Stiffness Characteristics Analysis of a Novel 3-DOF Parallel Kinematic Machine Tool

Transcription

1 Stffness Chrtersts Anyss of Nove 3-DOF Pre Knet Mhne oo Hqng Zhng nd Hrong Fng Abstrt A nove 1R wth three degrees of freedo redundnty tuted nd overonstrned RPU-SPR pre nputor s here presented s n terntve pproh for hgh speed hnng n erospe fed. Frsty, the tuton nd onstrnts of the pre nputor posed by pssve jonts re nyzed n ters of the srew theory, nd the degree of freedo of the pre nputor s further derved. Seondy, the knet nyss s rred out, the nverse poston nd geoetr onstrnt equtons of the pre nputor re estbshed, nd the over Jobn trx ws expty derved. Subsequenty, the stffness trx of the hn s dedued onsderng the est deforton of the nk, nd the stffness trx of the pre nputor s estbshed by the dfferent ppng retonshp between the tuted hns nd the ovng ptfor. he ner nd ngur stffness, egensrew deoposton, nd xu nd nu stffness egenvues re ntrodued to evute the stffness hrtersts of the nputor. Fny, through soe nuer expes, dstrbutons w of the perforne ndes of redundnty tuted nd overonstrned -RPU-SPR pre nputor re ustrted n dets. he resuts deonstrte tht the three degree of freedo redundnt tuton pre nputor proposed n ths pper hs uh better stffness perforne thn the -RPU-SPR pre nputor, nd hs uh ore extensve prospet n engneerng pptons. Index ers Redundnty tuted, overonstrned, pre nputor, egensrew, stffness. I. INRODUCION In reent yers, ower degree of freedo (DOF) pre nputor espey 1R wth three DOFs pre nputor s the n body of the hgh-end ntegent equpent s the fous of the urrent trend, whh hs been deonstrted by bundnt engneerng pptons n the erospe fed for opex oponent hnng, suh s Sprnt Z3 spnde [1], rept hybrd hne too [], Exehon hybrd hne too [3], et. In prt pptons, to nrese the workspe of 1R pre nputor, hybrd struture s genery derved by ntegrtng ser ode wth pre nputor. At the se te, to prove the orentton pbty of the end Mnusrpt reeved Deeber 5, 017; revsed Mrh 1, 018. hs work ws supported by the Centr Unverstes under Grnt No.017YJS158. Hqng Zhng s wth Shoo of Mehn, Eetron nd Contro Engneerng, Bejng Jotong Unversty, Bejng, P.R. Chn (e-: @bjtu.edu.n). Hrong Fng s wth Bejng Jotong Unversty nd Robots Reserh Center Leder, P.R. Chn (e-: hrfng@bjtu.edu.n). effetor, two or three degrees of freedo rottng hed n be tthed on the ovng ptfor, so the ut-degree of freedo hybrd hne too n be onstruted. In vew of strutur oponent wth rge denson nd opex freedo surfe n erospe fed, nove 1R ower DOF pre nputor wth hgher stffness nd hgher orentton pbty s of gret portne kerne ssue by ddng two trks n X-Y xs to for fve xs ser-pre hybrd hne too to opete hnng ng wth hgh effeny nd hgh ury. herefore, overonstrned pre nputor s spe ower DOF pre nputor e nto beng n ths ode, whh n effetvey vod the sngurty, nrese the workspe, nd prove knets nd dyn hrtersts, enrge stffness nd drvng stbty, nd so on. Wht s ore, t hs been suessfuy reeved extensve ttentons n dfferent engneerng nd tehnoog res s spe ower DOF pre nputor [4]-[6]. Up to now, ost of the nvestgtons n be fous on stffness hrtersts ssue of the pre knet hne (PKM). Doest nd foregn shors hve done nuerous efforts on the stffness of the pre nputors, nd the n ethods nude nyt ethod nd fnte eeent ethod (FEM). For expe, Gossen frsty put forwrd stffness ode for fu degree of freedo pnner nd spt ehns bsed on the prnpe of vrtu work, but ths ethod ony onsdered the stffness of tuton jont, not onsdered onstrnt fore nd oent posed by pssve jont [7]. Cnton epoyed sub-strutur trx ethod to estbsh the stffness nyt ode for Gough-Stewrt pre nputor to evute ts stffness perforne [8]. A oprehensve stffness odeng ethod ws frst proposed by Robert n vrtue of the srew theory, who estbshed the stffness ode nudng tenson/opresson, torson, nd defeton s we oupng, nd the whoe stffness ode ws obtned by equvent stffness for ner onneted sprngs n seres [9]. Zho nvestgted the over stffness trx bsed on the vrtu work prnpe by onsderng the tuton fore, onstrnt fore nd vrtu jont [10]. Zhng dopted vrtu jont ethod to forute the stffness trx of onstrned pre nputor n whh weghted funton oud be xzed n ters of the n dgon eeents of the stffness trx. However, the vue of tre of the stffness trx does not defntey predt the stffness perforne of the nputor [11]. he stffness of the Stewrt pre hne hs been ntensvey nvestgted by Khswneh [1], DOI: /IJE.018.V

2 ong others, whose pproh s bsed on the generzed Jobn trx n ters of the nu nd xu sngur vue of the stffness to reve the dstrbuton, nd sutneousy the stt stffness ode of the end effetor t dfferent postons ws estbshed by usng the fnte eeent softwre. Wng perfored preterzton ode n vrtue of oer fnte eeent softwre ANSYS, reserhed fnte eeent odeng ethod of vrous jonts, nd deonstrted the stffness perforne of the fve degrees of freedo rvrnt hybrd robot [13]. he stffness odeng of redundnt tuted nd overonstrned pre nputor s st rrey seen. Yn nd L ony nyzed the strutur hrtersts nd freedo, nd nverse nd forwrd knets of the two overonstrned RPU&SPR pre nputor [14]. Zhng utzed the vrtu jont ethod (VJM) to estbsh the stffness ode of hns nd jonts, nd then the sub-struture synthess ethod ws epoyed to synthesze the stt stffness nyt ode of the overonstrned Exehon pre odue, nd the stffness dstrbuton over the presrbed workspe ws further studed [15]. Cu estbshed the hn stffness nd ehns stffness of 3RPS-UPS pre nputor bsed on the srew theory nd ustrted the stffness proveent owng to the tuton redundny [16]. In sury, the struture of the pper s s foows: he struture of nove redundnty tuted nd overonstrned RPU-SPR pre knet hne too s desrbed, nd the degree of freedo s further obtned onsderng the tuton fore nd onstrnt fore/oent n ters of the srew theory n Seton II. he knet nyss of the pre nputor, s we s the nverse poston souton nd the onstrnt equtons, re rred out n Seton III. he over stffness of the pre nputor s strghtforwrd obtned onsderng tenson opresson, bendng nd torson deforton of the nks wthout onsderng the jonts deforton n Seton IV. he stffness perforne ndes nudng ner nd ngur stffness, stffness egensrew deoposton, nd xu nd nu stffness egenvue re ntrodued, the theoret ode proposed n ths pper s verfed by ens of the FEA ode, nd the stffness hrtersts of the pre nputor ws evuted n Seton V. Fny the onusons re drwn, nd deonstrte the erts of the proposed nputor. wo RPU hns dstrbute syetry, nd re oted n the pne. Sry, the two SPR hns re so syetr dstrbuton nd oted n the pne.he spnde too s tthed on the end of ovng ptfor for the hgh speed ng hnng. Fg. 1. he vrtu prototype. II. DEGREE OF FREEDOM ANALYSIS OF HE RPU-SPR PARALLEL MANIPULAOR A. Arhteture Desrpton of the Mnputor he pre knet hne too for hgh speed hnng ng onsdered n ths pper s shown n Fg.1 nd the topoog struture of ts ore odue nove redundnty tuted nd overonstrned -RPU-SPR pre nputor, s shown n Fg.. he pre odue s oprsed of the fxed ptfor tthed to the ovng ptfor through two dent revoute- prst- unvers (RPU) jonts n seres nd two dent spher- prstrevoute (SPR) jonts n seres respetvey nd the prst jont s tve jont whh s tuted by the ner servo otor. Fg.. he shet dgr of the pre nputor o ftte nyss, the bsoute oordnte syste B xyz nd the retve oordnte syste Auvw re estbshed s shown n Fg.. Wheren B s the dpont of the fxed ptfor, the X xs s ondes wth the vetor BB 1, the Y xs s onde wth the vetor BB, nd the Z xs s perpendur to the fxed ptfor upwrds. Srty, A s the dpont of the ovng ptfor, the u xs s ondes wth the vetor AA 1, the v xs s ondes wth the vetor AA, w xs s perpendur to the ovng ptfor upwrds. In the SPR hns (tkng the frst hn s n expe), the frst rotton xs s 11 of the spher jont s 347

3 pre to the Z xs, the seond xs s 1 s perpendur to s 11, nd the thrd rotton xs s 31 s perpendur to s 41 nd pre to the rotton xs s 51. Srty, n the RPU hns (tkng the fourth hn s n expe), the rotton jont xs s 14 s pre to the X xs, the unvers jont onssts of two vert R jonts, nd the frst rotton xs s 34 s pre to s 14, nd the seond rotton xs s 44 s perpendur to s 34 nd pre to v xs. B. Degree of Freedo Anyss of the Mnputor Aordng to the srew theory twst nd wrenh observton ethod, the 1,, 3, 4 hn provde tuton fore F, nd ts dreton ong the nk 0. he 1 nd 3 hn produe onstrnt fore F, whose dreton s pre to v xs pssng through the spher jont (=1, 3). he nd 4 hn produe onstrnt fore F nd onstrnt oent, whose onstrnt fore dreton s pre X xs pssng through the unvers jont nd onstrnt oent dreton s nor to the rotton xs of the unvers jont (=, 4). he dreton of onstrnt fore posed by pssve jonts denotes s f, nd dreton of onstrnt oent represents s [17]. It s known tht the nstntneous onstrnt fore nd onstrnt oent don t work on the enter of the ovng ptfor n ters of srew theory, tht s, F f v F (( ) f ) w 0( 1,3) (1) F f v F ( f ) w 0(,4) w 0(, 4) () Rewrtng Eqs.(1) nd () n trx for resuts n where f (( ) f ) v v J f f 0 (3) 031 J n be expnded s the foows f1 (( 1 1 ) f1) f f f3 (( 3 3 ) f3) J (4) f4 4 f Due to the spe onfgurton of the revoute jonts, there re three nery ndependent tes n the onstrnt Jobn trx J, so the ehns hs three redundnt onstrnts, nd the foru s bsed on the degree of freedo [18] g F d( n g 1) f v (5) where F represents the degree of freedo of the ehns, n represents the nuber of the oponents, g represents the nuber of the knet jonts, d 6 represents the order of the ehns, f represents the degree of freedo of the -th knet jont, v represents the redundnt onstrnts 1 of the ehns, nd represents the o degree of freedo. Nether onstrnt oupe n the se dreton, nor onstrnt fore n oner ong the onstrnt srew n the pre nputor, therefore, there s no oon onstrnt, tht s, 0. Due to wthout o degree of freedo, so 0. We n see fro the shet of the ehns, the nuber of the oponent s 10, the nuber of the knet jont s 1, nd the retve freedo of the knet jonts n the ehns s 18, the degree of freedo of the -PRU-SPR pre nputor n be thety uted by ppyng the odfed G-K equtons, tht s F 6 (10 1 1) (6) Aordng to the onstrnt fore nd oent, the ndependent degree of freedo s trnston tht perpendur to the onstrnt fore F nd two rottons tht perpendur to the onstrnt oent. Beuse the ehns hs four tve prst jont, so the ehns beongs to redundnty tuted nd overonstrned pre nputor. III. KINEMAIC ANALYSIS A. Poston Inverse Anyss Z-Y-X Euer nges re dopted to desrbe orentton trx of the ovng oordnte syste wth respet to the bsoute oordnte syste, frst rottng the ovng ptfor bout z-xs by nge, then bout y-xs of the new oordnte syste by nge,nd fny bout x-xs of the new oordnte syste by nge. hus, the orentton trnsforton trx R n be wrtten s foows, z, y, x R R R R s s s s s s s s s s s s s s s where s nd re the bbrevton of sne nd osne, respetvey. p x y z (7) represents the poston vetor of the orgn pont A n the bsoute oordnte syste. nd b represent the poston vetor n the bsoute oordnte of jonts A nd B, B1 B B3 B 4 nd A1 A A3 A 4 re both squre whose rurdus re nonted s r, r b, nd the oordnte of eh jont n the bsoute oordnte syste n be respetvey expressed s R 0, b r r s r r s b b (8) ( 1) where, 1,,3,4 4 In vrtue of the rrngeent of the revoute jonts, the four onstrnt ondtons n be strutured s R p b ( 1, 3) p (, 4) 0 (9) 348

4 Seetng preters,, z s three ndependent preters, prst oton n be rrnged s x 0, zs y, rtn s s s s s (10) he ose-oop vetor ethod s used to estbsh the equton of vetor AB n the bsoute oordnte syste B xyz L = p+ - b (11) Dot-utpyng Eq.(11) wth tsef, we yeds by tkng squre root ( p + - b )( p + - b ) (1) B. he Jbobn Mtrx of the Mnputor If the veoty vetor v nd ngur veoty vetor w of the ovng ptfor referene pont A re known, the veoty vetor v of the jont pont A tht onneted the tuted hn nd ovng ptfor n be expressed s v v ( 1,,3,4) (13) hen the veoty of the -th ner tutor n be expressed s Where v ( v ) (14) 0 0 he Eq.(14) n be wrtten n the trx for J v J, J 0 ( 0) (15) presents the tuton Jobn trx of the pre nputor. hus, obnng Eq.(4) nd Eq.(15)n be rewrtten n the trx for v J J, J 0 J 0 (16) where J 0 s the generzed Jobn trx of the pre nputor tht retes the veoty of jont to the veoty of the ovng ptfor. Aordng to the du retonshp between the veoty ppng nd the fore ppng, the reton between the hns nd the ovng ptfor n be obtned by Eq.(17) J, F M, f f f (17) 0 f Where presents the extern fore F nd extern oent M tng on the referene pont t the ovng ptfor, nd f presents drvng fore f nd onstrnt fore f of the knet hns. IV. SIFFNESS MODELING OF HE MANIPULAOR Wthout the oss of generty, when onstrutng the stffness nyt ode of redundnty tuted nd overonstrned pre nputor, t ws expty ssued tht the ovng nd fxed ptfors re perfety rgd, gnorng the deforton of the rotton jont, spher jont nd unvers jont, nd ony onsderng the est deforton of the nks. he tuton fore, onstrnt fore nd ptude exerted to the ovng ptfor n be denoted s $, f nd $ r, f r (=1,,3,4)respetvey. he seond nd fourth hn n provde the ovng ptfor wth onstrnt oent nd ptude $, f (=, 4), t n be so deoposed nto two onstrnt oents ong the drvng nk nd perpendur to the drvng nk dreton [19]. A. Stffness Modeng of the Chn he hn w produe tense deforton under drvng fore srew $, f, tht s k, k f EA (18) where E s the estty oduus, A s the ross-seton of the nk, nd k s the tenson/opresson stffness oeffent of the nk. he defeton r of the nk ong the onstrnt fore xs under onstrnt fore srew $ r, fr n be expressed s f r r r 3EI z k, kr (19) 3 where I z s the seton nert oent, nd k r s bendng stffness oeffent. he defeton 1 ong the nk xs under onstrnt oent srew $ j, f j n be denoted s f k, k 0 n1 n GI p (0) where G s sher oduus, I p s por nert, e,nd e R oent, e1 e Sry, the defeton of perpendur the nk xs under onstrnt oent srew $ j, f j n be ndted s EI z f ( e1 0) kt, kt (1) A trx for n be wrtten s where, f fr f j K0 r j1 j K 0 0 K 0 K K K dg k k k k, K r1 r r3 r4 =1,,3, 4, j=,4) (), =dg k k k k K, K K, 349

5 k n t K 1, 0 ( e1 0 ) 0 ( e1 0 ) K kn4 k ( e1 40 ) 440 4( e1 40 ) B. Stffness Mode of the Pre Mnputor he ppng retonshp between tuton hns deforton nd ovng ptfor dspeent X n be expressed s r j1 j v X J (3) J Where Jv Jr, Jr J (1: 4,1: 6), J ( e1 0 ) J ( e1 40 ) Substtutng Eqs. () nd (3)to Eq.(17), we n obtn the equton J0K0J vx (4) he stffness trx of the pre nputor n be rewrtten s K J K J (5) 0 0 V. SIFFNESS PERFORMANCE INDICES A. Lner Stffness nd Angur Stffness If the strutur preters nd pose re gven, the stffness trx of the pre nputor w be deterned. Soe stffness perforne ndes n be defned to evute the stffness hrtersts. Here we tret the dgon orrespondng eeent of the stffness trx s the ner stffness nd ngur stffness, whh n be defned n dets s foows [0]-[1]. kx K(1,1) k y K(, ) (6) kz K(3,3) kw K(6,6) where k x, k y nd k z re the ner stffness ong X-,Y-, nd v k Z-xs, respetvey; kw s the torson stffness bout Z-xs. One the preters of geoetry, onfgurton nd phys re gven n be I, the stffness trx under the onfgurton n be derved s foows Eq.(5) ABLE I: RELAED PARAMEERS OF MANIPULAOR ype Preter Vue r (/) 78 Struture r b(/) 565 (/) (750,150) Pose Phys (/rdn) 0 (/rdn) 0 z(/) 853 E(/P).06x10 11 G(/P) 7.69x10 10 A(/ ) 7.06 x10-4 I z(/ 4 ) 3.97 x10-8 he stffness trx of the hoe poston onfgurton n be obtned K (7) where the unts of ters re N/ for K 11, K, nd K 33, nd N/rd for K 44, K 55,nd K 66. Wth the hep of oer fnte eeent softwre AnsysWorkbenh, the vdty of the stffness ode s verfed, fnte eeent nyss of the PKM s onduted t the spefed onfgurton []. For ftte nyss, the fnte eeent ode of the vrtu prototype s onstruted for the hoe poston n the workspe, the defortons of the pre nputor under fore or oent re shown n Fg. 3. Fg. 3-3 ustrted the deforton of the pre nputor under fore ong the dreton of X-xs, Y-xs, nd Z-xs, respetvey. Fg.3d deonstrted the deforton of the pre nputor under the oent bout the dreton of Z-xs. 8 () Deforton ong X-xs under F=[0N,0,0] (b) Deforton ong Y-xs under F=[0,0N,0] 350

6 () Deforton ong Z-xs under F=[0,0, 0N] (d) Deforton bout Z-xs under M=[0,0, 0N] Fg. 3. Deforton wth fore/oent posed t the pont A. Bsed on the FEA nyss, the ner nd ngur stffness n be nuery uted by the rto the fore or oent nd the dspeent or nge. be II shows the oprson of the nyt ode nd the FEA ode. ABLE II: A COMPARISON WIH ANALYIC MEHOD AND FEA MEHOD Preter Anyt FEA k x (N/u) k y (N/u) k z (N/u) k w x10 4 (N/rd) It n be seen tht the resuts fro the nyt ethod th we wth those evuted by FEA ethod bsed on the hypotheses. herefore, the estbshed nyt ode of the whoe nputor stffness s effetve nd n be epoyed to evute the stffness perforne of the proposed nputor. For the purpose of nyss, spefed workspe s defned s 40 40, 40 40, nd z=853. Now the stffness dstrbutons of the PKM re ustrted n Fg.4. () Stffness dstrbuton n X-xs dreton (b) Stffness dstrbuton n Y-xs dreton () Stffness dstrbuton n Z-xs dreton (d) Stffness dstrbuton bout Z-xs dreton Fg. 4. Stffness dstrbuton w n presrbed workspe. 351

7 As shown n Fg.4-4d, the ner stffness k x nd k y re dstrbuted xs-syetry n nture nd the gntude of k x nd k y s very pproxte, wht s ore, the ner stffness n z dreton s rger thn tht n x nd y, whh just stsfed the hnng ng requreents for hgh poston ury. Aong wth the vryng of orentton preters of the PKM, the tendeny of vrton on the ner stffness n the workspe s deresed n Fg.4. he vrton of k z s ontrry to tht of k w. B. Egensrew Deoposton of Stffness Mtrx In the srew, the end deforton X n be usuy expressed n the xs oordnte syste, whe the wrenh srew n be genery ndted n the ry oordnte syste [3]-[5]. In order to ensure the onssteny of the oordnte syste, the twst srew nd wrenh srew re denoted n the se oordnte syste, the xs oordnte syste s onverted to the ry oordnte syste by epoyed the trx,.e., tht s, 033 I I 03 3 (8) where I denotes n dentty trx, nd the trx nterhnge the frst nd the st three eeents. herefore, the deoposton of the stffness trx s onverted nto the deoposton of the trx K, tht s, where nd e s the egenvues nd the egenvetors of the trx K t gven poston, respetvey. he egensrew deoposton of the stffness trx n be strghtforwrd obtned s 6 K kww, k, h 1 h w w, w K ee (9) dg([ ]) 10 h dg([ ]) w 1 n n hn (30) where k s the sprng onstnt, e s the -th egenvetor of the trx K, w s the unt srew of the egenvetor e, h expressed the pth of vetor w, n s the dreton vetor of the vetor w, s the poston vetor tht w retve to the orgn oordnte syste. he egensrew deoposton s pped to the stffness trx K s desrbed n Eq.(7). By sortng to sove the egensrew probe n Eq.(9), the sx egenstffness, the sx egensrew pthes h, nd the sx orrespondng unt egensrew w n be derved n ore det n Eq.(31) (31) he nterpretton of stffness trx K bsed on egensrew deoposton s eborted n be III, whh ndtes tht stffness trx K n be nterpreted by body suspended by sx-denson srew sprngs wth dretons ong the egensrew of the stffness trx K s shown n Fg. 5. p denotes the pth of he jont used n the srew sprng ABLE III: HE EQUIVALEN SCREW SPRING n p ( h ) Sprngs k / [ , , ] [0.0640, , ] [0.0780, , ] [0.0640, , ] [0.139, 0.988, ] [ , , ] [0.584, , ] [-0.390, , 0.863] [0.139, , ] [ , , ] [0.584, , ] [-0.390, , 0.863] Fro the be III nd Fg. 5, we n see tht the egensrew deoposton of the stffness trx n be dedued nto sx spe srew sprngs superposton, nd dved nto three groups of sprngs. Eh group hs two sprngs t one oon pont wth the se sprng stffness, nd the pth s opposte. Sne the pre nputor own two dfferent hns, the nputor doesn t exhbt one ertn syetry, nd the dstrbuton of the sprngs s not regur. C. he Mxu nd Mnu Egenvue of Stffness Mtrx In order to evute the stffness vue of soe postons n presrbed workspe, the xu nd nu egenvues of the stffness trx K re usuy epoyed s the evuton ndes of pre knet hne. 35

8 Fg. 5. he phys nterpretton of the stffness of the PKM. Fg. 6 nd Fg. 7 ustrted the ts of the xu vue nd the nu vue of the RPU-SPR pre nputor wth dfferent heght n presrbed workspe, fro the fgures, the xu nd nu egenvues of the stffness trx deresed wth the nreent of z. he owest hghest vue of xu stffness ours round the boundry of the workspe, so does the hghest vue of the nu stffness, sne the nputor pprohes sngur when t oes ner the workspe boundry. he prenry oprson between the redundnty tuted nd overonstrned RPU-SPR pre nputor nd the RPU-SPR pre nputor wthout tuton redundny ustrtes tht the forer own hgher stffness thn tht of the tter n se postons s shown n Fg.8. In ters of the engneerng ppton of hnng ng, the proposed nputor hs better stffness vues nd exhbts desrbe stffness hrtersts to stsfy the requreents for hgh poston ury Fg. 6. he xu egenvue of the stffness trx. Fg.7. he nu egenvue of the stffness trx. VI. CONCLUSION In order to opsh the hgh-speed hnng of erospe strutur oponents wth rge denson nd wth opex freedo surfe, ths pper proposed nove 1R redundnty tuted nd overonstrned pre knet hne too, whh n ntegrte two X-Y trks to onstrut fve xs hybrd hne too. Fro the nvestgton, the foowng onusons n be drwn: (1) he tuton fore nd onstrnt fore/oent of the proposed nputor re nyzed n det by sortng to the srew theory, nd the freedo of the pre nputor s further deterned. () he stffness ode of the redundnty tuted nd over-onstrned pre nputor ws foruted under the hypothess tht the n deforton soures re onentrted on the nks by sutneousy onsderng the tuton nd onstrnts fore, nd ths theoret ode s verfed by the FEA suton ethod. (3) he stffness dstrbutons of the proposed nputor re ustrted. he gebr hrtersts suh s the ner stffness, ngur stffness, egenvue nd egensrew of the stffness trx re usuy epoyed s the perforne ndex to evute the stffness of the pre nputor. he resuts ndte the proposed nputor hs uh hgher stffness thn the RPU-SPR pre nputor wthout tuton redundny, whh s gret ert nd hs wde engneerng pptons n the feds of ndustr robot nd pre knets hne toos. For the further work, ore perforne ndues suh s dexterty, oton-fore trnssson, knet ury nd rebty w be onsdered to enhne the bty of the proposed pre knet hne, nd then re-prototype w be fbrted. Fg. 8. he xu egenvue of the stffness trx. ACKNOWLEDGMEN hs reserh s sponsored by the Fundent Reserh Funds for the Centr Unverstes under Grnt No.017YJS

9 REFERENCES [1] N. Hennes nd D. Ster, Appton of PKM n erospe nufturng-hgh perforne hnng enters ECOSPEED, ECOSPEED-F nd ECOLINER, n Pro. 4th Chentz Pre Knets Senr, Chentz, Gerny, 004, pp [] B. Sno, he rept robot: Inverse knets, npubty nyss nd osed-oop dret knets gorth, Cbrdge Unversty Press, vo.197, no. 4, pp , [3] Z. M. B nd Y. Jn, Knet odeng of Exehon pre knet hne, Robots nd Coputer Integrted Mnufturng, vo. 7, no. 1, pp ,011. [4] H.. Lu,. Hung, nd A. Keskeéthy, Fore/oton trnsssbty nyses of redundnty tuted nd overonstrned pre nputors, Mehns nd Mhne heory, vo. 109, pp , 017. [5] B. Hu nd Z. Hung, Knetostt ode of overonstrned ower obty pre nputors, Nonner Dyns, vo. 86, no. 1, pp , 016. [6] X. Lu, Y. Xu, nd J. Yo, Contro-fed dyns wth deforton optbty for 5-DOF tve over-onstrned spt pre nputor 6PUS UPU, Mehtrons, vo. 30, pp , 015. [7] C. M. Gossen, Stffness ppng for pre nputors, IEEE rnstons on Robots nd Autoton, vo. 6, no. 3, pp , [8] C. M. Cnton, G. M. Zhng, nd A. J. Wverng, Stffness odeng of stewrt-ptfor-bsed ng hne, rnston of NAMRI/SME, vo.115, pp , [9] R. G. Robert, Mn rezton of n rbtrry spt stffness trx wth pre onneton of spe nd opex sprngs, IEEE rnstons on Robots nd Autoton, vo. 16, no. 5, pp ,000. [10] Y. Zho, Y. Jn, nd J. Zhng, Knetostt odeng nd nyss of n exehon pre knet hne (PKM) odue, Chnese Journ of Mehn Engneerng, vo. 9, no. 1, pp , 016. [11] D. Zhng, C. M. Gossen, Knetostt odeng of n-dof pre ehnss wth pssve onstrnng eg nd prst tutors, ASME Journ of Mehn Desgn, vo.13, no. 3, pp , 001. [1] E. Khswneh nd P. M. Ferrer, Coputton of stffness nd stffness bounds for pre nk nputors, Internton Journ of Mhne oos nd Mnufture, vo. 39, no., pp ,1999. [13] Y. Wng, H. Lu, nd. Hung, Stffness odeng of the rept robot usng the over Jobn trx, ASME Journ of Mehnss nd Robots, vo. 1, no., pp ,009. [14] Q. Yn, B. L, nd Y. L, Knets oprtve study of two overonstrned pre nputors, Mthet Probes n Engneerng, vo. 6, pp. 1-1, 016. [15] J. Zhng, Y. Zho, nd Y. Jn, Estodyn odeng nd nyss for n exehon pre knet hne, Journ of Mnufturng Sene nd Engneerng, vo. 138, no. 3, pp. 1-14, 016. [16] X. L. Cu, W. Y. Chen, nd X. G. Hn, Stffness proveent of 3RPS PKM by Redundnt tutng eg, Chnese Journ of Mehn Engneerng, vo.6, no.3,pp ,015. [17] Y. S. Zho, Y. D. Xu, nd J.. Yo, A fore nyss ethod for overonstrned pre ehnss, Chnese Journ of Mehn Engneerng, vo. 5, no.6, pp , 014. [18] Z. Hung, J. F. Lu, nd D. X. Zeng, A gener ethodoogy for obty nyss of ehnss bsed on onstrnt srew theory, Sene n Chn Seres E: ehnoog Senes, vo. 5, no. 5, pp , 009. [19] Y. Lu, X. Zhng, nd C. Su, Knets/stts nd workspe nyss of 3-eg 5-DoF pre nputor wth UPU-type oposte tve onstrned eg, Robot, vo. 31, no., pp , 013. [0] H. Ngtu nd M. Gur, Dyn nd stffness odeng of new 3DOF PKM for hgh speed hnng ppton, Internton Journ of Engneerng nd ehnoogy Senes, vo.3, pp , 014. [1] Y. G. L, H.. Lu, nd X. M. Zho, Desgn of 3-DOF PKM odue for rge strutur oponent hnng, Mehns nd Mhne heory, vo. 45, no. 6, pp , 010. [] K. K. Kur, A. Srnth, nd B. Sddhrth, Suton nd nyss of pre nputor for noeuvrng prosop er-cad bsed pproh, Internton Journ of Engneerng nd ehnoogy, vo. 7, no. 1, pp , 015 [3] Y. L nd Q. Xu, Stffness nyss for 3-PUU pre knet hne, Mehns nd Mhne heory, vo.43, no., pp , 008. [4] S. Hung nd J. M. Shes, he egensrew deoposton of spt stffness tres, IEEE rnstons on Robots nd Autoton, vo.16, no., pp , 000. [5] G. Chen, H. Wng, nd Z. Ln, he prnp xes deoposton of spt stffness tres, IEEE rnstons on Robots, vo. 31, no.1, pp , 015 Hqng Zhng s PhD student n the shoo of Bejng Jotong Unversty, Bejng, Chn fro 015. He ws born n He reeved the ster degree n ehn engneerng fro Hebe Unversty of Engneerng n 015 nd the bheor degree n ehn desgn nd theores fro Ynt Unversty n 01. Hs prry reserh nterest s fous on robots n oputer ntegrted nufturng, pre knets hne too, redundnt tuton robots, overonstrned pre nputors, nd ut-objetve optzton desgn, et. Hrong Fng reeved the bheor degree n ehn engneerng fro Nnjng Unversty of Sene nd ehnoogy n 1990, ster degree n ehn engneerng fro Shun Unversty n 1996, nd PhD degree n ehn engneerng fro Bejng Jotong Unversty n 005, respetvey. She worked s n ssstnt professor n the Deprtent of Engneerng Mehns t Bejng Jotong Unversty, Bejng, Chn, fro 010 to now. Currenty, she s fu professor n the Shoo of Mehn Engneerng nd Dretor of Robots Reserh Center. Her prry reserh nterests n pre ehnss, dgt ontro, robots nd utoton, hne too equpent, nd green nufturng. 354