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International Journal of Machine Tools & Manufacture 43 (2003) 15611569Sensitivity analysis of the 3-PRS parallel kinematic spindleplatform of a serial-parallel machine toolKuang-Chao Fana, Hai Wangb, Jun-Wei Zhaoc, Tsan-Hwei ChangdaDepartment of Mechanical Engineering, National Taiwan University, 1, Sec. 4, Roosevelt Road, Taipei, Taiwan, ROCbDepartment of Mechanical Engineering, Ming-Chi Institute of Technology, Taipei, Taiwan, ROCcSchool of Mechanical Engineering, Jiao Zhou College of Engineering, Henan, ChinadMechanical Industry Research Laboratory, ITRI, Taichung, Taiwan, ROCReceived 1 April 2003; accepted 9 July 2003AbstractThe special configuration of a serial-parallel type machine tool possesses five degrees of freedom and provides more flexibilityin NC machining. The parallel spindle platform plays the key role in manipulating three directions of movement. In this study, theinverse kinematics analysis of the platform is derived first. A sensitivity model of the spindle platform subject to the structureparameters is then proposed and analyzed. All the parameters influencing the position and orientation of the spindle platform arediscussed based on the sensitivity model and the simulated examples. Critical parameters to the accuracy of the cutter location canthen be found, which are the slider position, the strut length, and the tool length. 2003 Elsevier Ltd. All rights reserved.Keywords: Kinematic analysis; Sensitivity analysis; 3-PRS spindle platform; Serial-parallel machine tool1. IntroductionTo obtain higher accuracy and greater dexterity,machine tool manufacturers are developing parallel typestructures for the next generation of machine tools. It isalways a goal to pursue an interesting development inthe machine tool industry that holds great promise forimproving accuracy. Machine tools with their high rigid-ity and accuracy are ideally suited for precision appli-cations. Parallel manipulator offers a radically differenttype of machine structure relative to the traditional serialtype machines. It is believed that the inherent mechan-ical structure of the parallel type machines provides highdexterity, stiffness, accuracy and speed compared to theconventional multi-axis structure 1,2.In the past, many comprehensive studies and workshave been made in the area of parallel manipulators. Thekinematic behavior of the parallel mechanism withextensible strut has been discussed in many literaturesCorresponding author. Tel.: +02-23620032; fax: +02-23641186.E-mail address: .tw (K.-C. Fan).0890-6955/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0890-6955(03)00202-535. Some reports dealt with the kinematics of three-legged parallel manipulators 6,7. Most of these articlesfocused on the discussion of both the analytical and thenumerical methods to solve the kinematics of pure-paral-lel mechanisms 68. Some papers also discussed theaccuracy analysis of the Stewart platform manipulators911. A new type of serial-parallel machine tool hasbeen developed by the Mechanical Laboratory of ITRIin Taiwan. Simulation analysis and experimental testshave shown the necessity of its accuracy enhancement1214.Even though many researches have made contri-butions to the theory and design of the parallel kinematicmechanism, however, commercial applications of thispotential parallel machine have not been widespread yet.The main obstacle to the applications of this kind ofmachine tool is their unsatisfactory accuracy. Similar tothe traditional machine tool, there are several major fac-tors, such as the structural imperfection, the elasticdeformation and the thermal deformation, that willdegrade the machine accuracy. In order to find out theessential error sources, sensitivity analysis of the spindleplatform movement and error identification are neces-1562K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569sary for the purpose of accuracy enhancement of themachine tool.In this paper, the kinematics model of parallel mech-anism is derived in the first part. Sensitivity model ofthe spindle platform subject to structural and kinematicparameters is then proposed. All parameters influencingthe position and orientation of the spindle platform areanalytically estimated. Simulated example are given toidentify the most critical error sources.2. Machine tool configurationThe machine tool under investigation is called a serial-parallel type machine tool. The configuration is shownin Fig. 1. It consists of a three-degree-of-freedom spindleplatform and a conventional two-degree-of-freedom XY table to form a five-axis structure. The spindle isassembled in the platform, which is connected to threestruts of constant length by means of ball joints (or U-joint) that are equally spaced at a nominal angle of 120.The other end of each strut is connected to a slider witha rotational joint. Each slider can move up and downalong the corresponding vertical slideway fixed to a col-umn that is also spaced at nominal 120 angle from oneanother. The platform, such constructed, has one linearmotion in the Z-axis and two angular rotations (a andb) in the X- and Y-axes, respectively. The XY table thatsupports the workpiece provides the linear motions ofthe workpiece in two horizontal directions. The featuresof this configuration are easy to manufacture and control,Fig. 1.Structure of the serial-parallel type machine tool.more accurate and larger workspace, compared to theStewart platform-based parallel machine tool.3. Inverse kinematic analysisIn order to analyze the kinematics of the parallelmechanism,threerelativecoordinateframesareassigned, as shown in Fig. 2. A static Cartesian coordi-nate frame XYZ is fixed at the base of the machine toolwith the Z-axis pointing to the vertical direction, the X-axis pointing toward B1,and the Y-axis pointing alongthe B2B3line. The movable Cartesian coordinate frame,XYZ, is fixed at the center of the XY table with thesame axes directions as the XYZ coordinate frame. Thethird coordinate frame, xyz, is assigned to the tool tip,with the z-axis coinciding with the spindle axis. The balljoint b1is located in the plane xoTz.Let R and r be the radii of circles passing throughjoints Riand bi(i = 13), respectively. The positions ofRireferenced to the coordinate frame XYZ can beexpressed byR1?32R 0 H1?TR2?0?32R H2?T(1)R3?0 ?32R H3?Twhere Hiis the height of Rialong the Z-axis.The positions of the ball joints biwith respect to thecoordinate frame xyz areb1? r 0 hTb2?12r?32r h?T(2)b3?12r ?32r h?Twhere h is the distance from the tool tip oTto the centerof the platform.The coordinate frame xyz with respect to the coordi-nate frame XYZ, can be described by the homogeneoustransformation matrix T.1563K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569Fig. 2.Schematic diagram of the serial-parallel structure.T ?k1m1n1xTk2m2n2yTk3m3n3zT0 00 1?(3)?CbCg?CbSgSbxTSaSbCg ? CaSg?SaSbSg ? CaCg ?SaCb yT?CaSbCg ? SaSg CaSbSg ? SaCgCaCbzT0001?,where xTyTzTT=op is the position vector of the originof the frame xyz (tool tip position), and the orientationunit vectors k = k1k2k3T, m = m1m2m3T, and n= n1n2n3Tare the directional cosines of the axes x,y and z with respect to the coordinate frame XYZ. Sa,Sb, Sg, Ca, Cb and Cg indicate sin a, sin b, sin g, cosa, cos b and cos g, respectively. a, b and g are therotational angles (also called the Euler angles) of thespindle platform with respect to the X-, Y- and Z-axes,respectively.The Cartesian position of the ball joints biwith respectto the frame XYZ can be expressed by?bi1?o?ooTRop01?bi1?oT? T?bi1?oTi ? 1,2,3(4)Substituting those vector forms of Eq. (2) into Eq.(4) yields?b11?o?X1Y1Z11?rk1? hn1? xTrk2? hn2? yTrk3? hn3? zT1?(5)?b21?o?X2Y2Z21?r2k1?3r2m1? hn1? xT?r2k2?3r2m2? hn2? yT?r2k3?3r2m3? hn3? zT1?(6)?b31?o?X3Y3Z31?r2k1?3r2m1? hn1? xT?r2k2?3r2m2? hn2? yT?r2k3?3r2m3? hn3? zT1?(7)Obviously, as each strut is constrained by the corre-sponding rotational joint and ball joint, strut 1 can onlymove in the plane ?1: Y = 0; strut 2 in the plane ?2:1564K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569Y = ?3(X?R/2); and strut 3 in the plane ?3: Y =3(Y?R/2), as shown in Fig. 3. According to Eqs. (5)and(7),threeconstraintequationscanthusbeobtained asrk2? hn2? yT? 0(8)?12rk2?32rm2? hn2? yT?(9)?3?12rk1?32rm1? hn1? xT?R2?12rk2?32rm2? hn2? yT?(10)?3?12rk1?32rm1? hn1? xT?R2?Summing up Eqs. (9) and (10), and then simplified byEq. (8), it yieldsm1? k2(11)Subtracting Eq. (9) from Eq. (10), it yieldsr2(k1?m2)?hn1?R2? xT(12)Substituting m1and k2from Eqs. (3) and (11), the g canbe obtained asg ? f(a,b) ? ?arctan?sinasinbcosa ? cosb?.(13)Eq. (13) indicates that g is a function of a and b. Eqs.(12) and (8) indicate that xTand yTare both functionsFig. 3.Illustration of the three constraint planes.of directional cosines. xTand yTcan further be expressedin terms of the Euler angles as:xT?r2(cosbcosg ? sinasinbsing?cosacosg)(14)?hsinb ?R2yT? hsinacosb?rsinasinbcosg?rcosasingEqs. (13) and (14) show that if a, b, and zTare known,xTand yTcan be determined. The transformation matrixT can then be obtained by Eq. (3), and the Cartesianposition of each ball joint with respect to the frame XYZcan be calculated using Eq. (4). As the strut length liisconstant, the position component of the rotational jointin the Z-axis can be calculated using the inverse kin-ematics asHi? RiZ? ?l2i?(RiX?biX)2?(RiY?biY)2(15)? biZi ? 1,2,3Eq. (15) has two solutions, but the position of rotationaljoint is always above the ball joint. Only the positivesolution is effective.The inverse kinematics problem involves the compu-tation of the position of each of the rotational joint (theslider) through the corresponding ball joint if the spindleplatforms position and orientations are known.4. Sensitivity analysis of the spindle platformOn the basis of the machine tool configuration, itsstructure actually consists of three parallel structuralloops. For each loop, a closed form of vector chain ofthe linkage can be drawn, as shown in Fig. 4. The relatedequation can be presented in the following vector form.Fig. 4.Close loop of linkage i of the machine tool structure.1565K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569oLi?obi?op?oRi?ooTRoTbi?op?oRii(16)? 1,2,3where the matrixooTR describes the relative orientationof the spindle platform coordinate frame xyz to the basecoordinate system XYZ, the left upper script o denotesthe vector with respect to the base coordinate systemXYZ, and the left under script oTdenotes the vector withrespect to platform coordinate system xyz.Since the position and the orientation of the spindleplatform depend on the structural configuration of themachine tool, all these structural parameters will influ-ence the positional accuracy of the spindle platform. Inthis section, the influences of different structure para-meters on the position and orientation of the spindle plat-form are studied. An analytical method is proposed toinvestigate the sensitivity of different structure para-meters to different positions and orientations of the plat-form.For simplicity, all the scripts are omitted. Eq. (16) canthus be expressed by the following matrix form:?LiXLiYLiZ?k1m1n1k2m2n2k3m3n3?bixbiybiz?xTyTzT?RiXRiYRiZ?(17)Rewrite Eq. (17) into three components asLiX? k1bix? m1biy? n1biz? xT?Rix? cosbcosgbix?cosbsingbiy? sinbbiz? xT?RixLiY? k2bix? m2biy? n2biz? yT?RiY? (sinasinbcosg? cosasing)bix? (?sinasinbsing ? cosacosg)biy?sinacosbbiz? yT?RiYLiZ? k3bix? m3biy? n3biz? zT?RiZ?(?cosasinbcosg ? sinasing)bix? (cosasinbsing? sinacosg)biy? cosacosbbiz? zT?RiZHence, the length of strut i can be calculated byl2i? |Li|2? L2iX? L2iY? L2iZ(18)Introducing the following trigonometrical formulaesinf ?2tan(f/2)1 ? tan2(f/2);cosf ?1?tan2(f/2)1 ? tan2(f/2)and letting the variables a = tan(a/2), b = tan(b/2); c= tan(g/2) we havesina ?2a1 ? a2;cosa ?1?a21 ? a2sinb ?2b1 ? b2;cosb ?1?b21 ? b2(19)sing ?2c1 ? c2;cosg ?1?c21 ? c2According to the kinematics analysis described in theearlier session, those three degrees of freedom of thespindle platform are defined as one linear motion zTinthe Z direction and two angular motions, and , aboutthe X- and Y-axes, respectively. In other words, para-meters xT, yT, and g are dependent on the orientationangles of the platform, as clearly seen in Eqs. (13) and(14). Now, substituting Eq. (14) and Eq. (19) into Eq.(18), a new algebraic equation can be obtained as fol-lows:F ? F(zT,a,b,c,R,r,bix,biy,biz,RiX,RiY,RiZ,li) ? A12? A22? A32?(1 ? a2)2(1 ? b2)2(1 ? c2)2l2i(20)? 0whereA1 ? (1 ? a2)(1?b2)(1?c2)?bix?r2?2c(1 ? a2)(1?b2)biy? 4rabc?r(1?a2)(1 ? b2)(1?c2)2?(1 ? a2)(1 ? b2)(1 ? c2)?R2?RiX?A2 ? 4ab(1?c2) ? 2c(1?a2)(1 ? b2)(bix?r) ?(1?a2)(1 ? b2)(1?c2)?8abcbiy?(1 ? a2)(1 ? b2)(1 ? c2)RiYA3 ? ?2b(1?a2)(1?c2) ? 4ac(1 ? b2)bix?4bc(1?a2) ? 2a(1 ? b2)(1?c2)biy? (1?a2)(1?b2)(1 ? c2)biz? (1 ? a2)(1 ? b2)(1 ? c2)(zT?RiZ)It is noted from Eq. (13) that the variable c in Eq.(20) is also a function of a and b. In addition, as shownin Fig. 2, the radius R of the outer circle passing throughpoint Riis a function of RiXand RiY(i = 1 to 3), and theradius r of the inner circle passing through joints biis afunction of bixand biy(i = 1 to 3). Therefore, variablesR and r in Eq. (20) can be expressed byR ? f(RiX,RiY) ?RiX?13?3iRiX?2?RiY?13?3iRiY?2?1/2r ? f(bix,biy) ?bix?13?3ibix?2?biy?13?3ibiy?2?1/2Eq. (20) shows that a complete kinematics model ofthe parallel mechanism composes of two sets of para-1566K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569meters, namely the structure parameters (li, bix, biy, biz,Rix, Riy, Riz) and the platforms position parameters (zT,a, b). The influences of the structure parameters on theposition parameters reflect the structural sensitivity, orthe positional sensitivity of the platform. This can berealized mathematically by the operation of partial dif-ferentiation as follows.Differentiating Eq. (20) with respect to all structureparameters and position parameters for each structureloop yieldsdF ? AidT ? BidPi? 0i ? 1,2,3(21)whereAidT ?Fa?ida ?Fb?idb ?FZT?idZTBi?FRixFRiyFRizFbixFbiyFbizFli?R17dPi? dRixdRiydRizdbixdbiydbizdliT?R71Integrating the three loops of Eq. (21) together and sepa-rating the position parameters and structure parameterstodifferentsidesyieldsthefollowingsimplifiedmatrix form?AdT ? BdP(22)whereA ?Fa?1?Fb?1?FZT?1?Fa?2?Fb?2?FZT?2?Fa?3?Fb?3?FZT?3?R33B ?B1000B2000B3?R321dP ? dP1dP2dP3T?R211ThendT ? (?A?1)BdPLetC ? (?A)?1B)?R321(23)Matrix C represents the sensitivity matrix of positionparameters with respect to the structural parameters withthe coupling effect of three independent structure loops.The differential terms areFa? 2A1A1a? 2A2A2a? 2A3A3a?4a(1 ? a2)(1 ? b2)2(1 ? c2)2l2iFb? 2A1A1b? 2A2A2b? 2A3A3b?4b(1 ? a2)2(1 ? b2)(1 ? c2)2l2iFc? 2A1A1c? 2A2A2c? 2A3A3c?4c(1 ? a2)2(1 ? b2)2(1 ? c2)l2iFRix?FRRRix?FA1A1RixFRiy?FRRRiy?FA2A2RiyFRiz?FA3A3RizFbix?Frrbix?FA1A1bix?FA2A2bix?FA3A3bixFbiy?Frrbiy?FA1A1biy?FA2A2biy?FA3A3biyFbiz? 2A3 ? (1?a2)(1?b2)(1?c2)Fli? ?2li? (1 ? a2)2(1 ? b2)2(1 ? c2)2ca? ?bcb? ?a5. Examples and discussionsBased on the sensitivity matrix of Eq. (23), simulatedexamples are given in terms of different platform poses(a and b = 0, 5, 10, 15, 20, 25). The calculationis performed according to the obtained data of the proto-type machine tool from experiments as: (R = 349.368mm, r = 199.950 mm, l1= 1107.592 mm, l2=1107.664 mm, l3= 1107.526 mm). The results are listedin Tables 13. It is seen that when the platform pose isat =0, the parallel structure is symmetric so that thedifference in sensitivity values between three structurebranches is slight. The sensitivities of all position para-meters with respect to specific structure parameter at dif-ferent platform poses are summarized in the following.1. With spindle platform pose a = b = 0, becausethe structure branch #1 coincides with the X-axis,thesensitivitiesofalltheparameters(R1xR1yR1zb1xb1yb1zl1) have no influence on theorientation parameter of a.1567K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569Table 1Sensitivity analysis of the position zT(zT= 300 mm)a = 0, b = 0a = 5, b = 5a = 10, b = 10a = 15, b = 15a = 20, b = 20a = 25, b = 25zT/R1x0.03020.02100.01180.0023?0.0077?0.0180zT/R1y0.00000.00000.00000.00000.00000.0000zT/R1z0.33330.23110.12820.0244?0.0783?0.1744zT/b1x?0.0453?0.01120.00480.0030?0.0160?0.0495zT/b1y0.0000?0.0202?0.0222?0.00620.02580.0692zT/b1z?0.3333?0.2294?0.1243?0.02280.06910.1433zT/l1?0.3364?0.2326?0.1288?0.02450.07860.1753zT/R2x?0.0264?0.0384?0.0525?0.0699?0.0910?0.1159zT/R2y0.04570.06650.09100.12100.15770.2008zT/R2z0.33330.47470.61990.76800.91631.0581zT/b2x0.02260.07380.15040.25190.37600.5166zT/b2y?0.0392?0.0977?0.1815?0.2883?0.4132?0.5462zT/b2z?0.3333?0.4711?0.6012?0.7166?0.8092?0.8693zT/l2?0.3364?0.4819?0.6344?0.7939?0.9589?1.1231zT/R3x?0.02640.02320.0197?0.0159?0.0121?0.0084zT/R3y?0.0457?0.0402?0.0341?0.0276?0.0210?0.0145zT/R3z0.33330.29410.25190.20760.16190.1163zT/b3x0.02260.04560.06090.06790.06650.0571zT/b3y0.03920.0089?0.0138?0.0281?0.0338?0.0317zT/b3z?0.3333?0.2919?0.2443?0.1937?0.1430?0.0955zT/l3?0.3364?0.2966?0.2543?0.2100?0.1643?0.1184Table 2Sensitivity analysis of the orientation angular a (zT= 300 mm)a = 0, b = 0a = 5, b = 5a = 10, b = 10a = 15, b = 15a = 20, b = 20 = 25, b = 25a/R1x0.00000.00000.00000.00000.00000.0001a/R1y0.00000.00000.00000.00000.00000.0000a/R1z0.00000.00010.00020.00030.00050.0007a/b1x0.00000.00000.00000.00000.00010.0002a/b1y0.00000.00000.00000.00010.00020.0003a/b1z0.00000.00010.00020.00030.00040.0006a/l10.00000.00010.00020.00030.00050.0007a/R2x0.00020.00020.00030.00030.00040.0004a/R2y0.00040.00040.00050.00050.00060.0008a/R2z0.00290.00300.00310.00330.00360.0040a/b2x0.00020.00050.00080.00110.00150.0020a/b2y0.00030.00060.00090.00130.00160.0021a/b2z0.00290.00290.00300.00310.00320.0033a/l20.00290.00300.00320.00350.00380.0043a/R3x0.00020.00020.00020.00020.00020.0002a/R3y0.00040.00040.00040.00040.00040.0004a/R3z0.00290.00290.00300.00300.00310.0033a/b3x0.00020.00050.00070.00100.00130.0016a/b3y0.00030.00010.00020.00040.00070.0009a/b3z0.00290.00290.00290.00280.00280.0027a/l30.00290.00290.00300.00310.00320.00332. The sensitivity values of all structure parameters todisplacement parameter (zT) are larger than the orien-tation parameters (a, b). It is because the unit of angleis in degree and the unit of displacement is in mm.3. With the spindle platform pose a = b = 0, branches#2 and #3 are symmetric in the coordinate system, sothe sensitivities are equal in magnitude but could be dif-ferent in the sign.4. Table 1 shows that the sensitivities of the para-meters RiX, RiY, bix, biyand biz(i = 1 to 3) influencingthe position zTof the spindle platform are varied withthe platforms angles a and b in different cases of 0,15 and 25. For example, the sensitivity of the para-meter RiXvaries from 0.0302 at the pose (a = b = 00)to ?0.018 at the pose (a = b = 250). Compared to RiZ,the sensitivities of the parameters RiXand RiYto zTare1568K.-C. Fan et al. / International Journal of Machine Tools & Manufacture 43 (2003) 15611569Table 3Sensitivity analysis of the orientation angular b (zT= 300 mm)a = 0, b = 0a = 5, b = 5a = 10, b = 10a = 15, b = 15 a = 20, b = 20a = 25, b = 25b/R1x0.00030.00030.00030.00030.00030.0003b/R1y0.00000.00000.00000.00000.00000.0000b/R1z0.00330.00330.00330.00330.00330.0032b/b1x0.00050.00020.00010.00040.00070.0009b/b1y0.00000.00030.00060.00080.00110.0013b/b1z0.00330.00330.00320.00310.00290.0026b/l10.00340.00330.00330.00330.00330.0032b/R2x0.00010.00010.00010.00020.00020.0002b/R2y0.00020.00020.00020.00030.00030.0003b/R2z0.00170.00170.00170.00170.00170.0016b/b2x0.00010.00030.00040.00060.00070.0008b/b2y0.00020.00030.00050.00060.00080.0008b/b2z0.00170.00170.00160.00160.00150.0013b/l20.00170.00170.00170.00180.00180.0017b/R3x0.00010.00010.00010.00010.00010.0001b/R3y0.00020.00020.00020.00020.00020.0002b/R3z0.00170.00160.00160.00160.00160.0016b/b3x0.00010.00030.00040.00050.00060.0008b/b3y0.00020.00000.00010.00020.00030.0004b/b3z0.00170.00160.00150.00150.00140.0013b/l30.00170.00160.00160.00160.00160.0016significantly less. Similarly, the sensitivities of the para-meters bixand biyinfluencing the position zTare also lessthan biz(h). It means the cutters height position is moresensitive to the height related joint parameters.5. Tables 2 and 3 show that the sensitivities of all theparameters RiX, RiY, RiZ, bix, biyand biz(i = 1 to 3) influ-encing the orientation of the spindle platform are alsovaried with the platforms angles a and b in differentcases of 0, 15 and 25. Obviously, the Z-coordinateparameters also have larger influence on the orientationangles of the spindle platform, compared to the X- andY-coordinate parameters. In other words, the positioningaccuracy of the sliders (RiZ) and bizare more importantto the accuracy of the orientation angles of the spindleplatform.6. The data in Tables 13 clearly show that parameterli(i = 1 to 3) have larger influence on the platform pos-ition zTand orientation angles a and b and at differentplatforms poses. Therefore, the nominal strut length (li)is a very sensitive parameter to the accuracy of the cutterposition zTas well as the platforms angles.7. As a and b increase the platform is tilted to theloop 2 direction. The sensitivity of this loops structureparameters is more significant.8. The computational results are consistent with thephysical phenomenon of the machine tool. The sensi-tivity investigation of the position and the orientation ofthe spindle platform in the workspace can also be calcu-lated by using Eq. (23), while the platforms angles aand b are either assigned positive or negative.6. ConclusionsInverse kinematics model of the parallel mechanismhas been derived to compute the positions of therotational joints Riand the ball joints bi(i = 1 to 3) atgiven values of a, b and zT. The sensitivity analysis ofthe accuracy of the cutter location with respect to somestructure parameters is also proposed in this study. Fromsimulated examples it shows that the sensitivity of struc-ture parameters will change as the platform varies itsorientations. Among all parameters, the Z-coordinateparameters have larger influence on the position and theorientation of the spindle platform in different poses.The sensitivity analyses of platforms position withrespect to the structure parameters have demonstratedthat the positioning accuracy of the parallel sliders, theoriginal strut length, and the tool length are the mostsensitive parameters influencing the position and orien-tation of the spindle platform, or the cutter location, inpractice.References1 J. Wang, O. Masory, On the accuracy of a Stewart platformPart I: The effect of manufacturing tolerances, in: Proc. of IEE