Jeha Ryu* and Jongeun Cha. Jeha Ryu* is with Department of Mechatronics, Kwangju Institute of Science and Technology (K-JIST),

Transcription

1 Volumetrc Error Analyss and Archtecture Optmzaton for Accuracy of HexaSldeype Parallel Manpulators Jeha Ryu* and Jongeun Cha Jeha Ryu* s wth Department of Mechatroncs, Kwangju Insttute of Scence and echnology (K-JIS), Oryong-dong, Puk-gu, Kwangju 5-72, Republc of Korea (telephone: , e-mal: ryu@kjst.ac.kr). Jongeun Cha s wth Department of Mechatroncs, K-JIS(e-mal: gaecha@gegur.kjst.ac.kr). Abstract hs paper presents a volumetrc error analyss and an archtecture optmzaton method for accuracy of parallel manpulators. A comprehensve volumetrc knematc error model that relates all knematc error sources n the manpulator s archtecture to the pose errors of the end-effector s derved for HexaSlde (PUS) type parallel manpulators. he error model results n the total error transformaton matrx from whch three Error Amplfcaton Factors (EAF) are derved and used as desgn crtera for accuracy n the optmum desgn formulaton wth constrants on workspace and desgn varable lmts. hen, desgn optmzaton for accuracy has been performed by usng a nonlnear optmzaton technque. Optmzaton results have been valdated by Monte Carlo statstcal smulaton technque. Key Words: Parallel Manpulators, Accuracy, Error Analyss, Monte Carlo Smulaton, Optmum Desgn. O I. INRODUCION ptmal knematc desgn of robot manpulators s very mportant for achevng desred knematc performances. As parallel manpulators have advantages of hgher stffness, better accuracy, etc. over seral manpulators, optmal archtecture desgn of parallel manpulators, therefore, has drawn much attenton n recent years. In the optmal desgn, mzaton of the statc knematc performance measures such as dexterty, stffness, workspace, sotropy, accuracy, workng speed, and payload has been studed. Among them, dexterty has been consdered mportant because t s a measure of a manpulator s ablty to arbtrarly change ts poston and orentaton or to apply forces and torques n arbtrary drecton. Many researchers[-6], therefore, have performed desgn

2 optmzaton focusng on the dexterty of parallel manpulators by mnmzaton of the condton number of the Jacoban matrx. Workspace has also been consdered as an mportant desgn crteron n desgnng some parallel manpulators[7-] because they usually have smaller workspace compared to seral manpulators. Other researchers[2-5] consdered dexterty, workspace, and stffness smultaneously n the desgn of parallel manpulators. For example, Hong and Km[5] used a performance measure combnng the condton number wth the manpulablty ellpsod volume n desgnng a rapd prototypng machne called Eclpse M. Accuracy of a parallel manpulator s also an mportant desgn crteron for very fne moton tasks such as coordnate measurement, medcal operaton, machnng operaton, and mcro-manpulaton. Accuracy of a manpulator can be affected by many factors such as actuator control errors, nstallaton errors, manufacturng tolerances and clearances, archtecture desgn, etc. In order to nvestgate effects of manufacturng and actuaton errors on the platform pose accuracy, Wang and Masory[6] presented a knematc error model for a Stewart platform usng the D-H conventon. Ropponen and Ara[7] presented, for the modfed Stewart platform, the closed knematc error model that ncludes the jont poston and actuaton errors by dfferentatng the nverse knematc equaton. hey then computed the error ellpsods and the poston errors of the end-effector. Patel and Ehmann[8] derved a more complete knematc error model for the Stewart platform machne tool that accounts for all relevant error sources such as manufacture error, thermal and elastc deformaton error, control error, etc. hey then plotted error gan senstvty as a desgn tool for tolerance allocaton durng manufacture. B et al. [9] presented an accuracy analyss for a seral-parallel mcromoton manpulator and derved error effects that are useful for the desgn and manufacture of the manpulator. Km and Cho[2] presented forward and nverse error bound analyses as well as assocated egenvalue problems for the Stewart platform. X and Mechefske[2] derved an error model for hexapods wth fxed leg-lengths only focusng on the effect of the actuaton errors and evaluated the platform pose errors by applyng several matrx norms. hey observed that the error transmsson between the movng axes and the movng platform s closely related to the condton number of the Jacoban matrx. Pasek[22] analyzed the manpulator s accuracy based on the Monte Carlo statstcal experment approach and a Jacoban approxmaton. he aforementoned research about the accuracy, however, s manly for evaluatng the end-effector pose errors based on some knematc error models. Whle there have been many researches on the desgn optmzaton for dexterty, sotropy, load carryng capablty, workng speed, and workspace, there are, however, only a few researches on desgn optmzaton for accuracy. For 2

3 achevng good accuracy, we present an archtecture desgn optmzaton method for parallel manpulators n ths paper. Our approach s based on a comprehensve volumetrc error model that ncludes all knematc error sources such as actuaton, nstallaton, and manufacturng errors. he error model s derved from a dfferental nverse knematc equaton and a total error transformaton matrx rather than the Jacoban matrx s derved for relatng all error sources to the end-effector pose error. A detaled geometrc nterpretaton s gven to each error source that ncludes the actuator axs straghtness error. hen, as the man performance crteron, global error amplfcaton factors (EAF) that are derved from the sngular value decomposton of the total error transformaton matrx are used as desgn crtera for accuracy n the desgn optmzaton. he optmzaton problem s then formulated consderng constrants on workspace and desgn varable lmts. We appled the proposed desgn optmzaton technque, as an example, to a HexaSlde (PUS) type hexapod manpulator that has sx movng slders on the ground and sx fxed leg-lengths connected to a platform. he Monte Carlo smulaton evaluates the accuracy mprovement by the proposed desgn optmzaton method. hs paper s organzed as follows; n secton II, the geometry and nomenclature of the HexaSlde manpulator are descrbed. Secton III derves a comprehensve volumetrc model that ncludes all error sources. From the error model, a total error transformaton matrx s defned for optmal desgn formulaton. In secton IV, accuracy desgn crtera n terms of error amplfcaton factors are defned from the sngular value decomposton of the total error transformaton matrx. In secton V, an optmzaton problem s formulated for the accuracy desgn. In secton VI, the results of the optmzaton and valdaton by Monte Carlo smulaton are presented. In secton VII, conclusons are gven. II. DESCRIPION OF HE HSM A HexaSlde manpulator (HSM) shown n Fg. s composed of sx constant-length lnks, a moble platform, and sx prsmatc actuators fxed to the base. he lnks are connected to the actuators and to the platform by unversal and sphercal jonts par. hs manpulator controls the pose of the platform by movng the sphercal jonts A, from A, to, A wth sx actuators. he base coordnate frame XYZ s placed on the center O of the base. he orgn of the platform coordnate frame xyz s attached to the center C of the movng platform. Note that the pont C may be a tp of machnng tool or a tp of the touch probe n a coordnate measurng machne. All vectors and matrces wll be denoted n bold letters. he pose of the platform can be expressed by a poston vector C and a rotaton matrx R as 3

4 [ C C C] X, Y, z, C= X Y Z R = R θr ψ R φ () where θ, ψ, and φ are three Euler angles that are chosen wth respect to the global X, Y, and the local z axes, respectvely. III. VOLUMERIC ERROR ANALYSIS Knematc errors n the manpulator archtecture, such as manufacturng tolerances and clearances, jont nstallaton errors, and actuator control errors, can lead to the pose errors of the moble platform. In ths secton, a geometrc relatonshp between the knematc errors and the pose errors s derved. Fg. 2 shows the -th chan and knematc parameters of the HSM. In ths fgure, vectors a and n are the unt vectors along the actuaton and constant length lnk drectons, respectvely, and λ s the artcular varable that s the dstance from A, to A ponts. A closed-loop vector equaton of the -th chan s then gven by λ a = C RB A l n (2), where B s the poston vector from A dfferental error model s obtaned by dfferentatng Eq.(2) as B to C and s represented n the platform coordnate frame. δλ a + λ δ a = δ C δ RB Rδ B δ A lδ n δ ln (3), where the dervatve of the rotaton matrx s gven by δθ z δθ y δ R = δθ R = δθ z δθ xr (4) δθ y δθ x where = x y z δθ δθ δθ δθ s a vector of small rotatons. Usng Eq.(4), Eq.(3) can be rewrtten as 4

5 δλ a + λ δ a = δ C δ θ B Rδ B δ A lδ n δ ln (5), where B s represented n the base coordnate frame (.e., B = RB ). Multplcaton of n on both sdes of Eq.(5) gves ( ) l ( ) naδλ = nδ C+ n B δθ nrδb nδ A δ n λ δ a (6), snce nn and n δ n =. Consderng all 6 lnks, Eq.(6) can be represented n a matrx form as = J δx δλ MδL NδB N δa N δa (7) = where [ ] [ X Y Z ] C C C X Y z δ X= δc δ θ = δ δ δ δθ δθ δθ (8) [ ] δλ = δλ δλ δλ δλ δλ δλ (9) [ l l ] 6 δl = δ L δ () δ B δ B = R δ B 6 8 M () δ A δ A = R δ A6,, 8 M (2) λδa δ A = R λδ 6 a6 8 M (3) J ( n B ) n an an 6 6 = M M R (4) n6 ( n6 B6) an 6 6 an 6 6 5

6 L an M = M O M R L an (5) nr L an N = M O M R nr 6 L an (6) n L an N2 = M O M R n6 L an (7) Fg. 3 descrbes an error dagram that shows each knematc error source s contrbuton to the pose error of the platform. In ths fgure, crcles denote error boundary about the nomnal ponts, dotted lnes denote nomnal closed loops, and sold lnes denote actual closed loops due to knematc errors. Note that the terms shown n the fgure have the followng geometrc meanngs: δ X s the platform pose error vector, δ Λ s an actuator error vector that ncludes artcular varable control and backlash of the gearng, δ L s a length error vector of the lnks due to manufacturng tolerance and mperfect nstallaton between sphercal and unversal jonts, δ B and δ A are jont nstallaton error vectors of pont B and A,, respectvely, and δ A s an actuator axs straghtness error vector. he errors δ L, δ B, δ A and δ A are called nonactuaton errors that can be partly compensated n a controller model by some calbraton methods. If J s nvertble, Eq.(7) can be rewrtten as δx = Eδε (8) where 6 48 [ ] E= J I M M M N M N M N R (9) 2 2 6

7 48 M M M M (2) δε = δλ δl δb δa δa R where I s the 6 6 dentty matrx. he matrx E s defned as the total error transformaton matrx and the vector δ ε s defned as the error source vector. Important to note that Eq. (8) provdes transformaton of all error sources not only through the Jacoban matrx but also through other transformaton matrces, M, N, and N 2, all dependng on the confguraton of the manpulator. he dagonal matrces M, N, and N 2 mply geometrcally that nonactuaton errors (length, jont nstallaton, and actuator axs straghtness errors) are mapped nto actuator errors. hs mappng s ndependently done for each chan because of dagonal matrces. Notce also that when an,.e., when the actuator axs s perpendcular to the lnk axs, all matrces become sngular. Besdes, near sngular confguratons, platform pose errors are domnated more by nonactuaton errors than by actuator errors because nonactuaton errors are amplfed greatly by the / an factor n M, N, and N 2 matrces. herefore, very accurate knematc calbraton s requred for the nonactuaton errors. = IV. PERFORMANCE MEASURES ON ACCURACY hs secton dscusses transformaton propertes of the total error transformaton matrx E that transforms the error source vector δ ε to the platform pose error vector δ X. Based on ths dscusson, we wll deduce some performance measures on accuracy for the optmum archtecture desgn. o analyze the error output characterstcs of the transformaton equaton, the unt-norm error nputs are often used. However, the unt-norm error nputs may not permt the actual error range of each error source because the mum errors may sgnfcantly vary for dfferent sources. herefore, ndvdual error sources are normalzed usng the mum values of error sources as ) δλ = δλ δλ ) δl= δl δl ) δb = δb δb ) δa = δa δa ) δa = δa δa (2) 7

8 where ) over a vector denotes the normalzed vector and denotes the mum value of each error source that can be specfed by takng control and calbraton accuracy nto account. Note that the mum actuator axs straghtness error can be defned as A = a, where δ λ δ λ s the mum actuator length. Moreover, snce the Jacoban matrx J s dmensonally nhomogeneous, the concept of characterstc length[4] should be ntroduced for makng dmensonally homogeneous Jacoban matrx J ). hen, Eq. (8) can be rewrtten as δx = E ) δε ) (22) where ) ) ) ) ) ) E = δλ J M δl JM M δb JN M δa JN 2 M δa JN 2 ) ) ) ) ) ) δε δλ δl δb δa δa = M M M M Consder Eq. (22) as a general transformaton[23]: y = Ax (23) where x and y are n nput and m output vectors, respectvely, and A s a m n transformaton matrx. o nvestgate how the transformaton matrx A affects the magntude relatonshp between x and y, assume that x les nsde a unt sphere such that 2 x (24) then the transformed x les nsde an ellpsod defned by 2 + Ay (25) 8

9 where + A s a pseudonverse of A. he matrx A can be decomposed nto three matrces as A = UΣV (26) where U = [ u L u ] and V = [ v v ] m L are the egenvector matrces of n AA and AA, respectvely, and ( L σ ) s a matrx composed of sngular values of A ( 2 L ). Usng Eq.(26), Σ = dag σ, σ2,, m the pseudonverse of A can be wrtten as σ σ σ m + + A = VΣ U (27) where + dag ( σ, σ 2,, σ m ) Σ = L s a pseudonverse of Σ. Insertng Eq.(27) nto Eq.(25) gves ( )( ) ( Uy) dag ( σ σ2 L σ m )( Uy) VΣ Uy = yuσ V VΣ Uy =,,, (28) Eq.(28) depcts that y les nsde an error ellpsod whose prncpal axes are σ,, u L σmum. From the above dervaton, the error transformaton can be characterzed manly by sngular values that can be regarded as error amplfcaton factors (EAF). hree dfferent error amplfcaton factors have been consdered n the desgn of robot manpulators; () the volume of the error ellpsod defned n Eq.(28), () the mum sngular value, () the condton number. he volume of the error ellpsod has smlar concept as the manpulablty measure proposed by Yoshkawa[24], whch s proportonal to the ellpsod volume of a manpulator Jacoban matrx. he mum sngular value can be understood as the mum error amplfcaton factor. he condton number can be understood as a measure of the relatve error amplfcaton of the computed results upon solvng a lnear system of equatons assocated wth that matrx. hs number also characterzes sotropy (shape of the error ellpsod). From the total error transformaton matrx E ), three error amplfcaton factors can therefore be defned as 9

10 EAF )) det ( ) 6 = ( EE ) = σ ) E = (29) EAF = ( σ ) E ) (3) 2 EAF σ ) σ ) (3) 3 = ( ) /( ) E E mn where ( σ ) E s the sngular value of the total error transformaton matrx E ). he smaller the EAF, the smaller the ) pose error. herefore, one of the EAFs can be used as a crteron on accuracy for optmum desgn problem. Because all EAFs are pose-dependent, global EAFs over the whole workspace, whch are pose-ndependent, should be used for global optmzaton of the accuracy. However, n the most actual operaton, the manpulator s end-effector does not span the entre workspace because the sngular confguratons are very close to the workspace boundary. herefore, a central regon around a nomnal pose may be consdered n desgn optmzaton and EAF should be averaged n the central regon of the workspace[2]. he averaged global EAF for accuracy crteron can then be defned as GEAF EAFdV V = (32) V where V s the volume of the chosen central regon. V. FORMULAION OF HE OPIMIZAION PROBLEM In ths secton, an optmzaton problem s formulated for the accurate archtecture desgn that reveals mnmum platform pose error for gven knematc errors. As dscussed n secton IV, GEAF s selected as a cost functon and s to be mnmzed. Regardng constrants, we ncluded the desred workspace constrant and lmts on the desgn varables. o completely determne the HSM geometry, sxty desgn varables ( A,, A,, B, l ) should be determned. In order to smplfy the desgn problem, however, we reduced the number of the desgn varables by explotng the axs symmetrcty of the manpulator. It s assumed that A, and A, ponts are located on crcles. Fg. 4 shows the reduced set of desgn varables where R A and R B are the rad of the crcles passng through, A ponts and B

11 ponts, respectvely, H s the dstance between adjacent A, ponts, α and β are the horzontal and vertcal slant angles of the prsmatc actuator ral axes, λ s the mum actuaton length from, A to A,, l s the constant lnk length, and ψ s the separaton angle between adjacent B ponts. hese eght desgn parameters ( R A, R B, H, α, β, λ, l, ψ ) can completely defne the symmetrc geometry of the HSM. However, for further smplfcaton, R A and λ are predetermned to suffcently large values that can nclude translatonal workspace requrements n our desgn process. herefore, the number of fnal desgn varables s sx. Notce that the platform radus R B s used as a characterstc length that makes the Jacoban matrx dmensonally homogeneous. For normalzaton n Eq. (2), we used the mum error budget n able. Note that the mum error budget for nonactuaton errors can be chosen by consderng the mum allowable error after accurate knematc calbraton. he central regon we chose s a parallelepped(see Fg. 4(b)) whose center les at one thrd, from the bottom, of the dstance between the upper and the lower lmts of the whole translatonal workspace. In addton, for smplcty of desgn and analyss, the global EAF s averaged only at the center and at each of the eght vertces of the parallelepped workspace whle the orentaton of the platform s changng between the mum and mnmum orentaton angles at each pont. he optmzaton problem can then be formulated as Mnmze GEAF subject to Desgn varables lmts (33) Workspace constrants he fmncon functon n MALAB 5.2 Optmzaton oolbox[25] was used to solve the optmzaton problem. hs mnmzaton functon uses a Sequental Quadratc Programmng (SQP) method, n whch a Quadratc Programmng (QP) subproblem s solved at each teraton. In ths mnmzaton algorthm, an estmate of the Hessan of the Lagrangan s updated at each teraton usng the BFGS formula. VI. OPIMIZAION RESULS For three EAFs as dscussed n secton IV, Fg. 5 shows the ntal and optmzed archtectures of the HSM, whle able 2 shows the numercal values of the desgn parameters. he correspondng translatonal workspaces are shown n Fg. 6. As are seen n those fgures and able 2, the error ellpsod volume and the mum sngular value crtera produced smlar optmal archtectures and workspaces whle the condton number crteron generated

12 dfferent optmal archtecture (Desgn parameters for H and α are at ther mnmum lmts and R B has a smaller value than that from other two crtera). In order to estmate how much the pose error of the platform s reduced from the ntal desgn by the proposed optmzaton method, the manpulator s accuracy has been evaluated by the Monte Carlo statstcal experment approach[2]. In the Monte Carlo smulaton, we generated random vectors for δ B, δ A, δ A wth unform error dstrbuton nsde an error sphere wth the center at the nomnal locaton of a pont and wth a radus that s the mum allowable nstallaton error, say μm radus (See Fg. 7). For δ Λ and δ L, we assumed a mum allowable steady state control error and a lnk manufacturng tolerance. All of these length errors are assumed to follow unform dstrbuton wth mum lmts specfed by an error budget gude n able. he resultant pose errors are determned by the error model n Eq.(8). he average pose error over the central parallelepped workspace s obtaned at the ponts where the objectve functon was averaged. he average value s obtaned by 5 random values for meanngful Monte Carlo smulaton[2]. We selected three tmes of standard devaton as the mum pose error that are shown n able 3 for the ntal and the optmzed desgn. From able 3, the followng results can be observed; () the poston errors n the X and Y drectons are sgnfcantly reduced by about 4-56% for all EAFs, whle the poston error n Z drecton s slghtly reduced (about 9-7%). () he orentaton errors n all drectons are sgnfcantly reduced by about 8% for all EAFs. () he error ellpsod volume and the mum sngular value crtera generated less orentaton errors than poston errors and showed smlar results n terms of level of accuracy mprovement. Meanwhle, the condton number crteron reduced poston errors more than orentaton errors. VII. CONCLUSIONS In ths paper, an optmal accuracy desgn based on the volumetrc error model had been performed for a HexaSlde manpulator. Possble knematc errors were dentfed and geometrcally nterpreted from the dfferental error model and the pose error of the platform was obtaned from the knematc error sources through the total error transformaton matrx. hen usng the error equaton, three dfferent error amplfcaton factors were used as desgn crtera for accuracy. A desgn optmzaton of the HSM had been performed by mnmzng the error amplfcaton factors wth consderng the workspace and desgn varable lmts. Fnally the pose error was evaluated by Monte Carlo smulaton technque and t was shown that the optmzed manpulator had substantally smaller errors 2

13 compared wth the ntal desgn. ACKNOWLEDGEMENS hs work was supported by the Bran Korea2 (BK2) Project n 22. REFERENCES [] C. Gosseln and J. Angeles, he Optmum Knematc Desgn of a Sphercal hree-degree-of-freedom Parallel Manpulator, ASME rans. J. Mech., ransmssons., automat. Desgn, Vol., No. 2, pp , 989 [2] C. Gosseln and J. Angeles, A Global Performance Index for the Knematc Optmzaton of Robotc Manpulators, ASME rans. J. Mech Desgn, Vol. 3, No. 3, pp , 99 [3] R. Kurtz and V. Hayward, Multple-Goal Knematc Optmzaton of a Parallel Sphercal Mechansm wth Actuator Redundancy, IEEE rans. Robotcs Automat., Vol. 8, No. 5, pp , 992 [4] O. Ma and J. Angeles, Optmum Archtecture Desgn of Platform Manpulator, Proc. IEEE Int. Conf. on Robotcs Automat., pp. 3-35, June, 99 [5] K. H. Pttens and R. P. Podhorodesk, A Famly of Stewart Platforms wth Optmal Dexterty, J. Robotc Sys., (4): , June, 993 [6] K. E. Zanganeh and J. Angeles, Knematc Isotropy and the Optmum Desgn of Parallel Manpulators, Int. J. Robotcs Res., Vol. 6, No. 2, pp , 997 [7] J.-P. Merlet, Desgnng a Parallel Manpulator for a Specfc Workspace, Int. J. Robotcs Res., Vol. 6, No. 4, pp , 997 [8] R. Boudreau and C. M. Gosseln, he Synthess of Planar Parallel Manpulators wth a Genetc Algorthm, J. Mech. Desgn., Vol. 2, No. 4, pp , 999 [9] X. J. Lu, J. S. Wang and F. Gao, On the Optmum Desgn of Planar 3-dof Parallel Manpulators wth respect to the Workspace, Proc. IEEE Int. Conf. on Robotcs Automan.. pp , 2 [] M. Ceccarell and E. Ottavano, An Analytcal Desgn for CAPAMAN wth Prescrbed Poston and Orentaton, Proc. DEC, pp. -8, Baltmore, Maryland, Sept, -3, 2 [] J. Schönherr, Evaluaton and Optmum Desgn of Parallel Manpulators havng Defned Workspace, Proc. DECC, pp. -9, Baltmore, Maryland, Sept, -3, 2 [2] R. S. Stoughton and. Ara, A Modfed Stewart Platform Manpulator wth mproved Dexterty, IEEE rans. Robotcs Automat., Vol. 9, No., pp , 993 [3] A. remblay and L. Baron, Geometrcal Synthess of Star-Lke opology Parallel Manpulators wth a Genetc Algorthm, Proc. IEEE Int. Conf. on Robotcs Automat., pp , 999 [4] X. J. Lu, Z. L. Jn and F. Gao, Optmum Desgn of 3-dof Sphercal Parallel Manpulators wth respect to the Condtonng and Stffness ndces, Mech. Mach. heory, Vol. 35, No. 9, pp , 2 3

14 [5] K-S Hong and J-G Km, Manpulablty Analyss of a Parallel Machne ool: Applcaton to Optmal Lnk Length Desgn, J. Robotc. Sys., Vol. 7, No. 8, pp , 2 [6] J. Wang and O. Masory, On the Accuracy of a Stewart Platform Part I he Effect of Manufacturng olerances, Proc. IEEE Int. Conf. on Robotcs Automat., pp. 4-2, 993. [7]. Ropponen and. Ara, Accuracy Analyss of Modfed Stewart Platform Manpulator, Proc.995 IEEE Int. Conf. Robotcs Automat., :52-525, 995 [8] A. J. Patel and K. F. Ehmann, Volumetrc Error of a Stewart Platform-based Machne ool, Annals CIRP, Vol. 46, pp , 997 [9] S. B, G. Zong, R. Lu and S. Wang, Accuracy Analyss of the Seral-Parallel Mcromoton Manpulator, 997 IEEE Int. Conf. Computatonal Cybernetcs and Smulaton, Vol. 3, pp [2] H-S. Km and Y-J. Cho, he Knematc Error Bound Analyss of the Stewart Platform, J. Robotcs Systems, Vol. 7, No., pp , 2. [2] F. X and C. Mechefske, Modelng and Analyss of Errors n Hexapods wth Fxed-Leg Lengths, PKM 2 Conf., pp , U. of Mchgan, Sept, 4-5, 2 [22] Z. J. Pasek, Statstcal Approach to PKM Accuracy Analyss, PKM 2 Conf., pp , U. of Mchgan, Sept, 4-5, 2 [23] G. Strang, Lnear Algebra and Its Applcaton, Academc Press, 98. [24]. Yoshkawa, Manpulablty of Robotc Mechansms Int. J. Robotcs Res., Vol. 4, No. 2, pp. 3-9, 985 [25] MALAB Optmzaton oolbox User s Gude Verson 2 4

15 (a) Overvew A 3, A 4, Z A 2, A, O A 2 A Y X A 3, A 2, A, A 3 A 6, A4 A 4, A A 5 5, A 6 base A 6, A 5, B z 3 x B 4 moble platform B 2 C o y B5 B B 6 (b) Schematc Dagram Fg. HexaSlde Parallel Manpulator A, A, O a λ n A A, l C ' RB B C Fg. 2 Schematc Dagram of One Chan 6

16 δ A, O δλ δa = λδa δ l δ X δ B Fg. 3 Error Dagram 7

17 β H l R B ψ A3, A4, α λ cos β A 3, Y A 4, R A O X (a) (b) Fg. 4 (a) Desgn Varables, (b) Workspace constrant 8

18 z z z z y x 5 y x 5 (a) Intal Desgn (b) Volume y x 5 y x 5 (c)max. Sngular Value (d) Condton No. Fg. 5 Intal and Optmum Archtectures 9

19 (a) Intal Desgn (b) Volume (c) Max. Sngular Value (d) Condton No. Fg. 6 ranslatonal Workspaces at Intal and Optmum Desgns δ A, δ A, Fg. 7 Allowable Installaton Error Sphere 2

20 ABLE MAXIMUM ERROR BUDGE (μm) δ B δλ δ l δ A δ A ABLE 2 INIIAL AND OPIMUM DESIGN VALUES Intal Volume Optmum Max.Snular Value Condton No. H (mm) β ( ) ψ ( ) l (mm) R B (mm) α ( ) Desgn Varable Lmts : 7 H, β 9, ψ 5.56, l, R B, 8 α 8 2

21 ABLE 3 ACCURACY COMPARISON Optmum Intal Volume Max.Snular Value Condton No. δ x (μm) δ y (μm) δ z (μm) δθ x ( ) δθ x ( ) δθ z ( )