SERIAL 5DOF MANIPULATORS: WORKSPACE, VOID, AND VOLUME DETERMINATION

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1 Prceedngs f DEC99: 999 ASME Desgn Engneerng echncal Cnferences September 2 5, 999, Las Vegas, Nevada DEC99/DAC-868 SERIAL 5DOF MANIPULAORS: WORKSPACE, VOID, AND VOLUME DEERMINAION K Abdel-Malek Department f Mechancal Engneerng, he Unversty f Iwa, Iwa Cty, IA el. (9) amalek@engneerng.uwa.edu Harn-Ju Yeh Mcrtek, Inc. awan hjyeh@mcrtek.cm.tw Nada Kharallah Faculty f Engneerng and Archtecture Amercan Unversty f Berut Berut, Lebann malak@cybera.net.lb ABSRAC Algrthms fr dentfyng clsed frm surface patches n the bundary f 5DOF manpulatr wrkspaces are develped and llustrated. Numercal algrthms fr the determnatn f three- and fur-dof manpulatr wrkspaces are avalable, but frmulatns fr determnng euatns f surface patches bundng the wrkspace f fve-dof manpulatrs were never presented. In ths wrk, cnstant sngular sets n terms f the generalzed varables are determned. When substtuted nt the vectr functn yeld hyperenttes that exst nternal and external t the wrkspace envelpe. he appearance f surfaces parametrzed n three varables wthn the wrkspace reures further analyss pertanng t a cupled sngular behavr and s als addressed. Prevus results pertanng t bfurcatn pnts that were unexplaned are nw addressed and clarfed. Numerus examples are presented. I. INRODUCION Numercal methds fr determnng bundares f wrkspaces f mechansms and manpulatrs have been develped by a number f authrs n recent years. hese studes have als extended t the feld f cmputer-aded desgn (Abdel-Malek and Yeh 997a) where the cmputatn f the swept vlume s mprtant t sld mdelng. Smlarly, n manufacturng, the NC verfcatn f machnng prcesses reures the cmputatn f the wrkspace generated by the tl path n the wrkpece. Mre recently, Cecarell (995) used an algebrac frmulatn f a wrkspace bundary t frmulate desgn euatns f three-revlute (R) jnted manpulatrs and 4R manpulatrs (Cecarell and Vncuerra 995). he beneft f ths methd s shwn n the ablty t determne hles and vds n the accessble utput set. he same example treated by Cecarell wll be addressed here fr valdatn purpses. Other wrks that have dealt wth manpulatr wrkspace are reprted by Kumar (985), and Emrs (99) and Zhang et al. (996). Recently, Haug et al. (996) frmulated numercal crtera t fnd the wrkspace (called the accessble utput set) f a general mult-degree-f-freedm system usng a cntnuatn methd t trace bundary curves sutable fr the study f bth pen- and clsed-lp manpulatrs. he ntal crtera fr ths cmputatnal methd were presented by Haug et al. (992) and Wang and Wu (99). he algrthm cmputes tangent vectrs at bfurcatn pnts f cntnuatn curves that defne the bundary f manpulatr wrkspaces. A crsssectn f the wrkspace s btaned and bundary cntnuatn curves are traced. he methd was demnstrated fr a clsed-lp mechansm called the Stewart Platfrm (Luh et al. 996). hese curves are then assembled nt a mesh that s envelped by apprprate surface patches. hs methd has prved vald fr determnng the general shape f the accessble utput set. he man dffculty s n determnng the Cpyrght 999 by ASME

2 status f a sngularty at pnts alng cntnuatn curves. Althugh sngular behavr ccurrng at pnts alng the curves s dentfed, ths methd s cmpletely numercal and nly traces bundary curves. It des nt result n analytcal surfaces bundng the accessble utput set. A cmparsn between the numercal apprach ntrduced by Haug and clleagues and the apprach f the methd presented here fr lwer DOF was dscussed by Abdel-Malek et al. (997). he am f ths paper s t generalze the frmulatn fr determnng the wrkspace f the -DOF system presented elsewhere (Abdel-Malek and Yeh 997b), t a general 5-DOF system and t explan prevus results that remaned unexplaned. A frmulatn pertanng t a rank-defcency crtera f the pstn Jacban, ncludng the effect f jnt lmts, s presented n sectn 2. Frst- and secnd-rder crtera are ntrduced t delmt sngular sets. hese sets when substtuted nt the pstn vectr f the end-effectr yeld parametrc euatns f surfaces. he appearance f surfaces parametrzed n three generalzed crdnates s addressed thrugh secnd-rder analyss n Sectn. he frmulatn s demnstrated thrugh the analyss f spatal 5- DOF examples n Sectn 4. II. FORMULAION Defne R n as the vectr f n-generalzed crdnates characterzng a manpulatr cnfguratn. he vectr functn generated by a pnt n the end-effectr f a seral arm wrtten as a multplcatn f rtatn matrces and pstn vectrs s expressed by f ( )" = n j= x( ) = ( ) [ R ] p ( )# = j g j () = j= h! $ where x( ) =,..., n, and bth p j and R j are defned usng the Denavt-Hartenberg representatn methd (Denavt and Hartenberg 955 and Fu et al. 987) such that csθ csα snθ snα snθ R = snθ csα csθ snα csθ 0 snα csα (2) and ( ) p = acsθ asnθ d () where θ s the jnt angle frm x axs t the x axs, d s the shrtest dstance between x and x axes, a s the ffset dstance between z and z axes, and α s the ffset angle frm z and z axes. he generalzed varable s = d f the jnt s prsmatc and = θ f the jnt s = x y z and [ ] revlute. he vectr functn x( ) characterzes the set f all pnts nsde and n the bundary f the wrkspace. he am f ths wrk s t determne the bundary t ths set and t analytcally represent t. At a specfed pstn n space, E. can be wrtten as a cnstrant functn W( ) = f ( ) x g( ) y h( ) z = 0 (4) Jnt lmts mpsed n terms f neualty cnstrants n the L U frm f, where =,... n are transfrmed nt euatns (Haug et al. 996) by ntrducng a new set f generalzed crdnates l = [ λ,..., λn ] such that L U U L = ( + ) 2 + ( ) 2 sn λ =,..., n (5) hese generalzed crdnates λ are called slack varables n the feld f ptmzatn. In rder t nclude the effect f jnt lmts, t s prpsed t augment the cnstrant euatn W( ) wth the parametrzed neualty cnstrants such that f ( ) x ( ) H ( ) = g y = 0 h( ) z a bsn λ =,... n (6) where = [ l ] s the vectr f all generalzed crdnates. Nte that althugh n new varables (λ ) have been added, n euatns have als been added t the cnstrant vectr functn wthut lsng the dmensnalty f the prblem. he Jacban f the cnstrant functn H ( ) at a pnt 0 s the ( + n) 2 n matrx H( ) = H (7a) where the subscrpt dentes a dervatve. Wth the mdfed frmulatn ncludng the parametrzed neualty cnstrants, the Jacban s expanded as x 0 " H = (7b)! I l$# where the ntatn f dentes the partal dervatve f f wth respect t, and and [ x ] =! f f... f 2 g g... g 2 h h... h 2 n n n " $ # (8) 2 Cpyrght 999 by ASME

3 U L -(( - ) 2)csλ 0 U L 0 -(( 2-2 ) 2)csλ 2 [ l ] = 0 0! " U L... -(( - ) 2)cs $ # n n λ n s a dagnal blck matrx, 0 s a ( n ) zer matrx, and I s the dentty matrx. he bundary t the wrkspace W (wrkspace envelpe) s a subset f the wrkspace at whch the Jacban f the cnstrant functn f E. 7 s rw rank defcent (Kumar and Waldrn 98, Haug et al. 996, Abdel- Malek and Yeh 997b);.e., W { Rank H( ) < k, fr sme wth H( )= 0} (9) where k s at least ( + n ). Fr an n -DOF system, the Jacban H( ) s rw-rank defcent f and nly f ne f the fllwng cndtns are satsfed. N Jnts Reach ther Lmts If n jnts have reached ther lmts, the dagnal sub-matrx [ λ ] s full rw rank. herefre, the nly pssblty fr [ H ] t be rw-rank defcent s when the blck matrx [ x ] s rw rank defcent. Defne tw ndependent subvectrs f as p and u, as If u = p u,where p, u and p u= φ R m then p R ( n m ). (0) ype I sngularty set can be defned as S ( ) Jp : Rank [ x ] <, fr sme cnstant subset f L () where p s wthn the specfed jnt lmt cnstrants. he m n blck matrx [ ] s rw rank defcent at least ne, ( ) x where m = fr spatal and m = 2 fr planar manpulatrs. he rank f the ( m n) matrx s defned t be the rder f the largest nn-sngular suare sub-matrx whch can be frmed by selectng rws and clumns f the upper crner matrx [ x ]. In rder t make the sub-matrx [ x ] rank defcent f rder (d) where d = abs( m n), t s necessary t determne all sub-jacbans. Fr a rank-defcency (d), the largest suare sub matrx cannt have a larger dmensn than b= max( n d, m d). herefre, there must be m! [( m b)! b!] pssble ndependent rws that can be cnsdered n a sngle suare sub-matrx. Smlarly, there are n! [( n b)! b!] pssble cmbnatns f clumns. Hence, there exsts n! m! η = (2) ( n b)! b! ( m b)! b! sub-jacbans. Euatng the determnants t zer yelds η - number f euatns t be smultaneusly slved. Slutns t the η euatns are the sngular sets f ype I. hs crtern s used t btan suare sub-jacbans. Slutns f the resultng η euatns f all cndtns are sets f cnstant parameters dented by p and characterzed by det( h h jh k ) p = : = 0, fr, j, k =,..., n and j k det( h h jh k ) η = 2,,...,β () where h dentes a clumn f [ x ] = hk,..., hm. Fr each p, remanng varables are u. Sme Jnts Reach her Lmts When certan jnts reach ther lmts, e.g., lmt lmt lmt, j, k =, j, k, the crrespndng dagnal elements n the matrx [ λ ] wll be eual t zer. Fr example, f = mn (r max ), the dagnal element f [ l] wll be zer (.e., b csλ s zer fr ether =,..., 2n then has reached a lmt). herefre, the crrespndng [ W(, )] s subjected t the rankdefcency crtern. x... x x x... x " j k n [ H ]~ # (4) ! Slvng the rw rank defcency cndtn fr E. 7 s euvalent t slvng the rank defcency fr x [ x, x, x ] j, wth k lmt lmt lmt =, =, = (5) j j k k $ Cpyrght 999 by ASME

4 where the ntatn f represents the exclusn f the rght matrx frm the left matrx and t represents the sub-matrx f [x ] when jnt cnstrants are at ther lmts. Frm the fregng bservatn, the secnd type f sngular sets are frmulated. Defne a new vectr lmt lmt lmt lmt =,, whch s a sub-vectr f j k where dm 2 7 lmt ( n ) (6) Fr the case f dm 2 lmt 7= ( n 2 ), t s nted that the slutn f E. 7 s readly avalable as wll be dscussed n the fllwng paragraphs. he jnt crdnates can be parttned as = w, lmt, and w lmt = φ (7) hen, f [x w ( w, lmt )] s rw rank defcent, the sub- Jacban [ ξ ] s als rank defcent. Let the slutn fr ths cndtn be dented by $p, whch s a cnstant sub-vectr f w, and w = u, p$. he type II sngularty set s defned as ( 2) lmt lmt S Ÿ [ $ Jp = p ]: Rank[ ξ ( w, )] <, (8) lmt fr sme p$ ³ w, dm( ) ˆ( n -) B As fr the case f dm 2 lmt 7= ( n 2 ),.e., nly tw jnt varables are allwed t vary n ther ranges, dm( w ) = 2, and the sub-matrx [x w ] wll be f dmensn ( 2),.e., already rw rank defcent. Fr ths case, there are n slutns, but the type-iii sngularty s S ( ) ( n 2 p R ) : p lmt = [ lmt, lmt,...] = j B (9) Hyperenttes Substtutng the set p characterzed by the sets btaned n Es., 8, and 9, nt the accessble set x( ) yelds a hyperentty parametrzed n terms f the remander varables as () % where p S f E. () lmt ( ) x( p, u ) p [ p$ = =, ] 2 &K where S f E. (8) (20) lmt ( ) where p = S f E. (9) 'K subject t the neualty cnstrants f the generalzed crdnates a b sn λ. he vectr functn characterzed by E. 20 s a hypersurface that s parametrzed ether n tw parameters u = k r three parameters l s = k l m such that % dm( p) =, x( u) x( ) = & dm( p ) = 2, x( s ) ' ( ) (2) Hypersurfaces that are parametrzed n tw varables ( x( u )) have ther lmts prescrbed by he neualty cnstrants and can readly be depcted. Hwever, hypersurfaces parametrzed n three varables ( x( s ) ) represent cuplng between the jnts and reure further analyss. III. COUPLED SINGULAR BEHAVIOR Fr a 5-R manpulatr and f tw jnts are lcked, the remanng -DOF s end-effectr may have a vlume as ts wrkspace. Hwever, n sme cases and because f the cupled sngular behavr, ths wrkspace s nly a surface parametrzed n three varables ndcatng a cuplng between (e.g., a Gmbal). It was ntced that when ths case ccurs, at least tw jnts f the reduced rder manpulatr are cupled, and at least ne cnfguratn has has a rank defcency f tw. Snce the rank-defcency f ths surface s f rder ne, t yelds tw cnstant generalzed crdnates,.e., neparameter gemetrc enttes. Curves nstead f surfaces that dentfy the bundary wll be determned. Fr ths case, x ( s ): s R R, and the Jacban f the hypersurface s x e = xs s (22) e where ε s the crrespndng vectr f slack varables lmt e = λ λ λ. Fr a jnt at ts lmt ( ) l k m, the secnd blck matrx f E. 22 s rank defcent. An elementary matrx f rw peratns E appled t s ε yelds a rw echeln frm such that E sε = ERE (2) 0 0" where E = 0 0! 0 0 0$ #, 0 0" E 2 = ", and E = 0 0 ;! 0 0 0$ #! 0 0 0$ # where the subscrpt dentes the jnt number, and E RE s a rw echeln frm. hs same matrx appled t ξ s yelds E x = L 0 (24) s where L = L L2 L (25) where dm( Λ ) = ( 2 ) and dm( Λ j ) = ( 2 ) fr j = 2,,. Applyng E. 2 t Λ wth m =, n = 2 varables, and a rank defcency f d = yelds η = 4 Cpyrght 999 by ASME

5 euatns t be slved smultaneusly. Slutns t the three euatns are sngular sets dented by γ such that % D " ( γ = &K D γ γ )K D # = 2 0, fr s, β = φ (26) 'K! $ where D = LL 2, D 2 =ΛΛ 2, and D =ΛΛ subject t the jnt cnstrants and where γ s a subset f s such that s = γ β, where β s the remanng varable such that x ( g, β ) = G ( β ) (27) where Γ ( β ) s a parametrc curve. Dm( G( β )) = ( ) represents bundary curves t the hypersurface x( s ). he wrkspace s characterzed by the hypersurfaces x( u) and x( s ) subject t Γ( β ). he determnatn f ts bundary s addressed n the fllwng sectn. IV. PERURBAION MEHOD O DEERMINE HE BOUNDARY Snce hypersurfaces extend nternal and external t the wrkspace envelpe, t s necessary t dentfy regns (surface patches) f these hypersurfaces that are n the bundary, whether the external bundary r a vd. he curves resultng frm the ntersectn f hypersurfaces dvde each surface nt many regns. An algrthm develped by the authrs (Abdel-Malek and Yeh 996, 997c) s mplemented t dentfy these regns whereby curves f ntersectn are traced. In fact, these curves represent sngular trajectres f the end-effectr at whch the manpulatr lses at least tw degrees f freedm (cupled sngulartes). he ntersectn f tw sngular curves dentfy the s-called bfurcatn pnt. determne f a regn s nternal, a perturbatn methd s emplyed. Cnsder a pnt c n a hypersurface but nt L c U n the bundary,.e., < <. At x( c ), the velcty f the end-effectr s gven by [ & x] = [ x ][ ][ l & ] (28) l On a sngular surface, the term [ x ] K l, l s rankdefcent. Multplyng bth sdes f E. 28 by N (the bass f the null space f x ) yelds l N & c & x = N x ( ) λ = 0 (29) l Snce N s a cnstant vectr at c, the left hand sde f E. 29 characterzes the euatn f a plane n R as Nx & + Ny 2 & + Nz & = 0 where N = N N2 N s ndeed a vectr nrmal t the tangent plane f the sngular surface at c. Indeed, fr any value f &, all resultng velcty vectrs x & wll le n a plane whch has N as ts nrmal. w bservatns can be made: () he bass f the null space f x l s the vectr nrmal t the sngular surface at c. () Fr any gven jnt velcty vectr &, the velcty f the end-effectr s ether tangent t the sngular surface r zer,.e., the nrmal cmpnent f the end-effectr velcty s always zer,.e., v n = N x& = N & x = 0 (ths result s reprted by Haug et al. 996 usng a dfferent apprach). Fr ths nrmal, tw pnts alng the nrmal n each sde f the surface can be fund as p2, c x = x( ) ± N ϑ (0) where ϑ s a small varatn frm x( c ) (e.g., ϑ = 0. ). If bth pnts ξ p and ξ p 2 satsfy the cnstrant euatn, then the regn fr whch x( c ) belngs s nternal t the bundary. Perfrmng ths test n all regns yelds bundary surface patches defned by the euatns f x( c ) and bund by the numercal curves. V. BIFURCAION POINS, VOID DEERMINAION, AND COMPUING HE VOLUME A crss sectn f the wrkspace at any elevatn can be cmputed by numercally ntersectng each hypersurface wth a plane. he same algrthm used abve t dentfy regns s emplyed t trace ntersectn curves. Bfurcatn pnts dentfed by Haug et al. (996) can nw be explaned as the ntersectn f at least tw hypersurfaces. Indeed, cntnuatn lnes traced n that wrk are n effect the curves f ntersectn between tw hypersurfaces. If three hypersurfaces ntersect at a pnt, the rank defcency f the Jacban at ths pnt s f rder tw. he cmputatn f tangents at these pnts becmes mre dffcult as the number f surfaces augments. Vd determnatn usng the presented methd becmes a smple task as the perturbatn methd perfrmed n an nternal regn f a hypersurface reveals whether ths regn s a bundary. If t s a bundary whle nsde the external bundary, then ths regn envelps a vd. An example presented belw llustrates the determnatn f vds. Because hypersurfaces are n clsed frm, ther ntersectn wth a cuttng plane can be cmputed at any elevatn. A number f cuttng planes are ntrduced, and ther ntersectn wth the hypersurfaces cmputed. he vlume s cmputed by 5 Cpyrght 999 by ASME

6 determnng the area enclsed by the bundary and dscretely ntegratng ver the crss sectnal areas. VI. EXAMPLES EXAMPLE : A 5-DOF RRPRR MANIPULAOR Cnsder the 5-DOF manpulatr shwn n Fg.. = π 4 = 5 and = π 2@ ; = 0and =π 2@,p 2 :; = 5and 5 =π / 2@ p 9 2 π and 4 : = / 4, = 0, = π / 2 p : /,, / p 5 : / Nte that hypersurfaces due t p and p 2 are parametrzed n three varables (, 2, and 4) ndcatng cupled sngular behavr. A crss sectn f the wrkspace f the 5-DOF manpulatr depctng all sngular surfaces s shwn n Fg. 2. y 0 p x 0 x 6 z 6 Fg. A 5-DOF RRPRR manpulatr Jnts are cnstraned as 0 2π, π / 4 2 5π / 4, 0 5, 0 4 2π, and π / 2 5 π / 2. Usng Denavt-Hartenberg, the pstn vectr f pnt p n Fg. s determned as x =! -5sn5cssn 2sn 4-5sn5sncs4+ - 5sn 5snsn 2sn4+ 5sn 5cscs4+ 5cs sn sn + 5sn cs cs cs cs + 5cs cs + cs cs 5sncs2cs5+ 5sn cs2+ sn cs2 5sn + sn he crtera ntrduced n Sec. (2) s appled t E. () yeldng the fllwng sngular sets. p 2 = π / 2, 4 = 0, and 5 = π / 2@ p 2 :; = 2 π / 4, = 4 0, and = 5 0@, : = π / 4, = π / 2, and = π / 2 = π 4 = π 2 and = π 2@ ; = 0 = 0 = 0@ p p p 5 :, 4, and 5 p 6 : = 2 π / 2, = 0, and = 4 0 : /, /, :; = 5, = 0, and = 0@ :; = π / 2, = 5, and = 0@, p p " $ # Fg. 2 A crss-sectn f the wrkspace f the RRPRR Manpulatr Example 2: 4-DOF Manpulatr-Calculatng the Wrkspace Vlume Cnsder the 4-DOF manpulatr shwn n Fg.. z z z 2 x x 2 x y x () 5 Fg. here are p ; =... 6 sngular sets. Crss sectns at dfferent z-values are shwn n Fg. 4. z 5 5 z 4 6 Cpyrght 999 by ASME

7 Fg. 4. (a) at z = 2 (b) at z = 0 (c) at z = 0 (f) at z = 28 he area fr each crss-sectn s cmputed and entered n able. z 2 ABLE he cmputed area fr each crss sectn z A z A z A z A x Fg. 5. A 5-DOF RPPRR manpulatr he perturbatn methd appled t each surface regn dentfes the bundary as shwn n Fg. 6, where an nternal bundary t a vd s als dentfed..0 Z he crss-sectns are cnnected usng a mesh based n a trapezdal algrthm and the vlume s cmputed t Vtrapez = u, where u s a unt, whle based n splne algrthm, the vlume s cmputed t Vsplne = 7249 u EXAMPLE : A 5-DOF RPPRR wth Vds Cnsder the 5-DOF manpulatr shwn n Fg. 5, havng jnt lmts as 5 28, 0 0, 0 45., 0 28., and X 7 Cpyrght 999 by ASME

8 .0 Z 2.0 Vd Fg. 6. (a) Crss-sectn f the RPPRR manpulatr at =π 2 (b) Vd determnatn Example 4: he 5R General Manpulatr Cnsder the 5R seral arm shwn n Fg. 8 where α = π, d = where =,..., 5, a =, a 2 = 4, a =, a 4 = 4 and a 5 =. he crss sectn at z = 00. s shwn n Fg. 9. z x 0 z d a y 0 d 2 x z 2 a 2 2 d x 2 a z x a 4 z 4 4 x 4 Fg. 8 he general 5-revlute Manpulatr 5 X x 5 z 5 Fg. 9 A crss sectn f the general 5R manpulatr VII. CONCLUSIONS A general frmulatn fr determnng bundary surface patches n clsed frm t 5DOF manpulatrs has been presented. he wrkspace cnstrant functn was frmulated n terms f generalzed crdnates ncludng neualty cnstrants mpsed n each jnt. It was shwn that Jacban rank-defcency cndtns usually appled n rbtcs analyss t determne degenerate cndtns, are emplyed here t generate cnstant sngular sets and t dentfy cupled sngular behavr. It was als shwn that hypersurfaces n parametrc frm based n these sngular sets exst nternal, external, r may extend frm the nternal t the external f the wrkspace envelpe. It s emphaszed that these hypersurfaces are characterzed by parametrc euatns where the parameters are jnt varables. It was bserved that thse surfaces that are parametrzed n three varables exhbt sngular behavr that can be used t determne ther bundares n terms f parametrc space curves. Cupled sngular behavr fr hypersurfaces parametrzed n three varables was addressed and has been shwn t prvde slutns t a class f prblems dentcal t the case f the Gmbals mechansm. Results pertanng t the determnatn f the wrkspace, envelpe vlume, bfurcatn analyss, and crss sectnal vews f the wrkspace were presented. Valdatn examples, were llustrated t demnstrate the applcablty f the frmulatn t a wde range f prblems. One may cnclude that the slutn t these examples are pssble nly because hyperenttes are dentfed n clsed frm. the authrs belef, these are the nly reprted results that have yelded clsed frm euatns t surface patches f the wrkspace envelpe. VIII. ACKNOWLEDGMENS hs research was funded by the US Army ank Autmtve Research Center (ACOM) thrugh the ARC (Department f 8 Cpyrght 999 by ASME

9 Defense cntract number DAAE07-94-C-R094) at the Iwa Center fr Cmputer-Aded Desgn. IX. REFERENCES Abdel-Malek, K. and Yeh, H.J., 996, Determnng Intersectn Curves Between Surfaces f w Slds, Cmputer Aded Desgn, Vl. 28 (6/7) pp Abdel-Malek, K. and Yeh, H.J., 997a Gemetrc Representatn f the Swept Vlume Usng Jacban Rank-Defcency Cndtns, Cmputer Aded Desgn, v29(6), pp Abdel-Malek, K. and Yeh, H.J., 997b, Analytcal Bundary f the Wrkspace fr General -DOF mechansms, Internatnal Jurnal f Rbtcs Research. v6(2), pp Abdel-Malek, K and Yeh, H J, 997c, On the Determnatn f Startng Pnts fr Parametrc Surface Intersectns, Cmputer-Aded Desgn, Vl. 29() pp Abdel-Malek, K., Adkns, F., Yeh, H.J., and, Haug, E.J, 997, On the Determnatn f Bundares t Manpulatr Wrkspaces, Rbtcs and Cmputer-Integrated Manufacturng, Vl., N., pp.-5. Ceccarell, M. and Vncguerra, A., 995, On the Wrkspace f General 4R Manpulatrs, Internatnal Jurnal f Rbtcs Research, Vl. 4, N. 2, pp Ceccarell, M., 995, A Synthess Algrthm fr hree- Revlute Manpulatrs by Usng an Algebrac Frmulatn f Wrkspace Bundary, ASME J. f Mechancal Desgn, Vl. 7, N. 2(A), pp Denavt, J., and Hartenberg, R.S., 955, A Knematc Ntatn fr Lwer-par Mechansms Based n Matrces, Jurnal f Appled Mechancs, ASME, Vl. 22, pp Emrs, D.M., 99, Wrkspace Analyss f Realstc Elbw and Dual-elbw Rbt, Mechansms and Machne hery, Vl. 28, N., pp Fu, S., Gnzalez, J, and Lee, S., 987, Rbtcs: Cntrl, Sensng, Vsn, And Intellgence. McGraw-Hll, Inc., New Yrk. Haug, E.J., Wang, J.Y., and Wu, J.K., 992, Dextrus Wrkspaces f Manpulatrs: I. Analytcal Crtera, Mech. Struct. and Mach., Vl. 20, N., pp Haug, E.J., Luh, C.M., Adkns, F.A., and Wang, J.Y., 996, Numercal algrthms fr mappng bundares f manpulatr wrkspaces, ASME Jurnal f Mechancal Desgn, 8, pp Kumar, V., 985, Rbt Manpulatrs-Wrkspaces and Gemetrc Dexterty, Masters hess, he Oh State Unversty. Luh, C. M. Adkns, F. A. Haug, E. J. and Qu, C.C., 996, Wrkng Capablty f Stewart Platfrms, ASME Jurnal f Mechancal Desgn, Vl. 8, pp Pennck, G.R. and Kassner, D.J., 99, he Wrkspace f a General Planar hree-degree-f-freedm Platfrm-ype Manpulatr, ASME Jurnal f Mechancal Desgn, Vl. 5, pp Spans, J. and Khl, D., 985, Wrkspace Analyss f Regnal Structures f Manpulatrs, Jurnal f Mechansms, ransmssns, and Autmatn n Desgn, 07: Wang, J.Y. and Wu, J.K., 99, Dexterus Wrkspaces f Manpulatrs, Part 2: Cmputatnal Methds, Mechancs f Structures and Machnes, v2(4), pp Zhang, S.J., Sanger, D.J., and Hward, D., 996, Wrkspaces f a Walkng Machne and ther Graphcal Representatn. Part I: knematc wrkspaces, Rbtca, 4, pp Cpyrght 999 by ASME