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To measure the accuracy offive-axis machine tools, curves on a spherical test surface are defined as tool paths. The tool orientation is defined in the surface normaldirection. The center of the spherical test surface coincides with the center of the measuring ball. With this path and orientation inputto CNC controller, the 3D probe moves relative to the measuring ball on the spherical test surface. The overall positioning errors of therelative motion are measured by the 3D probe and are used to justify the volumetric accuracy of the five-axis machine. 2003 Elsevier Science B.V. All rights reserved.Keywords: Error modeling and measurement; Five-axis machine tool; Accuracy test1. IntroductionFive-axis CNC machine tools are used widely in the ma-chining of a workpiece with a sculptured surfaces. In addi-tion to conventional three linear positioning axes, five-axismachines have generally two extra rotary axes. All five axescan be controlled simultaneously to adjust the cutting tooloptimally with respect to the surface of the workpiece. Thetechnological advantages of five-axis machine tool includea much higher metal-removal rate with improved surfacefinish and significantly lower cutting time 1.In past decades, much work has focused on machinetool accuracy under the influence of geometrical errorsand/or thermal deformation 25. Many measurement de-vices have been developed to measure the individual errorcomponent and to test the accuracy of a multi-axis ma-chine tool as a whole. The most powerful and time savingdevice is the six-degrees-of-freedom laser measurement de-vice, which can be used to measure the six motional errorcomponents of a linear motion carriage at one time 6.Further, the double-ball bar (DBB) is frequently used todetermine out dynamic errors of a feed drive system suchas gain mismatch, lost motion and stick-slip 7. To extendthe measurement range of DBB, the so-called laser-ball bar(LBB) has been developed to measure positioning errors inCorresponding author.E-mail address: wtleipme.nthu.edu.tw (W.T. Lei).a three-dimensional working space 8. The grid encoder9 is on the other hand especially suitable for measuringdynamic path error around a sharp corner.Although these measurement devices have been usedsuccessfully to measure the accuracy of three-axis CNCmachine tools, no measurement device is available to testthe volumetric accuracy of five-axis CNC machine tools.In this paper, a new measurement device, the probeball,is presented, which is capable of measuring the overallpositioning errors of five-axis machine tools.2. Probeball measurement device2.1. Design featuresThe probeball is shown in Fig. 1. It consists of a 3Dprobe, an extension bar and a base plate with a measuringball on one side. The 3D probe has a standard tool holdertaper and is capable of three-degrees-of-freedom deviationmeasurement. The 3D probe uses an optical encoder asthe displacement sensor. Other displacement sensors suchas the linear variable displacement transducer (LVDT) orthe capacitance sensor are also possible. The extension barhas a socket at the free end and forms a ball joint with themeasuring ball. A permanent magnet is integrated in thesocket so that the extension bar and the measuring ball canbe connected together with magnetic force. The base plat0924-0136/03/$ see front matter 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0924-0136(03)00193-6128W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133Fig. 1. The probeball measurement device.is fixed onto the turntable of the five-axis machine withalignment of orientation.To measure the overall positioning errors between the tooland the workpiece, the probe is placed in the tool holderand the base plate is fixed on the turntable. After installingthe probeball measurement device, the kinematic chain ofthe five-axis machine is thus closed. The test path can beany curve on a spherical test surface. The tool orientationis defined in the surface normal direction. The center of thespherical test surface coincides with the center of the mea-suring ball. The radius of the spherical surface is set equalto the distance between the origin of the 3D probe sen-sors and the center of the measuring ball. The extension barcan have different length to define corresponding test space.With this path and orientation input to the CNC controller,the 3D probe is driven on the spherical test surface with themeasuring ball as center. The overall positioning errors aremeasured by the 3D probe.Because of the symmetrical character of spherical surface,it is advantageous to mount the measuring ball, thus thecenter of the spherical test surface, with an offset from theaxis of the turntable. With this arrangement, the measuringball keeps rotating with the turntable during the accuracytest so that all five axes are driven simultaneously. Therefore,the measured errors contain an error contribution from allaxes. The offset of the measuring ball and the length of theextension bar determine the test range of the driven axis.To ensure that the probeball device itself is not a partof the error sources, it is necessary to undertake accuratecalibration procedures before its usage. These proceduresinclude the initialization of the 3D probe sensors and themeasurement of the exact position of the measuring ball on acoordinate measurement machine (CMM). The 3D probe isso initialized that the outputs are set to zero when the centerof the ball joint is adjusted so as to be in the center line ofthe 3D probe taper. During the accuracy test, the outputs ofFig. 2. Test paths.3D probe represent the deviations of measuring ball fromthe center of the spherical test surface. It is to be emphasizedthat the probeball device does not measure the positioningerrors in the workpiece coordinates as it appears to do.2.2. Test pathsAs mentioned above, the test path can be any path onthe spherical test surface. Fig. 2 shows some examples oftest path. The path A is along the longitude of the sphericalsurface. With this path, only A-, Y- and Z-axis are driven.The A-axis is the only actively driven axis. In contrary, theY- and Z-axis are passively following axes. In other words,while the A-axis is driven, the Y- and Z-axis follow to keepthe kinematic chain closing. This path is suitable to test thestatic and dynamic errors of A-axis. The path C is along onthe equator of the spherical surface. In this case, the C-axisis actively driven, while the X-, Z-axis follow. Likewise, thepath C is special for the error test of C-axis. The path F is ahelix-like curve on the spherical test surface and covers thewhole spherical volume. All machine axes can be drivensimultaneously in this case. The measured errors provideenough information in describing the overall volumetricerrors of the target five-axis machine tool. The path S isa circle on the spherical test surface. In this case, all axesare driven to and fro and show points of velocity reversal.The path S is therefore especially suitable for testing thedynamic errors of rotary A- and C-axis.The probeball can be used for various purposes. If it isusedtotesttheoverallpositioningerrors,thepathFisagoodchoice. If it is used to identify or to estimate the error com-ponents of single axis, it is better to select simple test pathssuch as path A or C, because only limited error componentsare dominant in the measurement results. In the following,the detailed relationship between the test paths and the kine-matics of the target five-axis machine tool will be derived.3. Kinematic transformationBecause the test paths are defined in workpiece coordi-nates, the CNC input for the accuracy measurement withW.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133129Fig. 3. The target five-axis milling machine.the 3D probeball is independent of the kinematics of thefive-axis machine. Without loss of generality, the simulta-neous axis motion during the probeball test is explainedby a five-axis machine tool of the type ZX?Y?A?C?. The ma-chine structure is characterized by the integration of thetwo-degrees-of-freedom rotary block on the X- and Y-table,as shown in Fig. 3. The coordinate frames are shown inFig. 4.The transformation from the machine coordinate frameto the workpiece coordinate frame is conventionally calledthe forward transformation. On the other hand, the transfor-mation from the workpiece coordinate frame to the machinecoordinate frame is called the backward transformation.The forward transformation of a five-axis machine tool isalways explicitly solvable and has only one solution. In con-Fig. 4. Coordinate frames of the five-axis milling machine.trast, the backward transformation has always two solutionsregarding the position of the rotational axes. In following,we derive the relationship between the machine coordinateframe and the workpiece coordinate frame with the help ofthe homogeneous transformation matrix (HTM) 10.Assume that (Xm,Ym,Zm) is a point in machine coordi-nates and (Xw,Yw,Zw) is the same point but in workpiececoordinates. To derive the forward transformation, the originof the machine coordinates is firstly shifted to the intersec-tion of the two rotational axis with the vector (X1,Y1,Z1),then the axis A is rotated with aand axis C with cso thatthe turntable stays vertical. Finally, the machine coordinatesis shifted with the vector (X0,Y0,Z0) to the origin of theworkpiece coordinate frame. The transformation sequencescan be expressed asXmYmZmv1= Trans(X1,Y1,Z1)Rot(x,a)Rot(z,c)Trans(X0,Y0,Z0)XwYwZw1(1)whereTrans()andRot()representtheHTMsresultingfromthe translation and rotation operations, respectively.The forward kinematic transformation equations of thefive-axis machine can be derived from Eq. (1) and can beexpressed asXw= sincsina(Zm Z1) sinccosa(Ym Y1)cosc(Xm X1) X0(2)Yw= coscsina(Zm Z1) cosccosa(Ym Y1)+sinc(Xm X1) Y0(3)Zw= cosa(Zm Z1) sina(Ym Y1) Z0(4)I = sinasinc(5)J = sinacosc(6)K = cosa(7)where (Xw,Yw,Zw) is the tool center position and (I,J,K)the normalized tool orientation in workpiece coordinates.The backward transformation equations derived fromEq. (1) areXm= (X0+ Xw)cosc (Y0+ Yw)sinc+ X1(8)Ym= (X0+ Xw)cosasinc+ (Y0+ Yw)cosacosc(Z0+ Zw)sina+ Y1(9)Zm= (X0+ Xw)sinasinc+ (Y0+ Yw)sinacosc+(Z0+ Zw)cosa+ Z1(10)130W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133The rotational angles aand care solved from the givenorientation vector (I,J,K). There are always two solutionsfor aand c:Case 1 (K ?= 1). Solving Eq. (7) yields:a= cos1(K),(11)Solving Eqs. (5) and (6) with respect to the cyields:c= tan1?IJ?(12)Since ahas two solutions, chas two solutions also. Thesolution is either (a,c) or (a,c+ 180).Case 2 (I = J = 0, K = 1). From Eq. (7):a= 0In this special case, a= 0 and ccan be any value. Inpractice, cis calculated from:c= tan1?IJ?(13)The solution is either (0,c) or (0,c+ 180).Because there are always two solutions after the backwardtransformation, a strategy is necessary to select a suitableone. A simple criteria is the driving energy needed. The onewith a smaller distance to move will be selected. Of coursethe possibility of collision must be considered.4. Test paths and error model4.1. Test paths in workpiece coordinatesAs described above, the probeball device uses any pathon the spherical test surface as test path to examine theaccuracy of the five-axis machine tool. In the following,the descriptions of test paths in workpiece coordinates arederived.Fig. 5. Parameters of test path F.Fig. 6. Parameters of test path S.Fig. 5 shows the parameters to define the path F. To min-imize the test time, the rising angle of the path F is set to90. The effect is that the tool arrives the top position afterthe C-axis rotates 360. The path description in workpiececoordinates is thus:XwYwZw =Rwcos4cosRwcos4sinRwsin4(14)where Rwis the radius of the spherical test surface and thecircular angle. Similar to this, a path description for otherrising angles can also be derived.Fig. 6 shows the parameters to define the path S. The testcircle is defined symmetrical to the XwZwplane and is onthe XcYcplane of the coordinate frame XcYcZc. Thetransformation from circle coordinate frame Xc Yc Zcto workpiece coordinate frame Xw Yw ZwisXwYwZw =wTcXcYcZc(15)The matrixwTcdefines the transformation from circle coor-dinate frame to workpiece coordinate frame and is given aswTc= Tx(Rw)Ry?2 p?Tx(Rp)(16)where Rwis the radius of the spherical test surface, ptheangle between the plane of the circle and Xw-axis, and Rpthe radius of the circle.From Eqs. (15) and (16), the description of the circularpath in workpiece coordinates isXwYwZw =Rpsinp(cos 1) + RwRpsinRpcosp(1 cos)(17)where is the driving angle of the circle.W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133131Fig. 7. The command values of test path F.Because the tool orientation is always in the surface nor-mal direction, the normalized tool orientation (I,J,K) canbe expressed as(I,J,K) =?XwD,YwD,ZwD?(18)whereD =?X2w+ Y2w+ Z2w(19)4.2. Test paths in axis coordinatesWith the help of backward kinematic transformation, thetest path and orientation in workpiece coordinates are trans-formed into machine or axis coordinates. Figs. 7 and 8 showthe axis command values for paths S and F. In case of pathF, the rotating axes C and A are driven linearly, while otheraxes followed to keep the kinematic chain closing. In case ofpath S, all axes are driven to and fro and return to the startingpoint. The velocity reversal points can be identified clearly.As is known in the double-ball bar measurement technique,these velocity reversal points offer necessary conditions toFig. 8. The command values of test path S.show up dynamic motional errors such as stick-slip, lost mo-tion and backlash. For the parameters of Fig. 8, the velocityreversal points appears at 180for axis A and 120, 210foraxis C. It can be seen also that some axes have their velocityreversed at the same time, for example axes C and X. Onecan use a double-ball bar to identify the dynamic errors ofthe linear axis first. From the test results of the probeballdevice, the dynamic errors of rotational axis A or C can beidentified later.4.3. Error modelTo interpret the probeballs measurement results, it isnecessary to build an error model of the probeball measure-ment.Theerrormodeldescribestherelationshipbetweenthemeasured overall positioning errors and the error sources ofeach component in the kinematic chain of the five-axis ma-chine tool. The homogenous transformation matrix (HTM)1 method provides a good methodology for this theoreti-cal task. The geometric components can be classified intotwo categories. The first one is associated with the inaccu-rate motion of one servo-controlled axis. The second one isassociated with the errors of a link component. For each lin-ear or rotary axis, there are in general 6 motional errors inthe HTM. The errors of a link component include the per-pendicularity errors between axes and offset errors of blockcomponents such as the main spindle and the rotary block.The coordinate frames are defined in Fig. 3. The error modelcan be obtained through a sequential product of all HTMsof each kinematic component. The spatial relationship be-tween the workpiece coordinate frame and the reference co-ordinate frame isrTw=rTyyTxxTaaTccTttTw(20)where the indices w, t, c, a, x, y, r represent the abbreviationof workpiece, turntable, c-axis, a-axis, x-axis, y-axis andreference system, respectively.Similarly, the spatial relationship between the probe co-ordinate frame and the reference coordinate frame isrTp=rTzzTssThhTp(21)where the indices p, h, s, z represent the abbreviation ofprobe, tool holder, spindle block and z-axis, respectively.The center of the central ball Pb=?XbYbZb?devi-ates from the center of the ball socket Ps=?XsYsZs?due to geometric errors. The Pband Psare computed fromfollowing equations:?Pb1?T=rTw?0001?T(22)?Ps1?T=rTp?00R1?T(23)where R is the radius of the test surface.The position error vector Pe,rin the reference coordinateframe is given asPe,r= Pb Ps(24)132W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133Since the displacement measurement occurs in the probecoordinate frame, it is necessary to transform the positionerror vector Pe,rfrom the reference coordinate frame intothe probe coordinate frame:?Pe,r0?T= (rTD)1?Pe,r0?T=?Xp?Yp?Zp0?T(25)where the ?Xp, ?Ypand ?Zpare the deviations in the probecoordinate frame.5. Experimental resultsFig. 9 shows the accuracy test on the target five-axismilling machine with the 3D probeball. Figs. 1012 showsome test results. The initial errors are valid if the 3D probeand the coordinates of the measuring ball are properly ini-tiated. Fig. 10 shows the results of static test, whereby theerrors are sampled after positioning to predefined measuringFig. 9. Probeball device in measurement.Fig. 10. Static test errors of path F.Fig. 11. Dynamic test errors of path F, feedrate = 30mm/min.Fig. 12. Dynamic test errors of path F, feedrate = 150mm/min.point is finished. Figs. 11 and 12 show the results of dy-namic tests, in which the errors are sampled while the axesare driven with the input feedrate. Due to the bad dynamicsof rotating A-axis, the errors in the y-direction deterioratedramatically when the feedrate increases.In another research undertaking intended to identify andto estimate all error terms from the measured data of the 3Dprobeball, it was shown that the major error sources of thefive-axis milling machine are the perpendicularity errors ofthe two rotating axes.6. ConclusionsIn this paper, a new measurement device called the 3Dprobeball, is presented. It is capable of measuring theoverall positioning errors of five-axis CNC machine tools.The error measurements are based on the principle ofclosed chain measurement. During the accuracy test, the 3DW.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133133probeball closes the kinematic chain of the target five-axismachine tool. Because of the kinematic constraints, suitabletest paths are paths on a spherical test surface. The measuredpositioning errors are defined in the probe coordinate frameand can be transformed into the reference coordinate frameto predict the accuracy of the target five-axis machine. Withthe 3D probeball available, further investigations withthe aim of enhancing the machine tools accuracy can beconducted, including the estimation and compensation ofgeometric errors.References1 E.E. Sprow, Manuf. Eng. 111 (5) (1993) 55.2 V.B. Kreng, C.R. Liu, C.N. Chu, Int. J. Adv. Manuf. Technol. 9(1994) 79.3 V.S.B. Kiridena, P.M. Ferreira, Int. J. Mach. Tools Manuf. 34 (1)(1994) 85.4 V.S.B. Kiridena, P.M. Ferreira, Int. J. Mach. Tools Manuf. 33 (3)(1993) 417.5 A.K. Srivastava, S.C. Veldhuis, M.A. Elbestawit, Int. J. Mach. ToolsManuf. 35 (9) (1995) 1321.6 K. Lau, Q. Ma, X. Chu, Y. Liu, S. Olson, Technical Reportof Automated Precision Inc., Gaithersburg, MD 20879, USA,2002.7 H. Pahk, Y.S. Kim, H.H. 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