moleculr Biology (ISMB), pges , [17] J. C. Trinkle nd R. J. Milgrm. Complete pth plnning for closed kinemtic chins with sphericl joints. Int. J. Root. Res., 21(9): , 22. [18] D. Xie nd N. M. Amto. A kinemtics-sed proilistic rodmp method for high dof closed chin systems. In Proc. IEEE Int. Conf. Root. Autom. (ICRA), pges , 24. [19] J. H. Ykey, S. M. LVlle, nd L. E. Kvrki. Rndomized pth plnning for linkges with closed kinemtic chins. IEEE Trnsctions on Rootics nd Automtion, 17(6): , 21. REFERENCES [1] J. Cortes, T. Simeon, nd J. P. Lumond. A rndom loop genertor for plnning the motions of closed kinemtic chins using PRM methods.