Journal of Intelligent and Robotic Systems Vol. 35, pp

Transcription

1 PLANAR CABLE-DIRECT-DRIVEN ROBOTS: DESIGN FOR WRENCH EXERTION Rbert L. Wllams II 1 Department f Mechancal Engneerng Oh Unversty Athens, Oh Pal Gallna 2 Dpartment d Innvazne Meccanca e Gestnale Unversty f Padva Padva, Italy Jurnal f Intellgent and Rbtc Systems Vl., pp Keywrds: cable rbt desgn, cable-drect-drven rbts, actuatn redundancy, cable nterference, wrenchexertn wrkspace ABSTRACT Cable-drect-drven rbts and haptc nterfaces are appealng because f ther structural smplcty, hgh stffness, and hgh exerted wrench-t-weght rat. A majr drawback s that cables can nly exert tensn. Therefre, actuatn redundancy s requred t apply general wrenches (frce/mment vectrs). Even wth actuatn redundancy, nt all desred wrenches can be appled n sme cnfguratns due t ne r mre negatve cable frces requred. In addtn, cable nterference can be a serus prblem fr these devces. The bjectve f ths artcle s t present the best desgn fr planar cable-drect-drven rbts and/r haptc nterfaces wth ne degree f actuatn redundancy, wth regard t general wrench exertn and cable nterference. Results ndcate that the cable nterference cnstrant dmnates whch suggests the need fr future desgn wrk t allevate ths nterference. 1 Asscate Prfessr, Crrespndng Authr 2 Assstant Prfessr 1

2 INTRODUCTION Several cable-drect-drven rbts (CDDRs) and haptc nterfaces (CDDHIs) have been studed n the past. An early CDDR s the Rbcrane develped by NIST fr use n shppng prts (Albus, et. al., 1993). Ths devce s smlar t an upsde-dwn sx-degrees-f-freedm (df) Stewart platfrm (Stewart, 1966), wth sx cables nstead f hydraulc-cylnder legs. In ths system, gravty s an mplct actuatr that ensures cable tensn s mantaned at all tmes. Anther CDDR s Charltte, develped by McDnnell-Duglas (Campbell, et. al., 1992) fr use n Internatnal Space Statn. Charltte s a rectangular bx drven n-parallel by eght cables, wth eght tensnng mtrs munted n-bard (ne n each crner). Three strnged haptc nterfaces have been bult and tested, the Texas 9-strng (Lndemann and Tesar, 1989), the SPIDAR (Ish and Sat, 1994), and the 7- cable master (Kawamura and It, 1993). CDDRs and CDDHIs can be made lghter, stffer, safer, and mre ecnmcal than tradtnal seral rbts and haptc nterfaces snce ther prmary structure cnssts f lghtweght, hgh lad-bearng cables. On the ther hand, ne majr dsadvantage s that cables can nly exert tensn and cannt push n the mvng platfrm. All f the devces dscussed abve are desgned wth actuatn redundancy,.e. mre cables than wrench-exertng degrees-f-freedm (except fr the Rbcrane, where tensnng s prvded by gravty) n attempt t avd cnfguratns where certan wrenches requre an mpssble cmpressn frce n ne r mre cables. Despte actuatn redundancy, there exst subspaces n the ptental wrkspace where sme cables can lse tensn. Rberts et al. (1997) develped an algrthm fr CDDRs t predct f all cables are under tensn n a gven cnfguratn whle supprtng the rbt weght nly. Nne f these prevus artcles have presented CDDR r CDDHI desgn fr ptmal wrench exertn. Anther ptental prblem wth CDDRs and CDDHIs s cable nterference that further restrcts the wrkspace. A haptc nterface s a devce that can exert wrench (frce/mment) and/r tactle feedback t the human frm vrtual realty and/r remte envrnments. The current artcle fcuses n wrench feedback. Many researchers have been develpng haptc nterfaces fr varus applcatns. A cmprehensve study f frce-, wrench-, and tactle-feedback haptc nterfaces s prvded by Burdea (1996). 2

3 The bjectve f the current artcle s t present the best desgn fr CDDRs and CDDHIs wth ne degree f actuatn redundancy, wth regard t best general wrench exertn and wth regard t avdng cable/endeffectr nterference. Specfcally, ths artcle fcuses n the planar 3-df, 4-cable CDDR that s requred t exert general wrenches n the envrnment. Ths wrk equally apples t CDDHIs wth ne degree f actuatn redundancy that must exert general wrenches n the human peratr s hand. Ths artcle begns wth a descrptn f the planar CDDR, fllwed by CDDR cable nterference determnatn, statcs mdelng, a methd fr attemptng t mantan pstve cable tensns, and then desgn and results fr planar CDDRs wth ne actuatn redundancy, cnsderng bth pstve cable tensns and cable nterference. 3

4 LIST OF SYMBOLS x, y, φ Cartesan parameters f end-effectr {}, {H} Fxed base and mvng end-effectr Cartesan crdnate frames a, b, c end-effectr wdth, end-effectr heght, and square grund lnk sde L, Lˆ th cable length, unt vectr n the L drectn W fc Cable-nterference-free wrkspace h, T th cable end-effectr cnnectn pnt, th cable grund cnnectn pnt v and [ R] v unt drectns f tw cnsecutve edges f end-effectr H Orthnrmal rtatn matrx relatng {H} t {} f th cable frce { } { F M } T W = Cartesan wrench exerted n the envrnment by end-effectr R R R [ A ], [ A ] Statcs Jacban matrx, Mre-Penrse pseudnverse f [ A ] n th cmpnent f the kernel vectr f [ A ] α Scalar fr tensn ptmzatn 4

5 PLANAR CABLE-DIRECT-DRIVEN ROBOT (CDDR) Ths sectn descrbes the planar 4-cable CDDR, ncludes a bref dscussn n CDDR knematcs, and presents a general mdel t determne cable/end-effectr nterference. A CDDR cnssts f a rgd end-effectr supprted n-parallel by n-cables cntrlled by n tensnng actuatrs. Fgure 1 shws the planar 4-cable CDDR knematcs dagram. Fr 3-df planar peratn, there must be at least 3 cables. Snce cables can nly exert tensn n the end-effectr, there must be mre cables t avd cnfguratns where the end-effectr can g slack and lse cntrl. Our wrk s lmted t CDDRs wth ne degree f actuatn redundancy,.e. 4 cables n planar devces wth 3 Cartesan degrees-f-freedm. Ths scenar represents actuatn redundancy but nt knematc redundancy. That s, there s ne extra mtr whch prvdes nfnte chces fr applyng 3-df wrench vectrs, but the end-effectr has nly 3 Cartesan degrees-ffreedm (x, y, φ; angle φ s shwn n Fg. 1 and x, y are the cmpnents f the vectr frm the rgn f {} t the rgn f {H}, expressed n {}). Fgure 1 shws the defntns fr reference frame {} and mvng endeffectr frame {H}. In Fg. 1, the desgn parameters fr the planar 4-cable CDDR are a (end-effectr wdth), b (end-effectr heght), and c (square grund lnk sde). The length f each cable s dented as L, = 1,2,3,4. Planar CDDR Knematcs Ths sectn brefly dscusses the requred knematcs slutns. Assumng all cables always reman n tensn, CDDR knematcs s smlar t n-parallel-actuated rbt knematcs (e.g. Tsa, 1999; Gsseln, 1996). In CDDR smulatn fr desgn, the nverse pse knematcs slutn s straght-frward (gven the pse, calculate the cable lengths). The frward knematcs prblem requres the slutn f vercnstraned cupled nnlnear equatns and s mre dffcult. A Newtn-Raphsn numercal slutn s emplyed, where the vercnstraned Mre-Penrse pseudnverse s used n the teratn. The CDDR nverse velcty Jacban matrx s clsely related t the Newtn-Raphsn Jacban matrx and the statcs Jacban matrx. These knematcs slutns are presented n (Wllams, 1998). 5

6 c Y L 4 L 1 Y H b a X H L φ 2 L 3 c L^ - v v h Y H X H X Fgure 1. Planar 4-Cable CDDR Knematcs Dagram T Fgure 2. Cable/End-effectr Interference Dagram Planar CDDR Cable Interference A ptental wrkspace-lmtng prblem fr CDDRs s that f cable nterference. Ths can take three frms: cable/cable nterference, cable/wrkspace bject nterference, r cable/end-effectr nterference. In the planar case we can easly avd the frmer case by desgnng the devce s all cables are n dfferent parallel planes. Als, the cable/ wrkspace bject nterference wll nt be a prblem fr the planar case f all bjects are belw (r abve the plane f the CDDR). We cannt always avd by desgn the latter nterference and thus we present n ths sectn a methd fr determnng cable/end-effectr cllsns. Let us defne cllsn- free wrkspace W fc as the subset f wrkspace that can be reached wthut cllsn between the end-effectr and the cables. In the fllwng we assume that the end-effectr s a cnvex plygn and that all cables are attached t the plygn vertces. The am f ths sectn s t develp a methd fr checkng whether a end-effectr cnfguratn { y φ} x belngs t W fc. Let us defne φ as the angular value that the end-effectr can rtate antclckwse befre a cllsn. Cnversely, let us defne φ as the angular value that the end-effectr can rtate clckwse befre a cllsn. Nte that these lmtng angles depend n the pstn { x y} and they take nt accunt all cables. 6

7 Let us cnsder the th cable n the cable/end-effectr dagram f Fg. 2. The unt vectr n the drectn f the th cable s Lˆ. The cable s cnnected t the end-effectr at pnt h (Fg. 2 shws the vectr h t ths pnt, referenced t the mvng {H} frame). Fr each pnt h we defne tw vectrs v and v that represent the unt drectns f tw cnsecutve edges f the plygn. A cllsn ccurs when the cable drectn algns wth ne f the neghbrng cnsecutve plygn faces,.e.: Lˆ = v r Lˆ = v (1) The negatve sgns n (1) ndcate that at cllsn the cable drectn s algned but ppste n drectn t the plygn face. Defne T as the grunded pnt fr the th cable and { x y} T n {}. Then the cllsn cndtns (1) can be expressed as: T G = as the rgn f {H}, expressed H H [ H R] h G H H = λ [ H R] v r T [ H R] h G = [ H R] v where: H = T [ R] h G λ and [ ] H By ntrducng a new vectr m { m m } = T G x y T H λ (2) csφ sn φ R = sn φ csφ =, equatns (2) yeld: / / v v x y λ h λ h x x = csφ m x = sn φ m sn φ m x y csφ m y (3) where the symbl slutns f (3) are gven by: / vx ndcates that there are tw pssbltes, ne wth vxand ne wth vx. All pssble AB ± C B C A C φ = ± arccs (4) 2 2 B C 7

8 where: A = h x / vx h y / vy B = m x / vx m y / vy x C = m x / vy x m y / vx y f /, / v x v y / v x A = h B = m C = m f = A = h B = m C = m f = y y x / v y Frm (4) we btan 4 values fr each f the tw / vx, but nly ne f them satsfes (3) and at the same tme yelds pstve λ. It s straght-frward t mplement ths cable/end-effectr algrthm n a cmputer prgram. Let us ndcate these slutns as φ f we cnsder vx and φ f we cnsder vx. Ntce that φ has t be pstve whle φ has t be negatve. In cnclusn: φ = mn φ = max { φ1,..., φ n } { φ,..., φ } 1 n (5) gves the lmtng angles fr cable/end-effectr nterference cnsderng all cables = 1,2, L, n ; where n = 4 n ths artcle. 8

9 CDDR STATICS In ths artcle, the wrkspace wheren all cables are under pstve tensn whle exertng all pssble wrenches s called the statcs wrkspace. The wrench s a frce and mment appled t the envrnment, centered at the rgn f the end-effectr frame {H}; ths s dfferent frm a wrench defntn where the mment s abut the base frame {}. We assume velctes and acceleratns n the end-effectr are small and thus the devce may be cntrlled n a pseudstatc manner. Statcs mdelng and attemptng t mantan pstve cable tensn are presented n ths sectn. We use a smple methd t determne the extent f the statcs wrkspace,.e. the wrkspace wheren all pssble wrenches can be appled wth nly pstve cable tensn. Statcs Mdelng Ths sectn presents statcs mdelng fr planar CDDRs. Fr statc equlbrum the sum f external frces and mments exerted n the end-effectr by the cables must equal the resultant external wrench exerted n the envrnment. Fgure 3 shws the statcs free-bdy dagram fr the planar 4-cable CDDR. f 4 M R F R f 3 h 3 f 1 f 2 The statcs equatns are: Fgure 3. Planar 4-Cable CDDR Statcs Dagram 4 = 1 f = fl = F 4 = ˆ R m ([ H R] h ) f = M R = 1 = = 1 (6) In ths artcle gravty s gnred because t s assumed t be perpendcular t the CDDR plane; we assume the end-effectr s supprted n a plane. The defntn f frames {} and {H} are gven n Fg. 1. In (6), f s the 9

10 cable tensn appled t the th cable (n the negatve cable length unt drectn Lˆ because tensn); [ R] f must be n H s the rtatn matrx relatng the rentatn f {H} t {}; h s the pstn vectr frm the rgn f {H} t the th cable cnnectn, expressed n {H} (nly h 3 s shwn n Fg. 3); and F R and M R are the resultant vectr frce and mment (taken tgether, wrench) exerted n the envrnment. Agan, let us emphasze that ths wrench acts at the end-effectr frame {H} rgn. Substtutng the abve terms nt (6) yelds: where { } { } T f f1 f2 f3 f4 [ ]{ f } { } A = (7) W R T = s the vectr f scalar cable frces, { W } { F M } = { F F M } T R = s the resultant external wrench vectr exerted n the envrnment by the end-effectr (expressed n {} crdnates but felt at the rgn f {H}), and the 3x4 Statcs Jacban matrx [ A ] (expressed n {} crdnates) s: R R x y z [ A] Lˆ = ˆ L1 Lˆ Lˆ Lˆ [ ] ˆ [ ] ˆ [ ] ˆ [ ] H R h1 L2 H R h2 L3 H R h3 L4 H R h4 (8) The statcs equatns (7) can be nverted n an attempt t exert general wrenches whle mantanng pstve cable tensn. Ths wrk s presented n the next subsectn. Mantanng Pstve Cable Tensn Fr CDDRs wth actuatn redundancy, (7) s undercnstraned whch means that there are nfnte slutns t the cable frce vectr { f } t exert the gven wrench { W } 1 R. Rberts et al. (1997) present a methd fr determnng f a vectr f nly pstve cable frces exsts fr CDDRs under gravty ladng nly. Ths algrthm culd be extended t CDDRs wth general wrench exertn, but fr the planar 4-cable case wth nly ne degree f actuatn redundancy, a smpler methd s used here, based n Shen et al. (1994). Frst, t nvert (7) (slvng the requred cable tensns { f } gven the desred wrench { W } R ) we adapt the well-knwn partcular and hmgeneus slutn frm rate cntrl f knematcally-redundant manpulatrs: { f } [ A] { W } ([ I ] [ A] [ A] ){ z} = 4 (9) R

11 T ( ) 1 where [ I 4 ] s the 4x4 dentty matrx, { z } s an arbtrary 4-vectr, and [ ] T = [ A] [ A][ A] A s the 4x3 Mre- Penrse pseudnverse f [ A ]. The frst term f (9) s the partcular slutn t acheve the desred wrench, and the secnd term s the hmgeneus slutn that maps { z } t the null space f [ A ]. Fr actuatn redundancy f degree ne, an equvalent expressn fr (9) s: { f } f f = f f P1 P2 P3 P4 n1 n2 α (1) n3 n 4 where the partcular slutn [ A ] { W R } s the frst term n (1) and the hmgeneus slutn s expressed as the kernel vectr f [ A ] ( { n n n n } T N = ) multpled by an arbtrary scalar α. The methd we adapt frm Shen et al. (1994) t determne f a gven cnfguratn les wthn the statcs wrkspace fr a gven CDDR desgn s smple. T ensure pstve tensns f n all cables = 1,2,3,4 fr all pssble exerted wrenches, t s necessary and suffcent that all kernel vectr cmpnents (n, = 1,2,3,4) have the same sgn. That s, fr a gven cnfguratn f a gven CDDR desgn t le wthn the statcs wrkspace, all n > OR all n < ( = 1,2,3,4). If ne f these tw cndtns s satsfed, regardless f the partcular slutn, we can fnd a scalar α n (1) whch guarantees that all cable tensns { f } are pstve by addng (r subtractng) enugh hmgeneus slutn. Nte a strct nequalty s requred; f ne r mre n =, the cnfguratn n questn des nt le wthn the statcs wrkspace. Ths methd s smple but pwerful snce we needn t cnsder specfc wrenches, but t wrks fr all pssble wrenches. It shuld be nted that ths methd s necessarly cnservatve,.e. there may exst cmbnatns f pstve and negatve partcular and hmgeneus slutn cmpnents that may yeld pstve cable tensns by prper chce f α fr a certan wrench, but ur gal n desgn s t ensure that all pssble wrenches be satsfed n the statcs wrkspace, nt merely sme wrenches. It shuld als be nted that whle we demnstrate ths methd fr the planar 4-cable CDDR, t s 11

12 applcable t any CDDR wth ne degree f actuatn redundancy. Als, there s n guarantee that scalar α n (1) exsts t guarantee that all cable tensns { f } are pstve; n fact, when such α ceases t exst, ths defnes the bundary f the statcs wrkspace. Fr desgn purpses we need t calculate ths kernel vectr repeatedly, fr many CDDR desgns and many cnfguratns wthn each desgn. Equatn (11) gves the methd t calculate each kernel vectr cmpnent n, where A s the determnant f the 3x3 submatrx f [ A] wth clumn remved. ( 1 ) 1 n = A = 1,2,3,4 (11) Fr n-lne cntrl purpses, nce a gven wrench at a gven cnfguratn s determned t be feasble va the abve-descrbed methd, the tensns fr cntrl are calculated by (1), chsng α s that ne cmpnent f { f } s zer (r, a small pstve value) and the remanng terms are pstve. Of curse, f the partcular slutn happens t cntan all pstve terms then α can be chsen t be ; alternatvely n ths case, α can stll be chsen s that ne cmpnent f { f } s zer (r, a small pstve value), whch results n the same Cartesan wrench but lwer cable tensns. 12

13 PLANAR CDDR DESIGN RESULTS Ths sectn presents the parameters, desgn prcess, and results fr determnng the best CDDR desgn wth regard t wrench exertn and cable nterference. In prevus cmputer smulatns (Wllams, 1998), t was dscvered that the planar crssed-cable case (Fg. 1) was far superr t the planar nn-crssed-cable case (nt shwn), partcularly n exertng mments. Therefre we nly cnsder crssed cables n ths artcle. Planar 4-Cable CDDR Parameters Assumng a square grund lnk, there are nly three desgn parameters fr the 4-cable planar CDDR: endeffectr rectangular dmensns a and b, plus square grund lnk sde c (refer t Fg. 1). If we nrmalze wth c = 1 (the results may be scaled as needed), we have tw desgn parameters a and b. In ths artcle we lmt a and b t the same ranges, a, b [..4]. Nte a = b = s nt a feasble desgn but we allw ths case n ur wrk fr cmpleteness; the results wll shw ths case s n fact the least ptmal desgn. Fr the Cartesan plane we cnsder φ rtatn at a grd f XY pnts cverng, Y [.2.8] X, determned by fttng the largest desgn [a, b] = [.4.4] n the crners f the grund lnk square wth c = 1. Cnsderng ptch ranges f the human wrst (when the hand grasps a cylndrcal end-effectr munted perpendcular t the rectangular [a, b] end-effectr at the center), a nmnal desred rtatn range s φ = ±45. We wsh t satsfy ths statcs wrkspace desgn requrement at all XY pnts fr all appled wrenches. Example Statcs and Cable Interference Wrkspaces An example statcs wrkspace result s shwn fr planar CDDR desgn lcatn (a, b) = (.2,.3), c = 1 n the cntur plts f Fgs. 4. Fgures 4 were generated usng the kernel vectr methd descrbed n the last sectn. Fgure 4a shws the pstve lmtng angles φ T (the subscrpt T ndcates tensn) reachable at all XY lcatns cnsderng nly pstve cable tensns, fr all pssble exerted wrenches. Fr certan wrenches, φ T wll be larger at certan XY lcatns; hwever, we can guarantee all pssble wrenches may be satsfed n the Fg. 4a statcs wrkspace, agan fr nly the pstve cable cnstrant. Nte that Fg. 4a demnstrates plar symmetry, as all such plts fr any (a, b) desgn des: chse any value φ T n the XY plane and yu wll fnd the 13

14 dentcal value at the same radus frm the center pnt (X, Y) = (.5,.5) and drectly ppste the chsen pnt. Als fr ths type f plt, the maxmum pssble φ T always ccurs at the center pnt (X, Y) = (.5,.5) Y Y X X a. φ T Cntur (deg) b. -φ T Cntur (deg) Fgure 4. Example Statcs Wrkspace: Lmtng Angle φ T Results ver XY Plane a =.2, b =.3, c = 1, Fr All Pssble Wrenches Cmparng Fg. 4b t Fg. 4a, the -φ T statc wrkspace results can always be btaned by rtatng the φ T results abut the vertcal lne X =.5; thus we need nly cnsder the φ T attrbute fr desgn purpses and then - φ T wll have the same values, mrrred abut X =.5. Frm plts such as Fg. 4a, we can recrd the maxmum, average, and mnmum lmtng angles ver all XY. We wll then use these measures t cmpare dfferent (a, b) desgns. Fr the example n Fg. 4a, MAX(φ T ) = pnts), AVG(φ T ) = 66.6, and MIN(φ T ) = 9 (nt vsble n Fg. 4a snce these ccur nly at dscrete Due t the vertcal-mrrred symmetry, these measures are dentcal fr Fg. 4b (where MAX nw ndcates the maxmum negatve angle): MAX(-φ T ) = vsble n Fg. 4b snce these ccur nly at dscrete pnts), AVG(-φ T ) = , and MIN(-φ T ) = 9 (agan, nt The MAX result s fr a sngle XY pnt, always the mddle f the XY wrkspace; due t symmetry, the MIN result ccurs at tw pnts, the lwer-left and upper-rght XY crners fr φ T, swtchng t the upper-left and lwerrght XY crners fr -φ T ; the AVG result s fr all XY.

15 Fr CDDR desgn, Fgs. 4 are mprtant t determne feasble statcs wrkspaces wheren all pssble wrenches can be exerted usng nly feasble (pstve) cable tensns. Hwever, ths type f plt des nt gve the full stry snce cable nterference may be mprtant n CDDR desgn as well. As mentned n the cable nterference thery sectn, we nly cnsder cable/end-effectr nterference snce cable/cable nterference may easly be avded fr planar CDDRs by arrangng all cables n dfferent planes, and cable/human nterference s nt a prblem n the planar case. Fr the same planar CDDR desgn as Fgs. 4 ((a, b) = (.2,.3), c = 1), Fg. 5 shws the pstve lmtng angles φ C (the subscrpt C ndcates cllsn) ver all XY fr cable/end-effectr nterference. Fgure 5 reprts a knematc cnstrant f nterference between the end-effectr and any ne f the fur cables and hence s nt related t the exerted wrenches. Fgure 5 was generated usng the cable/end-effectr nterference methd presented earler. Fgure 5 demnstrates the same plar symmetry that Fgs. 4 have. Agan, the maxmum φ C s n the mddle f the wrkspace ((X, Y) = (.5,.5)), as was the case fr Fgs. 4. Als, the negatve lmtng angles -φ C may be btaned by expltng the same symmetry between Fgs. 4b and 4a: rtate the φ C results abut the vertcal lne X =.5 t get the -φ C cntur (nt shwn). Thus, we need cnsder nly the φ C attrbute fr desgn. MAX(φ C ) = Agan we recrd the MAX, AVG, and MIN values ver all XY fr desgn purpses: frm Fg. 5, 36.9, AVG(φ C ) = 24.4, and MIN(φ C ) = 7.1. Due t the vertcal-mrrred symmetry, the negatve angle measures are dentcal, wth a negatve sgn. Agan, The MAX result s always fr the mddle f the XY wrkspace; the MIN result ccurs at the lwer-left and upper-rght XY crners fr φ C, swtchng t the upper-left and lwer-rght XY crners fr -φ C ; the AVG result s agan fr all XY. Cmparng Fgs. 4a and 5, t s clear that the cable/end-effectr nterference cnstrant dmnates and ts asscated MAX, AVG, and MIN lmtng angle values φ C are much less than thse f φ T. Hwever, the statcs wrkspace cncept s stll crucal fr CDDRs snce we are lmted t feasble (pstve) cable tensns. Therefre, we wll cnsder bth statcs wrkspace and knematcs nterference wrkspace n CDDR desgn. 15

16 Y X Fgure 5. Example Interference Wrkspace: Lmtng Angle φ C Results (deg) ver XY Plane a =.2, b =.3, c = 1 Planar 4-Cable CDDR Desgn Curves We nw present desgn curves t ad n selectng the best planar CDDR desgn, cnsderng bth pstve cable tensns fr all pssble exerted wrenches and knematc cable/end-effectr nterference. Fgures 6 shw the MAX, AVG, and MIN φ T values ver the desgn plane (a,b). Remember, due t symmetry, we needn t cnsder -φ T separately. Fgures 6 cnsder nly the statcs wrkspace cnstrant that all cable tensns must reman pstve fr exertng all pssble wrenches. Each (a,b) lcatn n Fgs. 6 reprt the MAX, AVG, and MIN φ T (respectvely), each nstance cnsderng all XY lcatns. That s, fr each (a,b), a cntur plt smlar t Fg. 4a was generated and the cmputer prgram memrzed the MAX, AVG, and MIN φ T values fr later plttng aganst the a,b desgn plane. The prgram then terated ver all (a,b) lcatns under cnsderatn. Desgn parameters a and b were each vared frm t.4 (wth c = 1), n steps f.2. At each (a,b) lcatn, X and Y were each vared frm.2 t.8, n steps f.2 (these were als the parameters used n Fgs. 4). Nte a, b, c, X, and Y all have unts f length; the results may be scaled as needed fr CDDR desgn, based n the nrmalzed c = 1. 16

17 b a a a. MAX φ T (deg) b. AVG φ T (deg).4 b b a c. MIN φ T (deg) Fgure 6. Desgn Curves, Tensn Only: Lmtng Angle φ T Results ver a, b Parameters The MAX φ T results n Fg. 6a demnstrate a dfferent type f symmetry than Fgs. 4 and 5, abut the a = b lne. The AVG φ T results f Fg. 6b almst share ths symmetry, but are ff due t the lack f symmetry n the MIN φ T results f Fg. 6c. The MIN results are nt symmetrc due t the end-effectr vertces rtatng utsde f the XY plane under cnsderatn when the end-effectr s placed at the lwer-left and upper-rght crners (where the MIN φ T values ccur). Cnsderng the statcs wrkspace cnstrant nly, the best desgns

18 appear t be alng the a = b lne tward the center f the a,b range. Fr gd CDDR desgn, the AVG measure s mre mprtant than the MAX and MIN measures, snce the latter tw nly nvlve ne r tw XY pnts, whle the frmer nvlves all XY pnts. We must als cnsder cable/end-effectr nterference n planar CDDR desgn. Anther set f desgn curves was generated n the same manner as fr Fgs. 6, but ths tme cnsderng nly the knematc cable-endeffectr nterference cnstrant. Three cntur plts ver all desgn parameters a,b were generated, shwng the MAX, AVG, and MIN φ C lmtng angles (agan, -φ C needn t be cnsdered separately due t symmetry). These cntur lnes n all three cases were vertcal; that s, fr a gven value f a, the lmtng values φ C reman unchanged fr all pssble b values. Ths behavr s expected snce the end-effectr heght b des nt affect the nterference but the end-effectr wdth a des. Ths set f desgn curves fr cable/end-effectr nterference nly s nt shwn because they are s smlar t the next set f desgn curves shwn: the lmtng cases are fund ver all a, b cnsderng bth statcs wrkspace and nterference cnstrants. That s, the statcs-wrkspace-nly desgn curves f Fg. 6 were cmpared wth the knematcs-nterference-nly desgn curves (nt shwn) and the lesser value f the lmtng angles reprted (the smaller f φ T and φ C at each a, b desgn pnt), called φ TC t ndcate the lmtng angle cnsderng bth tensn and cllsn. Fgures 7 shw these Overall Desgn Curves, ncludng the MAX, AVG, and MIN lmtng angles φ TC ver all a, b desgn parameters under cnsderatn. Wth the exceptn f the lwer-left and lwer-rght crners f the a, b desgn plane, the nterference cnstrant dmnates. Fgures 7 shw the lmtng angle φ TC MAX, AVG, and MIN values that can be btaned by planar CDDRs at the a, b desgn chces shwn, cnsderng bth statcs wrkspace and knematcs cable/end-effectr nterference cnstrants. T chse the best and wrst planar CDDR desgns we use the fllwng prcedure. Over all a, b we use a weghted sum f the MAX, AVG, and MIN φ TC values, each nrmalzed t have a largest value f 1. t make each equal n mprtance; further chsng an equal weghtng between MAX, AVG, and 18

19 MIN yelds the Fnal Desgn Curve shwn n Fg. 8. A perfect scre wuld be 3. (hghest MAX, AVG, and MIN φ TC all ccurrng at the same a, b value), whle the wrst scre wuld be.. b a a a. MAX φ TC (deg) b. AVG φ TC (deg) b b a c. MIN φ TC (deg) Fgure 7. Overall Desgn Curves cnsderng bth Tensn and Cllsn Frm Fg. 8, the best planar CDDR desgns ccur at a = and b >.1 where the scre s apprxmately 2.7. The wrst desgn, as predcted, s at a = b =, whch s an nfeasble desgn; here the scre appraches. The next-wrst desgn les alng a =.32, where the scre s apprxmately 1.2. Therefre we chse: 19

20 BEST desgn: a =. b =.1 WORST desgn: a =.32 b =.5 b a Fgure 8. Fnal Desgn Curve cnsderng bth Tensn and Cllsn and Equally Weghtng MAX, AVG, and MIN Lmtng Angles 2 Earler we sad that the AVG φ TC value s the mst mprtant fr planar CDDR desgn snce t represents all pssble XY lcatns, whle the MAX s fr the center pnt nly and the MIN s fr tw crner pnts nly. If ne were t weght the results t use AVG nly and gnre MAX and MIN, ne wuld btan nearly the same results as gven abve (see Fg. 7b t verfy ths fact). Nte that the general case chsen fr detaled demnstratn n Fgs. 4 and 5 (a =.2, b =.3) was nether the best nr wrst case, wth an ntermedate scre f apprxmately 1.8.

21 BEST and WORST Results Fgure 9a shws the BEST planar CDDR desgn and Fg. 9b shws the asscated achevable lmtng angles φ TC cnsderng bth statcs wrkspace and knematcs cable/end-effectr nterference cnstrants. Fr cmparsn, Fg. 1a shws the WORST planar CDDR desgn and Fg. 1b shws the asscated achevable lmtng angles φ TC cnsderng bth statcs wrkspace and knematcs cable/end-effectr nterference cnstrants. Cnsderng bth statcs wrkspace and knematcs cable/end-effectr nterference cnstrants, the BEST desgn was nt able t satsfy ur desgn gal f a mnmum lmtng angle f φ = 45 ver the entre XY TC wrkspace. It des cme clse t 45 n the mddle f the wrkspace, frm the upper-left crner t the lwerrght crner, but ths s the maxmum lmtng angle rather than the mnmum lmtng angle. The WORST desgn angle results (Fg. 1b) are nt greatly dfferent frm the BEST desgn angle results (Fg. 9b); hwever, the BEST case s clearly preferable. In Fgs. 9b and 1b the cable/end-effectr nterference cnstrant dmnates. Cnsderng nly the statcs wrkspace cnstrant, there s a much larger dfference between the best and wrst cases (nt shwn snce ths s mpractcal). Many cases can exceed the desred angle f 45 f we gnre the nterference cnstrant (fr nstance, the general case f Fgs. 4 exceeds ths gal n mst f the XY statcs wrkspace). Interestngly, the BEST case dentfed abve fr bth statcs wrkspace and cable/end-effectr nterference cnstrants s nearly the wrst case cnsderng statcs wrkspace nly! Hwever, we cannt gnre the nterference cnstrant and s we prpse the results gven abve as the BEST and WORST cases. It s pssble that, thrugh clever desgn t mnmze the chance f cable/end-effectr nterference, ne culd ncrease the effectve verall planar CDDR wrkspace. It was the ntent f ur wrk t reman cnservatve n desgn. Reducng cable/end-effectr nterference s a tpc fr further wrk. 21

22 Y.6.4 Y X Fgure 9a. Best Desgn a =, b = X Fgure 9b. Best Lmtng φ TC Results (deg) Y.6.4 Y X Fgure 1a. Wrst Desgn a =.32, b = X Fgure 1b. Wrst Lmtng φ TC Results (deg) The BEST and WORST results presented are nt general but subject t ur desgn parameters (c=1 and a,b lmted t dscrete values as explaned prevusly) and ur desgn crtera (we want the largest statcs wrkspace pssble; ver all XY we wsh the largest lmtng φ angle wheren all cable tensns may reman nly pstve). 22

23 CONCLUSION Ths artcle presented desgn f planar cable-drect-drven rbts (CDDRs) wth ne degree f actuatn redundancy, wth regard t best statcs wrkspace fr general wrench exertn and wth regard t knematc cable/end-effectr nterference. The results may be extended t ther cable-drect-drven rbts and haptc nterfaces (CDDHIs) wth ne degree f actuatn redundancy that must exert varus wrenches n the envrnment and the human hand. Statcs wrkspace s defned as that wrkspace n whch all pssble wrenches may be appled wth nly pstve cable tensns. The well-knwn partcular and hmgeneus slutn frm knematcally-redundant manpulatr rate cntrl was adapted t calculate the tensn cntrl fr exertng cmmanded wrenches. A smple methd was presented t determne the lmts f the statcs wrkspace. Usng ths methd and the cable/end-effectr nterference determnatn methd presented, the best and wrst planar 4- cable CDDR desgns were fund, subject t ur desgn parameters and desgn crtera. The cable/end-effectr nterference cnstrant was fund t dmnate. Future wrk plans nclude develpng desgns t mnmze the cable/end-effectr nterference prblem and t mplement and evaluate ur CDDR and CDDHI desgns n hardware. Ptental applcatns f ths technlgy nclude ndustral rbt tasks (assembly, weldng, pantng, etc.), large-scale utdr rbts (e.g. cnstructn and shpyards), and haptc nterfaces. 23

24 REFERENCES J.S. Albus, R. Bstelman, and N.G. Dagalaks, 1993, The NIST ROBOCRANE, Jurnal f Rbtc Systems, 1(5): G.C. Burdea, 1996, Frce and Tuch Feedback fr Vrtual Realty, Jhn Wley & Sns, New Yrk. P.D. Campbell, P.L. Swam, and C.J. Thmpsn, 1995, "Charltte Rbt Technlgy fr Space and Terrestral Applcatns", 25 th Internatnal Cnference n Envrnmental Systems, San Deg, SAE Artcle C.M. Gsseln, 1996, "Parallel Cmputatn Algrthms fr the Knematcs and Dynamcs f Planar and Spatal Parallel Manpulatrs", Jurnal f Dynamc Systems, Measurement, and Cntrl, 118(1): M. Ish and M. Sat, 1994, A 3D Spatal Interface Devce Usng Tensed Strngs, Presence- Teleperatrs and Vrtual Envrnments, MIT Press, Cambrdge, MA, 3(1): S. Kawamura and K. It, 1993, New Type f Master Rbt fr Teleperatn Usng a Radal Wre Drve System, Prceedngs f the IEEE/RSJ Internatnal Cnference n Intellgent Rbts and Systems, Ykhama, Japan, July 26-, 55-. R. Lndemann and D. Tesar, 1989, Cnstructn and Demnstratn f a 9-Strng 6-DOF Frce Reflectng Jystck fr Telerbtcs, NASA Internatnal Cnference n Space Telerbtcs, (4): R.G. Rberts, T. Graham, and J.M. Trumpwer, 1997, "On the Inverse Knematcs and Statcs f Cable- Suspended Rbts", IEEE Internatnal Cnf. n Systems, Man, and Cybernetcs, Orland, FL. Y. Shen, H. Osum, and T. Ara, 1994, Manpulablty Measures fr Mult-wre Drven Parallel Mechansms, IEEE Internatnal Cnference n Industral Technlgy, D. Stewart, 1966, "A Platfrm wth Sx Degrees f Freedm", Prceedngs f the Insttute f Mechancal Engneers (Lndn), 1(15): L.W. Tsa, 1999, Rbt Analyss: The Mechancs f Seral and Parallel Manpulatrs, Wley, New Yrk. R.L. Wllams II, 1998, "Cable-Suspended Haptc Interface", Internatnal Jurnal f Vrtual Realty, 3(3):