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International Journal of Machine Tools & Manufaca,Engineerin25angle measurement instrument, and the conventional errors), but conventional instruments can only measuretable could then be ignored from the measurement results.The instrument usually recorded one time when the targetrotary table was rotated clockwise and the reference rotaryARTICLE IN PRESSCorresponding author. National Formosa University, Department ofAutomation Engineering, No. 64 Wenhua Rd., Huwei, Taiwan, ROC.Tel.: +88656315402; fax: +88656311500.E-mail addresses: .tw (W. Jywe),table was rotated counterclockwise. In general, one rotarytable calibration for a 3601 full circle requires 36 recordingif the sampled period of measurement system is 101.Ifa0890-6955/$-see front matter r 2007 Elsevier Ltd. All rights reserved.doi:10.1016/.tw (C.J. Chen), .tw (W.H. Hsieh),.tw (P.D. Lin), .tw (H.H. Jwo), (T.Y. Yang).instruments are the rotary encoder, the laser interferom-eter, the autocollimator and the precision level. A rotaryencoder 1 is commonly used in indexing measurement in arotary machine, e.g. a rotary table of the multi-axismachine tool, the joint of a robot, the spindles of machinetools and the indexing of a ball screw. However, the rotaryencoder is only suitable for the indexing error measure-ment. A laser interferometer 2 has often been used tomeasure a small angle, but it can only obtain indexing erroreither one dimensional (1D) error or 2D errors. Thecomplete calibration procedure of a rotary table requires 6DOF measurement for a 3601 full circle, but the measure-ment range of most measurement systems is smaller than101. Therefore the measurement range of the laserinterferometer and autocollimator are not enough and, inaddition, they are expensive. The conventional calibrationtechnique of the rotary table for a 3601 full circle requiresone reference rotary table, which must have high accuracyand high repeatability. The error of the reference rotaryC3and the reference rotary table could be obtained. The system calibration, stability test, system verification and full circle test werecompleted. 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 separation1. IntroductionA rotary table is frequently used in industry in suchthings as machine tools, CMM and assembly lines.Therefore, the calibration of the rotary table is veryimportant. The calibration of the rotary table requires anduring an indexing test. An autocollimator 3 is frequentlyused to measure small angles and it can be applied to twodimensional (2D) angle measurement (pitch error and yawerror), but its measurement range is small and it requireone standard polygon mirror. A rotary table has 6 DOFerrors (3 linear position errors and 3 angular positionreference rotary table, but with good repeatability is needed. After two full circle tests, the 4-DOF errors of both the target rotary tableA novel simple and low cost 4 degreecalibrating technique forW. Jywea,C3, C.J. Chenb, W.H. HsiehaNational Formosa University, Department of AutomationbNational Cheng-Kung University, Department of MechanicalReceived 30 October 2006; received in revised formAvailable onlineAbstractFor calibrating an angular rotary table, either a high precision standardemployed at high cost. This paper establishes a novel, simple and lowof a rotary table (three angular position errors and one linear positionone 1 dimensional (1D) grating and two 2 dimensional (2D) position-sensing-detectorsture 47 (2007) 19781987of freedom angular indexinga precision rotary tableP.D. Linb, H.H. Jwoa, T.Y. Yangag, No. 64 Wenhua Rd., Huwei, Taiwan, ROCEngineering, No. 1, University Rd., Tainan, Taiwan, ROC1 February 2007; accepted 13 February 2007February 2007table or a laser interferometer and related optics are normallycost technique to calibrate the 4-degrees-of-freedom (DOF) errorserror) for a 3601full circle by employing one reference rotary table,(PSD). With this technique, no highly /locate/ijmactoolARTICLE IN PRESSmore complete test is implemented, the calibration processwill takes a long time.In general, the rotary table includes the index error,wobble error and eccentricity. But conventional rotarytable calibration techniques (laser interferometer or auto-collimator) only calibrate the index error and the wobbleerror. However, the high precision rotary table must becalibrated in more details. Through the complete rotarytable calibration, the errors of rotary table can becompensated. In this paper, the errors of rotary table weredefined by 6 DOF, i.e. three linear position errors (dx, dy,dz) and three angular position errors (ex, ey, ez). The indexerror was represented by ez, the wobble error wasrepresented by exand ey, the eccentricity was representedby dxand dy.In recent years, angular measuring techniques havefocused on the interferometric methods. In 1992, Huanget al. 4 developed a small angle measurement systemwhich was based on the internal reflection effect in a glassboundary and Fresnels law. In Huangs system, theresolution was 0.2arcsec and the measuring range was3arcsec. In 1996, Xiaoli et al. 5 established a 2D smallrotation angle-measurement system using two differentparallel interference patterns (PIP) that were orthogonal toeach other. The standard deviation of Xiaolis system was0.6arcsec. In the following year Xiaoli et al. 6 improvedtheir system so that its resolution was 0.2arcsec andmeasuring range was 730arcmin. In 1997, Chiu et al. 7established a modified angle measurement technique with aresolution of 0.333arcsec and a measuring range of75.61.At its optimum performance, the systems resolution was0.288arcsec. In 1998, Zhou and Cai 8 established anangle measurement technique which was based on thetotal-internal reflection effect and heterodyne interferome-try. The system resolution was better than 0.3arcsec,depending on the refractive index selected. In 1998, Huanget al. 9 established a method of angle measurement, basedon the internal reflection effects, that used a single right-angle prism. They demonstrated that angle measurementwith a range of 7500arcmin, a nonlinearity error of70.1%, and a resolution of 0.1arcsec could be readilyachieved. In 1999, Guo et al. 10 developed an opticalmethod for small angle measurement based on surface-plasma resonance (SPR), and a measurement resolutionof 0.2arcsec was achieved experimentally. In 2003, Geand Makeda 11 developed an angle-measurement tech-nique based on fringe analysis for phase-measuringprofilometry. The measurement range was 72160arcsecand the deviation from linearity was better than 70.02arcsec. In 2004, Chiu et al. 12 developed an instru-ment for measuring small angles using multiple totalinternal reflections in heterodyne interferometry, and theangular resolution was better than 0.454arcsec over themeasurement range C02.121pyp2.121 for 20 total-internalreflections.W. Jywe et al. / International Journal of MachineMost angle-measurement technique research focuses on 1Dangle measurement and interferometric angle measurement,and 2D measurement also focuses on interferometrictechniques. However, interferometric systems are expensiveand complex, and cannot be used extensively in industry.Therefore, the low cost and multiple DOF measurementsystem is needed for rotary table calibration. The positionsensing detector (PSD) could be used to measure the rotarypart error, the speed of rotary part, the rotation directionof rotary part, the angular position, and the indexing error13,14. Jywe et al. employed two PSDs and one reflectivegrating to test rotary table performance 15, but itsmeasurement range was small (o11). In 15, no full circletest was implemented and no analytic solution wasprovided. However, for the general rotary table calibra-tion, the 3601 full circle test is necessary. This paper bothdescribes the building of one 4-DOF measurement systemand establishes a novel technique for rotary table full circletest. The 4-DOF system presented in this paper comprisesone 1D reflection grating, one laser diode, four PSDs andone reference rotary table.The laser interferometer and the autocollimator weremost used rotary table measurement system. However, inrotary table calibration process, the laser interferometerand the autocollimator need a high accuracy referencerotary table and a polygon mirror, respectively. Therefore,using the laser interferometer or autocollimator to calibraterotary table is expensive. Because , the cost of 1D reflectiongrating, PSD, signal conditioning unit of PSD and laserdiode and rotary table is about15of one laser interferometersystem or12of one autocollimator system. Moreover, in thepresented method, no high accurate reference rotary table,but with good repeatability is needed. Even the indexingerror and the geometric error of the reference rotary tableis large, they will be obtained by the presented method.2. The 4-DOF measurement systemIn this paper, the 4-DOF measurement system includesone reference rotary table, one 1D grating, one laser diode,two PSDs, two PSD processors, one A/D card and onepersonal computer (PC). Fig. 1 shows the schematicdiagram. The reference rotary table was placed on thetarget rotary table then the 1D grating was mounted on therotary table by the fixture. The laser diode and PSDs wereplaced near the 1D grating. The laser beam from the laserdiode was projected onto a 1D grating and then the 1Dgrating produced many diffraction light beams. In thispaper, the +1 order and C01 order diffraction light beamare used, and two PSDs were used to detect the diffractionlight beam. Generally six geometric errors are defined on arotary table, namely three linear position errors and threeangular position errors (pitch, roll, and yaw). The threelinear position errors are dx, dyand dz, and the threeangular position errors are ex, eyand ez, respectively. Inaddition, there are eccentricity between the grating and theaxis of the rotary table, which are defined as Dxand Dy.Tools & Manufacture 47 (2007) 19781987 1979The distance from the light point on the grating to therotary table origin point is h0.ARTICLE IN PRESStheThe outputs of PSDs were effect by the Dy, dy, ex, eyandez. Therefore, the PSD A x-axis output isPAx l1siny 2C15zC0sinyC138Dy dytany l1sinycos2C15z cosysin2C15zC0 sinyC138Dy dytany, 1where y is the diffraction angle of the grating, l1is thedistance between the PSD and the grating. The diffractionequation of the grating isnl dsiny C6 siny0, (2)Fig. 1. The schematic diagram ofW. Jywe et al. / International Journal of Machine1980where n is the order of diffraction, d is the grating constant,l is the wavelength of the laser source, y0is the incidentangle and y is the diffraction angle. In this paper, d is1/600mm, l 650nm and n 1. Therefore, the diffractionangle y is 22.9541.The PSD Ay-axis output isPAyh0sinC15x l1tan2C15xl1siny h0sin C15ytanC15y.(3)The PSD Bx-axis output isPBx l1siny C0 siny C0 2C15zC138 C0 Dy dytany l1siny C0 sinycos2C15z cosysin2C15zC138C0Dy dytany. 4The PSD By-axis output isPByh0sinC15x l1tan2C15xC0l1siny C0 h0sinC15ytanC15y.(5)From the above equations, the four geometric errors can bederived. ezisC15z12sinC01PAx PBx2l1cosyC18C19, (6)orl1PAx PBx2cosysin2C15z. (7)From Eq. (7), the distance between the grating and PSDcan be calculated, if the ezis known. The linear error in they direction isDy dyPAxC0 l1sinycos2C15z cosysin2C15zC0 sinyC138tanyl1siny C0 sinycos2C15z cosysin2C15zC138C0 PBx. 84-DOF measurement system.Tools & Manufacture 47 (2007) 19781987tanyIn a full circle test, Dyis constant, dyis the function valueof the rotary angle and the summation of dyis zero.Therefore, Eq. (8) can be rewritten asdyl1siny C0 sinycos2C15z cosysin2C15zC138C0PBxtanyC0 Dy. (9)From Eqs. (3) and (5), eyisC15y tanC01PAyC0 PBy2l1sinyC18C19. (10)The summation of the PSD Ay-axis and the PSD By-axisisPAy PBy 2h0sinC15x l1tan2C15x 2h0sinC15ytanC15y.(11)Because h0sin C15x5l1, Eq. (11) can be written asC15x12tanC01PAy PByC0 2h0sinC15ytanC15y2l1C18C19. (12)From Eqs. (6), (9), (10) and (12), the dy, ex, eyand ezcan beobtained throughout the PSD A and PSD B outputs.3. The model of the full circle testThe measurement range of most instruments is lessthan 101, so the complete calibration of a rotary tablerequires a special method. In normal rotary table calibra-tion, the autocollimator uses one polygon mirror and thelaser interferometer uses one reference rotary table. In thispaper, the technique also requires one reference rotarytable, but the requirement of the reference rotary table isonly that the errors of reference rotary table must berepeatable. In 1994, Lin 16 established a rotary tablethe first test, after the target rotary table and referencerotary table were set at 01 the first set of sample was takenby personal computer. Then, the target rotary table wasrotated 301 clockwise and the reference rotary table wasrotated 301 counter-clockwise and the other sets of samplewere taken by personal computer. From the aboveexperiment process, the following relationship can bederived:C15z11 C15zt1C0 C15zr1,C15z12 C15zt2C0 C15zr2,ARTICLE IN PRESS37777777777777777775W. Jywe et al. / International Journal of Machine Tools & Manufacture 47 (2007) 19781987 1981calibration technique which could measure the indexingerror of the rotary table for a 3601 full circle. However,the technique could only measure the indexing error.Consequently, an improved technique is established inthis section. When the errors of the reference rotarytable were considered, the geometric errors of the rotarytable areC15x C15xt C15xr; dx dxt dxr,C15y C15yt C15yr; dy dyt dyr,C15z C15ztC0 C15zr; dz dzt dzr, 13where ezis the index difference between the target rotarytable and the reference rotary table, and it accumulativelyvaries during the calibration procedure. The ex, ey, dx, dyand dzare not accumulative. Because one full circle testneeds two tests, the repeatability of the target rotary tableand the reference rotary table must be good, otherwise themeasured results will not repeat.The basic requirement of the calibrating technique isthat the target rotary table under calibration can berotated the same step size as the reference rotary table indifferent orientations, say on for clockwise and the othercounter-clockwise. Each sector of the table under testhas been compared with every sector of the referenceone in order to build the first set of data. For example,one rotary table was tested at 12 angular positionpoints around 3601 (i.e. at 01,301,601,y,3301), whichwere equally spaced segmented in the target rotarytable and the reference rotary table. At the start in1000 C1C1C1 C010000C1C1C1 00100 C1C1C1 0 C010 00C1C1C1 00010 C1C1C1 00C0100C1C1C1 0.1000 C1C1C1 0 C010 00C1C1C1 00100 C1C1C1 00C0100C1C1C1 00010 C1C1C1 000C010C1C1C1 0.266666666666666666640 0 0 0C1C1C11C01 0 0 0 0C1C1C10.C15z1n C15ztnC0 C15zrn, 14where ez1nis the first set of angular readings and n is thenumber of increments over 3601. The subscript t of thesymbol ezt1means the error of the target rotary tableand the subscript r means the error of the referencerotary table.In the second test of full circle test, the target rotarytable and reference rotary table was set to 01 again andthe reference rotary table was incremented by onenominal step (ex. 301). After the rotation of the referencerotary table, the first set of sample was taken. Then,the target rotary table was rotated 301 clockwise andthe reference rotary table was rotated 301 counter-clock-wise and the other sets of sample were taken. Fromthe above experiment process, the results of second testwere obtained. Then, the flowing relationship can bederived:C15z21 C15zt1C0 C15zr2,C15z22 C15zt2C0 C15zr3,.C15z2n C15ztnC0 C15zr1, 15where ez2nis the second set of angular readings and n is thenumber of increments over 3601. The two sets of measureddata can then be rearranged as follows:C15zt1C15zt2C15zt3.C15zr1C15zr2C15zr3.2666666666666666666437777777777777777775C15z11C15z12C15z13.C15z21C15z22C15z23.2666666666666666666437777777777777777775(16)C15zrnC15z2nand the original augmented matrix is shown as:1000 C1C1C1 C010000C1C1C1 C15z110100 C1C1C1 0 C010 00C1C1C1 C15z120010 C1C1C1 00C0100C1C1C1 C15z13.1000 C1C1C1 0 C010 00C1C1C1 C15z210100 C1C1C1 00C0100C1C1C1 C15z220010 C1C1C1 000C010C1C1C1 C15z.0C1C1C11.C0.0C1C1C1.C15z2n2666666666666666666437777777777777777775.(17)An augmented matrix of the reduced system can then bederived as follows:Since Eq. (18) is linear-dependent, more equations arerequired. An assumption is again made to presume that noclosing error exists within the reference rotary table andconsequently the following equation can be derived:C15zr1 C15zr2 C15zr3C1C1C1C15zrnC01 C15zrn 360C14. (20)ARTICLE IN PRESSTable 1Components of the prototype 4-DOF measurement systemPSD UDT SC-10D, active area 100mm2PSD signalprocessorOn-Trak OT-301PC Intel Pentium4 2.0G 256MB RAM 40G HDA/D Card Advantech PCI-1716, 16 bit, sampling range710V, Max. sampling frequency 250kHzLaser diode l 635nm, 5mW1D Grating Rolled diffraction grating, 600grooves per mm,Autocollimator NewPort LDS Vector, measurement range:2000mradW. Jywe et al. / International Journal of Machine Tools & Manufacture 47 (2007) 1978198719