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International Journal of Machine Tools fax: +88656311500. E-mail addresses: jywe@sunws.nfu.edu.tw (W. Jywe), table was rotated counterclockwise. In general, one rotary table calibration for a 3601 full circle requires 36 recording if the sampled period of measurement system is 101.Ifa 0890-6955/$-see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.02.004 pmc2@sunws.nfu.edu.tw (C.J. Chen), allen@nfu.edu.tw (W.H. Hsieh), pdlin@mail.ncku.edu.tw (P.D. Lin), schong@nfu.edu.tw (H.H. Jwo), jeyang@ (T.Y. Yang). instruments are the rotary encoder, the laser interferom- eter, the autocollimator and the precision level. A rotary encoder [1] is commonly used in indexing measurement in a rotary machine, e.g. a rotary table of the multi-axis machine tool, the joint of a robot, the spindles of machine tools and the indexing of a ball screw. However, the rotary encoder is only suitable for the indexing error measure- ment. A laser interferometer [2] has often been used to measure a small angle, but it can only obtain indexing error either one dimensional (1D) error or 2D errors. The complete calibration procedure of a rotary table requires 6 DOF measurement for a 3601 full circle, but the measure- ment range of most measurement systems is smaller than 101. Therefore the measurement range of the laser interferometer and autocollimator are not enough and, in addition, they are expensive. The conventional calibration technique of the rotary table for a 3601 full circle requires one reference rotary table, which must have high accuracy and high repeatability. The error of the reference rotary C3 and the reference rotary table could be obtained. The system calibration, stability test, system verification and full circle test were completed. The angular stability of this system was less then 2arcsec, while the displacement stability was less than 1.2mm. r 2007 Elsevier Ltd. All rights reserved. Keywords: Rotary table calibration; Full circle test; Grating; Position sensing detector; 4 Degree of freedom measurement; Error separation 1. Introduction A rotary table is frequently used in industry in such things as machine tools, CMM and assembly lines. Therefore, the calibration of the rotary table is very important. The calibration of the rotary table requires an during an indexing test. An autocollimator [3] is frequently used to measure small angles and it can be applied to two dimensional (2D) angle measurement (pitch error and yaw error), but its measurement range is small and it require one standard polygon mirror. A rotary table has 6 DOF errors (3 linear position errors and 3 angular position reference rotary table, but with good repeatability is needed. After two full circle tests, the 4-DOF errors of both the target rotary table A novel simple and low cost 4 degree calibrating technique for W. Jywe a,C3 , C.J. Chen b , W.H. Hsieh a National Formosa University, Department of Automation b National Cheng-Kung University, Department of Mechanical Received 30 October 2006; received in revised form Available online Abstract For calibrating an angular rotary table, either a high precision standard employed at high cost. This paper establishes a novel, simple and low of a rotary table (three angular position errors and one linear position one 1 dimensional (1D) grating and two 2 dimensional (2D) position-sensing-detectors ture 47 (2007) 1978–1987 of freedom angular indexing a precision rotary table P.D. Lin b , H.H. Jwo a , T.Y. Yang a g, No. 64 Wenhua Rd., Huwei, Taiwan, ROC Engineering, No. 1, University Rd., Tainan, Taiwan, ROC 1 February 2007; accepted 13 February 2007 February 2007 table or a laser interferometer and related optics are normally cost technique to calibrate the 4-degrees-of-freedom (DOF) errors error) for a 3601full circle by employing one reference rotary table, (PSD). With this technique, no highly accurate ARTICLE IN PRESS more complete test is implemented, the calibration process will takes a long time. In general, the rotary table includes the index error, wobble error and eccentricity. But conventional rotary table calibration techniques (laser interferometer or auto- collimator) only calibrate the index error and the wobble error. However, the high precision rotary table must be calibrated in more details. Through the complete rotary table calibration, the errors of rotary table can be compensated. In this paper, the errors of rotary table were defined by 6 DOF, i.e. three linear position errors (d x , d y , d z ) and three angular position errors (e x , e y , e z ). The index error was represented by e z , the wobble error was represented by e x and e y , the eccentricity was represented by d x and d y . In recent years, angular measuring techniques have focused on the interferometric methods. In 1992, Huang et al. [4] developed a small angle measurement system which was based on the internal reflection effect in a glass boundary and Fresnel’s law. In Huang’s system, the resolution was 0.2arcsec and the measuring range was 3arcsec. In 1996, Xiaoli et al. [5] established a 2D small rotation angle-measurement system using two different parallel interference patterns (PIP) that were orthogonal to each other. The standard deviation of Xiaoli’s system was 0.6arcsec. In the following year Xiaoli et al. [6] improved their system so that its resolution was 0.2arcsec and measuring range was 730arcmin. In 1997, Chiu et al. [7] established a modified angle measurement technique with a resolution of 0.333arcsec and a measuring range of75.61. At its optimum performance, the system’s resolution was 0.288arcsec. In 1998, Zhou and Cai [8] established an angle measurement technique which was based on the total-internal reflection effect and heterodyne interferome- try. The system resolution was better than 0.3arcsec, depending on the refractive index selected. In 1998, Huang et al. [9] established a method of angle measurement, based on the internal reflection effects, that used a single right- angle prism. They demonstrated that angle measurement with a range of 7500arcmin, a nonlinearity error of 70.1%, and a resolution of 0.1arcsec could be readily achieved. In 1999, Guo et al. [10] developed an optical method for small angle measurement based on surface- plasma resonance (SPR), and a measurement resolution of 0.2arcsec was achieved experimentally. In 2003, Ge and Makeda [11] developed an angle-measurement tech- nique based on fringe analysis for phase-measuring profilometry. The measurement range was 72160arcsec and the deviation from linearity was better than 70.02 arcsec. In 2004, Chiu et al. [12] developed an instru- ment for measuring small angles using multiple total internal reflections in heterodyne interferometry, and the angular resolution was better than 0.454arcsec over the measurement range C02.121pyp2.121 for 20 total-internal reflections. W. Jywe et al. / International Journal of Machine Most angle-measurement technique research focuses on 1D angle measurement and interferometric angle measurement, and 2D measurement also focuses on interferometric techniques. However, interferometric systems are expensive and complex, and cannot be used extensively in industry. Therefore, the low cost and multiple DOF measurement system is needed for rotary table calibration. The position sensing detector (PSD) could be used to measure the rotary part error, the speed of rotary part, the rotation direction of rotary part, the angular position, and the indexing error [13,14]. Jywe et al. employed two PSDs and one reflective grating to test rotary table performance [15], but its measurement range was small (o11). In [15], no full circle test was implemented and no analytic solution was provided. However, for the general rotary table calibra- tion, the 3601 full circle test is necessary. This paper both describes the building of one 4-DOF measurement system and establishes a novel technique for rotary table full circle test. The 4-DOF system presented in this paper comprises one 1D reflection grating, one laser diode, four PSDs and one reference rotary table. The laser interferometer and the autocollimator were most used rotary table measurement system. However, in rotary table calibration process, the laser interferometer and the autocollimator need a high accuracy reference rotary table and a polygon mirror, respectively. Therefore, using the laser interferometer or autocollimator to calibrate rotary table is expensive. Because , the cost of 1D reflection grating, PSD, signal conditioning unit of PSD and laser diode and rotary table is about 1 5 of one laser interferometer system or 1 2 of one autocollimator system. Moreover, in the presented method, no high accurate reference rotary table, but with good repeatability is needed. Even the indexing error and the geometric error of the reference rotary table is large, they will be obtained by the presented method. 2. The 4-DOF measurement system In this paper, the 4-DOF measurement system includes one reference rotary table, one 1D grating, one laser diode, two PSDs, two PSD processors, one A/D card and one personal computer (PC). Fig. 1 shows the schematic diagram. The reference rotary table was placed on the target rotary table then the 1D grating was mounted on the rotary table by the fixture. The laser diode and PSDs were placed near the 1D grating. The laser beam from the laser diode was projected onto a 1D grating and then the 1D grating produced many diffraction light beams. In this paper, the +1 order and C01 order diffraction light beam are used, and two PSDs were used to detect the diffraction light beam. Generally six geometric errors are defined on a rotary table, namely three linear position errors and three angular position errors (pitch, roll, and yaw). The three linear position errors are d x , d y and d z , and the three angular position errors are e x , e y and e z , respectively. In addition, there are eccentricity between the grating and the axis of the rotary table, which are defined as D x and D y . Tools d x ? d xt t d xr , C15 y ? C15 yt t C15 yr ; d y ? d yt t d yr , C15 z ? C15 zt C0 C15 zr ; d z ? d zt t d zr , e13T where e z is the index difference between the target rotary table and the reference rotary table, and it accumulatively varies during the calibration procedure. The e x , e y , d x , d y and d z are not accumulative. Because one full circle test needs two tests, the repeatability of the target rotary table and the reference rotary table must be good, otherwise the measured results will not repeat. The basic requirement of the calibrating technique is that the target rotary table under calibration can be rotated the same step size as the reference rotary table in different orientations, say on for clockwise and the other counter-clockwise. Each sector of the table under test has been compared with every sector of the reference one in order to build the first set of data. For example, one rotary table was tested at 12 angular position points around 3601 (i.e. at 01,301,601,y,3301), which were equally spaced segmented in the target rotary table and the reference rotary table. At the start in 1000 C1C1C1 C010000C1C1C1 0 0100 C1C1C1 0 C010 00C1C1C1 0 0010 C1C1C1 00C0100C1C1C1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 C1C1C1 0 C010 00C1C1C1 0 0100 C1C1C1 00C0100C1C1C1 0 0010 C1C1C1 000C010C1C1C1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 0 C1C1C11 C01 0 0 0 0 C1C1C1 0 . . . C15 z1n ? C15 ztn C0 C15 zrn , e14T where e z1n is the first set of angular readings and n is the number of increments over 3601. The subscript ‘t’ of the symbol e zt1 means the error of the target rotary table and the subscript ‘r’ means the error of the reference rotary table. In the second test of full circle test, the target rotary table and reference rotary table was set to 01 again and the reference rotary table was incremented by one nominal step (ex. 301). After the rotation of the reference rotary table, the first set of sample was taken. Then, the target rotary table was rotated 301 clockwise and the reference rotary table was rotated 301 counter-clock- wise and the other sets of sample were taken. From the above experiment process, the results of second test were obtained. Then, the flowing relationship can be derived: C15 z21 ? C15 zt1 C0 C15 zr2 , C15 z22 ? C15 zt2 C0 C15 zr3 , . . . C15 z2n ? C15 ztn C0 C15 zr1 , e15T where e z2n is the second set of angular readings and n is the number of increments over 3601. The two sets of measured data can then be rearranged as follows: C15 zt1 C15 zt2 C15 zt3 . . . C15 zr1 C15 zr2 C15 zr3 . . . 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ? C15 z11 C15 z12 C15 z13 . . . C15 z21 C15 z22 C15 z23 . . . 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (16) C15 zrn C15 z2n and the original augmented matrix is shown as: 1000 C1C1C1 C010000C1C1C1 C15 z11 0100 C1C1C1 0 C010 00C1C1C1 C15 z12 0010 C1C1C1 00C0100C1C1C1 C15 z13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 C1C1C1 0 C010 00C1C1C1 C15 z21 0100 C1C1C1 00C0100C1C1C1 C15 z22 0010 C1C1C1 000C010C1C1C1 C15 z23 . . . 0 . . . 0 . . . 0 . . . 0 C1C1C11 . . . C01 . . . 0 . . . 0 . . . 0 . . . 0 C1C1C1 . . . C15 z2n 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (17) An augmented matrix of the reduced system can then be derived as follows: Since Eq. (18) is linear-dependent, more equations are required. An assumption is again made to presume that no closing error exists within the reference rotary table and consequently the following equation can be derived: C15 zr1 t C15 zr2 t C15 zr3 tC1C1C1tC15 zrnC01 t C15 zrn ? 360 C14 . (20) ARTICLE IN PRESS Table 1 Components of the prototype 4-DOF measurement system PSD UDT SC-10D, active area 100mm 2 PSD signal processor On-Trak OT-301 PC Intel Pentium4 2.0G 256MB RAM 40G HD A/D Card Advantech PCI-1716, 16 bit, sampling range 710V, Max. sampling frequency 250kHz Laser diode l ? 635nm, 5mW 1D Grating Rolled diffraction grating, 600grooves per mm, Autocollimator NewPort LDS Vector, measurement range: 2000mrad W. Jywe et al. / International Journal of Machine Tools & Manufacture 47 (2007) 1978–19871982 1000C1C1C1 C010000C1C1C1 0 C15 z11 0100C1C1C1 0 C010 00C1C1C1 0 C15 z12 0010C1C1C1 00C0100C1C1C1 0 C15 z13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0000C1C1C1 1 C010 00C1C1C1 0 C15 z21 C0 C15 z11 0000C1C1C1 01C0100C1C1C1 0 C15 z22 C0 C15 z12 0000C1C1C1 001C010C1C1C1 0 C15 z23 C0 C15 z13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0000C1C1C1 10000C1C1C1 C01 P nC01 i?1 eC15 z2i C0 C15 z1i T 0000C1C1C1 C010000C1C1C1 1 C15 z2n C0 C15 z1n 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (18) From the last two rows in the reduced matrix, it can be shown that C15 zr1 C0 C15 zrn ? X nC01 i?1 eC15 z2i C0 C15 z1i T?C0eC15 z2n C0 C15 z1n T, (19) or X nC01 i?1 eC15 z2i C0 C15 z1i T?0. Fig. 2. Photograph of the 4DOF measurement system with 4 PSD. Fig. 3. Calibration results (b) standard deviation. Eq. (20) is then incorporated into the augmented matrix in Eq. (18) to give the following: 1000 C1C1C1 C010 0 00C1C1C1 0 C15 z11 0100 C1C1C1 0 C010 00C1C1C1 0 C15 z12 0010 C1C1C1 00C0100C1C1C1 0 C15 z13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 C1C1C1 0 C010 00C1C1C1 0 C15 z21 0100 C1C1C1 00C0100C1C1C1 0 C15 z22 0010 C1C1C1 000C010C1C1C1 0 C15 z23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0000C1C1C11 C010 0 00C1C1C1 0 C15 z2n 0000C1C1C1011111C1C1C1 1 360 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (21) Finally, using the Gaussian Elimination method, the actual individual angle e zti and e zri at each target position can be calculated. The calculation of e xti , e xri , e yti , e yri , d xti , d xri , d yti , d yri , d zti and d zri is different to e zti and e zri . For instance, C15 x11 ? C15 xt1 t C15 xr1 , C15 x12 ? C15 xt2 t C15 xr2 , . . . C15 x1n ? C15 xtn t C15 xrn e22T and C15 x21 ? C15 xt1 C0 C15 xr2 , C15 x22 ? C15 xt2 C0 C15 xr3 , . . . C15 x2n ? C15 xtn C0 C15 xr1 . e23T The summation of e xri is C15 xr1 t C15 xr2 t C15 xr3 tC1C1C1tC15 xrnC01 t C15 xrn ? 0 C14 . (24) ARTICLE IN PRESS W. Jywe et al. / International Journal of Machine Tools & Manufacture 47 (2007) 1978–1987 1983 Fig. 4. Stability test results (a)–(d). 4. Experimental results and discussion In this paper, the calibration of the 4-DOF measurement system, system stability, system verification and full circle test were accomplished. The photograph of this system was shown in Fig. 2. Components not shown in Fig. 2 include a desktop PC connected to the PSD signal processor via an A/D card. The component specifications were listed in Table 1. 4.1. System calibration System calibration was the first experiment. In this experiment, the NewPort autocollimator was used to provide the reference angular position. Its measurement range was 7410arcsec, resolution was 0.02arcsec and accuracy was 0.5arcsec. Fig. 3(a) shows the calibration result and Fig. 3(b) gives the standard deviations for ARTICLE IN PRESS Tools & Manufacture 47 (2007) 1978–1987 Therefore, the matrix of e xti and e xri is 1000 C1C1C1 C010000C1C1C1 0 C15 x11 0100 C1C1C1 0 C010 00C1C1C1 0 C15 x12 0010 C1C1C1 00C0100C1C1C1 0 C15 x13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 C1C1C1 0 C010 00C1C1C1 0 C15 x21 0100 C1C1C1 00C0100C1C1C1 0 C15 x22 0010 C1C1C1 000C010C1C1C1 0 C15 x23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0000C1C1C11 C010000C1C1C1 0 C15 x2n 0000C1C1C1011111C1C1C1 10 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (25) Similarly, 1000 C1C1C1 C010 0 00C1C1C1 0 C15 y11 0100 C1C1C1 0 C010 00C1C1C1 0 C15 y12 0010 C1C1C1 00C0100C1C1C1 0 C15 y13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 C1C1C1 0 C010 00C1C1C1 0 C15 y21 0100 C1C1C1 00C0100C1C1C1 0 C15 y22 0010 C1C1C1 000C010C1C1C1 0 C15 y23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0000C1C1C11 C010 0 00C1C1C1 0 C15 y2n 0000C1C1C1011111C1C1C1 10 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , (26) 1000 C1C1C1 C010000C1C1C1 0 d y11 0100 C1C1C1 0 C010 00C1C1C1 0 d y12 0010 C1C1C1 00C0100C1C1C1 0 d y13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 C1C1C1 0 C010 00C1C1C1 0 d y21 0100 C1C1C1 00C0100C1C1C1 0 d y22 0010 C1C1C1 000C010C1C1C1 0 d y23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0000C1C1C11 C010000C1C1C1 0 d y2n 0000C1C1C1011111C1C1C1 10 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (27) This technique can be used in the rotary table 6-DOF calibration, but in this paper, the measurement system could only measure 4-DOF errors, so this paper lists only four equations (Eqs. (21), (25)–(27)). The recorded count was based on the measurement range of the system. For example, the measurement range of Lin’s system (laser interferometer) [16] was about 101. W. Jywe et al. / International Journal of Machine1984 Therefore, one full circle test must record at least 36 points during the first and second tests, respectively. Fig. 5. Verification result (a) and (b). ARTICLE IN PRESS W. Jywe et al. / International Journal of Machine system uncertainty. Throughout the calibration process, it was clear that the linearity of e z was good and the uncertainty of e z was about 1.5arcsec. The angular Fig. 6. Full circle test Tools & Manufacture 47 (2007) 1978–1987 1985 position (e z ) measurement range of the 4-DOF measure- ment system was about 11 because almost all measurement r