A GENERAL FORMULA OF DEGREE-OF-FREEDOM FOR PARALLEL MANIPULATORS AND ITS APPLICATION

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1 Poceedngs of DETC/CE 006 AE 006 nenaonal Desgn Engneeng Techncal Confeences and Copues and nfoaon n Engneeng Confeence epee Phladelpha Pennsylvana UA DETC A GENERAL FORULA OF DEGREE-OF-FREEDO FOR PARALLEL ANPULATOR AND T APPLCATON Tng-L Yang E-al: yangl@pulc.p.s.cn NOPC Jnlng Peochecal Cop. Nanng-007 P R Chna Dong-Jn un E-al: kx@lpec.co NOPC Jnlng Peochecal Cop. Nanng-007 P R Chna Asac Ths pape pesens a new DOF foula fo echans. s an feaue s ha he calculaon of oly has a sngle value fo a gven echans whou he se of consan equaons each of paaees n he foula can e coecly deened y sple syol opeaon. The foula shows he ap elaonshp eween DOF and opologcal sucue of a echans. s eoded n he followng aspecs: Denson ype: so ha opologcal sucue of a echans can e epesened y syols. Oenaon and locaon chaacesc ax: so ha ank of a echans can e calculaed y syolc opeaon. Oenaon and locaon chaacesc equaon of seal echans and s syolc opeaon. Oenaon and locaon chaacesc equaon of paallel echans and s syolc opeaon. 5 The DOF calculaon ased on oenaon and locaon chaacesc equaons of seal and paallel echanss. The DOF foula pesened n hs pape has aleady een used fo opologcal analyss and synhess of paallel echanss and s advanages has een poven. Keywods: allel echans Topologcal sucue Rank yolc opeaon.. noducon Degee of feedo Degee-of-feedo (DOF) s a key opologcal paaee and one of he os fundaenal concepons of a echans. n he las 00 yeas seveal dozens of DOF foulas have een poposed n he leaue. The vaous DOF foulas can e gouped n wo caegoes: () DOF Foulas ased on seng up he kneac consan equaons and he ank calculaon y analycal ools such as scew syse heoy affne geoey Le [ 8 9 7] algea ec. The ao dawack of hese Foulas s ha he oly can no e deened easly whou seng up he kneac odel of he echans. The oly calculaed s an nsananeous oly whch can e dffeen fo geneal oly. () Foulas fo sple calculaon of oly havng a sngle value fo a gven echans whou he se of consan equaons [6765]. The ao dawack of hese Foulas n he leaue s ha a sple and coec ehod has no een pesened fo calculaon of key paaees n hese foulas. Theefoe hese foulas ae no suale fo soe paallel echanss. Ths pape pesens a new DOF foula fo paallel oo echanss. s an feaue s ha he calculaon of oly has a sngle value fo a gven echans whou he se of consan equaons; each of paaees n he foula can e coecly deened y sple syol opeaon. The foula pesened has een used fo opologcal analyss and synhess of paallel echanss [0-5 -] and has poven s advanages. The conen of hs pape s aanged as follows: econ : Fundaenal heoy fo DOF calculaon: yolc descpon of opologcal sucue. Oenaon and locaon chaacesc(n sho OLC) ax of he oupu lnk of a echans and s ank. OLC equaon of seal echanss (.e. sngle opened chan n sho OC) and s syolc opeaon. OLC equaon of paallel echanss and s syolc opeaon. econ : New DOF foula of echanss. econ : Ceon fo dvng pa selecon. econ 5: Ceon fo dle pa deenaon. econ 6: Type of DOF and s deenaon.

2 . Theoy Foundaon. Topologcal sucue of echans and s syol epesenaon.. Denson ype Defnon: Type of denson consan. Relave decon and poson eween axes of kneac pas on he sae lnk wll suec o consans of hs lnk (.e. denson consan). Thee ae 6 asc ypes of denson consan [9-] : () Axes of wo adacen kneac pas ae concden. s expessed as OC{-R/H-}. () Axes of seveal adacen R pas paallel o each ohe. s expessed as OC{-R//R// //R-}. () Axes of seveal adacen R pas nesec a a coon pon. s expessed as OC{ RR L R }. () Axes of wo adacen kneac pas ae pependcula o each ohe. s expessed as OC(-P -}. R (5) Axes of seveal adacen pas ae paallel o a coon plane. s expessed as OC{- (PP P)-}. (6) Any ohe case dffeen fo aove 5 cases. s expessed as OC{-R-R-}. Each of he aove sx ypes of denson consan s called a denson ype un and s called denson ype heeafe. The denson ype shall e deened dung echans desgn and shall only exs afe echans asselng. The denson ype s nvaale dung oon of echans. echans coposed of denson ype uns s called denson ype echans... yolc epesenaon fo opologcal sucue of echans The hee asc eleens of opologcal sucue ae: () Type of kneac pa: such as psac pa (P) evolue pa (R) helx pa (H) sphecal pa () cylndcal pa (C) and plana conac pa (E). () Denson ype uns and he conaons. () Connecon elaon (seal o paallel connecon) eween sucue uns (e.g. lnk o OC) Based on he hee asc eleens he opologcal sucue of a echans can e pesened y syols. Fo nsance he opologcal sucue of dencal anches of he echans n Fg. can e wen as: -OC{-R( P)//R- } RR -}. The opologcal sucue of ovng and he fxed plafo: R //R 6 ; R 6 //R ; R s no paallel o R 6 ; R R 5 R 9 R 0 R R 5 R 9 and R 0 nesec a one coon pon o'.. Oenaon and locaon chaacescs ax The oenaon and locaon ax of he oupu lnk of a echans can e expessed as [-] : x y z ech () α β γ Whee (x y z) s coodnae of he ogn of he ovng coodnae syse (aached on he oupu lnk) n he fxed coodnae syse. (αβγ) epesen decons of he axes of he ovng coodnae syse. Fo exaple hey ay e wo decon angles of axs z' of he ovng coodnae syse n he fxed coodnae syse and one oaon angle aound axs z' o hee Eula angles can e used. Defnon: f he followng condons ae sasfed equaon () shall e called as oenaon and locaon chaacesc ax (n sho OLC ax). ech () f a cean eleen of equaon () s a consan shall e denoed as. () f a cean eleen of equaon () s an ndependen oupu he denoaon of hs eleen s unchanged () f a cean eleen of equaon () s a non-ndependen eleen shall e placed n a ace.e. {non-ndependen eleen}. Veco fo of OLC ax whch s ndependen of he coodnae syse s used. Whee ( d) ech ( d) + { ( d) + { ech + ( d)} ( d)} () () s he ndependen anslaon oupu { ( d)} s he non-ndependen anslaon oupu ( d ) s he ndependen oaon oupu { ( d)} s he non-ndependen oaon oupu (d) s he decon of anslaon (oaon) oupu ) s he nue of ( ndependen anslaon (oaon) oupu ( 0 o) ech s he nue of ndependen oupu (.e. ank of he echans). The oecve fo opologcal desgn of ank-degeneaed -5) echans s o keep each unexpeced oon ( ech oupu as a consan whle he expeced oon oupus ae ealzed.e. ech Fo exaple OLC aces of P pa R pa and H pa ae: (// P) P 0 ( ( R ρ)) R (// R) (// H) + ( ( H ρ)) H (// H) nce he DOF of R pa s he ndependen oupu of he OLC ax can e seleced n wo dffeen ways: { ( ( R ρ))} R o (// R) ( ( H ρ)) { (// R)} laly he ndependen oupu of he OLC ax of H pa can e seleced n hee dffeen ways: { (// H) + ( ( H ρ))} H (// H) () (// H) + { ( ( H ρ))} { (// H)} + ( ( H ρ)) o { (// H)} { (// H)}. Oenaon and locaon chaacescs equaon of seal echans.. Oenaon and locaon chaacescs equaon The OLC ax of he OC s end lnk s also su of

3 he OLC ax Whee J of each kneac pa. s s he OLC ax of he h kneac pa. When he OC s coposed of seveal su-ocs equaon (5) can e ewen as Whee su s k su s he OLC ax of he h su-oc. Jus as enoned aove denson ype echans s coposed of denson ype uns. The OLC aces of denson ype uns ae lsed n ale. Equaon (5) o (6) s called oenaon and locaon chaacescs equaon (n sho OLC equaon) of seal echans. Especally Equaon (5) o (6) can e used fo deenng he ank of denson ype echans wh only sngle loop... Dependence deenaon of oenaon and locaon chaacesc ax. Based on he equaon () dependence of he OLC ax of he OC s end lnk can e deened accodng o dependence eween eleens of he OLC ax of he kneac pa. The dependence cea nclude: () Dependence eween wo oaons f R //R (o R /R ) hee s ~ ~ ~ + (// R ) (// R) (// R ) (5) (6) (5a) f R // // R R k hee s k ~ f k ; (//( R R )) ~ (// R) ~ f k. () Dependence eween wo anslaons Fo P and P (+) hee s (// P ) (// P + ~ ~ + f P * // (P * P k *) hee s (// P*) ~ + (// P ) ~ ) (// ( P P ~ (// ( P * P *) ~ k + )) f P // P + f P // P + (5) (5c) (// ( P * *)) Pk 5d ~ Whee P* ncludes anslaon of P pa and anslaon deved y R pa and H pa. f P * // (P * P k *) hee s (// P *) (// ( * *) P + Pk (5e) ~ ~ ~ () Dependence eween a anslaon and a oaon Roaon and devave anslaon of R pa (H pa) ae dependen on each ohe. Only one s ndependen eleen and he ohe one s a non-ndependen eleen. (5f) Tale Oenaon and locaon chaacescs aces of ype denson ype uns.. Opeaon of he OLC equaon Opeaon seps: () Expess he OC s opologcal sucue usng syolcal epesenaon. () Esalsh he fxed coodnae syse o-xyz on he fae lnk. () Deene ase pon o' of he end lnk (n ode o ake he OLC ax sple he ase pon always les n he pa axs of he end lnk). () Ls he su equaon of OLC ax of each kneac pa fo he fae o he end lnk n sequence. Bu he ndependen eleen of he ax s no aked ou. (5) Deene he axu ndependen se of oaon eleens ased on dependence cea fo oaons (equaon 5a 5 and 5f) accodng o he followng convenons

4 Roaon eleen whose devave anslaon s a consan shall e seleced fs. R pa shall e seleced po o H pa. Roaon ndependen of he axu dependen se shall e ndependen eleen and he devave anslaon of he sae kneac pa shall e non-ndependen eleen and shall e placed n a ace. (6) Deene he axu ndependen se of anslaon eleens ased on dependence cea fo anslaons (equaon 5c-5f) accodng o he followng convenons Tanslaon of P pa shall e seleced fs. Aong hose devave anslaons he anslaon along axs of H pa shall e seleced fs. Tanslaon eleen of he axu dependen se shall e ndependen eleen and he ohe eleens of he sae kneac pa shall e non-ndependen eleen and shall e placed n a ace. (7) Deene he dependence eween he non-ndependen eleen of he OLC ax and he axu ndependen se ased on equaon (5a)-(5f). (8) Deene he oupu oenaon and locaon chaacescs and he ank of he echans.. Oenaon and locaon chaacescs equaon of paallel echans.. Oenaon and locaon chaacescs equaon The OLC ax of paallel echans can e deened y he equaon (7) as n n ( d) ( d) Whee pa s OLC ax of paallel echans s OLC ax of he h anch... The nesecon opeaon ules The nesecon opeaon ules n equaon (7) ae as follows: (// R) (// R) P (7a) () [ ] [ ] [ ] [ (// P) ] [ ] [ (// P) ] () [ (// ( R R ))] [ ] [ (// ( R R ] )) [ (// ( P P ))] [ ] [ (// ( P P ))] () [ ] [ ] [ ] [ ] [ ] [ ] [ (// R) ] f R // R [ (// R) ] [ (// )] R 0 () [ ] //. f R R [ ] (// P ) f P // P [ (// P )] [ (// P )] 0 [ ] f P // P. (7) (7) (7c) (7d) (5) (6) [ (// R )] [ ( R R )))] [ (// P )] [ ( P P )))] [ (// ( R R)) ] [ (// ( R R ))] [ (// R R )] [ ] [ (// ( P P )) (// ( ))] P P [ (// P P )] [ ] (// R) [ 0] [ (// P )] [ 0] f [ ( R R ) ( R R)))] f f [ ( P P ) ( P P )))] f.. Opeaon of he OLC equaon Opeaon seps: f f R R R)). f f R R R)); P P P )); P P P )). ( ( R R)) R R)); ( ( R R)) R R )) ( ( P P )) P P )); ( ( P P )) P P )). (7e) (7f) () Esalsh he fxed coodnae syse o-xyz on he fae lnk. () Deene he ase pon o' of he ovng plafo. () Deene he OLC ax of each anch y equaons (5) and (6) and he opeaon ules. () Deene he OLC ax of paallel echans y equaons (7) and s opeaon ules. (5) Deene he oupu oenaon and locaon chaacescs and he ank of he echans.. Geneal DOF Foula DOF of a echans s: F f (8) Whee F s DOF of he echans f s DOF of he h kneac pa s nue of kneac pas s ank of he echans s dsplaceen equaons se (.e. nue of ndependen dsplaceen equaons). aee n equaon (8) can e susued y anohe expesson v n. L Ω Then he geneal and paccal DOF fo can e oaned as ν F f n. + Ω L (9) 0 Ω L ( + ) L f λ + f λ ( + ) ( + ) 0 < 0 (9a) λ f (9) whee ν s nue of ndependen loops (he paallel echans shall have ( ν + ) anches: he fs wo L

5 anches consue he fs asc loop he (+) h anch consues he h h asc loop he ( ν + ) anch consues he ν h asc loop). n{ } eans he ν asc loops wh nu ae seleced. L L s ank of he h loop foed y he (+) h anch (he loop wh nu ank s seleced aong all loops conanng he (+) h anch) and can e deened y equaon (5 o 6). Ω s nue of edundan consans (.e. nesecon of anks ( ) of υ asc loops). L s he ank of su of OLC ax L paally() of su paallel echans foed y he fs second and he h anch and OLC ax ( +) of he (+) h anch whee pa() can e oaned y equaon (7) and he su of pa() and ( +) can e oaned y equaon (5 6). Essenally L s ank of loop foed y equvalen OC (of he su-paallel echans) and he (+) h anch. λ s consan degee of he (+) h anch. shall ( +) e poned ou ha: afe he asc loops ae deened he nue of edundan consans s ndependen of he way n whch he asc loop s nueed. Bu f he h OC has a nuλ wll e oe convenen fo calculaon of edundan consans (efe o exaple fo deals ). + s nue of kneac pas n he (+) h anch f s DOF of he h kneac pa. Fo he Equaon (9) f he ν asc loops and he seal nue can e seleced andoly hen he os geneal DOF foula s ν F f + Ω L (0) Whee each of paaees n Equaon (0) s he sae n Equaon (9) and can e deened y he sae ehod. Alhough he equaon (9) and (0) can e used fo DOF calculaon of echans u equaon (9) s ease and oe convenen han equaons (0) (efe o exaple fo deals ). Fo he D.O.F. foula nue N ove of ove-consans of a spaal echans s Nove 6ν ν L + Ω () Whee ν s nue of asc loops of a echans; ohe paaees n Equaon () ae he sae n Equaon (9) and can e deened y he sae ehod. When a anch conans one loop o seveal loops s called hyd sngle-open-chan (HOC). f he OLC ax H of a HOC s equal o he OLC ax s of OC hs OC s called equvalen OC of he HOC. Geneally a HOC s coposed of a su-paallel echans and seveal kneac pas and lnks conneced n sees. Typcal 5 opologcal sucues of su-paallel echanss used n HOC and opologcal sucues of he equvalen OCs ae gven n ale. Fo DOF calculaon of paallel echanss wh HOC-anches each of HOC-anches shall e susued y s equvalen OC n ode o ake DOF calculaon easy and convenen. Equaon (9) and (0) show he ap elaonshps eween DOF and opologcal sucue of a echans. Tale Typcal sucues of su paallel echans n HOC No yol cheac daga DOF equvalen OC (R) (R) OC { P (U) (U ) (U) OC { P // R () When he () consues a nsananeous plane: () () () () OC { P R // R R (5R C) (5R C) (5R-C ) OC { P P Exaple Deene he DOF of he paallel echans shown n fg.. The syol epesenaon of s opologcal sucue as: The opologcal sucue of fou dencal anches: -OC{-R( P)//R- } RR -}. The opologcal sucue of ovng and he fxed plafo: R //R 6 ; R 6 //R.; R s no paallel o R 6 ; R R 5 R 9 R 0 R R 5 R 9 and R 0 nesec a one coon pon o'. Pocess: Fg. (T-R) paallel echans () Esalsh he fxed coodnae syse and selec he ase pon o' of he ovng plafo. () Accodng o known condons hee asc loops ae seleced:

6 LC {-R ( )//R P R R R -R 8 ( 7)//R P 6 -} 5 0R LC {-R 8 ( 7)//R P 6 //R ( )//R P -R R R -} 5 0R LC {-R ( )//R P //R 6 ( 7)//R P 8 - R R R -}. () Deene ank of each asc loop 9 0 5R Fo each asc loop whose opologcal sucue s known s ank can also e deened ased on he OLC equaon (equaon (6)) and ale. s easy o know ha OLC ax of each asc loop s: L OC{ R ( P ) // R } + OC{ R R R R } + OC{ R ( P )// R } ( R ) (// R ) ( R ) (// R ) ( ρ ) ( R + 6 ) (// R6 ) ( R ) L + L OC { R ( ) // // ( ) // } { } 8 P OC R R R R 7 R6 R P R Theefoe hee s 6 and 5 L L L. () Deene edundan consan of he echans 678 Basc loop : Foed y OC {-R ( )//R P - R -} and 678 OC {-R 6 ( 7)//R P 8 -R 9R 0 }. Basc loop : Foed y aachng OC {-R 6 ( 7)//R P R R -} eween he ovng plafo and he fxed 9 0 plafo. Accodng o equaon (9) hee s : 5 λ f L laly consan degee of he h anch 678 OC {-R ( )//R P - R -} s λ 0. R R 5 nce λ λ 0 accodng o equaon (9a) nue of edundan consan of hs paallel echans s Ω 0. (5) Deene DOF of he echans v F } + Ω 0 ( ) + 0. DOF: f n.{ L (6) Analyze oupu oenaon and locaon chaacescs of he echans Afe deene he ase pon o' of he ovng plafo he OLC ax of he echans s : n ( R ) ( R 6) ( R ) ( R 6) ( ( R R6 )) nce F and hee ae fou ndependen R oupu eleens: hee ndependen oupu oaon eleens and one ndependen oupu anslaon eleen along he decon pependcula o R and R 6. ohe non-ndependen oupu eleens ae consan. 6 6 (7) Dscusson By he equaon (0) fo DOF calculaon ν asc loops and he seal nue can e seleced andoly. Fo exaple asc loops of echans n fg. ae seleced as LC {-R P //R R R R 5 0R 9 -R 8 7 P //R 6 -} 6. L LC {-R 8 ( 7(//R P 6 //R ( )//R P -R R R -} 5 5 0R R L LC {-R }//R P -R 6 ( 7)//R P 8 -R R R -} 6 Fo he hd asc loop foed y OC {-R ( )//R P - R R 5 -} λ f nce λ < 0 L L 5 6 accodng o equaon (9a) he OLC ax of he su-paallel-echans foed y fs () second and hd anches need e deened. Accodng o equaon (7) hee s pa() ( R ) ( R 6) ( R R R 6) ( ( 6) pa Fo he hd asc loop foed y he su-paallel-echans and he foh anch accodng o equaon (5) deene s ank naely ( ( R + R6)) ( R () + ) pa () pa ( R ) Theefoe 5 and Ω 6 5. L L L By he DOF equaon (0) v F f + Ω 0 ( ) + L Aove dscusson ndcaes: fo he sae echans could ake he dffeen se of asc loops whch aye has he dffeen nue Ω of edundan consans. Alhough he equaon (9) and (0) can e used fo DOF calculaon u equaon (9) s ease and oe convenen han equaons (0). Exaple Deene he DOF of he paallel echans shown n fg.(a). The syol epesenaon of s opologcal sucue as The opologcal sucue of hee dencal anches -OC{-R//R//C-}. The opologcal sucue of ovng and he fxed plafo: Thee pas ehe on he ovng plafo o on he fxed plafo ae coplana.

7 Pocess: (a) () Fg. (T-0R) paallel echans () Esalsh he fxed coodnae syse and selec he ase pon o' of he ovng plafo. () Accodng o known condons wo asc loops ae seleced: LC {-R //R //C -C //R 5 //R 6 -} LC {-R //R //C -C 7 //R 8 //R 9 -} () Deene ank of each asc loop Fo each asc loop whose opologcal sucue s known s ank can also e deened ased on OLC equaon (equaon (6)) and ale. s easy o know ha OLC ax of each asc loop s: L OC { R // R // C } + OC { C // R5 // R6 } (// R) + (// R6 ) (// ( R R )) Theefoe hee s 5. laly we can know ha hee s 5. L L () Deene edundan consan of he echans Basc loop : Foed y OC {R //R //C -} and OC {-C //R 5 //R 6 -}. Basc loop : Foed y aachng OC {-C 7 //R 8 //R 9 -} eween he ovng plafo and he fxed plafo. Accodng o equaon (9) hee s λ f 5. L nce λ < 0 accodng o equaon (9a) OLC ax () of he su-paallel echans foed y he fs wo OCs need e deened. Accodng o OLC equaon (equaon (7)) hee s pa() (// R ) (// R6 ) Accodng o OLC equaon (equaon (6)) hee s pa() (// R ) 9 (// R9 ) pa() Theefoe. L 0 L 7 Accodng o equaon (9a) Ω 5. L (5) Deene DOF of he echans L v F f n.{ } + Ω (5 + 5) +. L (6) To analyze he oupu oenaon and locaon chaacescs of he echans Afe deene he ase pon o' of he ovng plafo accodng o equaon (7) he OLC ax of he echans s n ( R 0 ) ( R ) ( R7 ) nce F and 0 hee ae hee ndependen R oupu anslaon eleens. ohe non-ndependen oupu oaon eleens ae consan. f he hee pas on he ovng plafo o on he fxed plafo ae no n he sae plane. s no dffcul o pove ha DOF of hs echans s sll F. Exaple Deene he DOF of he paallel echans shown n fg. (a). The syol epesenaon of s opologcal sucue as The opologcal sucue of hee dencal anches : HOC { R( R) // R // P } as shown n fg.(a). The opologcal sucue of ovng and he fxed plafo: Thee pas ehe on he ovng plafo o on he fxed plafo ae o ae no n he sae plana. Pocess: () Esalsh he fxed coodnae syse and selec he ase pon o' of he ovng plafo. () All anches wh loops ae susued y s equvalen OC especvely as shown n fg. (). (a) () Fg. Dela echans Accodng ale s equvalen OC s ( R) OC{ R( P ) // R // P } and hee s. (// R) () nce he paallel echanss n Fg. () and Fg. ae he asc sae s no dffcul o pove ha DOF of hs echans s sll F and 0 () To analyze he oupu oenaon and locaon chaacescs of he echans nce F and 0 hee ae hee ndependen R oupu anslaon eleens. Ohe non-ndependen oupu oaon eleens ae consan. R

8 Exaple Deene he DOF of he paallel echans shown n fg.. The syol epesenaon of s opologcal sucue as The opologcal sucue of hee dencal anches: - OC { R P The opologcal sucue of ovng plafo and he fxed plafo: Thee R pas on he fxed plafo ae o ae no n he sae plana. v F } + Ω 5 (6 + 6) + 0. (5) DOF: f n.{ (6) Analyze he oupu oenaon and locaon chaacescs of he echans Afe deene he ase pon o' of he ovng plafo he OLC ax of he echans s n ( R ) ( R ) + L { } ( ρ) ( R + { } 7) ( ρ) ( R ) nce F hee ae only hee ndependen oupu eleens n.ovously hee ae hee dffeen schees: Pocess: Fg.. - OC{ R P echans () Esalsh he fxed coodnae syse and selec he ase pon o' of he ovng plafo. () Two asc loops ae seleced: LC { R P 6 P5 R LC { R P 9 P8 R7 () Deene ank of each asc loop. Accodng o equaon (6) hee s 6. L L () Deene edundan consan of he echans Basc loop : Foed y OC R P } OC { R P5 6 { and Basc loop : Foed y aachng OC R P }. { eween he ovng plafo and he fxed plafo. Accodng o equaon (9) hee s nce λ < 0 λ f 5 6 L. accodng o equaon (9a) OLC ax () of he su-paallel echans foed y he fs wo OCs need e deene. Accodng o OLC equaon (equaon (7)) hee s pa() ( R ) ( R + ) { ( ρ )} ( R ) Fo he second asc loop foed y he su-paallel-echans and he hd anch accodng o equaon (6) deene s ank naely pa() + ( R ) pa() ( R7) + + Theefoe 6 Ω L L L { ( ρ) } L 8 { ( R )} ; ( R ) + { ( R )} ; + { } ( R ) + { }. Ceon fo dvng pa selecon Geneally all dvng pas of a paallel echans shall e seleced o e on he sae plafo. Bu locaon of he dvng pa shall coply wh he followng ceon: Fo a echans wh DOF F pe-selec F kneac pas as he dvng pas and gdfy he. f DOF of he oaned echans s F * 0 (no local DOF>0 geneaed) hese F kneac pas can e used as dvng pas sulaneously. f F*>0 (o local DOF>0 geneaed) hese F kneac pas can no e used as dvng pas sulaneously. Exaple Take he paallel echans shown n fg.(a) as an exaple s DOF s F. f hee anslaons of cylndcal pas (C C and C 7 ) on he sae plafo ae seleced as dvng pas DOF of he oaned echans afe hee anslaons ae gdfed shall e F*?0. Accodng o he aove ceon hee anslaons of cylndcal pas can no e used as dvng pas sulaneously. Fo he paallel echans shown n fg.() whose hee C-pas ae no n he sae plane s no dffcul o pove ha he hee anslaons of hee C-pas can e used as dvng pas sulaneously. 5. Ceon fo dle pa deenaon () Fo paallel echans wh DOF 0 f wo lnks conneced y one kneac pa (o a cean DOF of he pa) have no elave oon hs kneac pa s called dle kneac pa ( o dle DOF of kneac pa). () Ceon fo dle pa deenaon Fo paallel echans wh DOF 0 supposng a cean kneac pa(o a DOF of kneac pa) s gdfed f DOF of he ognal echans s equal o DOF of he oaned echans afe he cean pa s gdfed hs kneac pa us e dle pa. Ohewse hee exss elave oon eween he wo lnks conneced y hs pa. dle pa (o dle DOF of kneac pa) ay help he ove-consaned echans o educe he nefeence caused y anufacung o nsallaon eos. Exaple Take he paallel echans shown n fg.5(a) as } exaple. has hee anches: OC{- R - -} R

9 678 OC{- R } and OC{- R - 9-}. R R and R 7 R 5 7 R 8 ae so allocaed ha axes of R R R R 5 R 7 and R 8 nesec a a coon pon o'. 6 and 9 ae allocaed aaly. n ode o deene ank pa wh equvalen hee evolue pas R whee axes of L of asc loops susue he R R and 6 9 R R R R R R R 5 R 7 and R 8 nesec a a coon pon o' as shown n fg.5(). Accodng o OLC equaon (equaon (5 o 6)) s easy o know ha ank of asc loop s 6 and Ω0. L Theefoe DOF of hs paallel echans s F f n{ } 5. (a) (c) L L () Fg.5 phecal echans wh dle kneac pa n ode o deene whehe hee s any dle DOF n 6 pa R R R R R and R gdfy all hese pas and a 9R sphecal echans s oaned (fg.(c)). Accodng o OLC equaon ((equaon (5 o 6)) ank of asc loop s L L and Ω0. Theefoe s DOF s F f n. { } 9-. Accodng o he L dle pa cea R R R R 9 R and R ae dle DOFs. Fo he aove analyss we know ha he echans n fg.5(a) s equvalen o he echans n fg.5(c). The dffeence s: only sx R pas need o nesec a one pon fo he echans n fg.5(a) u all nne R pas have o nesec a one pon fo he echans n fg.5(c). s ovous ha anufacung and asselng of he echans n fg.5(c) s oe dffcul. 6. Type of DOF and s deenaon 6. Topologcal sucue coposon of echanss 9 Any kneac chan KC (ncludng paallel echans) can e egaded as eng coposed y F dvng pas (DOF of each pa s ) and seveal BKCs [].e. KC Jd [ F] + { BKC} () Whee J d [F] s F dvng pas; BKC s asc kneac chan whch s a closed chan wh DOF0 and f any lnk s eoved fo s DOF wll ecoe F>0. 6. Type of DOF () Coplee DOF: f oenaon and locaon of any dven lnk elave o he fae s a funcon of all npu paaees hs echans has a coplee DOF. () al DOF: f oenaon and locaon of one dven lnk a leas elave o he fae s a funcon of pa of npu paaees hs echans has a paal DOF. () epaale DOF: f he echans can e decoposed no wo o oe ndependen pas and oenaon and locaon of he dven lnk n each pa s funcon of all npu paaees whn hs pa hs echans has a sepaale DOF. 6. Ceon fo DOF ype deenaon The DOF ype s deened anly y opologcal sucue dvng pa allocaon and fae lnk selecon of he echans. Fo ul-bkc kneac chan wh DOFF cea fo DOF ype deenaon nclude: () f all he F dvng pas ae n he sae BKC he echans has coplee DOF. () f F dvng pas ae dsued n dffeen BKCs he echans has paal DOF. () f F dvng pas ae dsued n dffeen BKCs and he echans can e decoposed no dffeen ndependen pas he echans has sepaale DOF. Fo exaple he echans n fg. 5(c) conans only one BKC. nce all dvng pas R R ) ae n he sae BKC ( 6 R 7 hs echans has coplee DOF.e. he oenaon and locaon of any dven lnk s a funcon of all hee npus. (a) Jd[F] () BKC (c) BKC Fg.6 (T-R) paallel echans Fo exaple he echans shown n fg. 6 conans wo BKCs. Dvng pas ( P P ) ae n BKC and dvng pas P ) ae n BKC hs echans has paal DOF. s ( 9 P easy o know ha poson of R 6 axs on ovng plafo

10 s a funcon of npus of P and P u s ndependen of npus of P 9 and P. These exaples eveal ha paal DOF s a necessay condon fo paallel echans o e decoupled eween npu and oupu [ 5 -]. 7. Concluson n hs pape a new geneal DOF foula has een pesened fo paallel echanss. s e s ha DOF calculaon has a sngle value fo a gven echans whou he se of consan equaons; each of paaees n he foula can e coecly deened y sple syol opeaon. The foula shows he ap elaonshp eween DOF and opologcal sucue of echans. s eoded n he followng aspecs: () Two new concepons The denson ype: o ha opologcal sucue of a echans can e epesened y syols. The OLC ax: o ha he oupu oenaon and locaon chaacescs and he ank of echans can e calculaed y syolc opeaon. () The OLC equaon of seal echanss and s syolc opeaon ules. () The OLC equaon of paallel echanss and s syolc opeaon ules. () The DOF calculaon ased on OLC equaons of seal and paallel echanss. The foula pesened n hs pape has een used fo opologcal analyss and synhess of paallel echanss and has poven s advanages. Acknowledgen The fnancal suppo of Naonal cence Foundaon of P. R. Chna (Gans NFC ) s gaefully acknowledged. Refeences [] Cheychev P.A. 85 The oe des e canses connus sous le no de paalle logaes e`e pae e oes pe sene s a`l Acade e pe ale des scences de an-pe esoug pa dves savans. [] Da J oly of Ove-consaned allel echanss AE J. ech. Des. 7 pp.. [] Fanghella P. Galle C. 99 oly analyss of sngle-loop kneac chans: an algohc appoach ased on dsplaceen goups ech. ach. Theoy 9pp [] Feudensen F. Alzade R. 975 On he degee-of-feedo of echanss wh vaale geneal consan Fouh Wold Congess on he Theoy of achnes and echanss Newcasle 975. [5] Gogu G. 005 oly of echanss: a Ccal Revew ech. ach. Theoy 0 pp [6] Gogu G. 005 oly and paaly of allel Roos Revsed va Theoy of Lnea Tansfoaons Euopean J. of echancs A/oldpp [7] Gu le. 96 Das Keu de Zwa nglau.gke de chauenkelen Fesschf O. uh. Zu 80 GuesagBeln. [8] Huang Z. L Q.C. 00 Type synhess of syecal lowe-oly paallel echanss usng he consan-synhess ehod n. J. Roocs Res [9] Hun K.H. 978 Kneac Geoey of echanss Oxfod Unvesy Pess Oxfod. [0] Jn Q. Yang T. L. 00 ucue ynhess of -DOF (-Tanslaon) allel Roo echanss Based on ngle-opened-chan Uns The AE 7 h Desgn Auoaon Confeence P sugh DETC /DAC-5. [] Jn Q. Yang T. L. 00ucue ynhess of a Class Fve-DOF allel Roo echanss Based on ngle-opened-chan Uns The 7 h Desgn Auoaon Confeence P sugh AE nenaonal DETC/DAC-5. [] Jn Q. Yang T. L. 00 ucue ynhess and Analyss of allel anpulaos wh -Denson Tanslaon and - Denson Roaon n: Poc. of AE 8h Desgn Auoaon Conf. oneal DETC 00/ECH 07. [] Jn Q. Yang T. L. 00 ynhess and Analyss of a Goup of DOF (T-R) Decoupled allel anpulao. n: Poc. of AE 8h Desgn Auoaon Conf. oneal DETC 00/ECH 0. [] Jn Q. and Yang T.-L. 00 "ynhess and Analyss of a Goup of -Degee-of-Feedo ally Decoupled allel anpulaos" AE J. ech. Des. 6 pp [5] Jn Q. and Yang T.-L. 00 "Theoy fo Topology ynhess of allel anpulaos and s Applcaon o Thee-Denson-Tanslaon allel anpulaos" AE J. ech. Des. 6 pp [6] Kuzach K. 99 echansche Leungsvezwegung he Geseze und Anwendungen aschnenau. Bee [7] Rco anez J.. Ravan B. 00 On oly analyss of lnkages usng goup heoy Tans. AE J. ech.des [8] Tsa L.-W. 000 echans Desgn: Enueaon of Kneac ucues Accodng o Funcon CRC Pess Boca Raon Floda. [9] Yang T. L. 98 ucual Analyss and Nue ynhess of paal echanss Poc. of he 6 h Wold Cong. On he Theoy of achnes and echanss New Delh [0] Yang T.L.986 Kneac ucual Analyss and ynhess of Ove-consaned paal ngle-loop-chans Poc. of he 9-h Bennal echanss Conf. Coluus AE pape 86-DET-89. [] Yang T.L. Yao F. H. 99.The Topologcal Chaacescs and Auoac Geneaon of ucual Analyss and ynhess of paal echanss -Theoy -Applcaon. Poc. of AE echanss Conf. Phoenx DE [] Yang T. L. Jn Q. ec al. 00 A Geneal ehod fo ucue ynhess of he Degeneae ank allel Roo echanss Based on he Unes of ngle-opened-chan Chnese J. of echancal cence and Technology No.pp.~5. [] Yang T. L. Jn Q. ec al. 00 ucual ynhess and Classfcaon of he -DOF Tanslaon allel Roo echanss Based on he Unes of ngle-opened-chan Chnese J. of echancal Engneeng Vol.8(8)pp.~6. [] Yang T. L. 00 Topology ucue Desgn of Roo echanss pulshed y echancal Engneeng Beng. [5] Zhang Q. X.96 udy on ucual Theoy of paal echanss Chnese J. of echancal Engneeng. Vol.9() 0