A Structural Model for Loading Analysis of a Hexapod Platform

Transcription

1 A Structura Mode for Loadng Anass of a Heapod Patform ABSTRACT *Hongang Sh, Yu She, and Xuechao Duan The Oho State Unverst, Coumbus, OH, USA Xdan Unverst, Chna *sh.7@osu.edu Ths paper proposes a structura anass of a heapod stage usng the Screw Theor. Ths Natona Insttute of Standards and Technoog (NIST) feura heapod stage s actuated b three panar stages and t can acheve hgh-precson motons n degrees of freedom (DOF). In ths paper, we present an anatca mode for dervng the stffness matr of the patform wth respect to the eterna oadng on the top patform. Ths smboc knematc mode s based on the geometrc parameters, the matera propertes and the topoog of the patform. Ths mode s used for structura anass and poston contro of the top patform where the eterna oadng s paced. KYWORD: moton stage, feure mechansm, heapod patform, stffness, knematc mode INTRODUCTION A nanopostoner s a hgh precson postonng devce used n moton contro wth nanometer precson. Most nanopostoners are made of feure mechansms, Awtar et a. (7), Venkteswaran and Su (), that are formed b mutpe (often dentca) feure pvots, eaf sprngs, or ther chans that are desgned to produce a defned moton upon the appcaton of an approprate oad, Lobontu (). These mechansms have the advantages of no backash and utra-hgh precson. Nanopostoners, Wu et a. (8), Sh and Su (), have been wde used n precson engneerng and pa an mportant roe n emergng nanotechnoog and medcne, Sh et a. () and Sh et a. (). The modeng and stffness anass of the parae heapod mechansm have been chaengng tasks due to ther nherent structura compet and dffcutes n modeng the moton of degrees of freedom (DOF), Da et a. (). A ot of pror work has been done regardng the stffness anass of DOF stages. F smuaton and phsca eperments are the two common used methods for anang the DOF stffness of a nanopostoner. Brouwer et a. () modeed and derved the stffness of a DOF manpuator b usng software SPACAR. Yang et a. () and Yang et a. () deveoped a method to measure n-pane stffness of a nanopostoner b usng atomc force mcroscop (AFM) and measured the stffness of the heapod nanopostoner. When beng compared wth the anatca modes, these methods are tme consumng, cost, and neffcent n the desgn process. There s reatve ess work done n the dervaton of the anatca DOF stffness matr due to the structura compet. Sh

2 and Su () but an anatca mode of the workspace of a meso-scae heapod nanopostoner based on the stffness of feures and nverse knematcs. J et a. () desgned a DOF nanopostoner and derved a contro mode based on the stffness, the comb drve force, and the ntegrated capactve dspacement sensor. Sh et a. () but a stffness based anatca contro mode for a MMS heapod nanopostoner. However, there s ess work done reated to the anass of appng eterna oadng on a heapod mechansm. In ths paper, we adopt the Screw Theor to derve a smboc mode of the DOF stffness matr of the heapod nanopostoner wth respect to the eterna oadng. Ths macro-scae monothc heapod nanopostoner was prevous but b the Natona Insttute of Standards and Technoog (NIST), USA. The dervaton foows a bottom-up procedure, whch starts from the three ocked actuatng stages to the end effector, the center of the top patform. Based on the mode, we anae the reatonshp between the dspacement of the top patform and the eterna oadng. The rest of the paper s organed as foows. Secton descrbes the geometrc and matera propertes of the heapod patform. Secton presents the Screw Theor and the stffness matr of snge wre feure. In secton, we ustrate the topoog of the heapod patform accordng to the eterna oadng. The anatca mode of the eterna oadng s derved and shown wth an eampe. The concuson s stated n Secton. KINMATIC MODL As shown n Fg., the heapod moton stage s composed of three man parts: base stages, struts, and top patform. Three X-Y mcro-postonng base stages, whch can generate moton n two orthogona drectons, are smmetrca postoned on the base pane. Fgure : Settng of the DOF NIST heapod moton stage.

3 In ths stud, we consder the near actuator s rgd enough and t s ocked before provdng near moton. We assume the three bottom stages are fu constraned n the pane so that the dspacement of the top patform s on caused b the deformaton of the compant parts. As shown n Fg., we bud the coordnate sstem on the center of the top surface of the top patform. The eterna forces and moments n three drectons are apped on the coordnate orgn. Fgure : Schematc drawng of the heapod mechansm For convenence, we defne the foowng parameters for descrbng the geometr of the knematc mode. The struts have a tota ength L and dameter D, and have a short feure jont of ength and dameter d at each end. Fgure shows the geometrca reatonshp of the tweve ponts. We denote the poston of the s ponts at the top patform and the s at the base stages b A, and b B, respectve. The dstance between the neghborng ntersectng ponts of the struts at the top patform s C. The dstance between the non-neghbor ntersectng ponts of the struts at the top patform s C. For the base stages, the dstance between the neghborng ntersectng ponts of the struts s C. The dstance between the non-neghborng ntersectng ponts of the struts at the base s C. These ponts on the movng patform and the stages can be descrbed n the goba coordnate frame as r a A = Z( ), =,,, t r b B = Z( ), =,,, H t ()

4 where Z s the -b- rotaton matr about the as. r a and r b are the rad of the strut attachment ponts (bottom pates n home poston). H s the heght of the heapod mechansm from the bottom stage to the top pane of the patform and t s derved as T the unknown b sovng equaton A B A B L. t s the thckness of the top patform. Anges, and, are tabuated n Tabe. A struts are made of aumnum wth Young s moduus of eastct = GPa, and Posson s rato =.8. Fgure : Geometrca aout of the feure center of the the top patform and the bottom stages. STIFFNSS MATRIX OF WIR FLXUR In order to anae the mpact of the eterna oad apped on the devce, we need to determne the reatonshp between deformaton and the oadng. Here, we app the Screw Theor approach to derve the stffness matr and anae the moton of the top patform. Ths methodoog s presented n Su et a. (), Su et a. () and Sh (). For convenence, a bref descrpton of ths approach s gven beow. We denote the deformaton of a feure mechansm b a genera twst,,,,, W ˆ F, F, F, W, W, W. ˆ and the oad s denoted b a wrench T The are reated b, W ˆ = K Tˆ, Tˆ= C Wˆ, C K = I, () where K and C are s b s stffness and compance matrces, respectve.

5 Fgure : Modeng of a wre feure based on the Screw Theor As shown n Fg., a feure wth crcuar cross secton s fed on one end. The coordnate sstem s but on the center of the free end. The stffness matr of t can be wrtten as I K w, () where / d, /, are non-dmensona constants determned b the geometres and the matera propertes. / I d s the area moments of nerta about as. s Young s modue and =.8 s the Posson s rato. After substtutng the parameters n Tabe, we obtaned the compance matr as = \ K C w w ()

6 Tabe. Geometrc dmensons of the heapod mechansm ra.9mm, rb 8.9mm, t.mm, H.mm, L mm, mm D mm, d.mm, c mm, c mm, c mm, c mm = 89, = 7, = 9, =, =, = =, =, =, =, =, = 9 STIFFNSS ANALYSIS OF TH NIST HXAPOD NANOPOSITIONR In ths secton, we derve the stffness matr based on the topoog of the mechansm accordng to the Screw Theor. Fgure : Schematc drawng of one strut. The compance matr of each wre feure s denoted b C w, shown n q. (). The compance matr of each strut s denoted as C s. As shown n Fg., the strut s modeed as a sera chan of a cndrca rod wth dameter D and two wre feure jonts wth dameter d at both ends. B the equaton of sera feure chan, we derve mathematca the overa compance matr of a sera feure chan as, s C = Ad C Ad, () where Ad s the so-caed -b- adjont transformaton matr, = R Ad =. () DR R Here R s a -b- rotaton matr. D s the skew-smmetrc matr defned b a transatona vector d. B q. () and Tabe, we obtan C = C = C w. As shown n Fg., the compance of the compant cndrca rod C s obtaned b substtutng the dameter D and ength L nto q. (). The three adjont matrces are defned b

7 ,,). (,,,), (, (,,),, = L d I R d I R d I R (7) Agan, we can obtan the stffness matr b substtutng the parameters n Tabe = C K s s (8) The struts are connected to the top patform n parae. The overa stffness matr of s parae feure chan s cacuated as = = s Ad K Ad K, (9) where the adjont matrces are the coordnate transformaton of each strut. As shown n Fg. 7, the coordnate transformaton of No. and No. struts to the center of the top patform s cacuated b three steps, so the adjont matrces are defned b the mutp of three matrces. No. and No. struts are obtaned b a rotaton of No.. No. and No. struts are obtaned b rotaton of No.. Therefore, the adjont matrces of these struts can be wrtten as. Fgure 7. Coordnate transformatons of the dervaton

8 Ad Ad Ad Ad Ad Ad = Ad = Ad = Ad = Ad = Ad = Ad Ad Ad Ad Ad Ad,,, Ad, Ad Ad,, () where the rotaton matrces and transatona vectors of jont matrces are defned b = Y ( ), d Z( X ( / ), d = Z( / ), d = Z( / ), d = Y ( ), d Z( / ), d / ), d X ( / ), d = Z( / ), d = Z( / ), d (,,), ( (c (,,), (,,), (,,), ( (c (,,), (,,), (,,), (,,), c ) Tan( / ) /, c c ) Tan( / ) /, c /, t), /, t), () where the speca rotaton anges are defned as = arcsn (( c c )/ L), = arccos( H/ Lcos ). () B substtutng the parameters n Tabe, we derved the stffness matr as K. () The compance matr s the nverse matr of the stffness matr and t s

9 C. () Here, we denote the oadng on the top patform b a wrench, Ŵ L = ( F, F, F, M, M, M ). () B q. (), the deformaton caused b the wrench can be derved as Tˆ L L CŴ = (,,,,, ). () For eampe, when the top patform s oaded wth a target sampe and ts weght s kg, the oadng wrst can be wrtten as Ŵ L = (,, 9.8,,,). (7) From q. (), the moton caused b the target sampe s Tˆ L = (,,,,,. ). (8) Ths means the heapod moves.- mm n the opposte drecton from the orgna poston before the heapod s actuated b the three panar stages. CONCLUSIONS A modeng s proposed for the dervaton of the DOF stffness matr of a heapod nanopostoner wth respect to the eterna oadngs. The dervaton process can be apped to the stffness anass of other DOF compant mechansms. B means of the Screw Theor, the topoog s anaed and the anatca mode s derved based on the geometrc dmensons, and matera propertes. Ths mode s used for budng the stffness-based contro. When combng the patform moton due to the eterna oadng and the poston contro from the three actuaton stages, a forward knematc contro can be derved, especa when poston sensors are not avaabe for feedback contro. The mode s aso usefu for the dervaton of the aowed oadng space wth respect to the workspace of the heapod nanopostoner. RFRNCS Brouwer, D. and Jong, B. de and Soemers, H. (), Desgn and modeng of a s DOFs MMS-based precson manpuator, Precson ngneerng, Vo. (), pp. 7 9.

10 Da, J. S. (), uer Rodrgues formua varatons, quaternon conjugaton and ntrnsc connectons, Mechansm and Machne Theor, Vo. 9, pp.. J, L. and Zhu, Y. and Moheman, S. O. R. and Yuce, M. (), A mcromachned DOF nanopostoner wth ntegrated capactve dspacement sensor, Proceedng of I Sensors, pp. 7. N. Lobontu (), Compant mechansms: desgn of feure hnges, CRC Press. S. Awtar, A. H. Socum,. Sevncer (7), Characterstcs of beam-based feure modues, ASM Journa of Mechanca Desgn, Vo. 9 (), pp. 9. Sh, H. (), Modeng and anass of compant mechansms for desgnng nanopostoners, Ph.D. dssertaton, The Oho State Unverst, Coumbus, OH, USA. Sh, H. and Duan, X. and Su, H.-J. (), Optmaton of the workspace of a MMS heapod nanopostoner usng an adaptve genetc agorthm, n Robotcs and Automaton (ICRA), I Internatona Conference on, Hong Kong, Ma - June 7, pp. 8. Sh, H. and Su, H.-J. (), Workspace of a feure heapod nanopostoner, n Proceedngs of ASM IDTC/CI, Chcago, Inos, August -. Sh, H. and Su, H.-J. and Dagaaks, N. and Kramar, J. A. (), Knematc modeng and cabraton of a feure based heapod nanopostoner, Precson ngneerng, Vo. 7(), pp Sh, H. and Su, H.-J. (), An anatca mode for cacuatng the workspace of a feure heapod nanopostoner, ASM Journa of Mechansms and Robotcs, Vo. (), p. 9. Sh, H. and Su, H.-J. and Dagaaks, N. (), A stffness mode for contro and anass of a MMS heapod nanopostoner, Mechansm and Machne Theor, Vo. 8, pp. -. Su, H.-J. and Sh, H. and Yu, J. (), Anatca compance anass and snthess of feure mechansms, n Proceedngs of ASM IDTC/CI, Washngton, DC, August 8-. Su, H.-J. and Sh, H. and Yu, J. (), A smboc formuaton for anatca compance anass and snthess of feure mechansms, ASM Journa of Mechanca Desgn, Vo.(), p. 9. T. Wu, J. Chen, S. Chang (8), A s-dof prsmatc-spherca-spherca parae compant nanopostoner, I Transactons on Utrasoncs, Ferroeectrcs and Frequenc Contro, Vo. (), pp.. Venkteswaran, V.K. and Su. H.-J. (), A parameter optmaton framework for determnng the pseudo-rgd-bod mode of cantever-beams, Precson ngneerng, Vo. (),. Yang, S. H. and Km, Y. and Purushotham, K. P. and Yoo, J.-M. and Cho, Y.-M. and Dagaaks, N. (), AFM characteraton of nanopostoner n-pane stffnesses, Sensors and Actuators A: Phsca, Vo. (), pp Yang, S. H. and Km, Y.-S. and Yoo, J.-M. and Dagaaks, N. G. (), Mcroeectromechanca sstems based Stewart patform wth sub-nano resouton, Apped Phscs Letters, Vo. (), pp. 99.