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1 Unvert of Wrw nttutonl repotor: h pper mde vlble onlne n ordne wth publher pole. Plee roll down to vew the doument telf. Plee refer to the repotor reord for th tem nd our pol nformton vlble from the repotor home pge for further nformton. o ee the fnl veron of th pper plee vt the publher webte. Ae to the publhed veron m requre ubrpton. Author():. Hung H.. Lu nd D.G. Chetwnd Artle tle: A Method to Formulte Dmenonll Homogeneou obn of Prllel Mnpultor Yer of publton: 2 Ln to publhed rtle: Publher ttement: 2 IEEE. Peronl ue of th mterl permtted. Permon from IEEE mut be obtned for ll other ue n n urrent or future med nludng reprntng/republhng th mterl for dvertng or promotonl purpoe retng new olletve wor for rele or redtrbuton to erver or lt or reue of n oprghted omponent of th wor n other wor. Ctton: Hung. et l. (2). A Method to Formulte Dmenonll Homogeneou obn of Prllel Mnpultor. IEEE rnton on obot Vol. 27 No. pp. 5-56

2 A method to formulte dmenonll homogeneou obn of prllel mnpultor Hto Lu n Hung nd Dere G Chetwnd Abtrt h pper preent generl nd temt pproh for formultng the dmenonll homogeneou obn n mportnt ue for the detert evluton nd dmenonl nthe of f-df (f 6) prllel mnpultor hvng med rottonl nd trnltonl movement pblte. Utlng f ndependent oordnte to derbe the pefed moton tpe of the pltform the f f dmenonll homogeneou obn derved dretl from the generled obn provded tht the mnpultor h onl one tpe of tutor. he ondton number of the new obn then emploed to evlute the detert of two tpl -DF prllel mnpultor n llutrton of the effetvene of th pproh. Inde erm Prllel mnpultor dmenonll homogeneou obn I. INDUCIN he obn onventonll defned the lner mp between tutor rte n the jont pe nd velot twt n the operton pe pl n mportnt role n nemt performne evluton nd optmton of mnpultor hvng dfferent rhteture. For thoe mnpultor hvng oupled trnltonl nd rottonl movement pblte t h been reogned tht the lgebr hrtert uh ondton number mmum/mnmum ngulr vlue et. of the obn vr wth the lng due to the nonten n phl unt of t term. h m ue erou problem [-] when the re emploed the ondtonng nde n nemt performne evluton nd degn optmton. herefore t rtll mportnt to formulte dmenonll homogeneou obn n whh ll entre hve the me phl unt. A revew of the etng lterture how tht two method mght be emploed to formulte dmenonll homogeneou Mnurpt reeved Deember 2. h reerh wor prtll upported b the Ntonl Nturl Sene Foundton of Chn (NSFC) under grnt 555 nd H. Lu wth the Shool of Mehnl Engneerng njn Unvert njn 72 Chn (e-ml: luhttju@hoo.om.n).. Hung wth the Shool of Mehnl Engneerng njn Unvert njn 72 Chn (phone: ; f: ; e-ml: htntju@publ.tpt.tj.n). He lo prtll wth the Unvert of Wrw Coventr CV4 7AL UK. (th@eng.wrw..u) D. G. Chetwnd wth the Shool of Engneerng Unvert of Wrw Coventr CV4 7AL UK. (e-ml: D.G.Chetwnd@wrw..u). obn. ne w the length-bed method tht dopt hrtert/nturl length to normle ll trnltonl element n the obn ubjet to otrop ontrnt [4-]. Degn optmton re then bed on the ondton number of the dmenonll homogeneou obn beng mnmed. h method uted to the degn of n otrop mnpultor beue the hrtert/nturl length vre wth the tem onfgurton. A more prtl pproh to ue the pont-bed method to heve dmenonll homogeneou obn b lnng the tutor rte wth the lner velote t everl pont on the end-effetor provded tht ll tutor re of dentl tpe. eerh long th tr n be tred b to the erl wor [] delng wth the optmum degn problem of plnr nd ptl erl mnpultor. he de w then etended to 6-DF prllel mnpultor reultng n 6 9 dmenonll homogeneou obn [2-4]. More reentl ung -PS or -PS mehnm n emple the pont-bed method w ten further to formulte the f f dmenonll homogeneou obn of f-df prllel mnpultor [5 6]. However th method uffer from rther hev omputtonl burden prtl dervtve hve to be etblhed on e-b-e b. Drwng on the need for nemt performne evluton nd optmton th pper preent generl nd temt omputtonl pproh for obtnng the f f dmenonll homogeneou obn of f-df prllel mnpultor (f 6). he method mplemented n two tep: () formultng the lner mp between the jont rte nd velot twt ung the generled obn [7]; (2) genertng lner mp between the velot twt nd f lner velote t et of eleted pont on the end-effetor provded tht the mnpultor h onl one tpe of tutor. n the b of th new obn ondton number then ued to evlute the detert of two -DF prllel mnpultor hvng oupled trnltonl nd rottonl moton pblte. II. FMULAIN F HE GENEALIZED ACBIAN h eton brefl revew the formulton proe of the generled obn [7] of prllel mnpultor hvng f-df ( 2 f 6). Wthout lo of generlt ume tht the mnpultor ompoed of l ( f l f ) lmb onnetng the pltform wth the be eh eentll ontnng n

3 2 ( 2 l ) -DF jont wth t mot one of them tuted. hu two fmle of prllel mnpultor n be ten nto ount. he frt fml over full prllel mnpultor hvng f ontrned tve lmb ( n 6 for ll lmb). he eond fml ompre thoe hvng f unontrned tve lmb (.e. n 6 for eh of thee f lmb) plu one properl ontrned pve lmb. For onvenene the properl ontrned pve lmb lw degnted b l f. An other prllel mnpultor not belongng to thee two fmle n be delt wth n mnner mlr to tht ued below. n the one hnd the et of ontnuou moton of the pltform form Le group SE() [8]. r SE S r GL 4 () where r the poton vetor of the orgn of the pltform n the fed bod frme nd the orentton mtr of wth repet to the referene frme. At n pont n SE() the Le lgebr of SE() 6-dmentonl vetor pe nd n be epreed α r e α o r (2) where δr the vrton of r nd δα ew-mmetr mtr wth t entre repreentng the ngulr vrton of the pltform. Note tht e() h vrtonl rther thn dfferentl truture of SE() o to enomp broder ene of the frt order nemt n term of velot poe error nd defleton. Note tht e() omorph to 6 v the mppng δα δr t 6 $ δr δα () Sne $ t h form of twt n rew theor nd thu e() referred to the twt pe denoted for mbol onten n the lter ue. n the other hnd the et of wrenhe eerted on the pltform form 6-dmenonl vetor pe W nown the wrenh pe wth $ w f τ beng t element where f nd τ re the ppled fore nd moment eerted on the pltform. ndw re pr of dul pe nown the tngent pe nd otngent pe of SE() repetvel [8] It h been hown [7] tht for n f-df prllel mnpultor n unquel be deompoed nto pr of omplementr ubpe.e. n f dmenonl ubpe nd 6-f dmenonl ubpe oted repetvel wth the permtted nd retrted ntntneou moton of the pltform nd thu nown the twt ubpe of permon nd retrton. Correpondngl W n be deompoed nto pr of omplementr ubpe.e. n f dmenonl ubpe W nd 6-f dmenonl ubpew oted repetvel wth tuton nd ontrnt wrenhe eerted on the pltform b the lmb nd thu nown the wrenh ubpe of tuton nd ontrnt. Note tht $ t nd $ w re epreed n the form of oordnte nd r-oordnte the vrtul wor done b $ w on $ t n be repreented b the nner produt tht equvlent to the reprol produt defned n rew theor [2-22]. δw $ $ = $ $ $ + $ w t w w t t $ $ $ $ $ $ $ $ w t w t w t w t It h been proved [7] tht $ w $ t $ w $ t $ w $ t δ nd $ w $ t δ $ w ( $ w ) doe not do wor on $ t ( $ t ) nd $ w ( $ w ) doe wor on $ t ( $ t ). hu the followng reltonhp hold: Let t j (4) rthogonlt: W W (5) * * Dult: W W (5b) $ ˆ $ ˆ w W ( j 2 n ) nd $ ˆ t j w $ ˆ W ( j 2 6 n ) be the b element of the four vetor ubpe oted wth the th lmb. he twt of the pltform n then be repreented b lner ombnton of the b element of nd ( 2 l )beue ll lmb hre the me pltform $ $ $ $ $ t t t t t n δρ 6 n $ ˆ δρ $ ˆ j t j j t j j j where $ ˆ t j nd δρ j ( $ ˆ t j nd δρ j ) re the j th ( j th ) unt rew of permon (retrton) nd t ntent. In rew theor $ ˆ t j none other thn the unt rew of the j th -DF jont wthn the th lmb hvng onnetvt of n 6 [22]. In velot nl where merel the del moton of the pltform re ondered t gve δρ q nd δρ uh tht j j n j 6 n t j t j t j j j (6) $ t $ t v ω $ q $ ˆ $ ˆ (7) where v nd ω re the lner velot of the referene pont on the pltform nd ngulr velot of the pltform nd q j the rte of the j th -DF jont n the th lmb.

4 v je j v P 4 e 42 v p P e 2 e 4 e p 22 4 e 2 P 2 p P 2 e 2 Pltform P p p P e e Fg.. Frme ettng on the pltform Fg.2. Pont nd e rrngement on tetrhedron Let $ ˆ w g be the unt wrenh of tuton oted wth the tuted jont whh lbelled g n the th ( 2 f ) lmb. Note tht $ ˆ w g dul to $ ˆ t g but orthogonl to t j $ ˆ ( j 2 n j g ) n ordne wth the properte gven n Eq.(5). Smlrl let $ ˆ w be the th unt wrenh of ontrnt n the th lmb. Alo note tht $ ˆ w orthogonl to $ ˆ t j ( j 2 n ). hu tng the nner produt on both de of Eq.(7) wth $ ˆ w g nd $ ˆ w repetvel reult n = where n $ ˆ $ ˆ $ ˆ $ ˆ $ ˆ $ ˆ t $ q (8) ˆ ˆ ˆ $ w g $ w g $ t g ˆ ˆ $ ˆ w g 2 $ w g.2 $ t g $ ˆ ˆ ˆ w g f $ w g f $ t g f w w t w2 w2 t2 $ ˆ $ ˆ $ ˆ w6 n w6 n t6 n l f f f q q = q q q q 2 l g g 2 2 g f f 6 n 6 mtr nown the generled obn of f-df prllel mnpultor whh h broder ene thn the overll obn developed b other men n the etng lterture [2]. Wth the d of Eq.(8) the formulton of f f dmenonll homogeneou obn of prllel mnpultor wll be dued n net eton. III. FMULAIN F HE f HMGENEUS ACBIAN f DIMENSINALLY heoretll the trnlton of three non-ollner pont on rgd bod re uffent to unquel dentf the moton of f the bod n term of trnlton nd rotton. Hene on the b of the generled obn gven n Eq.(8) omputtonl heme to formulte the f f dmenonll homogeneou obn of f-df prllel mnpultor propoed follow. Frtl ple referene frme t pont on the be nd bod fed frme on the pltform t pont. Alo et n ntntneou referene frme wth t orgn t pont whle t three orthogonl e remn prllel to thoe of hown n Fg.. Seondl epre the velot v t pont P on the pltform n term of velot v t pont nd ngulr velot ω of the pltform.e. v v ω p (9) Epreng ll vetor n wth p the poton vetor pontng from to P ple et of e t pont P nd let e j be the unt vetor long the jth uh. hen tng the dot produt on both de of Eq.(4) wth e j led to where nd j j g t v $ Ad $ () $ ej j p e j j e e nd j Ad g p p re the vetor of e p evluted n wth beng the orentton mtr of wth repet to ( ); $ j unt wrenh of ero pth meured n e j t pont P. j repreentng unt fore long In order to full derbe the dfferent moton tpe of prllel mnpultor hvng oupled DF t neer to elet et of pproprte pont on the pltform nd pef the relevnt e long whh the dot produt ten. Although tng three dtnt nd non-ollner pont on the pltform uffent to dentf t poe prtl gener w to do th to te 5 pont the nddte: e.g. the 4 verte of trngulr prmd (tetrhedron) hvng n equlterl be nd

5 4 ABLE I ENUMEAIN F MIN YPES F ~6 DF PAALLEL MANIPULAS Moton tpe Derpton p rotton bout the nd e or three e not oneutvel prllel nd trnlton long the nd e rotton bout the nd e or three e not oneutvel prllel nd 2 trnlton long the nd e 2 rotton bout the e or two e not oneutvel prllel nd trnlton long the nd e 2 rotton bout the nd e or two e not oneutvel prllel nd 2 trnlton long the nd e rotton bout the nd trnlton long the nd e 2 rotton bout the nd or two e not oneutvel prllel nd trnlton long the rotton bout the nd 2 trnlton long the nd e p $ $ 2 $ 2 $ 22 $ $ 2 p $ $ 2 $ $ 4 $ 42 p $ 2 $ 22 $ 2 $ 4 $ 42 p $ $ 2 $ 4 $ 42 p $ $ $ 2 $ p $ 2 $ 22 $ 2 p $ $ 2 $ ---otton ---rnlton three dentl oele fe plu the entre of t be for provdng l mmetre nd eer mplementton. For th rrngement of pont ume tht pont the entre of the be equlterl trngle lng n the plne of ten to be the plne ontnng the entre of jont onneted wth the pltform. In th w t uffent to hooe two orthogonl e t eh verte uh tht e e 2 e p e p ( 24 ) hown n Fg.2. hu nd 2 e n o e 2 P p o n for 2 e 4 e 42 p 4 p where the poton ngle of p ( 2 ) lng n the plne; p the heght of the tetrhedron whh equl to the rumrbng rdu of the be. And t pont e e 2 ewrte Eq.() n mtr form where p 2 42 p p g t $ $ $ e v Ad $ () v p v v2 v42 p n 6 mtr (.e. two for eh j e wth 24 j 2 nd three more for e j wth j 2 ). For n f-df prllel mnpultor wth pefed moton tpe n f lnerl ndependent row vetor of p n be eleted to form n f 6 mtr p. hu the lner mp 6 L : $ v n be obtned t where p f pv p q p p Ad g (2) p n f f mtr nown the dmenonll homogeneou obn of f-df prllel mnpultor. It e to ee tht p depend onl upon p nd e j. herefore the omputtonl burden to formulte the dmenonll homogeneou obn n be redued drmtll one the generled obn h been mde vlble ung the pproh gven n Seton II. It hould be noted tht the hoe of f lnerl ndependent row vetor of p not unque. In order to olve th problem p n be formulted b ether of two w. B ettng p one w to hooe p of whh the mnmum ngulr vlue te the mmum vlue. Although th temt pproh t rther tme onumng. B utlng the -2- potonng prnple n the jg degn more trghtforwrd w to elet f $ jtht enble to derbe the prerbed moton tpe of the pltform nd to heve the hghet feble degree of l mmetr. For emple t ver nturl to elet $ $ 2 $ 4 nd $ 42 to formulte p of 22 prllel mnpultor n term of two trnlton long nd two rotton bout the nd e. n th b ble I enumerte utble p for number of ~6 DF prllel mnpultor

6 (p) 5 hvng oupled DF. It n be een tht for eh e the eleted f lnerl dependent wrenhe relte to no more thn pont out of 5 nddte. here re ome -DF non-overontrned prllel mnpultor hvng oupled DF for whh the dmenonll homogeneou obn n dretl be generted f ether the trnltonl or the rottonl omponent n $ t n be ten the ndependent oordnte. he -DF module wthn the rept robot [24] tpl emple of th ondton. For uh e Eq.(8) rewrtten n prttoned form v q vv v ω v () nd then f for emple the lner velot of pont ten the ndependent oordnte ble II DIMENSINS F A -PS PAALLEL MECHANISM (UNI: mm) b q A ; B (=2). b ; q the ntl length of the th prmt jont A r A 2 A 5 4 v q p p vv v v (4) B 2 q 2 IV. EXAMPLES he detert nle of two tpl prllel mehnm re rred out ung Eq.(2) or (4) developed n Seton III to llutrte the effetvene of the propoed pproh. A. -PS prllel mehnm A hown n Fg. the -PS prllel mehnm ont of be pltform nd three dentl lmb eh onnetng the be wth the pltform n equene b revolute jont n tuted prmt jont P nd pherl jont S. herefore n 5 ( 2 ). he unt rew of permon ˆ$ t j ( j 2 5 ) n the th lmb n be obtned b q 2 $ ˆ t ˆ$ t 2 ˆ 4 $ t ˆ 5 $ t 4 $ ˆ t 5 (5) B Fg.. Shemt dgrm of -PS mehnm b B n.546 m.646 m.746 m where j unt vetor long the j th -DF jont of the th lmb; A nd q 2 B A. he jont e re rrnged uh tht 2 ; 4 nd 5 re ondent wth three rottonl e of the pherl jont wth 2. Utlng the properte gven n Eq.(5) the unt wrenh of ontrnt (tuton) ˆ$ w ( $ ˆ w 2 ) nd the unt rew of retrton $ ˆt ( 2 ) n equentll be determned b the obervton method [22]. $ ˆ w θ Fg.4. Dtrbuton of ( p) of the -PS mehnm o n $ ˆ 2 w2 ˆ q 2 n $ t (6) 2 n where n 2. Subttutng Eq.(9) nd (2) nto Eq.(8) reult n the generled obn of the -PS mehnm. θ = (7)

7 (p) q q q2 q q q Conderng the mehnm to hve three degree of freedom n term of one trnlton nd two rotton whh then produe other prt moton p n be formulted b eletng v2 long the e 2 t pont P ( A ) ( 2 ) the three ndependent oordnte.e. where $ $ $ (8) p $ e2 2 e 2 o n e he orentton mtr of wth repet to ( ) n be generted b (9) where nd re three Euler ngle of preeon nutton nd bod rotton repetvel wth ; nd denote ne nd one funton. hu ubttutng Eq.(7)-(9) nto Eq.(2) fnll reult n dmenonll homogeneou obn. A n llutrton onder -PS prllel mehnm hvng the geometr prmeter gven n ble II rottonl pblt of 4 throughout ~ 6 nd trnltonl pblt of 2 mm from =56.4 mm to =76.4 mm. Fg.4 plot the ondton number of p p p evluted throughout the entre t worpe. It n be een tht for gven -oordnte of p te mmum vlue t 4 when 2 24 ; t mnmum vlue our t. It how tht the nemt performne n be mproved lghtl b nreng the dtne between nd wthn the gven rnge of the troe. B. -UPS&UP prllel mehnm Fg.5 how the hemt dgrm of the -UPS&UP prllel mehnm whh form the mn bod of the rept robot [24]. he mehnm ompoed of three dentl unontrned tve UPS lmb nd one properl ontrned pve UP lmb. Eh UPS lmb onnet the be to the pltform n equene b unverl jont n tuted prmt A ; B (=2). H h ble III DIMENSINS F A ICEP B (UNI: mm) b H B2 2 q 5 b ; q the ntl length of the th prmt jont b 6 A 2 4 Fg.5 Shemt dgrm of the -UPS&UP mehnm wthn the rept robot r q 4 4 jont nd pherl jont whle the UP lmb onnet the be to the pltform b unverl jont followed b prmt jont. herefore n 6 ( 2 ) for the UPS lmb nd n4 for the UP lmb. he be of four vetor ubpe of the th UPS lmb nd UP lmb re gven n n etenve nl of th tem [7] o t generled obn el formulted. For th prtulr e the three omponent of the lner velot v of pont n be ten three ndependent oordnte. 24 B A 4 Fg.6 Dtrbuton of ( p) of the -UPS&UP mehnm A B.5 m.95 m.85 m

8 7 hu the dmenonll homogeneou obn be obtned dretl b ung Eq.(4). v q vv p 2 v 24 n24 p n p vv v v (2) v q q n n where j (=2) unt vetor long the j th -DF jont of the th lmb. q q q2 q wth q beng the rte of the th tuted UPS lmb long.here the jont e re rrnged uh tht nd A q4 4 n nd n Conder for llutrton -UPS&UP prllel mehnm hvng the geometr prmeter gven n ble III wth lndrl t worpe of heght h 2 mm nd dmeter mm. Fg.6 plot the ondton number of p p evluted throughout the entre t worpe. For gven -oordnte of the mmum vlue of p our t the worpe boundr; whlt t mnmum vlue our t. h how tht the nemt performne n be mproved b dereng the dtne between nd n the gven rnge of h. Fnll t mut be treed tht the unt of v p nd q re both length/tme for thee two emple ledng to p beng dmenonle. h men tht not onl the ondton number but lo the mmum/mnmum ngulr vlue of re dmenonle nd n thereb be ued the lol ondtonng nde. However for prllel mehnm tuted b revolute jont the unt of q ngle/tme nd p no longer dmenonle lthough t remn dmenonll homogeneou. In th e p utble onl for nemt performne evluton. he wort nlt e when prllel mehnm drven b mture of prmt nd revolute tutor. he method preented here no longer wor dretl nd o pel tretment wll need to be ued to heve the dmenonll homogeneou obn for nemt performne evluton. p IV. CNCLUSINS h pper propoe new pproh for dervng the f dmenonll homogeneou obn for detert nl of f-df ( 2 f 6) prllel mnpultor hvng onl one tpe of tutor.e. ether revolute or prmt. he followng onluon re drwn. ) he propoed pproh generl for f-df prllel mnpultor hvng oupled trnltonl nd rottonl moton pblte provded tht proper lner mppng n be mde between the twt of the pltform nd the ndependent oordnte n ordne wth the pefed moton tpe of the pltform. 2) he proedure to formulte the dmenonll homogeneou obn tndrded nd omputtonll effetve thn to the ue of the generled obn. ) Both ondton number nd ngulr vlue of the derved obn n be emploed lol ondtonng nde for prllel mehnm tuted b prmt jont; where onl the ondton number n be ued for thoe tuted b revolute jont. EFEENCES [] H. Lpn. Duff Hbrd twt nd wrenh ontrol for robot mnpultor ASME. Meh. rn. Automt. De. vol. pp une 988. [2] K. Dot C. Melhorr nd C. Bonevento A theor of generled nvere ppled to robot Int.. obot. e. vol. 2 no. pp. 9 Feb. 99. [] K. L. Dot C. Melhorr E. M. Shwrt nd C. Bonevento obot mnpulblt IEEE rn. obot. Autom. vol. pp une 995. [4]. M. Angele ptmum rhteture degn of pltform mnpultor he ffth Interntonl Conferene on Advned obot vol. 2 pp [5] M. ndr. Angele F. njbrn Chrtert pont nd the hrtert length of robot mnpultor ASME De. Eng. Dvon vol. 45 pp [6]. Angele F. njbrn nd. V. Ptel n the degn of the nemt truture of even-e redundnt mnpultor for mmum ondtonng n Pro. IEEE Int. Conf. obot nd Automton Ne Frne M 5 pp [7]. Angele Knemt otrop n humn nd mhne n Pro. IFoMM 9th World Congr. heor Mh. Meh. Mln Itl Aug. 29 Sept. 2 vol. pp. XLII XLIX 995. [8]. Angele I there hrtert length of rgd-bod dplement? Meh. Mh. heor vol. 4 no. 8 pp [9] W. A. Khn. Angele he netott optmton of robot mnpultor: he nvere nd the dret problem ASME. Meh. De. vol. 28 no. pp []. Angele Fundmentl of obot Mehnl Stem: heor Method nd Algorthm rd ed. New Yor: Sprnger-Verlg 2. [] C. M. Goeln he optmum degn of robot mnpultor ung detert nde. obot. Autom. St. vol. 9 no. 4 pp [2] S.-G. Km. u New dmenonll homogeneou obn mtr formulton b three end-effetor pont for optml degn of prllel mnpultor IEEE rn. obot. Autom. vol. 9 no. 4 pp [] M. Kong Y. Zhng Z. Du L. Sun A novel pproh to dervng the unt-homogeneou obn mtre of mehnm n Pro. 27 IEEE Int. Conf. Mehtron Autom. pp [4]. Alturr. Slgdo V. Petu A. Hernánde Pont-bed obn formulton for omputtonl nemt of mnpultor Meh. Mh. heor vol. 4 no. 2 pp f

9 [5] G. Pond. A. Crretero Formultng obn mtre for the detert nl of prllel mnpultor Meh. Mh. heor vol. 4 no. 2 pp [6] G. Pond. A. Crretero Quntttve deterou worpe ompron of prllel mnpultor Meh. Mh. heor vol. 42 no. pp [7]. Hung H. Lu D. G. Chetwnd Generled obn nl of lower moblt mnpultor Meh. Mh. heor ubmtted. [8]. Murr Z. X. L S. Str A Mthemtl Introduton to obot Mnpulton Bo ton FL: CC.994. [9] L. Hogben Hndboo of Lner Algebr Chpmn & Hll/CC. 27. [2] K. H. Hunt Knemt Geometr of Mehnm. New Yor ford Unvert Pre 978. [2]. K. Dvdon K. H. Hunt obot nd rew theor: pplton of nemt nd tt to robot ford Unvert Pre 24. [22] L.-W. obot nl: the mehn of erl nd prllel mnpultor New Yor: ohn Wle & Son In [2] S. A. oh L.-W. obn nl of lmted-df prllel mnpultor ASME. Meh. De. vol. 24 no. 6 pp [24] K. E. Neumnn obot US Ptent