, DE 82(1), DEXTERITY OF MANIPULATOR ARMS AT AN OPERATING POINT

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1 Abdel-Malek, K., 995,"Dexterty f Manpulatr Arms at an Operatng Pnt," Prceedngs f the st ASME Advances n Desgn Autmatn, DE Vl. 8(), pp DEXERIY OF MANIPULAOR ARMS A AN OPERAING POIN Karm Abdelmalek Department f Mechancal Engneerng and Center fr Cmputer Aded Desgn he Unversty f Iwa Iwa Cty, IA 5 ABSRAC hs paper presents a methd fr analyzng dexterty, hence functnalty f rbtc manpulatrs at an peratng pnt. Dexterty f Manpulatrs cmprsng sphercal wrsts s studed. he gal s t prvde the user wth knwledge f rentablty f the end-effectr at a target. Wrst accessble utput sets are determned analytcally by frst determnng manpulatr sngulartes, fllwed by back substtutng the sngulartes nt the cnstrant equatns t parametrze surfaces. Snce regns f a surface may exst nsde the wrst accessble utput set, and ther regns n the bundary, surfaces are segmented nt subsurfaces. he segmentatn f surfaces s carred ut usng ntersectn methds t cmpute hgher rder sngulartes. determne whether a subsurface s an nternal r bundary ne, each subsurface s studed fr exstence nsde the wrst accessble utput set by perturbng a pnt alng ts nrmal. Clsed-frm slutns f the bundary f the wrst accessble utput set are btaned. A servce sphere s lcated at an peratng pnt n the accessble utput set and the sphere s ntersected wth the bundary sub-surfaces. he ntersectn curves are cmputed and prjected nt the space f a cylnder, then the cylnder s unrlled t depct a map. Dexterty charts are ntrduced as maps depctng rentablty f the end-effectr at a target. he prblem f servce regn verlappng s slved by prjectng the slutns nt dexterty charts and supermpsng servce regns. INRODUCION hs paper s amed at btanng an mprved understandng f the functnalty f rbtc arms. In the past, researchers n the feld f rbtcs have studed functnalty n terms f spaces. Reachable wrkspaces have been addressed by Rth (975), sa and Sn (98), Yang and Lee (98), Gsseln (99), Emrs (99), and Haug et al. (99). Dexterus wrkspaces have been studed by Kumar and Waldrn (98), Yang and La (985), and Wang and Wu (99). Wrkspaces, hwever, d nt prvde adequate nfrmatn abut the functnalty f the rbt at specfc targets. Fr example, gven a requred dexterus wrkspace usng the current methds, ne cannt determne a sutable placement f the rbtc arm t attan mum functnalty. he health care system s a key area n whch a better understandng f functnalty wll have great mpact. In bran surgery, fr example, a surgen may requre the end-effectr f a manpulatr carryng a laser fr tssue destructn t be pstned n several dfferent rentatns near a malgnant tumr (Lavallee 99). hus, a knwledge f pssble rentatns f the end-effectr at the target s f great mprtance t the surgen. A measure f the amunt f dexterty, called the Dexterus Sld Angle (DSA), was analytcally defned by Abdel-Malek and Paul (99). Spne surgery s anther example n whch knwledge f the DSA s needed. hs type f surgery s ne f the mre cmplcated prcedures perfrmed by surgens. In partcular, a rbt can assst surgens n nsertng pedcle screws durng spnal fusn t crrect sclss. Determnatn f the DSA at each screw lcatn wll prvde the surgen wth mprved flexblty n manpulatn.

2 In preparng a cavty fr a prsthess, a rbt arm must fllw the cntur f a pre-planned trajectry defned by the prsthess gemetry. btan mum functnalty, the rbt peratr makes the chce f the lcatn f the rbt base. Wth the ad f dexterty charts, exhbtng the functnalty at specfc targets, the rbt peratr wll be able t better make ths decsn. In an assembly cell n the shp flr f a manufacturng envrnment, a manpulatr s needed t assemble a prduct n ne cell whle perfrmng anther task n anther cell. he questns f where t lcate the tw cells wth respect t the rbt fr mum functnalty may arse. set the backgrund fr ths paper, defntns f relevant terms are stated. Accessble Output Set (Haug et al. 99) he regn f space that can be reached by a pnt n the manpulatr, fr all cmbnatns f jnt crdnates. Dexterus Wrkspace (Kumar and Waldrn, 98) A subspace f the accessble utput set wthn whch a vectr n the end-effectr may assume all rentatns. Because f jnt lmts and gemetrc cnstrants (e.g., at wrkspace bundares), the dexterus wrkspace may nt nclude all f the accessble utput set. Wrst Pnt When a manpulatr hand (end effectr) may be mdeled as a rgd bdy that rtates abut a pnt W fxed n sme lnk f the manpulatr, W s referred t as the wrst pnt. Mst ften, W wll be lcated at the ntersectn f three successve revlute jnts, n a chan f three lnks. Wrst Accessble Output Set: he set f pnts that can be accessed by the wrst pnt. Servce Sphere: A sphere parametrzed as xss ( p) centered at p that s used t ndcate dexterty at an peratng pnt p. he servce sphere can assume any radus less than the length f the last lnk f the manpulatr. he terms Servce Sphere and Servce Angle were frst ntrduced by Vnagradv et al. (97). Servce Pnt: Pnt n the servce sphere thrugh whch the end-effectr may penetrate. Servce Regn: A Regn SR ( pq, ) n the servce sphere cntanng nly servce pnts, where q s the vectr f generalzed crdnates. Nte that there may be any number N R f servce regns, each f area A ( ) U A. = p. he ttal servce area s therefre ( p) Dexterus Sld Angle DSA(p) (Abdel-Malek and Paul 99) he rat f the ttal area f the servce regns t the area f the servce sphere at pnt p,.e., N R N R U ( p) π () DSA( p) = A h = where h s the radus f the servce sphere. Because DSA s nt an angle n the rdnary sense, and because t s smlar t the sld angle ω = N R U A ( p) h () = t was called the Dexterus Sld Angle (DSA) whch represents a measure f dexterty at an peratng pnt. Open chan mechansms such as rbtc manpulatrs cmprse a number f lnks cnnected va jnts; e.g. prsmatc, revlute, and sphercal. Generalzed crdnates q * n = [ q, q,..., qm ] R, where m s the number f degrees f freedm, are used t characterze the cnfguratn (pstn and rentatn) f each lnk n the manpulatr. A sx degree-f-freedm arm wth a sphercal wrst can be thught f as havng tw segments. he frst segment (frst three jnts) s respnsble fr pstnng the wrst pnt W. he secnd segment (sphercal wrst) s respnsble fr the rentatn f the endeffectr.. BOUNDARY PARAMERIZAION OF HE WRIS ACCESSIBLE OUPU SE

3 he cmbnatn f the frst three jnts, cupled wth jnt lmts and nternal sngulartes f the mechansm, may result n a cmplex accessble utput set. In rder t analytcally fnd expressns fr the bundary surfaces f ths set, t s necessary () t develp a set f analytcal crtera t btan the pstnng f the wrst n terms f the generalzed crdnates, () determne the bundary surfaces due t sngulartes asscated wth the set, and () determne the subset f these surfaces due t jnt lmts. he mathematcs f pstnng f the wrst pnt are readly avalable by the use f the Denavt- Hartenberg (D-H) representatn (Denavt and Hartenberg 955). he D-H representatn prvdes a systematc methd fr descrbng the relatnshp between adjacent lnks. he transfrmatn matrx descrbng a transfrmatn frm lnk (-) t lnk fr a revlute jnt s csθ csα snθ snα snθ a csθ snθ csα csθ snα csθ a = snθ A () snα csα d where θ, depcted n Fg., s the jnt angle frm x t the x axs, d s the dstance frm the rgn f the (-)th crdnate frame t the ntersectn f the z axs wth the x, a s the ffset dstance frm the ntersectn f the z axs wth the x axs, and α s the ffset angle frm the z axs t the z axs. Jnt - Jnt Jnt + Lnk - Lnk θ z z - a z y x w W O - (a) x - d // // θ O Fgure (a) DH Representatn, (b) Ntatn used n btanng the accessble utput set he hmgeneus transfrmatn matrx that specfes the cnfguratn f the th frame wth respect t the base crdnate system s the prduct f successve transfrmatn matrces f, j = = j j= where s the number f degrees-f-freedm and s f the frm x O... () = R p (5) Where R s the rtatn matrx between frame - and frame and p s the pstn vectr frm the rgn f the - frame t the th frame. Fr a sx axs manpulatr wth a sphercal wrst, the hmgeneus transfrmatn matrx relatng the end-effectr and the wrst t the reference frame s (b) x q x

4 6 = 6 (6) he vectr x descrbes the accessble utput set f the wrst pnt such that q xq R xw p = + (7) where x w s the vectr descrbng the wrst pnt (Fg. b), reslved n the reference frame f lnk. In rder t determne the bundary f the accessble utput set fr a mechansm, McKerrw (99) shwed that sngulartes (bth nternal and bundary) can be cmputed by prper manpulatn f the Jacban f the mechansm. Haug at al. (99) presented a numercal methd fr mappng bundares f accessble utput sets fr a general, mult degree-f-freedm mechansm. In ths paper, frst and hgher rder sngulartes are cmputed. Sngulartes are substtuted nt the cnstrant equatn t parametrze bundares f the wrst accessble utput set. Fr a gven cnfguratn f the manpulatr, the generalzed crdnates satsfy ndependent hlnmc knematc cnstrant equatns f the frm Φ( q) = xq R xw p = (8) n l Where Φ:R R s a smth functn, and l s the number f cnstrant equatns. In addtn, the generalzed crdnates q are subject t nequalty cnstrants representng jnt lmts. mn q q q (9a) mn mn q q q (9b) q q q (9c) he cnstrant Jacban f the cnstrant functn Φ( q ) f Eq. (8) fr a certan cnfguratn q s the matrx Φ Φq ( q ) = ( q ) () q j he wrst accessble utput set s thus n A= x R : Φ ( q) =, fr sme q () { q } he bundary f the wrst accessble utput set fr a manpulatr s a subset f the accessble utput set at whch the sub-jacban Φ q f the knematc cnstrant functn f Eq. (8) s rw-rank defcent (Haug et al. 99),.e., A { xq A: Rank Φq( q) < l, fr sme q} () Fr a three degree-f-freedm mechansm (wrst accessble utput set), equatng the determnant f the Jacban t zer wll result n the sngulartes f the system. It s mprtant t realze that sme f these sngulartes wll nt satsfy the nequalty cnstrants f the jnt varables q. mpse the nequalty cnstrants, t s cnvenent t parametrze Eq. (9) by ntrducng new generalzed crdnates λ such that fr an nequalty cnstrant f the frm mn q q q (a) can be parametrzed as q = a + b sn λ (b) mn mn where a = ( q + q ) and b = ( q q ) are the md pnt and half range f the nequalty cnstrant. he Jacban wth respect t the new crdnates can be wrtten as Φ q j = Φqqλ (a) q dλ j j

5 Sngulartes can be determned by equatng the determnant f the Jacban t zer such that Fx ( ) = Φ q q λ = (b) Slvng Fx ( ) and substtutng the results nt Eq. (b), a set f frst rder sngulartes µ (=,...,m) s generated, where m s the ttal number f sngulartes. Frst rder sngulartes generated by Eq. (b) are f tw types. Internal sngulartes are thse due t the assembly f the mechansm tself. Bundary sngulartes are due t nequalty cnstrants mpsed n jnts (e.g., space lmtatn, nterference, and actuatr capablty). Equatn b s used t fnd the bundary f the wrst accessble utput set n clsed frm. It s f nterest t parametrze the bundary f the wrst accessble utput set t later cmpute the ntersectns f the bundares wth the servce sphere t determne servce regns. Slutns are then prjected nt anther space t vsualze the servce regns. Substtutng each sngularty nt the accessble utput set Eq. (7), a set f surfaces Χ ι (µ ι ) are parametrzed such that Χ ( µ [ m ι ) = x ( µ ), x ( µ ),..., x ( µ m) ] (5) where =,...m. In determnng accessble utput sets, surfaces generated by sngulartes may ntersect each ther. Parts f a surface may be nternal whle ther parts may be bundary t the wrst accessble utput set. Intersectng curves between surfaces determne a dfferent type f sngularty, whch dvde the surface nt a number f subsurfaces. he set f generalzed crdnates resultng frm ths ntersectn are hgher rder sngulartes (the s-called bfurcatn pnts f a crss sectn f the accessble utput set). Pars f surfaces are ntersected such that j x ( µ ) x ( µ j ) = fr j (6) Eq. (6) wll result n a number f hgher rder sngulartes. he number f sngulartes s augmented t µ, =,..., m, m+,..., n; where (n-m) s the number f surface ntersectns resultng n new sngulartes. he matrx f subsurfaces s augmented t m m n Ψ ( µ ι ) = [ Ψ ( µ ), Ψ ( µ ),..., Ψ ( µ m), Ψ ( µ m+ ),..., Ψ ( µ n )] (7) Equatn (7) ncludes all subsurfaces due t nternal, bundary, and hgher rder sngulartes. It remans t determne whether these subsurfaces are nternal r bundary surfaces. hs can be perfrmed by perturbng a knwn pnt n the subsurface and determnng whether ths pnt satsfes the equatn f cnstrant (Eq. 8), subject t nequalty cnstrants f Eq. (9). Fr a subsurface Ψ ( q) due t a sngularty µ, the nrmal t the surface at a knwn pnt q, where q and q are generalzed crdnates, s gven as (Dcarm 976) Ψ Ψ q q nq $( ) = (8) Ψ Ψ q q Fr a small perturbatn t abut the pnt q n the subsurface Ψ ( q) alng the nrmal nq $( ), the crdnates f the perturbed pnts are x = Ψ ( q ) ± tn$( q ) (9) Fr the perturbed pnt t exst nsde the accessble utput set, t has t satsfy Eq. (8), subject t nequalty cnstrants f Eq. (9). A slutn s sught t the fllwng system f equatns. R x + p Ψ ( q ) m tn$( q ) = () w mn q q q (a) 5

6 mn q q q (b) mn q q q (c) he subsurface Ψ ( q) s an nternal surface f and nly f there exsts a slutn fr Eq. () fr bth perturbatns ± t, cnsstent wth the nequaltes f Eq. ().. DEERMINING SERVICE REGIONS determne servce regns at an peratng pnt p, a servce sphere parametrzed as x ss (,), st s lcated wth center at p and radus h (dstance between pnt p n the end-effectr and the wrst pnt W). he wrst pnt assumes a pstn n the surface f the servce sphere and has t satsfy Eq. (8). Usng ths defntn, the servce regn s defned as the pnts n the surfaces f the servce sphere and nsde the wrst accessble utput set. he servce regn SR ( pq, ) s a set that exsts n the surface f the servce sphere SR( pq, ) = { xss( p) =, fr sme q} () and belngs t the wrst accessble utput set such that SR ( pq, ) = { Φ ( q) =, fr sme q } () Snce the bundary f the wrst accessble utput set has been determned, t s pssble t ntersect each subsurface wth the servce sphere t determne the ntersectn curve. determne whether the servce regn s enclsed by the ntersectng curve, a ray s cast n the drectn f the nrmal passng thrugh p. he ray wll ntersect the sphere n tw pnts s and s. he servce regn s dentfed by determnng whch f these pnts satsfes the lcal cnstrant equatn. he lcal cnstrant equatn accunts fr the nequalty cnstrant asscated wth the sngularty f the subsurface. he ntersectn curve s the set f slutns t Λ such that Ψ ( q) x ss ( st, ) Λ = = () q a b sn λ he servce regn s dentfed by determnng ne f the pnts n the ray satsfyng the fllwng Ψ Λ = ( q) + tn$( p) xss( p) = (5) lcal q a bsn λ Numercal slutns usng the Newtn-Raphsn teratn methd are btaned. vsualze servce regns, t s cnvenent t ntrduce dexterty charts. Dexterty charts are prjectns f servce regns nt anther space. he ntersectn curve f each subsurface wth the servce sphere s parametrzed n terms f parameters s and t n the surface f the sphere. Usng a prjectn methd (aylr and Mann 97), the curve s then mapped nt a cylnder and the cylnder s unrlled. he sphere x ss (,)(wth st radus h) s mapped nt the cylnder x c ( uv, )(wth radus h) such that the arc length n the uv-plane s gven by dz dσ = du + dv = h dt + dz = h dt + dβ (6) dβ where θ s the lngtude angle and β s the lattude angle. he arc length n the sphere s gven by dσ = h cs βdt + h dβ (7) Dvdng Eq. (6) by Eq. (7) and ntegratng usng the requrement that z = when β = and z > when β > and substtutng β wth ( π s ), the peratr L s ntrduced such that L[ xss(,) s t ] xc (,) u v = (8) 6

7 x x where the peratr L s defned as fllws. L s π π hlg sec( s) + tan( s) u = = (9) t v ht. ILLUSRAING DEERMINAION OF HE WRIS BOUNDARY llustrate the fregng analyss, cnsder the manpulatr depcted n Fg.. he frst segment f the manpulatr cmprses ne prsmatc and tw revlute jnts (Fg. a). z z x x z W a z W (a) Fgure (a) hree jnts f a manpulatr, (b) A sphercal wrst (three ntersectng axes) Fr ths manpulatr, the three hmgeneus transfrmatn matrces (jnts,, and ) are csq sn q snq = csq a snq = () q csq sn q sn q csq = where q, q, and q are the generalzed varables representng jnt angles. Multplyng (b) = and extractng the rtatn matrx (Eq. 7) csq csq snq csq sn q R = snq csq csq sn q snq () sn q csq he pstn vectr s p = [ a csq asn q q] () Fr a wrst pnt lcated at x w = [ d w ], where d w s the dstance alng the z-axs f the wrst pnt wth respect t reference frame, the equatn f cnstrant (Eq. 8) f the wrst pnt s, 7

8 x dw csq csq a csq Φ( q) = y d sn cs sn = w q q qa () z d snq q w Fr the remander f ths dscussn let a = and d w = 5. hs manpulatr has jnt cnstrants as fllws q q = c + c sn λ () q 7 q = b + b sn λ (5) q q = d d 6 + sn λ (6) Where the generalzed crdnates λ were ntrduced accrdng t Eq. (). Evaluatng Eq. (b) sn( b + b snλb csλ cs( d + d sn λ) dw a sn( b + b sn λ ) b csλ Φ λ = cs( b + λ λ + λ + λ λ b sn b cs cs( d d sn ) dw acs( b b sn ) bcs ccsλ cs( b + b sn λ) sn( d + d sn λ ) d csλdw sn( b + b sn ) sn( d + d sn ) d cs d λ λ λ w (7) cs( d + d sn ) d cs d λ λ w Internal and bundary sgulartes are cmputed by evaluatng the determnant f the Jacban and equatng t zer Φ λ = c csλ sn( d + d sn λ ) d csλ d b csλ cs( d + d sn λ ) d + a = (8) ( ) w w subject t cnstrant Eq. (9). Sngulartes are determned by analyzng Eq. (8), as fllws. π π he frst term f Eq. (8), csλ =, ndcates that λ =,. Substtutng nt Eq. () results n tw sngulartes q =,. Fr sn( d + d sn λ ) =, sn( q ) =,.e., tw sngulartes π π q =, π. Fr csλ =, λ =,,.e., tw addtnal sngulartes q = 6,. π π Fr csλ = λ =,,.e., tw sngulartes q =, 7. Fnally fr a cs( d + d sn λ ) dw + a =, csq = wll exst f and nly f a < d w.. d w Nte that nly sngulartes that are cnsstent wth the cnstrants are taken. he sngularty q = π s nt cnsstent wth the cnstrants (des nt satsfy Eq. (6)) thus t s nt cnsdered. he ttal number f sngulartes s 7. Surfaces are parametrzed by substtutng each sngularty nt Eq. (). Fr example, the surface x due t sngularty q = s readly determned. cs q[ 5. dw + a] x ( q = ) = sn q[ 5. dw + a], q 7, 6 q (9) 866. dw + q and smlarly, 8

9 cs q[ 5. dw + a] x ( q = 6) = sn q[ 5. dw + a] where, q 7 and q " () 866. dw + q and fr surface x 5, the parametrzed surface s cs q[cs qdw + a] x 5 ( q = ) = sn q[cs qdw + a] q 7, 6 q () snqd w Fgure a depcts each surface generated by the set f sngulartes. he unn f these surfaces envelps the accessble set als shwn n Fg a. Fgure b s a crss sectn f the surfaces. rus x (q =) x 6, x 7 x 6 x x cylndrcal surface x (q =) cylndrcal surface x (q =) x x x 7 d w sn 6 rus x 5 (q =) x 5 (a) cylndrcal surface x (q =-6) Fgure (a) A sectn f the wrst accessble utput set (b) A crss sectn f the wrst accessble utput set Nte that surfaces x 6 ( q = ) and x 7 ( q = 7 ) are planar surfaces. llustrate the ntersectn f surfaces t determne hgher rder sngulartes, cnsder the ntersectn between the cylndrcal surface x ( q = 6 ) and the trus x 5 ( q = ). he ntersectn curve between the tw surfaces can be cmputed by slvng the equatn 5 x x = () carryng ut the algebra, cs q = 5. () fr q = 6, and q =, the curve s a crcle wth crdnates (. 5dw + a)csq (. 5dw + a)snq where q 7 () (b) 9

10 Smlarly, the secnd curve where q = 6, and q = d w sn 6, the curve s a crcle (. 5dw + a)csq (. 5dw + a)snq where q 7 (5) d w sn6 he frst set ( q = 6, and q = ) are sngulartes smlar t thse resultng frm Eq. (7). he secnd set ( q = 6, and q = d w sn 6 ) s a hgher rder sngularty set that has the effect f subdvdng the surfaces nt subsurfaces Ψ. Fgure depcts x havng tw subsurfaces: Ψ (shwn dtted) and Ψ (shwn sld). hs means that x s segmented t subsurface Ψ n the nterval q [ d w sn 6 ], and subsurface Ψ n the nterval q d w sn 6. [ ] Ψ Ψ 9 Ψ Ψ Ψ 8 Ψ 5 Ψ 6 Ψ Fgure A crss-sectn f subsurfaces f the wrst accessble utput set Usng ths methd f ntersectng surfaces t fnd hgher rder sngulartes, the 7 surfaces are dvded nt subsurfaces depcted n Fg.. Smlarly, x has tw subsurfaces: Ψ n the nterval q [ 6 ], and Ψ n the nterval q [ ] subsurfaces: Ψ 5 n the nterval q [ 6 ], Ψ 6 n the nterval q [ 6] Ψ 7. Surface x 5 has three, and Ψ 7 n the nterval q [ 6 ], whle the remander f the surfaces are nt subdvded (e.g., Ψ 8 = x, Ψ 9 = x ). determne whether each subsurface s a bundary r nternal subsurface t the wrst accessble utput set, the perturbatn methd (Eq. ) s perfrmed. Fr example, cnsder a pnt n Ψ n the md range f ts nterval such that mn q = q + q = + 7 = 5, q mn = ( q + q ) = ( + ) = 6, and the ( ) ( )

11 thrd cmpnent s the sngularty at q =. hus the pnt n the surface s q [ ] = 5 6 he unt nrmal t Ψ usng Eq. (8) s cs q (cs ) + cs cs q dw q qdwa n $ = sn q(cs q) dw + sn q csqdwa (6) snq csq d + snq d a w w he unt nrmal at the pnt q n the subsurface Ψ s evaluated [ ] nq $( ) = Usng Eq. (), the subsurface Ψ s an nternal surface f and nly f bth perturbatns ( t =±. ) f q have slutns f the augmented matrx f Eq. () and Eq. (). hat s cs q (cs ) ( ) $ qdw + a Ψx q tn x sn q (cs q d a ) ( ) tn$ w + Ψy q y sn qd q ( ) tn$ w + Ψz q z q snλ = (7) π π q snλ π π q sn λ 6 Fr t =. there exsts a slutn t Eq. (7) such that q = [ ]. Whle fr t =+. n slutn can be fund. hus Ψ s a bundary surface f the wrst accessble utput set and s shwn n Fg. 5b. Ψ Ψ 9 Ψ 8 Ψ 5 Ψ (a) Ψ 7 Fgure 5 (a) Bundary subsurfaces f the wrst accessble utput set (b) he wrst accessble utput set (b)

12 Fr subsurface Ψ, the pnt n the md-range f the nequalty cnstrants s q = nq $( ) = , [ 5 ]. he nrmal t ths surface at q s [ ]. ) there exsts a slutn t Eq. (7) such that q = [ ] and fr ( t =+. hus subsurface Ψ s an nternal subsurface. Wth knwledge f subsurfaces, the bundary f the wrst accessble utput set s determned (depcted n Fg. 5b). he abve methd was used t determne the accessble utput set fr a number f manpulatr cnfguratns. Fgure 6a,b,c,d,e, and f depct accessble utput sets fr a varety f cmbnatn f revlute (R) and prsmatc (P) jnts. (a) (b) (a) (b) (a) Fgure 6 wrst accessble utput set (a) RRR, (b) RPR, (c) RRP, (d) RPR, (e) PRP, and (f) RPR 5 ILLUSRAING DEERMINAION OF SERVICE REGIONS llustrate the abve frmulatn, cnsder the 6 DOF manpulatr depcted n Fg. 7. (b)

13 q W planar surface Ψ servce sphere lcated at p Ψ planar surface q cylndrcal surface Ψ q (a) Fgure 7 (a) knematc skeletn f a sx axs manpulatr, (b) servce sphere lcated at a target p and ntersectng the wrst accessble utput set he frst three jnts f the manpulatr have nequalty cnstrants q q 8 q he bundary f the wrst accessble utput set f ths manpulatr has been studed and s depcted n Fg. 7b. At a pnt p = [ ] n the accessble utput set, the servce sphere s lcated. determne the servce regns f ths manpulatr, and hence the dexterty charts, the ntersectn between each subsurface and the servce sphere s carred ut. Fr example, cnsder the curve between subsurface Ψ and the servce sphere. he nrmal t Ψ usng Eq. (8) s calculated n $ = [ csq sn q ] (8) and Eq. () may be wrtten as sn q hsn scst csq hsn ssnt q css Λ = = 5 5 (9) q sn λ π q 5. πsnλ q snλ Slvng the abve augmented matrx and prjectng the slutn nt the uv-plane usng Eq. (8), t s nw pssble t vsualze the bundary f the servce regn as depcted n Fg. 8. (b)

14 5.7 Fgure 8 Bundary f a regn due t subsurface Ψ determne whether the servce regn s enclsed by the bundary, the augmented matrx Λ s slved such that hsn scst wcsq hsn ssnt wsn q Λ = = (5) hcst q snλ he slutn t Λ results n tw pnts s and s. he slutn t xq ( q) s = and xq ( q) s = s cmputed t determne the generalzed crdnates q = [ 7 ] and q = [ 5 ] respectvely. Only s satsfes the lcal cnstrant. In addtn, s prjected nt the uv-plane exsts nsde the ntersectn curve, whch ndcates that the regn due t Ψ shaded regn depcted n Fg. 9. s the 5.7 u Fgure 9 Servce regn due t subsurface Ψ Smlarly, the prcedure s repeated fr all subsurfaces ntersectng the servce sphere. he prjectn f the ntersectn curve f Ψ wth the servce sphere and ts regn are depcted n Fg. a. he prjectn f the ntersectn curve Ψ and ts regn are depcted n Fg. b. v

15 u u (a) v Fgure (a) Regn due t subsurface Ψ, (b) Regn due t subsurface Ψ determne the servce regn due t the wrst accessble utput set, the regns are supermpsed (Fg. 9, Fg. a, and Fg. b). he resultant servce regn s depcted n Fg.. (b) v 5.7 u Servce Regn Fgure Servce regn at target p due t the wrst accessble utput set 6 SERVICE REGIONS OF HE SPHERICAL JOIN In rder t cmplete the determnatn f dexterty charts, the servce regns due t the jnt lmts f the sphercal jnt are cmputed. In rder t mtvate the dscussn, cnsder a typcal sphercal wrst mdeled as three revlute jnts ntersectng at a sngle pnt W (Peper 968). he pnt p may sweep regns n the surface f the sphere centered at W. Alternatvely, we have defned the servce sphere t be f radus h lcated at p, thus the wrst pnt may assume lcatn n regns n the servce sphere. hese regns are bund by arcs cnnectng the surface pnts W j. he crdnates f the wrst pnts W j are determned by substtutng mum and mnmum jnt lmts. At a target pnt p, the mechansm s assembled and the cnstrant equatn s p 6 R6 xe p6 = (5) where 6 x E s the vectr descrbng a pnt n the end-effectr wth respect t lnk 6. he sphercal wrst s cnstraned as mn q q q (5a) mn mn v q q q (5b) q q q (5c) 5

16 At a target pnt p, and at any cmbnatn f the abve jnt lmts, the crdnates f the wrst pnt s well defned fr eght cnfguratns (eght cmbnatns f q, q5, q6). cmpute the crdnates f the wrst pnt, the fllwng equatn s used. W= 6 R ( R6 xe + p6 ) p (5) Each par f wrst pnts s cnnected va an arc. be able t prject the arc cnnectng tw wrst pnts W and W nt the uv-plane as well, a lcal crdnate system s frmed at p wth ne f ts axes alng the vectr frm p t W such that v = W p (5) and the perpendcular t the plane f the arc s v = v ( W p) (55) Fnally the thrd rthgnal vectr t bth v and v s v = v v (56) he rtatn matrx relatng the new crdnate system t the glbal reference frame s R B [ v v v] (57) Wrst pnts n the arc can be generated alng the arc such that hcsβ A W= R B hsn β + p (58) where the angle β s an ncremental change such that β = β + ( Angle( v, v )). he bundary curve s then prjected usng the peratr L (Eq. 9) such that A A L( W (,) s t ) = W (,) u v (59) llustrate, cnsder the sphercal wrst f the PUMA arm wth the fllwng crdnate parameters. able Denavt Harteneberg parameters f the PUMA arm Jnt θ α a d Jnt lmts q 5 9 q q 66 It s requred t determne the servce regn at a target pnt p = [ ] he jnt varables ( q, q, and q) are calculated usng Eq. (5), and the crrespndng wrst pnts (Eq. 5) are presented n table. able Wrst pnts cmputed due t jnt lmts Input q, q5, q6 q q q Wrst pnt q =, q5 =.5 67 W = [ ] q = q5 = + W = , [ ] 6 6

17 q q =, q5 = W [... ] = =, q5 = W [... ] = A trad s created at p such that the rtatn matrx (Eq. 57) s R B = and the angle between v and v s cmputed ( Angle( v, v) = 79. ). he arc s then prjected nt the uv-plane usng Eq. (59), where the servce regn s depcted n Fg.. v 5.98 w w n-servce regns 5 w w Fgure Servce regn due t the sphercal jnt 7. CONCLUSION In ths paper, a methd fr evaluatng dexterty f rbtc manpulatrs s presented. Usng ths analytc methd, the knematcs f the manpulatr are segmented nt tw parts. he wrst accessble utput set s studed and analytcally determned usng the Jacban f the manpulatr. Sngulartes are cmputed and substtuted nt the cnstrant equatn t parametrze surfaces. A dffculty was encuntered n cmputng hgher rder sngulartes. Often tmes the cmputer prgram culd nt cmpute all sngulartes. he dffculty s currently beng addressed. At an peratng pnt n the wrkspace, the servce sphere s used t defne servce regns. he ntersectn curves resultng frm the ntersectn f the bundary surfaces f the wrst accessble utput set and the servce sphere are prjected nt dexterty charts. Dexterty charts are ntrduced n ths paper t ad n vsualzng manpulatr rentablty. Servce regns due t a sphercal wrst are als determned by prjectng the arcs cnnectng the wrst pnts due t sphercal jnt lmts. Prjectng the servce regns nt dexterty charts prvde a cmplete descrptn f the functnalty f a manpulatr at a target. 8. REFERENCES u 7

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