Complete Path Planning for Planar Closed Chains Among Point Obstacles

Transcription

1 Complete Path Plannng for Planar Closed Chans Among Pont Obstacles G.F. Lu, J.C. Trnkle Department of Computer Scence Rensselaer Polytechnc Insttute Troy, New York Emal: { lugf,trnk }@cs.rp.edu Abstract A method to compute an exact cell decomposton and correspondng connectvty graph of the confguraton space (C-space) of a planar closed chan manpulator movng among pont obstacles s developed. By studyng the global propertes of the loop closure and collson constrant set, a cylndrcal decomposton of the collson-free porton of C-space (C-free) s obtaned wthout translatng the constrants nto polynomals as requred by Collns method []. Once the graph s constructed, moton plannng proceeds n the usual way; graph search followed by path constructon. Expermental results demonstrate the effectveness of the algorthm. I. INTRODUCTION Because of ther speed and stffness, parallel manpulators have recently attracted the nterest of robotcs researchers and ndustral users. However, ther closed loop structure gves rse to jont varable dependences, whch manfest n a topologcally complex confguraton space. An mportant consequence of ths s that, n general, the C-space cannot be globally (and smoothly) parameterzed by a sngle set of d varables (for example a subset of the jont dsplacements), where d s the number of degrees of freedom of the manpulator. In other words, any d-dmensonal atlas of C-space wll contan multple charts. Ths fact generally makes moton plannng more challengng for parallel manpulators than t s for seral manpulators. A. Prevous Work It s well known that general exact moton plannng algorthms for seral manpulators are hghly complex [5], [6], [7], [8], [5]. In fact, the most effcent exact plannng algorthm s Canny s, whose worst-case tme complexty s exponental n the dmenson of C-space [6]. In prncple, exact algorthms can be appled to systems wth holonomc equalty constrants such as those mposed by the closed knematc loops n parallel manpulators by defnng convenent charts and managng them (see [7], page 4). However, the dffculty of mplementng exact algorthms for general systems fueled a paradgm shft to sample-based algorthms [9], [0], []. Sample-based algorthms buld a graph that approxmates the global structure of the collson-free porton of C-space (C-free). The graph has nodes that correspond to selected ponts of C-free and arcs between nodes that ndcate path connectedness between the correspondng ponts. The graph can be thought of a network of hghways, or a roadmap, of C-free. The roadmap becomes sutable for moton plannng when the followng two attrbutes are attaned: () there s a one-to-one correspondence between components of the graph and components of C-free, and () gven a pont n C-free, t s easy to fnd a path connectng t to the graph. At ths pont, moton plannng s essentally reduced to graph searchng. Sample-based algorthms have been qute successful for systems whose C-space can be parameterzed by a sngle chart wth number of coordnates equal to the number of degrees of freedom of the system, but less successful otherwse [], [6]. Even though one can always generate an ambent space parameterzable by a sngle chart by choosng more parameters than the dmenson of C-space, the number of sample ponts needed to construct a good roadmap grows exponentally wth the dmenson of the ambent space, because the number of connected components of the collsonfree porton of ths space grows exponentally wth ts dmenson. Second, for most parallel manpulators of nterest, parameterzaton of C-space usng the mnmal number of coordnates requres multple charts. These can be dffcult to defne and to choose sutable metrcs to obtan globally well dstrbuted sample ponts. The dffcultes assocated wth applyng sample-based moton plannng methods to parallel manpulators and the avalablty of new results n topology led to renewed nterest n exact plannng algorthms for closed knematc chans (see Fgure ) [7], [8], [9]. Trnkle and Mlgram derved some global topologcal propertes of the C-space (the number of components and the structures of the components) of sngle-loop closed chans wth sphercal jonts n a workspace wthout obstacles [7]. These propertes drove the desgn of a complete moton

2 plannng algorthm that works roughly as follows. ) Choose a subset A of the lnks that can be postoned arbtrarly, and yet the remanng lnks can close the loop; ) Move the lnks n A to ther goal orentatons along an arbtrary path whle mantanng loop closure; 3) Permanently fx the orentatons of the lnks n A; 4) Repeat untl all lnks are fxed. The man result that guded the algorthm s desgn s Theorem n [7]. In essence, the C-space s the unon of manfolds that are products of spheres and ntervals. The jont coordnates correspondng to the spheres are those that can contrbute to the subset A mentoned above and the structure of the C-space suggests a local parameterzaton (.e., convenent chart ) for each step. The plannng algorthm for closed chans n [7] was not desgned to handle obstacles. In [8], [9], pont obstacles were added to the workspace of a planar manpulator, but the closed chan constrant was relaxed. The porton of C-space correspondng to collsons between the manpulator and the obstacles (the C-obstacle) was analyzed n detal, to reveal that C-free locally fbres over a lower-dmensonal base manfold wth fbers composed of open ntervals (called local component sheaves). Glung together the sheaves produces cells of C-free, whose boundares are determned by ther crtcal ponts. B. Contrbuton In ths paper, the concepts used n [7], [8], [9] are brought together to desgn an algorthm to construct an exact cell-decomposton of C-free of an m-lnk planar closed chan movng among pont obstacles. The man steps are: ) Partton C-space nto two peces embedded n two (m-3)-dmensonal tor; ) Compute the boundary of the loop closure constrant varety to dentfy the reachable portons of C-space; 3) Compute the collson varetes n each torus and construct a connectvty graph of C-free whle gnorng the loop closure constrant; 4) Use the boundary of the loop closure varety to refne the graphs and to determne ther connectvtya. As a byproduct of the approach, t s trval to determne cell membershp and reachablty of an arbtrary pont n C-space. Gven ths fact and the graph, one can easly test moton plannng problems for path exstence and then construct a path f one exsts. II. BASIC NOTATION AND TERMINOLOGY Imagne a planar seral chan of m lnks connected by revolute jonts, wth one end free, and the other Fg.. dscs). o φ l l φ l 3 φ 3 l 6 φ l 4 4 l 5 φ 5 A closed 6-chan among pont obstacles (shown as small connected to ground. The ground s regarded as lnk m and s referred to as the base of the chan. Relatve to the base, the open chan has m degrees of freedom and ts C-space s smply a product of m crcles, (.e., C = (S ) m ). A closed m-chan can be constructed by attachng the dstal end of the open chan to the base as shown n Fgure. Mathematcally, ths attachment mposes two algebrac equalty constrants, causng the C-space of the closed chan to become a compact, closed, real, varety of dmenson m 3. Ths varety s a manfold as long as the dstance between the two base connectons s not equal to one of the m crtcal lengths [7]. To fx notaton, let {l,, l m } denote the fxed lnk lengths and {φ,, φ m } denote ther angles measured counterclockwse from the vector from the center of jont to the center of jont m. Snce our nterest s n motons of the closed chan relatve to the base, φ m s set to zero. Suppose that there s a fnte set, O of pont obstacles {p,, p n } that the closed m-chan may not touch. The set of confguratons for whch a lnk ntersects a pont obstacle forms an arrangement of (m- 4)-dmensonal varetes. The unon of these varetes s the C-obstacle C obst. If lnk-lnk collsons are also to be avoded, C obst becomes the unon of the (m-4)- dmensonal collson varetes and an (m-3)-dmensonal lnk-lnk collson set. C-free, denoted by C free, s the complment of C obst n C. Fnally, the path plannng problem can be stated as follows: gven φ nt = (φ,, φ m ) nt C free and φ fnal = (φ,, φ m ) fnal C free determne a contnuous map τ [0, ] (φ (τ),, φ m (τ)) C free such that φ(0) = φ nt and φ() = φ fnal. III. C-SPACE OF PLANAR CLOSED CHAINS Here we summarze three results from topologcal approaches to moton plannng that were crucal to the work presented here. The frst result s about the connectvty of the C-space of a planar closed chan. We need a concept called long lnks [7] whch s defned as a subset L of the lnks such that the sum of the lengths of every par of dstnct lnks n L s strctly greater than half of the sum of the lengths of all m lnks. Due to the

3 g S T T # T g T # T # T # T # T # T T Fg.. Constructon of C-space of closed chans va crtcal crcles of an open chan. Fg. 3. Fve types of connected C-spaces of closed 5-chans. strct nequalty n the defnton, the number of long lnks L must be 0,, or 3. If L s equal to 3, the C-space has two components; otherwse, t has one. The second result gves the topology of the C-space. It says that for gven lengths {l,, l m } and base length l m that s generc wth respect to those m lengths, the C-space s the boundary of a manfold wth boundary, whch s gven as the unon of sub-manfolds of the form (S ) k I m k [7], where I d denotes the nterval of dmenson d. To clarfy the above concluson, consder Fgure, whch shows a horzontal base lnk and three moveable lnks. The concentrc crcles on the rght are the crtcal crcles (not drawn to scale) of the open 3-chan. If the end pont of the 3-chan s anchored at any pont n the shaded annular regon, ts C-space s that of a closed 4-chan. If the anchor pont s nteror to one of the three reachable annul, that C-space s, one crcle, or two dsjont crcles as shown. Assume that the end pont of the open 3-chan s connected to an open chan based at the left end of the base lnk, and further, assume that as a result of the connecton, the end-pont of the prevously open 3-chan has one degree of freedom, thus effectvely creatng a closed 5-chan. Two possble workspaces for the endpont of the open 3-chan are shown: a crcle and a curve segment. The C-space of the newly-formed closed chan can be determned by glung together all the C-spaces at each pont. For example, begn at the left end of curve γ and traverse t to ts other end. Intally, the C-space over each pont of γ s empty, snce the open 3-chan cannot reach those ponts. At the ntersecton wth the outer-most crcle, the C-space of the closed chan s a pont, but the workspace segment lyng nsde the outermost annular regon generates a tube. At the pont where curve ntersects the next crtcal crcle, the C-space of the closed 5-chan s a fgure eght. Ths sgnfes a bfurcaton of the tube nto two tubes. The two tubes coalesce nto a sngle tube at the next crossng of the same crtcal crcle. Fnally, at end of the curve, C-space s a crcle. Thus the C-space of the mechansm wth the end of the open 3-chan constraned to follow the curve segment s a tube pnched closed at one end, open at the other, and wth a hole through the tube somewhere between the two ends. Applyng the same logc to the closed 5-chan that would result from connectng the end of the open 3-chan to the lnk shown as a dashed lne, one fnds that the C-space of the closed 5-chan s a sphere. Followng ths approach, one can show by constructng crcles and base lnks of varous szes that a closed 5-chan has up to sx types of C-spaces, among whch the shown n Fgure 3 are connected. The sxth type of C- space s the dsjont unon of two copes of T. Here S and T represent the two-dmensonal sphere and torus, respectvely, and # denotes the connected sum of two spaces. The thrd result pertans to the parametrzaton of the C-space. Snce the C-space of a generc closed m- chan s an (m-3)-dmensonal manfold, C-space can be locally parameterzed by a set of m 3 jont angles. However, fxng the orentatons of m 3 lnks (n addton to the fxed base), does not fx the confguraton of the closed chan. Returnng to Fgure, fxng φ 3, φ 4 and φ 5 stll allows elbow-up and elbow-down postures of lnks and. Ths last result suggests parttonng C-space nto an elbow-up pece and an elbow-down pece as follows. Break the closed chan at the thrd jont, thus creatng an open -chan and an open (m-3)-chan based at opposte ends of the base lnk. The C-space of the second open chan s the (m-3)-dmensonal torus. For an arbtrary pont n ths space, the chan can be closed n 0,, or confguratons of the -chan. When there are two confguratons, they are labeled elbow-up and elbowdown. Snce there are never more than two confguratons that close the loop, two copes of the torus suffce to represent the C-space of the closed chan. When there s only one confguraton, the elbow-up and elbow-down confguratons have converged, so at these ponts, the

4 tor are connected. These confguratons form a varety referred to as the boundary varety that s the subject of the next secton. IV. BOUNDARY VARIETY AND ITS DECOMPOSITION In ths secton, we outlne a recursve projecton method (smlar to the approach n [8], [9]) to determne the structure and a cell decomposton of the boundary varety. The k th level of recurson wll be denoted by appendng (k) to the expresson n queston. Refer to Fgure. As descrbed above, let us break the closed m-chan nto an open -chan CH () wth lnk lengths {l, l }, and an open (m-3)-chan CH () wth lnk lengths {l 3,, l m } based at the pont (l m, 0). Choose the jont angles of CH () as the parameterzaton of the elbow-up and elbow-down tor. Further, let f() be the forward knematc map of CH (). The boundary varety B() s the set of all confguratons for whch the endponts of the two open chans can be connected when the lnks of the open -chan are collnear. Wth the constrant of collnearty, the space of possble end pont locatons Σ() of the -chan n the workspace s a par of concentrc crcles of rad l + l and l l centered at the orgn. The boundary varety can now be defned as follows: B() = f() (Σ()). Note that B() s the unon of the C-space of the two closed (m-)-chans M () and M () wth lnk lengths {l + l, l 3,, l m } and { l l, l 3,, l m }. Also, B() s empty f and only f the ntersecton between Σ() and the annulus centered at (l m, 0) wth rad m =3 l and mn( l 3 ± ±l m ) s empty, n whch case, C-space s not connected. The process descrbed above s repeated for each of the two closed (m-)-chans. That s, each of these closed chans s assumed to have a ther thrd jonts removed, whch gves rse to crtcal crcles centered on the orgn wth rad l ± l ± l 3 and whose unon s Σ(). Smlar to B(), the boundary varety B(), s gven as B() = f() (Σ()), where f() s the forward knematc map of the {l 4,, l m } open (m-4)-chan CH () based at the pont (l m, 0). By constructon, t s clear that B() s the unon of the C-spaces of the four closed (m-)-chans each wth frst lnk length equal to one of the four crtcal rad. B() s also the set of crtcal values, or skeleton of the projecton of B() onto the (m-4)-dmensonal torus wth coordnates {φ 4,, φ m }. Recurson contnues untl B(m 3) s defned. In ths case, Σ(m 3) s the unon of m 3 (geometrcally) concentrc crcles centered at the orgn. The boundary varety B(m 3) s the set of values of φ m where the Fg. 4. A btorus drawn n a cubcal represenaton of C-space. The skeleton of ths btorus under a vertcal projecton map (drawn bold), s three crcles. crcle of radus l m centered at (l m, 0) ntersects the crcles of Σ(m 3). Wth {B(),, B(m 3)} defned, C-space can now be decomposed nto reachable and unreachable cells whch are (m-3)-dmensonal cylnders. Example: Consder a closed 6-chan, wth lnk lengths, {0.55,.9457,.3,.948, , 5.785}. The C-space of ths chan s connected, snce there are only two long lnks. It s contaned n two three-dmensonal tor that are connected through the boundary varety B(). B() s the unon of the C- spaces of two closed 5-chans M () and M () wth the last four lnk lengths {.3,.948, , 5.785} and the frst lnk of length equal to one of the crtcal lengths {.3945,.4969} of the crcles composng Σ(). Usng the approach descrbed n Fgure, the C-spaces of each of M () and M () s a btorus, T #T (see Fgure 3). Ths s consstent wth the fact that each of of M () and M () have two long lnks. To draw B(), t s convenent to represent a 3-torus as a three-cube wth edge length π and opposte faces dentfed. In ths space, a btorus (qualtatvely lke those composng B()) s shown n Fgure 4. Note that the bold grth curve and the two crcles where the btorus s cut (recall that the top and bottom crcles are dentfed) s the skeleton of the btorus under the projecton map onto the (φ 4 -φ 5 )-space. B() s the skeleton of B() under the vertcal projecton onto the (φ 4 -φ 5 )-face of the cube. In Fgures 6 and 7, B() s plotted n thn sold closed curves. Note that B() conssts of sx crcles; two pars of small concentrc crcles and one par of large concentrc crcles contanng the other two pars. The large crcles are the projectons of the grth curves of the btor of B(). The two pars of small crcles are the small crcular skeletal curves lke those shown n Fgure 4. The fact that these pars of crcles are concentrc mples that the two btor of B() are nested n C-space. The last step s to project the skeleton of B() to an edge of the cube. Ths s shown by the dashed vertcal lnes (three are covered by other vertcal lnes) n Fgures 6 and 7. Usng the ap-

5 proach dscussed n Fgure, one can show that sx of the eght crtcal crcles wth the rad {6.658, , 4.30, 3.96,.7668,.6644} are ntersected transversally by the workspace of the end pont of the open -chan CH (3). These correspond to the crtcal values of φ 5. V. COLLISION VARIETIES Let V j p, j =,, m, denote the (m-)- dmensonal varety correspondng to p lyng on lnk j. The unon of these varetes over all lnks gves the contrbuton of p to the C-obstacle: V p = m j= V j p. The unon of the V p over all pont obstacles n O s denoted by V : n V = V p. = for j = m,, 3, and =,, n), whch can be vewed as the contact varetes of CH () clpped by the B(). The portons of the constrant varetes of CH () lyng on the unreachable sde of B() are elmnated. Fgure 5 shows a cylnder wth rectangular cross secton n a three-dmensonal C-space. The cylnder s cut by V p, V p and two patches of B(), labeled B () and To study the global structure of these varetes, the closed chan s agan broken nto the open -chan CH () and the open (m-3)-chan CH (). The contact varetes of CH () are already understood from prevous work [9]. However, our nterest s n V j p B (). Assumng the top and bottom of the rectangular column are dentfed and gnorng B(), there are two cells n the cylnder. Assumng that the regon above the top patch and below the bottom patch of B() are unreachable, a porton of V p s clpped and there are three cells. The topologcal propertes of the remanng varetes, Vp and Vp, = {, n}, are determned by the technque developed n [7] and descrbed n the dscusson of Fgure. These varetes can be expressed as follows: Vp = f() (γ) () Vp = f() (γ) () where γj s the workspace of the end-pont (always a closed loop) of CH () when lnk j s n contact wth p and f() s the forward knematc map of CH (). Agan t s mportant to understand the ntersecton of these contact varetes wth B(). Snce the boundary constrant requres lnks and to be colnear, the ntersectons of Vp and Vp wth B() can be seen to be the C-spaces of two closed (m-)-chan formed by replacng lnks and by a sngle lnk of length l ±l and fxng ths lnk n contact wth p. Example contnued: We ntroduce two pont obstacles, p = (4, ) and p = (3, ). Note that t s mpossble for lnk or to touch ether pont, so the correspondng varetes are empty. Consequently, only the contact varetes of lnks 3, 4, and 5 appear n Fgures 6 and 7. Note that the two thckest vertcal lnes at φ 5 ±.38, defne extreme ponts of the boundary varety B(). By these fgures, one can also determne the structure of Vp 5, whch project to the two second thckest vertcal lnes n the Fgures. Take Vp 5 as an example. The projecton of Vp 5 s the lne φ whose ntersecton wth the nteror between the par of large concentrc crcles of B() s two separated lne segments, and wth the nteror of the nner par of crcles an nterval. Ths reveals that the cross-secton of Vp 5 n the horzontal plane of each 3-torus changes from two separated segments to one segment, and then back to two separated segments (recall that opposte faces of each 3-torus are dentfed). The boundary of the cross-secton s ether four or two separated ponts, the unon of whch gves B() Vp 5. Glung the two peces of Vp 5 n the elbow-up and elbow-down tor along B() Vp 5 yelds the surface T #T. Ths result can also be obtaned usng the approach descrbed n Fgure for the correspondng closed 5-chan of Vp 5 wth the lnk lengths {0.55,.9457,.3,.948,.45), whch has two long lnks. Usng the same analyss method, we can see that Vp 5 s a torus T (n the fgures, the lne of Vp 5 has no ntersecton wth the nteror between the nner par of crcles). VI. GRAPH REPRESENTATION OF C-FREE The recursve decomposton of the elbow-up and elbow-down tor makes the constructon of a connectvty graph straght forward. In essence, the graph constructon process s recursve startng wth the one-dmensonal crcle parameterzed by φ m and workng up to the full (m-3)-dmensonal C-space. Referrng to Fgure 6, the crcle n Tu parameterzed by φ 5 has dstnct crtcal ponts. Removal of these ponts from the φ 5 crcle defnes a set of ntervals. Some of the twodmensonal cells about these ponts could be dsconnected at these ponts, but whether or not ths s the case s only revealed as the method proceeds. Therefore, ntally, these ntervals and the cells above them are assumed to be dsconnected at the crtcal ponts. Thus, at ths stage, the graph of the space of φ m s smply dsconnected nodes. The same decomposton occurs n Td as shown n Fgure 7, but no attempt s made to connect the two graphs yet.

6 The next step s to lft the graphs so that they represent the cell structure of the two -tor parameterzed by φ m and φ m. The ntervals dentfed n the frst step serve as the base manfolds for the second step. If one fxes φ m, then φ m les on a crcle. For the crtcal values of φ m, the crcle s drawn as a dashed vertcal lne (9 of the are vsble). Between the crtcal crcles are two-dmensonal cylndrcal cells. For example, consder the two crtcal crcles straddlng φ 5 = n Fgure 6. Ths porton of C-space s a thckened crcle, or tube, that s cut n four places by projected skeleton of the boundary varety. Ths dentfes four two-dmensonal cells. The tube just to the rght s cut n only two places, yeldng two cells. However, n the (φ m,φ m )-space some of the cells n the two tubes are connected. After mergng the connected cells, three possbly dsconnected cell reman. The graphs are agan lfted, ths tme nto the (φ m, φ m, φ m 3 )-space. Fgure 5 shows a cell n the (φ m -φ m )-space. Above t s the (φ m, φ m, φ m 3 )-space wth one-dmensonal constrant and boundary varetes. The locus of crtcal ponts are those n the ntersecton of the surfaces. Its projecton onto the cell s the curve D, whch splts the cell nto two. Above each non-crtcal pont n the two cells, there are four constrant surfaces. Assumng that the space above B () and below B () volates the loop closure constrant, then the four surfaces defne two reachable cylndrcal cells above the cell (a,d) and three cells above (D,b). Further the two cells bounded below by B () are connected, as are those bounded below by the surface labeled. Mergng yelds three cells above the orgnal cell n (φ m,φ m )-space. Analyzng all adjacent cells n (φ m,φ m )-space n the same way yelds all connected components of the (φ m, φ m, φ m 3 )-space, and the process contnues untl all connected components of dmenson m 3 are dentfed. VII. EXPERIMENTAL RESULTS Our method for closed 5-chans and 6-chans was mplemented n Matlab and tested for many plannng problems. Typcally, a closed 5-chan closed chan movng among 4 pont obstacles requred about 60 seconds, and closed 6-chan movng among pont obstacles requred about 0 seconds. Returnng agan to the example, a connectvty graph of C-free was constructed accordng to the approach descrbed above. Then a moton plannng problem was specfed by the followng confguratons: φ nt = ( ,.83, 0.046,.946, 0.46, 0) φ goal = ( , , 0.046,.046, 0.346, 0). φ 3 B () B () B () B () φ 4 a a D D D b B () B () b B () B () B () B () Fg. 5. A rectangular cell n (φ 4,φ 5 )-space onto whch the crtcal ponts of constrant varetes n (φ 4,φ 5,φ 6 )-space are projected. One of the confguratons s n the elbow-up torus; the other s n the elbow-down torus. The computed path s projected onto the two two-dmensonal tor shown n Fgures. 6 and 7. (Note that n these lowerdmensonal spaces, one should not expect the path to jump from the elbow-up to the elbow-down spaces through the projecton of B()). Whle t s dffcult to see n a three-dmensonal plot, the path n the full C-space crosses through the boundary varety B(). Anmaton of the moton n ths example can be found n ftp://6bar:6bar@ VIII. COMPLEXITY ANALYSIS The complexty of any algorthm that performs cell decomposton on C-free s bounded from below by the number of connected components. Whle we have not proven ths, there s evdence that suggests that ths lower bound s Ω(n m 3 ), m 5. We also conjecture an upper bound, by consderng the polynomal representaton of the collson and loop closure varetes. Assumng that the hghest degree of these polynomals s δ, then Halpern s cell complexty results [0] appled to our approach mply that each connected component of C-free could be composed of O(n m 4 δ) cells. Snce each of these cells must be computed, the worst-case complexty of our decomposton algorthm s O(n m 7 ), m 5. To provde nsght nto our conjecture, we show how to obtan these results for the case of m = 5 wth n pont obstacles. The C-space s two-dmensonal and the contact varetes are one-dmensonal curves n C-space. An upper bound on the number of components can be obtaned by mposng three condtons: () all elbow-up and elbow-down confguratons are vald, () all contact φ 5

7 φ Elbow up rectangle T u Lnes determnng the regon of the C space Lnes passng through skeleton of B() B() Projecton of collson varety by lnk 5 Projecton of collson varety by lnk 4 Skeleton of collson varety by lnk 3 Intersecton between B() and varety by lnk 3 Projected path varetes are closed, and (3) all possble ntersectons among varetes occur and they occur n general poston. These condtons can be satsfed only f: l 5 = 0, l = l 4, l = l 3, l > l (3) whch defnes a degenerate closed chan wth base lnk of length zero. The result s effectvely a closed fourchan wth base lnk pnned at one end rather than both. One can see that t s always possble to move φ 4 and φ 3 arbtrarly and that for every φ 4 and φ 3 there are two confguratons of the other two lnks; elbow-up and elbow-down. Thus C-space s two copes of T, Tu and Td, glued along the two crcles correspondng to confguratons where all the lnks are colnear. φ = φ [ π, π] (4) φ = φ + π [ π, π], (5) φ 5 Fg. 6. The path (drawn as green ponts) projected to Tu. The two trangles ponted to the left and rght represents the ntal and goal confguratons, respectvely φ Elbow down rectangle T d Lnes determnng the regon of the C space Lnes passng through skeleton of B() B() Projecton of collson varety by lnk 5 Projecton of collson varety by lnk 4 Skeleton of collson varety by lnk 3 Intersecton between B() and varety by lnk 3 Projected path φ 5 Fg. 7. The path (drawn as green ponts) projected to T d..e., Tu Td are the dsjont unon of two crcles. Gven an m-dmensonal manfold Q, defne H k (Q) the k-dmensonal homology group of Q. The dmenson of H k (Q) can be nterpreted as the number of k- dmensonal generators of Q. For example, for T t can be shown that H 0 (T ) = R, H (T ) = R for T = S S, and H (T ) = R. The geometrc nterpretaton s that the torus has one connected component, two onedmensonal generators (T = S S ), and one twodmensonal generator (the space nsde the torus). Also, gven an (m )-dmensonal closed submanfold Q s of Q, the m-dmensonal relatve homology group of Q wth respect to Q s, defned as H m (Q, Q s ), gves the number of components of Q Q s. Each contact varety Vp j s a unon of two closed curves, each n one torus, wth two ponts n common. However, Vp and Vp 4 common are concdent crcles correspondng to all lnks are collnear, as do Vp and Vp 3. Thus the C-obstacle, V =,j Vp j, s the unon of 6n crcles. We then compute the number of ntersectons between these 6n crcles. Under the general poston assumpton about O, we can show that the total number of ntersectons s 4n n. Let V denote the dsjont unon of 6n crcles n V. The set of ntersecton ponts among the 6n crcles s denoted X. It s obvous that H (V ) = R 6n, H 0 (V ) = R 6n, H 0 (X) = R 4n n, and H 0 (V ) = R. What s left to compute s H (V ), for whch we can use the followng exact sequence: 0 H (V ) H (V, X) H 0 (X) H 0 (V ) 0 No three ponts are n the same lne, no two ponts and the orgn are n the same lne, and no two lnes each passng through a dfferent par of ponts ntersect at the same pont n the crcle centered at the orgn and wth the radus l.

8 from whch we obtan H (V, X) = H (V ) = R 4n n. Substtutng nto another exact sequence 0 H (C) H (C, V ) H (V ) H (C) 0 yelds H (C, V ) = R 4n n. Ths shows that the number of components of the C-free s 4n n. Snce n our algorthm each component s decomposed nto at least one cell, the number of cells n a graph wth the worst-case number of components s at least 4n n. Ths gves the asymptotc lower bound of Ω(n ). On the other hand, notce that the number of cells n our graph for C-free s bounded by the product of the number of crtcal ponts and the number of collson curves. Let δ be the degree of the polynomals of collson curves. Then the number of cells n our graph s bounded by (4n n + 6nδ) (6nδ) = O(n 3 ) s the asymptotc upper bound of the worst-case complexty. IX. CONCLUSION Ths paper presents a method for an exact cell decomposton of the reachable collson-free porton the confguratons space (C-space) of a planar closed chan wth m-lnks movng among pont obstacles. Ths method combnes the results on the topology of C- spaces of planar closed chans and the decomposton method proposed n [9]. Frst, the C-space s covered usng only two charts. Then, each pece s mbedded n an (m-3)-dmensonal torus. The structure of the collson set s used to decompose both tor nto cells of collson-free confguratons (C-free) and to construct graphs representng the structure of C-free. The global graph s then establshed by jonng graphs of the the two tor f they are connected, as ndcated by the analyss of the boundary of the loop closure constrant. A beneft of the decomposton method s that t s very easy to determne cell membershp of a gven confguraton. Ths fact combned wth the exact graph representaton of C- free allows one to check path exstence by graph search. If a path exsts, the cylndrcal structure of the free cells facltates easy path constructon. The proposed algorthm s complete, yet tme consumng. 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