Robotics and Computer-Integrated Manufacturing

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1 Robotcs and Computer-Integrated Manufacturng 27 (2) Contents lsts avalable at ScenceDrect Robotcs and Computer-Integrated Manufacturng journal homepage: Optmal desgn of a 3-PUPU parallel robot wth complant hnges for mcromanpulaton n a cubc workspace Yuan Yun, Yangmn Department of Electromechancal Engneerng, Faculty of Scence and Technology, Unversty of Macau, Av. Padre Tomás Perera Tapa, Macao SAR, Chna artcle nfo Artcle hstory: Receved 23 August 2 Receved n revsed form 24 March 2 Accepted May 2 Avalable onlne 26 May 2 Keywords: Parallel mechansm Flexure hnge Parastc rotaton Optmal desgn abstract In recent years, nanotechnology has been developng rapdly due to ts potental applcatons n varous felds that new materals and products are produced. In ths paper, a novel macro/mcro 3-DOF parallel platform s proposed for mcro postonng applcatons. The knematcs model of the dual parallel mechansm system s establshed by the stffness model wth ndvdual wde-range flexure hnge and the vector-loop equaton. The nverse solutons and parastc rotatons of the movng platform are obtaned and analyzed, whch are based on a parallel mechansm wth real parameters. The reachable and usable workspace of the macro moton and mcro moton of the mechansm are plotted and analyzed. Fnally, based on the analyss of parastc rotatons and usable workspace of mcro moton, an optmzaton for the parallel manpulator s presented. The nvestgatons of ths paper wll provde suggestons to mprove the structure and control algorthm optmzaton for the dual parallel mechansm n order to acheve the features of both larger workspace and hgher moton precson. & 2 Elsever td. All rghts reserved.. Introducton Nanotechnology, whch s an ntegraton of robotcs, control technque, materal scence, actuator and sensor technology, has been developng rapdly due to ts potental for changng the ways n whch materals and products are created. Nanotechnology nvolves the precse manpulaton and control of atoms and molecules to create novel structures wth remarkable propertes []. Ths nfluental technology requres a new feld termed nanomanpulaton to deal wth how to handle components and structures n nanometer scale by utlzng devces wth hgh postonng accuracy and dexterous manpulaton skll and controllng external forces wth sensory feedback, whch has been enabled by the nventon of bocell manpulaton, optcal fbers algnment, mcro devce assembly, and scannng probe mcroscopes [2]. Moreover, many applcatons n precson engneerng, such as wafer stepper lthography machnes, atomc force mcroscopes, space telescopes and nterferometers, laser communcaton systems and some other senstve optcal applcatons,would be mpossble to realze wthout a careful solaton of the nstrument from the vbraton envronment. These dsturbances n the form of multple degree-of-freedom (DOF) vbratons not only degrade the performance of the senstve nstruments and but also lead to falures n some bologcal experments and chemcal experments. To meet strngent accuracy and performance requrements, vbratons nherent to a flexble structure or generated by on-board dsturbance sources wll have to be controlled. Correspondng author. Tel.: þ ; fax: þ E-mal address: [email protected] (Y. ). In recent years, most of the nvestgatons on 2 or 3-DOF mcro parallel manpulators are focusng on the decoupled flexure parallel manpulators and ultra-precson manpulators [3,4]. In fact, for the parallel manpulators, the characterstcs of small workspace restrcted the development of parallel manpulator n the past decade. Takng nto account the requrement of a multple DOF mechansm for multple DOF postonng n a larger workspace and effectve vbraton attenuaton, many researchers utlze parallel manpulators wth complant parts for mcro postonng and actve vbraton solaton. A spatal complant parallel robot wth pseudo-elastc flexure hnges s proposed, whch uses a knd of shape memory alloy (SMA) as flexure hnges and ts workspace s larger than mm 3 []. Ouyang [6] proposed a new desgn of a parallel structure for macro mcro systems. The macro moton (DC motor) and mcro moton (PZT actuator) are connected by a parallel structure, the two motons are coupled under one complant mechansm framework. At the same tme, a knd of dual parallel mechansm called a 6-PSS parallel mechansm and a 6-SPS one s developed [7]. The expermental prototype s actuated by pezoelectrc motors whch can realze a centmeter-scale stroke wth postonng precson better than nm, and pezoelectrc ceramcs actuators wth a hgh resoluton about nm and non-backlash desgn ensure nanometer scale precson over the cubc centmeter workspace [8,9]. A XYZ flexure parallel mechansm (FPM) s also proposed wth large dsplacement and decoupled knematcs structure [4]. The large-dsplacement FPM has a large moton range over mm. Moreover, the FPM can acheve decoupled X-, Y- and Z-axs translatonal motons wth small cross-axs errors less than.9% and small parastc rotatons less than. mrad. The genetc /$ - see front matter & 2 Elsever td. All rghts reserved. do:.6/j.rcm.2..

2 978 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) algorthms and artfcal neural networks as an ntellgent optmzaton tool are used for the dmensonal synthess of the spatal sx DOF parallel manpulator []. Our motvaton n ths paper s to develop a novel 3-DOF spatal manpulator wth a cubc centmeter-scale workspace and the mcron or nanometer scale resoluton for 3-D manpulaton and vbraton solaton durng nanometer-scale experments. A dual parallel manpulator usng flexure hnges wll be developed, whch syntheszes both mcro postonng manpulaton n a large workspace and actve vbraton solaton for future possble use n crystal growth experment n mcro gravty envronment. Based on our prevous research results [ 4], an optmzaton on the parallel mcromanpulator wll be carred out through consderng the performance of the usable workspace, parastc rotatons and dexterty measures usng optmzaton toolbox n MATAB for multvarable nonlnear functon, PSO method and GA method. Comparng wth the exsted large-dsplacement FPMs for 3-DOF mcro manpulaton, the desgned manpulator has the advantages of smple structure, easy assembly, and large workspace. The nvestgatons of ths paper wll provde suggestons to mprove the structure and control algorthm optmzaton for the dual parallel mechansm n order to acheve the features of both large workspace and hgh moton precson. In the remander of the paper, a novel dual translatonal complant parallel mcromanpulator s desgned based on a 3-UPU prototype n Secton 2 and the performance analyss s gven n Secton 3. In Secton 4, the structure s optmzed to meet the requrement. At last, conclusons wll be gven n Secton. 2. Conceptual desgn of a manpulator wth large workspace A complant parallel manpulator has been desgned and fabrcated as shown n Fg. n Mechatroncs aboratory of Unversty of Macau. The parameters of the prototype are lsted n Table. The workspace of end-effector s only about several tens of mcrons along each axs. In order to enlarge the workspace and guarantee the accuracy, a redundant parallel mechansm combnng a 3-PUU parallel mechansm wth another spatal 3-UPU one s desgned as shown n Fg. 2. The parallel platform usng flexure hnges at all jonts conssts of a moble platform, three guded rals fxed to the base platform, and three lmbs wth dentcal knematc structure. Moreover, snce the flexure unversal hnge has a complcated structure whch s hard for manufacturng and qualty control, a Movng Platform PZT Actuator Feedback Pezo Input Traxal Accelerometer Flexure Hnge Fxed Base Fg.. A prototype of a 3-UPU parallel manpulator. Table Geometrc and materal parameters of the 3-UPU prototype. Item Radus of upper hnges dstrbuton r Radus of lower hnges dstrbuton R Radus of flexure hnge r f ength of flexure hnge l f Radus of strut wth PZT r s ength of strut wth PZT l s Modulus of elastcty of flexure hnge E f Modulus of elastcty of strut E s Stroke of PZT actuator Q c Sldng Block PZT Actuator Mcro moton knd of assembled flexure hnge s developed to overcome the dffcultes n puttng together all parts. 2.. Desgn of dual motons Snce the 3-PUU and 3-UPU knematcal structures wth conventonal mechancal jonts can be arranged to acheve only translatonal motons wth some certan geometrc condtons satsfed. In ths paper, the manpulator s formed by combnng together the two structures. Actually, the manpulator s a 3-PUPU parallel manpulator. For convenence, the moton of the end-effector can be dvded nto two parts, the macro moton and the mcro moton. The sldng-blocks are drven by three prsmatc actuators, respectvely, whch offer smooth precson moton along lnear guded rals to provde three translatonal macro motons n a cubc centmeter workspace. At the same tme, the mcro moton s provded by three precson actuators whch can ncrease the accuracy of the whole system Motor selectons Macro Moton Movng Platform Pezo Input Feedback Value Flexure Hnge 2 mm 9 mm.8 mm mm 6.3 mm 46 mm 26 GPa 2 GPa mm near Gude Ral Fg. 2. A descrpton of the redundant 3-PUPU parallel manpulator. As shown n Fg. 2, each lmb connects the moble platform to the fxed base by one prsmatc actuator, one flexure unversal (U) hnge and a pezoelectrc (PZT) actuator followed by another U jont n sequence, where the frst U jont s fxed at the sldng block and the second one connects to the movng platform. The end-effector s actuated by three PZT actuators to offer the mcro motons. PZT actuators have such merts as sub-nanometer resoluton, large force generaton, sub-mllsecond response, no

3 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) Table 2 Mechancal propertes of several common materals at room temperature for flexure hnges. Materal Yeld strength s (MPa) Young s modulus E (GPa) s=e T-6Al-4V Beryllum copper alloy CuBe Al alloy 77 T Steel ASTM A magnetc felds, extremely low steady state power consumpton, no wear and tear, vacuum and clean room compatblty. An ultrasonc motor (USM) s characterzed by the absence of noses durng operaton, wth hgh torque-weght rato, hgh accurate speed, and poston control, whch can provde unlmted stroke wth a sub-mcron level accuracy. The applcatons can be found n mcroscopy, precson moton, robotcs and so on. Ths knd of motor s sutable for the macro moton of rough postonng. Hence, three USMs are adopted as the prsmatc actuators to provde wth the macro moton for the mechansm and three groups of gude-way and way-block are adopted as the components. But the dynamc stall force of the USM s only about 3 N, the desgned manpulator adopts the horzontal drve arrangement to realze the macro moton n ths paper Flexure hnges In the past several years, more and more desgners adopt the flexure hnges nstead of conventonal mechansm jonts snce the backlash and frctons n the conventonal jonts affect the performances of hgh precson parallel mechansms remarkably. Although the adopton of flexure hnges n the parallel mechansm systems ncreases the hgh precson, short stroke actuators wth nanometer scale level precson result n a very small workspace. In ths paper, the flexure unversal hnge s a slender shaft confguraton wth very hgh torsonal stffness, whch s adopted as passve jont to ensure the large workspace of the whole system and hgh precson moton. As the wde-range flexure hnge s dffcult to manufacture for ts slender confguraton, a knd of assembled flexure hnge s developed, n spte of destroyng the monolthc feature [8]. Moreover, consderng that t s necessary to choose materals whch allow great elastc deformatons wthout undergong damagng plastc deformatons, the materals behavng the greatest ratos between the yeld strength and the Young s modulus (s=e) should be selected. The mechancal propertes of several common materals are lsted n Table 2. Snce the materal selecton of the flexure hnges wll also affect the natural frequency of the whole system, the beryllum copper alloy CuBe 2 wth hgh Young s modulus and s=e s adopted as the flexure hnge materal possessng well elastcty and hgh stffness. 3. Knematcs model As shown n Fg. 3, a reference frame o-xyz s attached to the fxed platform at the center o. A local coordnate system o x y z s attached to the movng platform at the center o. The coordnate system o x y z s establshed on the th lmb of the parallel mechansm. et l¼[l l 2 l 3 ] T be the vector of the three PZT actuated length varables and the vector r oo ¼½xyzŠ T of the reference pont o be the poston of the movng platform. In ƒ! ƒƒ! addton, let b be the vector oo and m be the vector o M. 3.. Knematcs model of macro moton Snce the PZT actuators are locked when the 3-PUPU for macro moton s actvated, the three PZT actuators can be treated as three struts n ths stuaton. It s obvous that the deformaton of the wde-range flexure hnges cannot be gnored for a mcro postonng applcaton. The knematcs model s establshed based on stffness equaton. The wde-range flexure hnges and struts are analyzed usng beam elements based on FEM theory. The sngle lmb s dvded nto four parts as shown n Fg. 3. et T r be the transformaton matrx of the th lmb from o x y z coordnate system to o-xyz. K o, K m and K u are the stffness matrces of the flexure hnge connectng to the base platform, the flexure hnge connectng to the movng platform and the strut n o x y z coordnate system, respectvely. p o, p, p 2 and p m are nodal load vectors of these nodes shown n Fg. 3 n o-xyz coordnate system. d o, d, d 2 and d m are nodal dsplacement vector n o-xyz coordnate system. The wde-range flexure hnges and struts can be treated as structural beams. et p, A, and be the axal force, cross sectonal area and length of a beam, respectvely, E, G, I, and J be the elastc modulus, shear modulus, prncpal moments of nerta and torson moment of nerta, respectvely. The element tangent stffness matrces n the local coordnate systems are composed of three components: lnear stffness K, the geometrc stffness matrx K G, and the large-dsplacements stffness matrx K D as follows []: K e ¼ K þk G þk D where K s gven by Fg. 3. Sngle lmb of the parallel mechansm. 2 EA EA 3 2EI 2EI EI 2EI GJ GJ 4EI 2EI 2 2 4EI 2 2EI 2 EA EA 2EI 2EI EI 2EI GJ GJ 6 2EI 4EI EI 2 4EI 2 ð

4 98 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) and the geometrc stffness matrx K G ¼ p The large-dsplacement stffness matrx could be neglected when the deformaton ncrement of the element s very small. Dvdng the stffness matrces nto 2 2 sub-matrces, the stffness matrx of the whole lmb n o x y z coordnate system can be acheved by assemblng the nodal stffness matrces: 2 K o K 3 o 2 K K ¼ o 2 K o 22 þk u K u 2 K u 2 K u 22 þk m K m 2 K m 2 K m 22 et K ¼ T T r K T r arbtrarly chosen d ðn ðd ðn d ðn þ ¼fðd ðn ð7 The acceleraton term can be gnored n statc analyss. Assume ðn K ðn t Dd ðn ¼ fðd ðn ðd ðd ðn, ðd ðn where F ðd ðn ¼K ðn d ðn system. Eq. () can be wrtten as 8 < : K ðn t ðn þ d Dd ðn ¼ fðd ðn ¼ d ðn Dd ðn =@d ðn ¼ n conservatve where K ðn t s the tangental stffness matrx of each lmb. The detaled flow chart for ths problem s descrbed n Fg. 4. A noteworthy fact s that the tangental stffness matrces of each lmb have to be updated n order to converge fast to solutons n each sub-step of the terated algorthm. It s a very huge amount of calculatons for a mult-body parallel system. Therefore, a knd of modfed Newton Raphson method s adopted, whch uses the orgnal tangental stffness matrx nstead of renewal one at the begnnng of each new teraton. Ths method generally requres more teratons and sometmes s less stable but t s less cost computatonally [6]. ð8 p ¼½p o p p 2 p m Š T d ¼½d o d d 2 d m Š T The agrange equaton of a sngle lmb can be formulated as p ¼ K d þm d ð2 where M s the mass matrx of sngle lmb, whch s smlar n form to the stffness matrx acheved by assemblng the element mass matrces M e, defned by 3.2. Knematcs model of mcro moton The prsmatc actuators for macro moton are self-locked when the mcro moton s avalable. Snce the dsplacement of wderange flexure hnges can be gnored for several hundreds of mcro strokes of actuators, the knematcs equaton can be establshed based on Pseudo-Rgd-Body (PRB) model easly. As shown n Fg., the flexure hnge can be regarded as a U jont and two torsonal massless sprngs wth stffness K b or K m wth respect to the o-xyz coordnate system. The torson moment of the flexure M e ¼ Z le N H N dx where N s the shape functon matrx and H s the generalzed densty matrx. In addton, let p be the external load vector actng on the movng platform, the force and moment equlbrum equaton can be wrtten as p X3 ¼ p m ¼ Snce the wde-range flexure hnges are adopted n ths parallel mechansm, the system cannot be expressed by a lnear equaton exactly. The stffness matrx and nodal load can be descrbed by a functon of nodal poston. Eq. (2) can be wrtten as ð3 ð4 START n= Input data of structure parameters Solve T Solve K n local coordnate system Solve K n global coordnate system n=n+ Solve T Solve K n global coordnate system Assemble K of each lmb Solve p K d Solve mbalance load vector p and d p ðd K ðd d M d ¼ fðd where fðd s the mbalance load vector. If d s an accurate soluton of Eq. (), we can obtan p ðd K ðd d M d ¼ fðd ¼ Therefore, the Newton Raphson method s utlzed, whch uses an teratve process to approach one root of a functon. The specfc root that the process locates depends on the ntal and ð ð6 Solve the ntal value of d Solve d Assemble K of each lmb Solve d END d p Y Output d Fg. 4. Flow chart of the terated algorthm for the desgned case. d N

5 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) dexterty measure whch can use the condton number as the ndex. The condton number and condtonng ndex can be defned as k ¼ JJJJJ J hnge caused by the angular dsplacement y b or y m can be derved by M b ¼ K b y b, M m ¼ K m y m ð9 For the th lmb, the vector-loop equaton can be wrtten as r oo þm b l k ¼ ð Dfferentatng Eq. () wth respect to tme _l k ¼ _r oo l o k ð where o s the vector of angular veloctes of the th lmb. Dotmultplyng both sdes of Eq. () by k, t can be assembled nto a velocty mappng equaton 2 3 _l _l 2 ¼ J_r oo ð2 _l 3 where 2 3 J ¼ ð3 k T k T 2 k T 3 Fg.. PRB model of the parallel mechansm for mcro moton. s the Jacoban matrx of top 3-UPU parallel manpulator, whch relates the output veloctes to the PZT actuated jont rates Objectve functon for optmzaton Consderng that the knematcs of the desgned parallel manpulator has hghly nonlnear characterstcs, the genetc algorthm (GA) and partcle swarm optmzaton (PSO) can be used to solve ths problem. The GA can be appled to solve a varety of optmzaton problems wth propertes of dscontnuous, non-dfferentable, stochastc, or hghly nonlnear, etc. The PSO s a relatvely new algorthm proposed by Kennedy and Eberhart, whch s a populaton-based stochastc optmzaton technque nspred by the socal behavor of brd flockng or fsh schoolng. PSO method can be employed to solve a varety of optmzaton problems such as dscrete optmzaton, artfcal neural network tranng, fuzzy system control and other stuatons where the evolutonary technques can be employed [7]. Both the GA and PSO are ntalzed wth random start values wthn the search space, whereas the PSO has no evolutonary operators of crossover and mutaton. As the knowledge of the authors, the applcatons of PSO are frst ntroduced to the ssues of manpulator optmal desgn by Xu and [3,4]. The objectve functon for optmzaton s defned as a mxed performance ndex that s a weghted sum of the usable workspace of mcro moton (V w2 ), the parastc rotaton (f max ) and the m ¼ =k ð4 where JJ denotes the 2-norm of the matrx. A global dexterty ndex s gven by R V m global ¼ m dv X m V N w ð w w A V where w s one of N w ponts, whch s unformly dstrbuted over the workspace. Therefore, the objectve functon s defned by followng equaton f ¼ w w2 þw V f f max þð w w2 w f ð6 w2 m global where w w2 A½,Š, w f A½,Š and w w2 þw f A½,Š. Consderngthe structure of the dual parallel manpulator, the desgnable varables are the length of the strut, the radus of the fxed platform and the length of the flexure hnge. Then, the complex method of the nonlnear programmng, GA and PSO method n MATAB toolbox are used to accomplsh the parametrc optmzaton by solvng the mnmum of the objectve functon f descrbed n Eq. (6). 4. Smulaton results and optmzaton analyss In ths case, a knd of PAS-seres pezoelectrc actuator (PAS) provded a convenent mountng package wth a mm stroke s adopted to provde mcron scale accuracy motons. A knd of USM named as HR8--S-3 (Nanomoton, Israel) s adopted as macro actuators wth the postonng precson better than nm, whch can bascally satsfy the macro moton demands. In addton, three groups of gude-way and way-block are adopted for the components of the pezoelectrc motors. The parameters of the parallel mechansm are shown n Table 3. In ths secton, the nverse solutons and parastc rotatons of movng platform are analyzed and smulated on a parallel mechansm. Whereafter, the reachable and usable workspace of the macro moton and mcro moton of the mechansm are plotted. Fnally, based on the analyss of usable workspace and parastc rotaton, an optmzaton of the parallel manpulator s presented. 4.. Parastc rotatons The nfluences of the torson ntroduced by the wde-range flexure hnges have to be consdered, whch may cause the parastc rotatons of end-effector. In Eq. (), each d b has one Table 3 The orgnal geometrc and materal parameters of the redundant 3-PUPU manpulator. Item Radus of upper hnges dstrbuton r Radus of lower hnges dstrbuton R Radus of flexure hnge r f ength of flexure hnge l f Radus of strut wth PZT r s ength of strut wth PZT l s Modulus of elastcty of flexure hnge E f Modulus of elastcty of strut E s Stroke of PZT actuator Q c Stroke of PZT motor Q m Mass of the movng platform M Mass of the lmb m Value 2 mm 9 mm.8 mm mm 6.3 mm 46 mm 26 GPa 2 GPa mm mm (7mm).4 Kg.28 Kg

6 982 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) Table 4 Input motons of actuators and parastc rotatons of movng platform. Dsplacement (mm) Input motons (mm) Parastc rotaton (mrad) [ 2] ( ) ( ) [ 2.] ( ) ( ) [ 2 ] ( ) ( ) [ 2 ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ 2 2 ] ( ) ( ) [2 2 ] ( ) ( ) nput element generated by actuator. d and d 2 are unknown vectors too, p and p 2 are zero vectors snce no loads act on node and 2, and d m can be solved by the dsplacement of the movng platform. Utlzng the mddle double sub-equatons of Eq. () for each lmb and Eq. (6) together are 42 equatons and 42 unknown elements ncludng three nput parameters and three parastc rotatons of movng platform. In Table 4, the nput motons of actuators and parastc rotatons of movng platform are gven by theoretcal stffness model. It ndcates that the parastc rotatons are zero when the movng platform only has the dsplacement of z-axs and the maxmal parastc rotatons of the checkng ponts are several mcro radans and gradually ncrease when the movng platform departures from ntal poston along x-axs and y-axs that cannot be gnored n the desgn of control algorthms for mcro/nanopostonng applcatons Workspace determnaton As the well elastc nature of the selected materal of wderange flexure hnges, the workspace of mcro moton only depends on the lmts of PZT actuators. Consderng the dfferent characterstcs between flexure hnge and conventonal mechancal jont, the parastc rotatons of movng platform have to be calculated and lmted n a neglgble order of magntude. In ths paper, the lmt of the rotaton angle s set to mrad. As shown n Fg. 6, the workspace of macro moton s about several mllmeters n each axs. As shown n Fg. 7, the reachable workspace of mcro moton, when the nputs of PZT motors are zero and selflocked at the ntal poston, s about several tens mcrons n each axs. It s assumed that the maxmal usable nscrbed workspace of the macro moton workspace s a column wth a radus of Wr, heght of H and symmetrcal about the ntal poston. The maxmal usable nscrbed workspace of the mcro moton workspace s a column wth a radus of Wr 2 and heght of H 2. The usable workspace s shown n Fgs. 6 and 7 denoted by W and W 2, respectvely. The W s about a radus of 2.46 and 2.4 mm heght column and the W 2 s about a radus of 79.2 and 36:6 mm heght column. Fg. 6. Reachable and usable workspace of the macro moton Poston accuracy Snce flexure hnges are adopted by ths manpulator at all jonts, the poston error of macro moton s only related to the attanable resoluton of the USMs. The error modelng s establshed wth the USMs poston resolutons usng knematcs modelng for the three lmbs parallel robot. An error of dd o n actuators movement wll produce a postonal error of dd m. The poston resoluton of the end-effector could be estmated by follows: 8 < : K ðn t ðn þ dd Dd ðn ¼ fðdd ðn ¼ dd ðn Dd ðn ð7 where Fg. 7. Reachable and usable workspace of the mcro moton. dd ¼½dd o dd dd 2 dd m Š T In accordance wth the Eq. (7) and the nm attanable resoluton of HR8--S-3, the poston error of macro moton s about 7 nm 3 as shown n Fg. 8, whch s covered n the mcro workspace as presented n the prevous subsecton.

7 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) x Poston error of macro moton Table Optmzaton results of ndependent runs by usng multvarable nonlnear method. Intal guess ½l s Rl f Š (mm) l s (mm) R (mm) l f (mm) f(n) t (mn) z (mm) [46 9 ] [6 8 ] [4 9 9.] [ 6 ] [6 7 ] [4 66 9] Table 6 Optmzaton results of ndependent runs by usng GA and PSO. 7 8 x y (mm) Fg. 8. The poston error of the macro moton. x (mm) x Method Number of teratons Mnmum global error gradent l s (mm) R (mm) l f (mm) f(n) t (mn) GA 86 e e e e e PSO 33 e e e Condtonng ndex y (mm) 4.4. Condtonng ndex From the Eq. (4), the condtonng ndex of the mcro moton wth the USMs for macro motons beng locked at the ntal poston can be plotted as shown n Fg. 9. The global dexterty ndex s about wth the 7 72 samplng ponts selected n the workspace. 4.. Optmzaton results.... Fg. 9. The condtonng ndex dstrbuton. The three desgnable varables are the length of the strut, the radus of the ntal radus of lower hnges dstrbuton and the length of the flexure hnge wth the lmt of: 4 mmrl s r2 mm, mmrrr2 mm and 9 mmrl f r2 mm. Consderng the emprcal numercal values of three desgnable varables, the varables w w2 and w f are.44 and.33 n order to attach equal mportance to the three desgnable varables. In ths paper, the tradtonal optmzaton toolbox n MATAB for multvarable nonlnear functon s adopted frst. The program s developed on a personal computer (Intel Pentum 4 CPU 3. GHz, GB RAM) wth sx ndependent runs. The ntal values are set as shown n the Table wth the correspondng results of optmzed varables and x (mm).. elapsed tme of the program. It s notceable that the optmzaton method s senstve to the ntal values. Snce the fmncon functon may only gve the local solutons, bg dscrepancy between number of results can be observed n Table. In addton, the convergent processes of GA and PSO are llustrated. Both the GA and PSO are ntalzed wth random startng guesses vectors wthn the gven parameters ranges. The populaton sze s assgned as 7. For the GA approach, the genetc operators are chosen to be non-unform mutaton wth rato of.8 and arthmetc crossover wth the rato of.8, respectvely. For the PSO, the ntal nerta weght and the fnal nerta weght are selected as.9 and.4, respectvely. The local and global acceleraton constants are assgned as 2. The maxmum generaton number, the mnmum global error gradent and the optmzaton results are lsted n Table 6 for contrast. The optmzaton results solved by GA are more acceptable n comparson wth the tradtonal optmzaton method. Wth reference to the GA, all of the three ndependent processes of PSO produce the same results. Wth reference to the three optmzaton approaches, the PSO gves the best ftness value of 8.78 and the GA gves the better ftness value of than the tradtonal optmzaton method (8.788). Moreover, the convergent processes of GA and PSO are llustrated n Fgs. and, respectvely. It s notceable that the convergence rate of the PSO method s much sharper than that of the GA method. Whereas, n vew of the calculatonal tme shown n Tables and 6, the PSO requres the longest tme to solve the fnal results. Choosng the result of the best ftness produced by PSO, the desgned dual parallel manpulator s optmzed to take the advantage of larger mcro moton workspace and smaller parastc rotaton wth certan sacrfce of the global dexterty ndex. The volume of maxmal usable nscrbed workspace of the mcro moton workspace s.24e 3 mm 3. The rado of the optmzaton for the volume of the usable workspace s 72.63%. The optmzaton ratos of parastc rotatons of the optmal parallel manpulator are lsted n Table 7. It s shown that the

8 984 Y. Yun, Y. / Robotcs and Computer-Integrated Manufacturng 27 (2) Ftness value gbest val optmzaton ratos of the maxmum parastc rotatons of test ponts are about %. But the best performances of the workspace and parastc rotatons are acheved wth certan sacrfce of the global dexterty ndex decreased about 33.46%.. Conclusons Best: Generaton Fg.. Convergent processes of GA = test_func ([.667,.4,.9 ]) epoch.. Dmenson In ths paper, a novel complant dual parallel platform for mcro postonng applcatons s desgned. The desgned manpulator has the advantages of smple structure, easy assembly and large workspace compared wth the exsted large-dsplacement FPMs for 3-DOF mcro manpulaton. The stffness model of the macro parallel mechansm wth wde-range flexure hnges s establshed. The roots of the equatons are obtaned by usng Newton Raphson method. Then the knematcs model s establshed for the mcro moton system. Referrng to the parameters of a 3-UPU prototype, the nverse knematc results and the parastc rotatons are gven on some checkng ponts n workspace. It s found that the parastc rotatons are zero when the movng Dmenson 3 Fg.. Convergent processes of PSO. Table 7 Optmzaton ratos of maxmal parastc rotatons. Partcle Dynamcs PSO Model: Common PSO Dmensons : 3 # of partcles : 24 Mnmze to : Unconstraned Functon : test_func Inerta Weght :.4 Green = Personal Bests Blue = Current Postons Red = Global Best Dsplacement (mm) Parastc rotaton (mrad) Rato (%) [ 2 ] ( ) 23.3 [ 2 ] ( ) 8.4 [ ] ( ) 7.39 [ ] ( ) 2.83 [ 2 2 ] ( ) 42. [2 2 ] ( ) 34.7 platform s movng along z-axs. If the movng platform departures from ntal poston along x-axs and y-axs, the maxmal parastc rotatons of the checkng ponts wll gradually ncrease, whch are about several mcro radans. Then the workspaces of macro and mcro motons are plotted. Fnally, a 3-DOF mcro parallel manpulator s optmzed by consderng the usable workspace of mcro moton, the parastc rotaton and the dexterty. Accordng to the objectve functon for optmzaton defned by a mxed performance ndex, an optmzaton result s derved by the optmzaton toolbox of MATAB usng multvarable nonlnear functon, GA and PSO method. Generally speakng, usng PSO method can acheve the best optmzaton results among the three methods, but the runtme for the PSO s much longer than the normal optmzaton method. For the optmzed parallel manpulator, the volume of the usable workspace and the maxmum parastc rotatons of test ponts are sgnfcantly mproved, but the best performances of the workspace and parastc rotatons are acheved after certan sacrfce of the global dexterty ndex. The nvestgatons of ths paper may provde suggestons for the structure optmzaton to acheve the features of both larger workspace and hgher moton precson. The results wll be useful n modfyng the structure of the prototype platform wth hgh dynamc propertes. A prototype of the upper 3-UPU parallel robot has been developed, and the expermental nvestgatons wll be carred out to verfy the postonng performance of the prototype. Acknowledgment Ths work s supported by the Macao Scence and Technology Development Fund under Grant no. 6/28/A and research commttee of Unversty of Macau under Grant no. U6/8-Y2/ EME/YM/FST. References [] Yang G, Ganes JA, Nelson BJ. Optomechatronc desgn of mcroassembly systems for manufacturng hybrd mcrosystems. IEEE Transactons on Industral Electroncs 2;2(4):3 23. [2] Ramadan AA, Takubo T, Mae Y, Oohara K, Ara T. Developmental process of a chopstck-lke hybrd-structure two-fngered mcromanpulator hand for 3-D manpulaton of mcroscopc objects. IEEE Transactons on Industral Electroncs 29;6(4):2 3. [3] Y, Xu Q. Desgn and analyss of a totally decoupled flexure-based XY parallel mcromanpulator. 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