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Journal of Materials Processing Technology 139 (2003) 127133 Error measurement of fi ve-axis CNC machines with 3D probeball W.T. Lei, Y.Y. Hsu Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC Abstract This paper presents a new measurement device and corresponding method for an accuracy test of fi ve-axis CNC machines. This device is named probeball, and consists of a 3D probe, an extension bar and a base plate with a measuring ball on one side. The 3D probe has a standard taper and is capable of three-degrees-of-freedom displacement measurement. The extension bar has a socket at its free end. A permanent magnet is integrated in the socket so that the extension bar and the measuring ball can be connected together with magnetic force. After installing the probeball device, the kinematic chain of the fi ve-axis machine tool is closed. To measure the accuracy of fi ve-axis machine tools, curves on a spherical test surface are defi ned as tool paths. The tool orientation is defi ned in the surface normal direction. The center of the spherical test surface coincides with the center of the measuring ball. With this path and orientation input to CNC controller, the 3D probe moves relative to the measuring ball on the spherical test surface. The overall positioning errors of the relative motion are measured by the 3D probe and are used to justify the volumetric accuracy of the fi ve-axis machine. 2003 Elsevier Science B.V. All rights reserved. Keywords: Error modeling and measurement; Five-axis machine tool; Accuracy test 1. Introduction Five-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, fi ve-axis machines have generally two extra rotary axes. All fi ve axes can be controlled simultaneously to adjust the cutting tool optimally with respect to the surface of the workpiece. The technological advantages of fi ve-axis machine tool include a much higher metal-removal rate with improved surface fi nish and signifi cantly lower cutting time 1. In past decades, much work has focused on machine tool accuracy under the infl uence of geometrical errors and/or thermal deformation 25. Many measurement de- vices have been developed to measure the individual error component and to test the accuracy of a multi-axis ma- chine tool as a whole. The most powerful and time saving device is the six-degrees-of-freedom laser measurement de- vice, which can be used to measure the six motional error components of a linear motion carriage at one time 6. Further, the double-ball bar (DBB) is frequently used to determine out dynamic errors of a feed drive system such as gain mismatch, lost motion and stick-slip 7. To extend the measurement range of DBB, the so-called laser-ball bar (LBB) has been developed to measure positioning errors in Corresponding author. E-mail address: .tw (W.T. Lei). a three-dimensional working space 8. The grid encoder 9 is on the other hand especially suitable for measuring dynamic path error around a sharp corner. Although these measurement devices have been used successfully to measure the accuracy of three-axis CNC machine tools, no measurement device is available to test the volumetric accuracy of fi ve-axis CNC machine tools. In this paper, a new measurement device, the probeball, is presented, which is capable of measuring the overall positioning errors of fi ve-axis machine tools. 2. Probeball measurement device 2.1. Design features The probeball is shown in Fig. 1. It consists of a 3D probe, an extension bar and a base plate with a measuring ball on one side. The 3D probe has a standard tool holder taper and is capable of three-degrees-of-freedom deviation measurement. The 3D probe uses an optical encoder as the displacement sensor. Other displacement sensors such as the linear variable displacement transducer (LVDT) or the capacitance sensor are also possible. The extension bar has a socket at the free end and forms a ball joint with the measuring ball. A permanent magnet is integrated in the socket so that the extension bar and the measuring ball can be connected together with magnetic force. The base plat 0924-0136/03/$ see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00193-6 128W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133 Fig. 1. The probeball measurement device. is fi xed onto the turntable of the fi ve-axis machine with alignment of orientation. To measure the overall positioning errors between the tool and the workpiece, the probe is placed in the tool holder and the base plate is fi xed on the turntable. After installing the probeball measurement device, the kinematic chain of the fi ve-axis machine is thus closed. The test path can be any curve on a spherical test surface. The tool orientation is defi ned in the surface normal direction. The center of the spherical test surface coincides with the center of the mea- suring ball. The radius of the spherical surface is set equal to the distance between the origin of the 3D probe sen- sors and the center of the measuring ball. The extension bar can have different length to defi ne corresponding test space. With this path and orientation input to the CNC controller, the 3D probe is driven on the spherical test surface with the measuring ball as center. The overall positioning errors are measured by the 3D probe. Because of the symmetrical character of spherical surface, it is advantageous to mount the measuring ball, thus the center of the spherical test surface, with an offset from the axis of the turntable. With this arrangement, the measuring ball keeps rotating with the turntable during the accuracy test so that all fi ve axes are driven simultaneously. Therefore, the measured errors contain an error contribution from all axes. The offset of the measuring ball and the length of the extension bar determine the test range of the driven axis. To ensure that the probeball device itself is not a part of the error sources, it is necessary to undertake accurate calibration procedures before its usage. These procedures include the initialization of the 3D probe sensors and the measurement of the exact position of the measuring ball on a coordinate measurement machine (CMM). The 3D probe is so initialized that the outputs are set to zero when the center of the ball joint is adjusted so as to be in the center line of the 3D probe taper. During the accuracy test, the outputs of Fig. 2. Test paths. 3D probe represent the deviations of measuring ball from the center of the spherical test surface. It is to be emphasized that the probeball device does not measure the positioning errors in the workpiece coordinates as it appears to do. 2.2. Test paths As mentioned above, the test path can be any path on the spherical test surface. Fig. 2 shows some examples of test path. The path A is along the longitude of the spherical surface. With this path, only A-, Y- and Z-axis are driven. The A-axis is the only actively driven axis. In contrary, the Y- and Z-axis are passively following axes. In other words, while the A-axis is driven, the Y- and Z-axis follow to keep the kinematic chain closing. This path is suitable to test the static and dynamic errors of A-axis. The path C is along on the equator of the spherical surface. In this case, the C-axis is actively driven, while the X-, Z-axis follow. Likewise, the path C is special for the error test of C-axis. The path F is a helix-like curve on the spherical test surface and covers the whole spherical volume. All machine axes can be driven simultaneously in this case. The measured errors provide enough information in describing the overall volumetric errors of the target fi ve-axis machine tool. The path S is a circle on the spherical test surface. In this case, all axes are driven to and fro and show points of velocity reversal. The path S is therefore especially suitable for testing the dynamic errors of rotary A- and C-axis. The probeball can be used for various purposes. If it is usedtotesttheoverallpositioningerrors,thepathFisagood choice. If it is used to identify or to estimate the error com- ponents of single axis, it is better to select simple test paths such as path A or C, because only limited error components are dominant in the measurement results. In the following, the detailed relationship between the test paths and the kine- matics of the target fi ve-axis machine tool will be derived. 3. Kinematic transformation Because the test paths are defi ned in workpiece coordi- nates, the CNC input for the accuracy measurement with W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133129 Fig. 3. The target fi ve-axis milling machine. the 3D probeball is independent of the kinematics of the fi ve-axis machine. Without loss of generality, the simulta- neous axis motion during the probeball test is explained by a fi ve-axis machine tool of the type ZX?Y?A?C?. The ma- chine structure is characterized by the integration of the two-degrees-of-freedom rotary block on the X- and Y-table, as shown in Fig. 3. The coordinate frames are shown in Fig. 4. The transformation from the machine coordinate frame to the workpiece coordinate frame is conventionally called the forward transformation. On the other hand, the transfor- mation from the workpiece coordinate frame to the machine coordinate frame is called the backward transformation. The forward transformation of a fi ve-axis machine tool is always explicitly solvable and has only one solution. In con- Fig. 4. Coordinate frames of the fi ve-axis milling machine. trast, the backward transformation has always two solutions regarding the position of the rotational axes. In following, we derive the relationship between the machine coordinate frame and the workpiece coordinate frame with the help of the 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 workpiece coordinates. To derive the forward transformation, the origin of the machine coordinates is fi rstly 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 that the turntable stays vertical. Finally, the machine coordinates is shifted with the vector (X0,Y0,Z0) to the origin of the workpiece coordinate frame. The transformation sequences can be expressed as Xm Ym Zm v1 = Trans(X1,Y1,Z1)Rot(x,a)Rot(z,c) Trans(X0,Y0,Z0) Xw Yw Zw 1 (1) whereTrans()andRot()representtheHTMsresultingfrom the translation and rotation operations, respectively. The forward kinematic transformation equations of the fi ve-axis machine can be derived from Eq. (1) and can be expressed as Xw= 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 from Eq. (1) are Xm= (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) 127133 The rotational angles aand care solved from the given orientation vector (I,J,K). There are always two solutions for 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 ? I J ? (12) Since ahas two solutions, chas two solutions also. The solution is either (a,c) or (a,c+ 180). Case 2 (I = J = 0, K = 1). From Eq. (7): a= 0 In this special case, a= 0 and ccan be any value. In practice, cis calculated from: c= tan1 ? I J ? (13) The solution is either (0,c) or (0,c+ 180). Because there are always two solutions after the backward transformation, a strategy is necessary to select a suitable one. A simple criteria is the driving energy needed. The one with a smaller distance to move will be selected. Of course the possibility of collision must be considered. 4. Test paths and error model 4.1. Test paths in workpiece coordinates As described above, the probeball device uses any path on the spherical test surface as test path to examine the accuracy of the fi ve-axis machine tool. In the following, the descriptions of test paths in workpiece coordinates are derived. Fig. 5. Parameters of test path F. Fig. 6. Parameters of test path S. Fig. 5 shows the parameters to defi ne the path F. To min- imize the test time, the rising angle of the path F is set to 90. The effect is that the tool arrives the top position after the C-axis rotates 360. The path description in workpiece coordinates is thus: Xw Yw Zw = Rwcos 4cos Rwcos 4sin Rwsin 4 (14) where Rwis the radius of the spherical test surface and the circular angle. Similar to this, a path description for other rising angles can also be derived. Fig. 6 shows the parameters to defi ne the path S. The test circle is defi ned symmetrical to the XwZwplane and is on the XcYcplane of the coordinate frame XcYcZc. The transformation from circle coordinate frame Xc Yc Zc to workpiece coordinate frame Xw Yw Zwis Xw Yw Zw = wTc Xc Yc Zc (15) The matrix wTc defi nes the transformation from circle coor- dinate frame to workpiece coordinate frame and is given as wTc = Tx(Rw)Ry ? 2 p ? Tx(Rp)(16) where Rwis the radius of the spherical test surface, pthe angle between the plane of the circle and Xw-axis, and Rp the radius of the circle. From Eqs. (15) and (16), the description of the circular path in workpiece coordinates is Xw Yw Zw = Rpsinp(cos 1) + Rw Rpsin Rpcosp(1 cos) (17) where is the driving angle of the circle. W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133131 Fig. 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) can be expressed as (I,J,K) = ?X w D , Yw D , Zw D ? (18) where D = ? X2 w+ Y2w+ Z2w (19) 4.2. Test paths in axis coordinates With the help of backward kinematic transformation, the test path and orientation in workpiece coordinates are trans- formed into machine or axis coordinates. Figs. 7 and 8 show the axis command values for paths S and F. In case of path F, the rotating axes C and A are driven linearly, while other axes followed to keep the kinematic chain closing. In case of path S, all axes are driven to and fro and return to the starting point. The velocity reversal points can be identifi ed clearly. As is known in the double-ball bar measurement technique, these velocity reversal points offer necessary conditions to Fig. 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 velocity reversal points appears at 180for axis A and 120, 210for axis C. It can be seen also that some axes have their velocity reversed at the same time, for example axes C and X. One can use a double-ball bar to identify the dynamic errors of the linear axis fi rst. From the test results of the probeball device, the dynamic errors of rotational axis A or C can be identifi ed later. 4.3. Error model To interpret the probeballs measurement results, it is necessary to build an error model of the probeball measure- ment.Theerrormodeldescribestherelationshipbetweenthe measured overall positioning errors and the error sources of each component in the kinematic chain of the fi ve-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 classifi ed into two categories. The fi rst one is associated with the inaccu- rate motion of one servo-controlled axis. The second one is associated with the errors of a link component. For each lin- ear or rotary axis, there are in general 6 motional errors in the HTM. The errors of a link component include the per- pendicularity errors between axes and offset errors of block components such as the main spindle and the rotary block. The coordinate frames are defi ned in Fig. 3. The error model can be obtained through a sequential product of all HTMs of each kinematic component. The spatial relationship be- tween the workpiece coordinate frame and the reference co- ordinate frame is rTw = rTyyTxxTaaTccTttTw (20) where the indices w, t, c, a, x, y, r represent the abbreviation of workpiece, turntable, c-axis, a-axis, x-axis, y-axis and reference system, respectively. Similarly, the spatial relationship between the probe co- ordinate frame and the reference coordinate frame is rTp = rTzzTssThhTp (21) where the indices p, h, s, z represent the abbreviation of probe, 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 from following equations: ? Pb1 ?T = rTw?0 001 ?T (22) ? Ps1 ?T = rTp?0 0R1 ?T (23) where R is the radius of the test surface. The position error vector Pe,rin the reference coordinate frame is given as Pe,r= Pb Ps(24) 132W.T. Lei, Y.Y. Hsu/Journal of Materials Processing Technology 139 (2003) 127133 Since the displacement measurement occurs in the probe coordinate frame, it is necessary to transform the position error vector Pe,rfrom the reference coordinate f