Transcription

1 Error Calbraton on Fve-axs Mach TtleDsplacement Measurement between Sp Dssertaton_ 全文 ) Author(s) Hong, Cefu Ctaton Kyoto Unversty ( 京都大学 ) Issue Date URL Rght Type Thess or Dssertaton Textverson author Kyoto Unversty

2 Error Calbraton on Fve-axs Machne Tools by Relatve Dsplacement Measurement between Spndle and Work Table 212 Cefu Hong

3 Table of Contents Chapter 1 Introducton Drect and ndrect measurement of error motons of machne tools Prevous studes for ndrect measurement for rotary axs Objectve and orgnal contrbuton of ths thess.. 7 Chapter 2 R-test devce and knematc model of fve-axs controlled machne tools Introducton Contact-type R-test devce Measurng prncple of contact-type R-test measurng nstrument Procedure to measure three-dmensonal dsplacement by contact-type R-test Contact-type R-test prototype developed n ths study Knematc model of fve-axs controlled machne tools Geometrc errors Sources of geometrc errors Poston-ndependent and poston-dependent geometrc errors of rotary axes Knematc model of fve-axs controlled machne tools Chapter 3 Calbraton of poston-ndependent and poston-dependent geometrc errors of rotary axes by statc R-test measurement Introducton Objectve and orgnal contrbuton of ths chapter Measurement procedure I

4 3.3.1 R-test measurement cycle Sphere poston Graphcal presentaton of R-test profles Separaton of squareness errors of lnear axes Identfcaton of poston-ndependent and poston-dependent geometrc errors of a rotary axs Objectve Separaton of the nfluence of squareness errors of lnear axes Identfcaton of poston-ndependent geometrc errors Identfcaton of poston-dependent geometrc errors of a rotary axs Expermental case study Graphcal presentaton of R-test profles Expermental setup Measurement result Observaton Separaton of squareness errors of lnear axes Identfcaton of geometrc errors assocated wth B-axs Concluson. 55 Chapter 4 Observaton of thermal nfluence on error motons of rotary axes by statc R-test Introducton Objectve and orgnal contrbuton of ths chapter Test procedure Analyss of R-test profles Graphcal presentaton of R-test profles Error parameter of the rotary table to be dentfed Calbraton procedure of geometrc errors of the rotary table... 6 II

5 4.5 Case study Expermental setups Measured temperatures Graphcal presentaton of R-test profles Calbraton of geometrc errors of the rotary table Concluson. 7 Chapter 5 Non-contact R-test for dynamc measurement on fve-axs machne tools Introducton Objectve and orgnal contrbuton of ths chapter Selecton of laser dsplacement sensor for non-contact R-test Objectve Laser dsplacement sensors used n ths study Expermental nvestgaton of measurement uncertanty of laser dsplacement sensors for profle measurement of sphere Constructon of algorthm to calculate three-dmensonal dsplacement of sphere center Dfference n algorthms for contact-type and non-contact R-tests Algorthm to calculate three-dmensonal dsplacement of sphere center Expermental verfcaton of the proposed algorthm Compensaton scheme of the measurement error caused by laser dsplacement sensor Objectve Interpolaton of the measurement error wth RBF Network Developed prototype of non-contact R-test devce Developed prototype of non-contact R-test wth LK-G Profle measurement of a sphere wth LK-G III

6 5.6.3 Expermental verfcaton of the proposed algorthm Case studes Calbraton of an error map of a rotary axs n statc measurement Dynamc measurement wth synchronous moton of rotary axs and lnear axes Concluson. 111 Chapter 6 Influence of geometrc errors of rotary axes on a machnng test of cone frustum Introducton Objectve and orgnal contrbuton of ths chapter Set-up of cone frustum machnng test Influence of poston-ndependent geometrc errors of rotary axes Influence of poston-dependent geometrc errors of rotary axes Analyss objectve and basc methodology Angular postonng error of B-axs Axal error moton of B-axs Lnear error moton of B-axs to Z-drecton Angular postonng error of C-axs Axal error moton of C-axs Perodc pure radal error moton of B-axs and C-axs Perodc concal tlt error moton of B-axs and C-axs Change n poston and orentaton of C-axs centerlne dependng on B-axs rotaton Perodc pure radal error moton and tlt error moton of C-axs dependng on B-axs rotaton Summary of analyss Expermental case study Objectve Influence of poston-ndependent geometrc errors IV

7 6.6.3 Influence of major poston-dependent geometrc errors Measurement of error motons of C-axs and ts nfluence Concluson. 136 Chapter 7 Conclusons Acknowledgements References V

8 Chapter 1 Introducton 1.1 Drect and ndrect measurement of error motons of machne tools Machne tools wth two rotary axes to tlt and rotate a tool wth respect to a workpece, n addton to three orthogonal lnear axes, are collectvely called fve-axs machne tools. As a typcal example of fve-axs machne tool confguraton n today s market, a tltng-rotary table type fve-axs machne tool s shown n Fg. 2-5 n Secton A fve-axs machne tool s typcally used n machnng components wth sculptured surface, such as an mpeller, where t s ndspensble to contnuously change the tool orentaton to the workpece to generate the desgned surface. A fve-axs machne tool s also often used when conductng mult-surface machnng, snce the chuck of a workpece could be smplfed as t owns the ablty to change the relatve orentaton between a tool and the workpece, whch potentally reduces non-cuttng tme and set-up tme. Other potental advantage wth the fve-axs machnng ncludes shorter tool extenson, whch reduces the tool deflecton (n three-axs machnng, a tool extenson must be sometmes longer to avod unntended nterference to the workpece). The ntroducton of fve-axs machne tools could potentally lead to cost down, and hgher effcency for a machnng workshop. Wth an ncreasng need for machnng components wth geometrc complexty n a hgh effcency, fve-axs machne tools are extensvely used n varous manufacturng applcatons requrng hgher machnng accuracy, e.g. de and mold makng. However, the ncreased number of the controlled axes makes t more dffcult to mechancally adjust the algnment of each axs. For example, t s a common practce for machne tool bulders to mechancally adjust the squareness of two lnear axes by modfyng the algnment of mechancal parts, e.g. gude ways on the machne bed. It s often more dffcult and 1

9 tme-consumng, or sometmes not possble wthout re-machnng structural parts, to mechancally adjust the squareness of an axs of rotaton to a lnear axs. Furthermore, error motons of each axs accumulate as the postonng error of the tool. For these reasons, t s generally harder to acheve hgher accuracy on a fve-axs machne tool compared to conventonal three-axs machne tools. The mprovement of ther moton accuraces s a crucal demand n the market [Veldhus, 1995, Khan, 21]. Note that the term error moton s defned n ISO 23-7:26 [ISO 23-7, 26] as unntended relatve dsplacement n the senstve drecton between the tool and the workpece. The geometrc error s defned as a change n the geometry of the machne s components present n the machne s structural loop from the nomnal [Schwenke, 28]. Note that a structural loop of a machne tool s defned as an assembly of mechancal components whch mantan a relatve poston between the tool and the workpece table. To mprove the machne s overall accuracy, t s clearly crucal to frst measure geometrc errors of each axs. In [Schwenke, 28], measurement methodologes to detect geometrc errors n machnes are dstngushed between drect and ndrect methods. The drect measurement of geometrc errors represents the analyss of sngle errors, such as lnear postonng errors, straghtness errors, and angular errors of each axs. For example, the lnear postonng error of a lnear axs s typcally measured by usng a laser nterferometer [ISO 23-2, 26]. One setup of ths measurement measures only the lnear postonng error of a sngle axs, whle mnmzng the nfluence of other error motons. A key s to set up the measurng nstrument such that only the target error moton nfluences measurement results. Many measurement methodologes accepted by machne tool bulders are drect measurement [Sartor, 1995]. Whle t s easer to ensure the measurement accuracy of drect measurement, the effcency of the drect measurement can be a crtcal ssue. For example, for orthogonal three-axs machnes, 3 lnear dsplacement errors, 6 straghtness errors, 3 squareness errors, and 6 angular errors must be measured by dfferent setups to construct the machne's knematc model (see Secton 2.3 for further detals). 2

10 On the other hand, the ndrect measurement focuses on the tool tp locaton as the superposton of these sngle errors. A typcal example of the ndrect measurement s the crcular test usng the double ball bar (DBB) descrbed n ISO 23-4 [ISO 23-4, 25], as shown n Fg In the crcular test, measured error profles are nfluenced by many error motons of two lnear axes, e.g. postonng errors, straghtness errors of each axs, and the squareness error of two axes. By best-fttng the machne s knematc model such that ts smulated TCP (tool center poston) trajectores concde wth measured trajectores, one can estmate many error motons by a sngle crcular test [Kakno, 1993]. Ths example llustrates a strong advantage for ndrect measurements. Z Y X Fg. 1-1 Crcular test n XY plane wth the DBB (Double Ball Bar): X, Y-axes are commanded a crcular nterpolaton, centered at the nomnal ball poston attached on the table. The change n the length of the telescopng bar connectng both balls s measured by a lnear encoder nstalled n t. The contourng error moton of XY-axes n a crcular can be measured. For rotary axes n fve-axs machne tools, many drect measurement methodologes are descrbed n ISO 23-1 [ISO 23-1, 1996] and ISO 23-7 [ISO 23-7, 26], and are also wdely done by machne tool bulders. For example, the measurement of angular postonng accuracy of a rotary axs by usng an autocollmator and a reference polygon (or a reference ndexng table) s one of drect measurements typcally done by many machne tool bulders. However, these drect measurement methodologes only evaluate error 3

11 motons of a sngle rotary axs. When multple rotary axes stacked to each other, t s n practce mportant to evaluate how error motons of one axs changes wth the rotaton of the other axs. It s dffcult to apply drect measurements to such evaluaton, snce t requres many setup changes, and thus sgnfcant measurng tme, effort, and cost. The objectve of ths thess s to propose an ndrect measurement methodology to evaluate error motons of multple rotary axes n fve-axs machne tools. 1.2 Prevous studes for ndrect measurement for rotary axs Recently, there have been many recent research works on the ndrect measurement of geometrc errors of rotary axes n the fve-axs knematcs. Ths thess studes the applcaton of one of these approaches, the R-test, to the ndrect measurement of a large class of error motons. To clarfy the orgnal contrbuton of ths thess, ths subsecton brefly revews prevous research works. (1) DBB (double ball bar) The DBB s a length measurng devce to measure the dstance between two spheres by usng a lnear encoder nstalled nsde a telescopng bar connectng them, as shown n Fg As was dscussed n the prevous subsecton, the crcular test for lnear axes by usng the DBB s descrbed n ISO 23-4 [ISO 23-4, 25] and s wdely accepted by machne tool bulders. Many research efforts have been reported on ts extenson to calbrate locaton errors of rotary axes [Kakno, 1994, Sakamoto, 1997, Mahbubur, 1997, Abbaszaheh-Mr, 22, Tsutsum, 23, L, 23, Zargarbash, 26, Le, 27, Uddn, 29, Ibarak(2), 21]. ISO/TC39/SC2 has been lately dscussng the ncluson of DBB n the revson of ISO [ISO/CD , 211]. For example, Fg. 1-2 llustrates a ball bar test descrbed n BK2 of ISO/CD [ISO/CD , 211]. When there exsts a squareness error of the C-axs average lne to the X-axs average lne, the measured dsplacement 4

12 profle n a polar plot (wth respect to C-axs angular poston) s shfted to the X-drecton, as shown n Fg. 1-1(b). A potentally crtcal ssue wth such a DBB-based approach s n ts effcency [Ibarak(2), 21]. Snce the ball bar measurement s one-dmensonal, t often requres at least a couple of dfferent setups to dentfy all locaton errors (see Secton 1.3 for the term locaton errors ). It requres an experenced operator to be always wth the measurement, and thus ts full automaton s dffcult. Z X Y C Actual C-axs average lne Influence of squareness of C- to X-axs (a) Schematc of a ball bar test Y Measured bar length C X (b) Polar plot of the measured bar length wth respect to C-axs angular poston Fg. 1-2 A ball bar test to calbrate the orentaton of C-axs average lne [Ibarak, 212]. (2) R-test Wekert [Wekert, 24], Brngmann and Knapp [Brngmann, 26] presented the R-Test, where the three-dmensonal dsplacement of a sphere attached to the spndle s measured by three (or four n [Wekert, 24, Brngmann, 26]) lnear dsplacement sensors nstalled on the table. Whle the 5

13 ball bar measures the sphere dsplacement only n one drecton (.e. the bar drecton), the R-test measures a three-dmensonal error trajectory n an automated measurement cycle. ISO TC39/SC2 has been also dscussng the ncluson of R-test n the revson of ISO [ISO/CD , 211]. Commercal R-test devces are now avalable from IBS Precson Engneerng [IBS] and Fda [Fda]. A smlar measurement devce composed of capactance sensors, called Capball, s proposed n [Zargarbash, 29]. In ths thess, we employ the R-test and study the extenson of ts applcaton. More detals on the R-test devce tself wll be presented n Chapter 2 and later. (3) Probng of artfact In recent years, hgh-accuracy touch-trgger probes for machne tools, are avalable n today s market (e.g. [Renshaw, Hedenhan]). From ts nature, such a probe has a good communcaton capablty wth a CNC system, whch potentally facltates the automaton of error calbraton and compensaton. Many tests by (1) the ball bar and (2) the R-test can be done by usng such a probe, when the tests are quas-statc. ISO 136-3:2 [ISO 136-3, 2] descrbes such a test for CMMs wth a rotary table as the fourth axs. Probng-based calbraton of offset errors of rotary axs average lne can be done n some commercal CNCs [US Patent, 27, Yamamoto(3), 211]. Its extenson to a set of locaton errors of rotary axes has been reported n the lterature [Erkan, 21, Erkan, 211, Irtan, 21, Matsushta, 21]. (see Secton 1.3 for the term locaton errors ) (4) Machnng tests The NAS (Natonal Aerospace Standard) 979 [NAS 979, 1969] descrbes a fve-axs machnng test of a cone frustum, whch s wdely accepted as a fnal performance test by machne tool bulders. Its ncluson n ISO/CD :211 [ISO/CD 171-7, 211] s currently under dscusson at ISO TC39/SC2. Some researchers n the lterature [Bosson, 27, Yumza, 27, Matsushta, 28, 6

14 Matsushta, 211, Hong, 211] have analyzed the senstvty of locaton errors of rotary axes on the geometrc accuracy of the machned cone frustum workpce. However, t s generally not possble to separately dentfy each locaton error by a sngle cone frustum machnng test [Matsushta, 211]. In other words, these research works clarfed that a sngle cone frustum machnng test cannot be used as an ndrect method to calbrate all locaton errors. Some researchers [Ibarak(1), 21, Yamamoto(1), 211, Yamamoto(2), 211] have presented new machnng tests as an ndrect measurement of the machne s geometrc error parameters. (5) Laser tracker The trackng nterferometer (the term n [ISO/FDIS 23-1, 211]), or the laser tracker, s a laser nterferometer wth a steerng mechansm to change the laser beam drecton to track a target retroreflector (typcally a cat s eye [Takatsuj, 1999]) [Lau, 1986, Kohama, 28, Schwenke, 25, Yano, 26, Schwenke, 29, Takeuch, 21]. Unlke many other ndrect schemes revewed n ths subsecton, the trackng nterferometer can potentally be appled to drect measurement of rotary axs error motons at arbtrary locatons, wthout synchronous moton of lnear axes [Schwenke, 29]. More studes wll be need. 1.3 Objectve and orgnal contrbuton of ths thess As was revewed n the prevous subsecton, many measurement nstruments have been proposed for moton error calbraton of rotary axes. Whle usng dfferent measurng nstruments, the majorty of these prevous works has a common objectve: to dentfy locaton errors of rotary axes. In ISO 23-7 [ISO 23-7, 26], locaton errors of a rotary axs are defned as axs shfts of the axs average lne,.e. the straght lne segment located wth respect to the reference coordnate axes representng the mean locaton of the axs of rotaton. In other words, locaton errors only represent the average error n the poston and orentaton of the axs of rotaton. For example, prevous works revewed n the prevous subsecton on the R-test, [Wekert, 24, Brngmann, 7

15 26, Zargarbash, 29] only presented ts applcaton to the dentfcaton of locaton errors. In past works, revewed n the prevous subsecton, ball bar tests [Kakno, 1994, Sakamoto, 1997, Mahbubur, 1997, Abbaszaheh-Mr, 22, Tsutsum, 23, L, 23, Zargarbash, 26, Le, 27, Uddn, 29, Ibarak(2), 21], probng tests [Erkan, 21, Erkan, 211, Irtan, 21, Matsushta, 21], machnng tests [Ibarak(1), 21, Yamamoto(1), 211, Yamamoto(2), 211] all presented ther applcaton to locaton errors calbraton only. Quas-statc tests for a rotary axs descrbed n ISO to -3 [ISO , 1998] also focus only on locaton errors. Clearly, as s well understood by many machne tool manufacturers, locaton errors of rotary axes are one of the most fundamental error factors n the fve-axs knematcs. From our experences, however, many latest commercal small-szed fve-axs machne tools have relatvely small locaton errors of rotary axes due to recent techncal advances n measurement and assembly adjustment schemes. In such a case, t s of more mportance to observe and calbrate not only the average of error motons of a rotary axs, but also how error moton changes wth the rotaton of a rotary axs. Such an error moton as a functon of the angular poston of a rotary axs, s represented by component errors n ISO 23-7 [ISO 23-7, 26], and s referred to as poston-dependent geometrc errors n ths study. The objectve of ths study s to propose an ndrect measurement method to calbrate geometrc errors (partcularly poston-dependent geometrc errors) of multple rotary axes on fve-axs machne tools. Compared to the ball bar test or touch-trgger probes, the R-test has a strong advantage n that t can measure the TCP dsplacement n all the three drectons (X, Y, and Z drectons) smultaneously. Therefore, by a sngle measurement cycle, the R-test can obtan sgnfcantly more nformaton at varous angular postons of rotary axes than other measurng nstruments, wthout requrng setup changes. For ths reason, R-test s chosen to be the measurement devce n ths thess. Compared to past researches revewed n the prevous subsecton, the 8

16 orgnal contrbuton of ths thess can be summarzed as follows: 1. The R-test devce tself was proposed n [Wekert, 24]. Past researches on the R-test [Wekert, 24, Brngmann, 26, Zargarbash, 29] have focused on ts applcaton to the calbraton of locaton errors of rotary axes. As wll be descrbed n detals n Secton 2.3, locaton errors are the most fundamental error motons. Ths thess wll present ts extenson to poston-dependent geometrc errors, or error map, of rotary axes. Furthermore, the procedure to graphcally present R-test results s also ths thess s orgnal contrbuton, to help users ntutve understandng of the machne s error motons. (Chapter 3) 2. The thermal nfluence can be one of domnant error factors n machne tool moton accuraces. As wll be dscussed n detal n Secton 4.1, conventonal thermal tests for machne tools n ISO standards only evaluate thermal nfluence on the postonng accuracy. Ths thess wll present a new test method to evaluate thermal nfluence on error motons of rotary axes by applyng the R-test. (Chapter 4) 3. In all past R-test researches, the R-test devce use contact-type lnear dsplacement sensors wth a flat-ended probe. For more accurate dynamc measurement, and safer measurement, ths thess wll propose a non-contact type R-test wth laser dsplacement sensors. It sgnfcantly complcates the algorthm to calculate the sphere dsplacement. Ths thess wll propose an algorthm to calculate t for the non-contact type R-test. (Chapter 5) 4. Past researches can be found on the analyss of the nfluence of the machne s error motons on the machnng accuracy. For example, for the cone frustum machnng test descrbed n NAS979 standard [NAS 979, 1969], researchers [Yumza, 27, Matsushta, 28, Uddn, 29] analyzed how locaton errors of rotary axes affect the machnng accuracy of the test pece. These past researches are, however, lmted to locaton errors only. Ths thess wll present a numercal analyss of geometrc errors (especally poston-dependent geometrc errors) of rotary axes on machnng accuracy. (Chapter 6) 9

17 Chapter 2 R-test devce and knematc model of fve-axs controlled machne tools 2.1 Introducton As was revewed n Secton 1.2-(2), the R-test devce tself was proposed by Wekert [Wekert, 24] and ts development s not a part of ths thess s orgnal contrbuton. Ths chapter wll frst revew the conventonal contact-type R-test devce tself, as well as ts computaton algorthm to measure the sphere dsplacement (Secton 2.2). Then, the last half of ths chapter (Secton 2.3) wll revew the knematc model for a fve-axs machne tool, as well as geometrc error parameters ncluded n t. The objectve of the ndrect measurement presented n ths thess s to dentfy the knematc model from a set of measured TCP profles measured by the R-test. The machne tool knematc model s thus the fundamental of ths thess. 2.2 Contact-type R-test devce Measurng prncple of contact-type R-test measurng nstrument Fgure 2-1 shows the schematcs of the conventonal contact-type R-test devce used n ths study ([Wekert, 24, Brngmann, 26, Zargarbash, 29, Ibarak, 211]). A ceramc precson sphere of the radus R s attached to a machne spndle. Three contact-type lnear dsplacement sensors wth a flat-ended probe are attached on a fxture (named by sensors nest ) that s fxed on a rotary table. Usng the pre-calbrated drecton vector of each dsplacement sensor, the dsplacements of the three probes can be drectly transferred to the dsplacement of the sphere center. More detaled algorthm to measure the sphere center dsplacement wll follow. 1

18 Spndle of the machne tool Sphere attached on the spndle Dsplacement sensor Sensors nest attached on the table Fg. 2-1 Schematcs of the R-test devce Procedure to measure three-dmensonal dsplacement by contact-type R-test (1) Basc relatonshp of sphere dsplacement and sensor outputs The relatonshp of sphere dsplacement and sensor outputs wth contact-type R-test was presented n [Brngmann, 27, Ibarak, 29, Oyama, 29]. As the fundamental n ths thess, t wll be brefly revewed n ths subsecton. To transfer the dsplacements measured wth three lnear dsplacement sensors to the three-dmensonal dsplacement of the sphere center, the unt drecton vectors of three probes, denoted by V = (u, v, w ) T ( = 1, 2, 3), are necessary. Fgure 2-2 shows the relaton between the dsplacement of sphere center and the -th sensor dsplacement. The orgn of the coordnate system, O = (,, ) T, s defned at the sphere center n ts ntal poston; the orentaton of the coordnate system s defned based on the machne coordnate system [Schwenke, 28]. When the sphere center s at O, the sphere and the -th probe contacts at P. After the sphere center moves to O j = (x j, y j, z j ) T (j = 1,..., N), they contacts at P. Denote the ntersecton of the lne O P and the probe surface by 11

19 P. By ths movement, suppose that ths -th sensor s dsplaced by the dstance d j. The dstance between O j and P = (P x, P y, P z ) T, the dstance between P and P (denoted by e), the dsplacement of the -th sensor (denoted by d j ), and the radus of the sphere, R, are related as follows: ( O P ) = ( P x ) + ( P y ) + ( P z ) j x d j 2 j + e 2 y = ( x The vector O P satsfes: j 2 j + y 2 j z 2 j + z ) j 2 = R 2 + e 2 (2-1) T O P = [ Px, Py, Pz ] = ( R + dj )[ u, v, w O = + + = ( + ) P Px Py Pz R dj T ] (2-2) From Eq. (2-1) and Eq. (2-2), the followng equaton can be obtaned: x u + y v + z w = d (2-3) j j j j Applyng Eq. (2-3) to all the three sensors, the dsplacements of three sensors are related to the sphere dsplacement as follows: u1 u2 u3 j 2 j 3 j j j j v1 v2 v3 (2-4) w 1 w2 w3 [ d, d, d ] = [ x, y z ] 1, ntal sphere poston O sphere poston after movng O j P P' e P' dj V flat-ended probe of the dsplacement sensor Fg. 2-2 Relaton between the dsplacement of sphere center and the -th sensor dsplacement. (2) Calbraton of unt drecton vector of each sensor To calculate the sphere dsplacement by the R-test, the unt drecton 12

20 vector of each sensor, V ( = 1, 2, 3), must be pre-calbrated. In dfferent nstallaton of the R-test devce on the machne s work table (wth dfferent orentaton of the R-test devce n the workpece coordnate sysetm), drecton vectors become dfferent n the machne coordnate system. Therefore, unt drecton vectors (denoted by V ( = 1, 2, 3)), must be calbrated wth each R-test setup. Smlarly as all the prevous R-test works (e.g. [Wekert, 24]), they are dentfed by usng the machne tool s postonng as the reference. Ther calbraton procedure s as follows: To calbrate the unt drecton vector of each sensor, the spndle-sde sphere center, denoted by (x, y, z), s commanded n ths study as follows: a. Intal poston s set to be (x, y, z) = (,, ); b. (x, y, z) = (+l,, ), where l R ; c. (x, y, z) = (-l,, ); d. (x, y, z) = (,, ) e. (x, y, z) = (, +l, ); f. (x, y, z) = (, -l, ); g. (x, y, z) = (,, ) h. (x, y, z) = (,, +l);. (x, y, z) = (,, -l); j. Repeat a to by three tmes; k. Command the sphere center to ts ntal poston, (x, y, z) = (,, ). Total 28 postons are commanded. At each commanded poston (x j, y j, z j ) (j = 1,, N = 28), the sphere center s stopped, and the dsplacement of the sensors, [d 1j, d 2j, d 3j ] T (j = 1,, N = 28), are logged. In the calbraton, the machne s postonng error s assumed to be suffcently small compared to the measurement uncertanty. In typcal commercal machne tools, the postonng error wthn the operaton above s expectedly a couple of mcrometers at most, when l = 1mm (as n case studes presented n ths thess). In actual R-test cycles to be presented n ths thess, the sensor dsplacement s usually much smaller than l = 1mm (typcally several ten mcrometers at most). For the measurement uncertanty proportonal to the 13

21 dsplacement, t can be sad that the nfluence of the machne s postonng error n the calbraton procedure s neglgbly small. When the sphere center dsplacement, [x j, y j, z j ] T (j = 1,, N), s known, the unt drecton vectors (.e. T [ u, v, w ] ( = 1, 2, 3)) can be dentfed wth solvng the followng problem by the least square method: u N 1 u2 u3 mn [ d ] [ ] 1 j, d2 j, d3 j x j, y j, z j v1 v2 v3 (2-5) w w w u, v, w j= 1 where represents the two-norm of a vector. 2 (3) Measurement of three-dmensonal dsplacements of sphere center In R-test measurement cycles, the objectve of R-test devce s to measure the sphere center dsplacement. When dsplacement sensors readouts, d j ( = 1, 2, 3), are gven, the dsplacement of sphere center, O j = (x j, y j, z j ) T (j = 1,..., N), can be calculated from Eq.(2-4) as follows: u1 u2 u3 j j j 1 j 2 j 3 j v1 v2 v3 (2-6) w 1 w2 w3 [ x y z ] = [ d d d ] 1 (4) Calbraton of offset of the sphere center from rotaton center of the spndle When the precson sphere s attached to the spndle, there may be an offset of the sphere center from the axs of spndle rotaton, as shown n Fg. 2-3(a). It s favorable to mechancally remove ths offset by usng e.g. a fxture to mnutely move the sphere. Instead, n our experments, ths offset s measured by usng the R-test devce n the followng procedure, and ts nfluence s removed numercally: a. After calbratng unt drecton vectors of three dsplacement sensors wth the procedure descrbed n Secton (2), the spndle s rotated for more than 36. The dsplacements of the sensors are logged and a profle of the sphere center dsplacement (x j, y j ) (j = 1,, N) s calculated wth Eq. (2.6). 14

22 Fgure 2-3(b) llustrates a trajectory of the sphere center when the spndle rotates. Due to the sphere offset from the spndle axs average lne, the trajectory of the sphere center wll be a crcle when the spndle rotates. b. From the measured trajectory, the center of the trajectory P S (p Sx, p Sy ) can be dentfed wth solvng the followng problem by the least square method: N psx, psy, RS j = 1 2 ( x p ) + ( y p ) 2 mn R (2-7) j Sx j where R S s the dentfed radus of the measured trajectory. Snce the offset of the sphere center s not the nherent error from the machne, the nfluence of the sphere center offset from the rotaton center of spndle s numercally elmnated from the R-test measurement result. The detals of the algorthm wll be presented n Secton Sy S 2 Center lne of rotaton of the spndle Trajectory of the sphere center when the spndle rotates Y Sphere Sphere center O X Rotaton of the spndle Offset of sphere center Center of rotaton of the spndle: P S (a) Offset of the sphere center (b) Calbraton of the offset Fg. 2-3 Calbraton of the sphere center offset from the rotaton center of the spndle Contact-type R-test prototype developed n ths study The developed contact-type R-test prototype s shown n Fg A ceramc precson sphere of the dameter 25.4 mm (ts major specfcatons are shown n Table 2-2) s attached to a machne spndle. Three contact-type lnear dsplacement sensors wth flat-ended probes (MT1281 from Hedenhan) are nstalled on a fxture (named by sensors nest ) that s fxed on the table. The 15

23 man specfcatons of the sensors are lsted n Table 2-1. Jg to fx the rotaton of the spndle Sphere wth radus of 12.7 mm R-test sensors nest (a) External vew (1) 14 mm (from bottom of sensors nest to sphere center) 6 (b) External vew (2) Fg. 2-4 Contact-type R-test prototype used n ths thess. Table 2-1 Specfcatons of the lnear dsplacement sensor (MT1281 from Hedenhan). Measurng prncple Measurement range System accuracy Gaugng force (vertcally upward) Sgnal perod Mechancally permssble traversng speed Photo-electrc scannng of an ncremental scale wth sprng-tensoned plunger 12 mm ±.2 μm.35 to.6 N 2 μm 3 m/mn 16

24 Table 2-2 Confguraton of the reference ball (from Mortex Co., Ltd) Dameter Sphercty Materal Accuracy grade 25.4 mm (±1.25μm) 1 μm AL23 AFBMA Grade 5 (wth sphercty tolerance of.13μm ) The sphere attached on the spndle should be fxed durng a fve-axs measurement cycle. Throughout ths study, we use a fxng jg attached between the tool holder and the spndle, as shown n Fg. 2-4(a). The detaled analyss of R-test measurement uncertanty s gnored n ths research, snce t can be found n [Wekert, 24]. 2.3 Knematc model of fve-axs controlled machne tools Geometrc errors Sources of geometrc errors In [Schwenke, 28], sources of geometrc errors n a machne tool are summarzed. Here, ts bref revew wll be presented. The accuracy of machne tools s affected by many error sources. Due to a change n geometry of the structural loop components, the actual poston and orentaton of the representatve tool center lne relatve to the workpece dffers from ts nomnal poston and orentaton. The followng major error sources affect the accuracy of the relatve end-effected poston and orentaton: (1) Knematc errors: Knematc errors are errors due to mperfect geometry and dmensons of machne components as well as ther confguraton n the machne s structural loop, axs msalgnment and statc errors of the machne s measurng systems. (2) Thermal-mechancal errors: Thermal-mechancal errors are errors due to the presence or changng of nternal and external heat/cold sources n machne tools, and very often sgnfcant expanson coeffcents and expanson coeffcent dfferences of 17

25 machne part materals. Among many error sources n machne tool knematcs, thermal-mechancal errors can be one of domnant error factors under extended usage of the machne [Ramesh, 2, Schwenke, 28]. (3) Dynamc errors: Dynamc errors are errors caused by moton control, acceleraton or deceleraton. In the analyss, they are often dstngushable from the errors caused by other error sources by applyng dfferent feed speeds and/or acceleratons for the same moton path. Ths thess wll present the methodology to calbrate (1) knematc errors (n Chapter 3), (2) thermal-mechancal errors (n Chapter 4), and (3) dynamc errors (n Chapter 5). Other error sources, such as errors caused by machnng forces or loads could be crtcal [Schwenke, 28]. However, ths study does not cover them. All of these error sources nfluence the relatve error motons between the tool and the workpece on the machne s knematcs, and are modeled as geometrc errors of the machne n ths thess Poston-ndependent and poston-dependent geometrc errors of rotary axes The objectve of the measurement schemes to be presented n ths thess s to numercally dentfy geometrc errors representng varous error motons descrbed above. In ths thess, geometrc errors are categorzed nto two sub-categores,.e. poston-ndependent geometrc errors and poston-dependent geometrc errors. Ther defnton wll be gven n ths subsecton. (1) Machne confguraton Frst, ths thess consders the fve-axs machne confguraton wth a tltng rotary table as shown n Fg The machne has three lnear axes (X-, Y-, and Z-axs) and two rotary axes (B-, and C-axs). It must to be emphaszed that the basc dea of ths thess can be straghtforwardly extended to any confguratons of fve-axs machnes. 18

26 Fg. 2-5 Confguraton of the fve-axs machne tool consdered n ths study. (2) Defnton of coordnate systems In ths study, the coordnate system fxed to the machne frame or bed s called the reference coordnate system. The coordnate system attached to the rotary table s referred as the workpece coordnate system. The orgn of both coordnate systems s set to be the ntersecton of nomnal B-axs and C-axs. The B-coordnate system s defned as the coordnate system fxed on the B-axs. In other words, the B-coordnate system s gven by rotatng the reference coordnate system around ts Y-axs by the nomnal B-axs angular poston, B. (3) Poston-ndependent geometrc errors For a rotary axs, as llustrated n Fg. 2-6, ts axs of rotaton s represented by a lne wth two orentaton parameters and two poston parameters. In ISO 23-6:26 [ISO 23-6, 26], the axs average lne s defned by a straght lne segment located wth respect to the reference coordnate axes representng the mean locaton of the axs of rotaton. Locaton errors assocated wth ths rotary axs [ISO 23-7, 26] are defned as the poston and the orentaton of the axs average lne of the rotary axs. In the example shown n Fg. 2-6, locaton errors E AC and E BC represent the tlt angle of the axs average lne around the X- and Y-axs from ts nomnal orentaton, respectvely, and E XC and E YC represent ts offset n the X- and Y-drecton, from ts nomnal poston, respectvely. Snce locaton errors represent average orentaton and poston, and thus not dependent on ts rotaton, locaton errors 19

27 are sometmes called poston-ndependent geometrc errors n ths thess. For the machne confguraton shown n Fg. 2-5, total eght locaton errors of rotary axes and three locaton errors of lnear axes, shown n Table 2-3, are suffcent [Tsutsum, 23, Inasak, 1997, Abbaszaheh-Mr, 22]. Table 2-3 also presents the notaton of locaton errors used throughout ths thess, as well as ther bref descrpton. In ths thess notaton (for example δx BY n Table 2-3), the frst (set of) characters represents the drecton of devaton (δx, δy, and δz for lnear devatons, and α, β, and γ for angular devatons). The frst character n the subscrpt (the B-axs for δx BY) represents the axs concerned (strctly, the coordnate system attached to ths axs). The devaton s defned n reference to the coordnate system attached to the axs represented by the last character n the subscrpt (the Y-axs for δx BY). The upper-scrpt represents locaton errors n ths thess. It s to be noted that the notaton n ISO 23-7 [ISO 23-7, 26, Schwenke, 28] defnes each geometrc error wth respect to the sngle machne (reference) coordnate system. In our model, geometrc errors of each axs are defned n a relatve sense wth respect to the axs on whch t s mounted. Table 2-3 also shows the correspondence of our notaton of error parameters wth that n ISO/FIDS 23-1:211 [ISO/FIDS 23-1, 211]. EAC C Y Axs average lne of C-axs EYC EXC Z EBC Rotary table (C-table) X Fg. 2-6 Locaton errors of a rotary axs (C-axs) [ISO 23-6, 27]. 2

28 Table 2-3 Poston-ndependent geometrc errors Symbol [Inasak, 1997] Symbol [ISO/FIDS 23-1, 211] Descrpton Locaton errors assocated wth rotary axes δx BY E X(Y)B Lnear offset of B-axs average lne n X-drecton. δy BY E Y(Y)B Lnear offset of the B-coordnate system n Y-drecton, whch s equvalent to the lnear offset of C-axs average lne n Y-drecton. δz BY E Z(Y)B Lnear offset of B-axs average lne n Z-drecton. α BY E A(Y)B Parallelsm error of B-axs to Y-axs around X-axs. β BY E B(Y)B Intal angular postonng error of B-axs. γ BY E C(Y)B Parallelsm error of B-axs to Y-axs around Z-axs. δx CB E X(B)C Lnear offset of C-axs average lne from B-axs average lne n X-drecton. α CB E A(B)C Squareness error of C-axs to B-axs. Locaton errors assocated wth lnear axes α YZ E A(Y)Z Squareness error of Z-axs to Y-axs. β XZ E B(X)Z Squareness error of Z-axs to X-axs. γ YX E C(Y)X Squareness error of X-axs to Y-axs. (4) Poston-dependent geometrc errors of rotary axes It s to be emphaszed that locaton errors only represent mean locaton and orentaton of axs of rotaton. The locaton and the orentaton may vary due to ts rotaton (descrbed by the term axs of rotaton error moton n ISO23-7 [ISO 23-7, 26]). A large class of error motons can be modeled as geometrc errors that vary dependng on the angular poston of a rotary axs. They are referred to as poston-dependent geometrc errors [Lee, 29] n ths study. For example, as s shown n Fg. 2-7(a), geometrc errors δx CB and δy CB, whch are defned as lnear offset of the C-axs average lne wth respect to the B-coordnate system, are constant and ndependent on the rotaton of C-axs by defnton. On the other hand, when they are parameterzed dependent on C-axs angular poston, denoted by δx CB (C) and δy CB (C), they can model a perodc pure radal error moton [ISO 23-7, 26], or run-out of C-axs, as s shown n Fg. 21

29 2-7(b). Smlarly, a perodc tlt error moton [ISO 23-7, 26] of C-axs, often called angular moton or conng [Schwenke, 28] n the ndustry, can be modeled by α CB (C) and β CB (C), as shown n Fg. 2-7(c). Nomnal C Locaton errors (δx CB, δy CB) B (a) Poston-ndependent geometrc errors Run-out of C-axs (δx CB (C), δy CB (C)) B (b) Perodc pure radal error moton (run-out) of C-axs Angular moton of C-axs (α CB (C), β CB (C)) C B (c) Perodc concal tlt error moton (angular moton) of C-axs Fg. 2-7 Examples of poston-ndependent and poston-dependent geometrc errors. Table 2-4 shows poston-dependent geometrc errors assocated wth rotary axes for the machne confguraton n Fg It s to be noted that parameters assocated wth B-axs are dependent only on the angular poston of B-axs, whle those assocated wth C-axs are dependent on both B- and C-axes angular postons. Ths s because that an error moton of C-axs may be affected 22

30 by B-axs angular poston (ts typcal causes nclude gravty-nduced deformaton of bearngs or mechancal structure). In ths study, we represent poston-dependent geometrc errors as follows: δ x ( B) = δx ~ BY + δx ( B) (2-8) BY BY where δx BY represents a constant term,.e. a locaton error. The symbol ~ represents a poston-dependent term. All the error parameters of B-axs (.e. δx BY (B), δy BY (B), δz BY (B), α BY (B), β BY (B), and γ BY (B)) are composed of a poston-ndependent term (see Table 2-3) and a poston-dependent term, analogous to Eq. (2-8). Table 2-4 Descrptons of poston-dependent geometrc errors of rotary axes Symbol Symbol Descrpton [ISO/FIDS 23-1, 211] δx BY (B) E X(Y)B Radal error moton of B-axs of rotaton n X-drecton wth B rotaton δy BY (B) E Y(Y)B Axal error moton of B-axs of rotaton n Y-drecton wth B rotaton δz BY (B) E Z(Y)B Radal error moton of B-axs of rotaton n Z-drecton wth B rotaton δx CB (C, B) E X(B)C Radal error moton of C-axs of rotaton n X-drecton wth C, B rotaton δy CB (C, B) E Y(B)C Radal error moton of C-axs of rotaton n Y-drecton wth C, B rotaton δz CB (C, B) E Z(B)C Axal error moton of C-axs of rotaton n Z-drecton wth C, B rotaton α BY (B) E A(Y)B Tlt error moton of B-axs around X-axs wth B rotaton β BY (B) E B(Y)B Angular postonng error of B-axs rotaton γ BY (B) E C(Y)B Tlt error moton of B-axs around Z-axs wth B rotaton α CB (C, B) E A(Y)C Tlt error moton of C-axs around X-axs wth C, B rotaton β CB (C, B) E B(B)C Tlt error moton of C-axs around Y-axs wth C, B rotaton γ CB (C, B) E C(B)C Angular postonng error of C-axs rotaton 23

31 2.3.2 Knematc model of fve-axs controlled machne tools The knematc model to compute the tool center poston (TCP) n relatve to the workpece s the bass of error calbraton schemes presented n ths thess. Although ts dervaton can be found n many prevous publcatons [Abbaszaheh-Mr, 22, Inasak, 1997, Soons, 1992, Srvastava, 1995], ths subsecton only brefly revews t. Assumng the machne confguraton shown n Fg. 2-5, the TCP n the reference coordnate system s calculated as follows: r q r Tt 1 = [ 1] T (2-9) where r r y x Tt = Ty Tx Tt x Tt = Da ( α YZ ) Db ( β y Tx = Dc ( γ XY ) Dx ( X r Ty = D ( Y + psy) y r 4 4 T t R XZ ) D ( Z + p z sx ) ) (2-1) denotes a HTM (Homogeneous Transformaton Matrx) representng the transformaton of the TCP n the tool coordnate system to the reference coordnate system. In the formulaton above, the tool coordnate system s attached to the TCP. To smplfy the formulaton, Eq. (2-1) only contans the nfluence of squareness errors of lnear axes shown n Table 2-3. Throughout ths thess, the left-sde superscrpt r represents a vector n the reference coordnate system, and t represents a vector n the tool coordnate system. D x (x), D y (y), 4 4 D z (z) R represent the HTMs for lnear motons n X-, Y-, and Z-drectons respectvely, and D a (a), D b (b), D c (c) 4 4 R represent the HTMs for angular motons about X-, Y-, and Z-drectons respectvely (see Eq. (2-15)). Command postons of X-, Y-, Z-, B-, and C-axes are gven by X, Y, Z, B, and C R, respectvely. Note that (p sx, p sy ) s the center offset of the sphere center wth respect to the spndle axs average lne. Then, defne the workpece coordnate system attached on the rotary table (C-axs). The TCP n ths workpece coordnate system, denoted by w q, s calculated as follows: 24

32 25 = = 1 ) ( q T q T q r w r r r w w (2-11) c b b r w r T T T = (2-12) ) ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( B D B z D B y D B x D B D B D B D T b BY z BY y BY x BY c BY b BY a b r δ δ δ γ β α = (2-13) ) ( )), ( ( )), ( ( )), ( ( )), ( ( )), ( ( )), ( ( C D B C z D B C y D B C x D B C D B C D B C D T c CB z CB y CB x CB c CB b CB a c b δ δ δ γ β α = (2-14) Throughout ths thess, the left-sde superscrpt w represents a vector n the workpece coordnate system. The HTMs of lnear motons and angular motons (.e. D x (x), D y (y), D z (z), D a (a), D b (b), and D c (c)) are gven n e.g. [Inasak, 1997, Soons, 1992, Srvastava, 1995, Slocum, 1992 ]: = ) ( x x D x, = ) ( y y D y, = ) ( z z D z, = 1 cos sn sn cos 1 ) ( a a a a a D a, = 1 cos sn 1 sn cos ) ( b b b b b D b, = 1 1 cos sn sn cos ) ( c c c c c D c (2-15) Equaton (2-11) formulates how error motons of each axs are related to the TCP n the workpece coordnate system. Snce the R-test devce measures the sphere dsplacement relatve to the work table,.e. the TCP n the workpece coordnate system, ths equaton s the bass of calbraton schemes to be presented n ths thess. It should be emphaszed that the formulaton presented n ths subsecton assumes that all the axes have a rgd-body behavor. Error motons of each axs caused by non-rgd body behavors, e.g. elastc deformaton caused by varous loads, are beyond the scope of ths thess.

33 Chapter 3 Calbraton of poston-dependent and poston-ndependent geometrc errors of rotary axes by statc R-test measurement 3.1 Introducton As was revewed n Chapter 1, all the prevous works on the R-test n the lterature [Wekert, 24, Brngmann, 26, Zargarbash, 29] only presented ts applcaton to the calbraton of locaton errors of rotary axes. Locaton errors only represent average poston and orentaton of rotary axes. A strong advantage of the R-test s n ts effcency to collect a large set of sphere dsplacement data at varous postons of rotary axes. The collected data can be used to observe not only average error motons of rotary axes, but also how error motons of rotary axes change wth ther rotaton. The objectve of ths chapter s to propose a new algorthm to analyze a profle of sphere dsplacement measured by the R-test to numercally calbrate error motons of two rotary axes at varous angular postons. As was dscussed n Secton 2.3, such a change n error motons can be parameterzed by poston-dependent geometrc errors. 3.2 Objectve and orgnal contrbuton of ths chapter Compared to prevous R-test works revewed n Secton 1.2, orgnal contrbutons of ths chapter can be summarzed as follows: (1) Whle prevous R-test studes focused on numercal parameterzaton of locaton errors from R-test results, t s dffcult for users to ntutvely understand error motons of rotary axes wth raw R-test trajectores. The frst contrbuton n ths chapter s on the demonstraton of an ntutve, graphcal presentaton method of R-test measurements to understand how error motons of rotary table changes n three-dmensonal space dependng 26

34 on the swvelng angle (Secton 3.4). (2) All the prevous R-test studes n the lterature requre that error motons of lnear axes must be suffcently small as a prerequste to calbrate error motons of rotary axes. When ths assumpton s not met, ther approaches are subjected to potentally a sgnfcant dentfcaton error [Brngmann, 29]. To partally address ths ssue, Secton 3.5 proposes a scheme to separate squareness errors of lnear axes by performng a set of R-test measurement cycles wth dfferent sphere postons. (3) By extendng prevous works on locaton errors dentfcaton, Secton 3.6 proposes the applcaton of R-test to the dentfcaton of poston-dependent geometrc errors of a rotary axs. Expermental case studes wll be conducted on the fve-axs machne tool to demonstrate these three contrbutons. 3.3 Measurement procedure R-test measurement cycle Ths secton frst presents the R-test statc measurement procedure. Ths procedure s bascally the same as the one presented n prevous R-test studes (e.g. [Brngmann, 26, Ibarak, 211, Zargarbash, 29]). In a R-test measurement cycle, the machne table s ndexed at each combnaton of gven B- and C-angular postons, B (=1,, N b ) and C j (j=1,, N c ). Measurement poses, B and C j, must be dstrbuted over the entre workspace of each rotary axs. The X, Y, and Z axes are postoned such that there s nomnally no relatve dsplacement of the sphere to the sensors nest. The nomnal sphere poston n the reference coordnate system, denoted by r q (B,C j ) 3 1 R, s gven by: r w q ( B, C j ) q = Db ( B ) Dc ( C j ) 1 1 (3-1) where w q R 3 1 represents the nomnal poston of the sphere n the workpece coordnate system. D() denotes the HTM, see Secton The rght-sde superscrpt represents the commanded poston. X-, Y- and Z-axes are 27

35 postoned at r q (B,C j ) for each B and C j. An example of R-test measurement cycle s llustrated n Fg C=9º, 12º,, 33º C C=º C=6º C=3º B= At B=, C axs s ndexed by e.g. every 3. The spndle-sde sphere s postoned by XY axes such that ts relatve dsplacement to the R-test sensors nest s nomnally zero. Actual dsplacement s statcally measured at each angle. (a) When B = C=9º, 12º,, 33º C C=º C=3º C=6º Analogous measurement s done at varous B angles (e.g. B=-9, -6,, 9 ). B=B (b) When B-axs s postoned at B Fg. 3-1 An example of R-test measurement cycle Sphere poston All the prevous R-test studes [Wekert, 24, Brngmann, 26, Zargarbash, 29, Ibarak, 211, Slamn, 21] suggested a setup where the sphere s located away from the axs average lne of C-axs rotaton. The basc R-test setup [ISO/CD , 21, Wekert, 24, Brngmann, 26] performs a sngle measurement cycle as llustrated by Setup 1-a (outer low) n Fg. 3-2(a) (For the smplcty, Fg. 3-2(a) only shows a measurement cycle at B =. Analogous cycles are performed at gven B s). When a sngle sphere s used, t s not possble to fnd the orentaton of 28

36 the rotaton axs of C-axs at each C j. To observe C-axs tlt error motons, the R-test measurement cycle must be repeated at two dfferent sphere postons, as depcted n Setups 1-a (outer low) and 1-b (outer hgh) n Fg. 3-2(a). Spndle Precson sphere Sensors nest of R-test Setup 1-b (outer hgh) C Setup 1-a (outer low) Rotary table (a) Setups 1-a (outer low) and 1-b (outer hgh) C Setup 2-b (center hgh) Setup 2-a (center low) Rotary table (b) Setups 2-a (center low) and 2-b (center hgh) Fg. 3-2 Setups for R-test measurement. These R-test measurement cycles requre the synchronous moton of lnear axes and a rotary axs. Therefore, the measured dsplacement profle s nfluenced by not only error motons of a rotary axs, but also those of lnear axes, as was quanttatvely dscussed n [Brngmann, 29]. When the sphere s located on the nomnal C-axs average lne (see Fg. 3-2(b)), no lnear axs moves wth C-axs rotaton at gven B. In many machnes, t s often dffcult, or not possble, to place the sphere at the ntersecton of B- and C-axes. Thus, when the cycle n Fgs. 3-2(a) and (b) s performed at dfferent B, lnear axes (X and Z-axes) must move to place the sphere on the nomnal C-axs average lne. In Setup 2-a (center low) and 2-b (center hgh) n Fg. 3-2(b), the postonng error wth ths operaton s only nfluence of error motons of lnear axes on measured 29

37 profles. By combnng Setups 2-a (center low) and 2-b (center hgh) n Fg. 3-2(b), all the tlt and lnear error motons of B- and C-axes can be observed, except for the angular postonng error of C-axs. To smplfy the measurement procedure, ths thess assumes that the angular postonng error of C-axs s pre-calbrated by a dfferent measurng nstrument and needs not to be measured by R-test. Note that the combnaton of Setups 2-a (center low) and 2-b (center hgh) s n prncple equvalent to the test descrbed n ISO 23-7 [ISO 23-7, 26]. 3.4 Graphcal presentaton of R-test profles (1) Background and objectve By graphcally presentng measured three-dmensonal trajectores, an experenced user can make many ntutve observatons on error motons of rotary axes, and possbly ther potental causes. Prevous R-test works n the lterature, revewed n Secton 1.2-(2), focused only on numercal parameterzaton of locaton errors from R-test results. No work has been reported on graphcal presentaton of R-test results. Ths secton wll present a procedure to graphcally present dsplacement profles measured by the R-test. At each stop poston wth B (=1,, N b ) and C j (j=1,, N c ), denote the measured sphere dsplacement n the workpece coordnate system by w w q ( B, C ). Note that q( B, C ) represents the dsplacement of the R-test j j sensors nest relatve to the spndle-sde sphere. It s hereby referred to as the measured sensors nest dsplacement. We dsplay the sensors nest dsplacement w w q( B, C ), nstead of the sphere dsplacement, q ( B, C ), snce the man j scope s n evaluatng error motons of rotary axes. The symbol wth the bar represents the measured dsplacement by the R-test. Note that, snce ncremental lnear dsplacement sensors are used for the R-test probes, the measured dsplacement must be reset at the ntal poston (.e. w q ( B, C ) = at B = C j = º), whch s called ntal resettng n [Ibarak, j 211]. After the ntal resettng, the sphere dsplacement measured by R-test j 3

38 probes s represented by: w q( B, C w w o ) = q( B, C ) q(, ) (3-2) o j j where w q(b,c j ) represents the sphere dsplacement gven by the knematc model, Eq. (2-11) n Secton In other words, the symbol wth the bar represents the sphere dsplacement under the nfluence of ntal resettng. It s dffcult to ntutvely understand rotary axes error motons from a raw R-test profle n the workpece coordnate system (see expermental data n Fg. 3-8 n Secton ). By convertng t to the reference coordnate system, ts ntutve understandng becomes much easer. Furthermore, measured R-test profles are nfluenced by not only the machne s error motons, but also many factors such as the ntal resettng or a calbraton error of the tool length. For more ntutve understandng of rotary axes error motons, the nfluence of such factors must be removed. Ths s the basc dea of the graphcal presentaton scheme presented n ths secton. (2) Proposed analyss procedure Ths secton proposes the followng procedure for the dsplay of measured w R-test profle, q ( B, C ). j a. Transformaton to the reference coordnate system: The measured sensor dsplacement n the workpece coordnate system, w denoted by q( B, C ), s transformed to the poston n the reference j r coordnate system, denoted by q ( B, C ) by usng nomnal angular postons of B- and C-axes as follows: j r q( B 1, C j ) = D b ( B ) D ( C c j ) w q( B, C 1 j ) (3-3) b. Compensaton of ntal resettng: The ntal resettng of dsplacement sensors (see Eq. (3-2)) makes t more dffcult to ntutvely understand the nfluence of an offset of the axs average lne of rotary axes. For example, when there exsts an offset n the C-axs 31

39 average lne from ts nomnal poston n the X-drecton, denoted by δx CY ( = δx CB +δx BY at B = n Table 2-3), and n the Y-drecton, denoted by δy CY ( = ˆ j δy w BY n Table 2-3), ts nfluence on the R-test profle, q ( B, C ), s gven by: 1 ) = Db ( B δx δy ) 1 δx δy w ˆ q( B, C j CY CY ) Dc ( C j CY CY (3-4) The symbol wth the hat ^ represents the estmate. The last term of Eq. (3-4) represents the ntal resettng, makng w ˆ o ( B =, C j o q = ) = [,,] T. By substtutng Eq. (3-4) nto Eq. (3-3), the nfluence of δx CY and δy CY to the sphere r poston n the reference coordnate system, q ( B, C ), s llustrated n Fg. 3-3(a) (at B = º, C j = º~36º). Ths suggests that the offset of C-axs average lne s represented by constant expanson or shrnkage of the R-test trajectory. j Y Actual sphere trajectory Nomnal trajectory Intal poston X ( δx CY, δy CY ) r (a) Wth resettng at ntal poston, q ( B, C ) j Nomnal trajectory Y Actual sphere trajectory δy CY X δx CY r (b) Wth compensatng ntal resettng, q ( B, C ) Fg. 3-3 An llustratve example of the nfluence of axs shft of C-axs average lne n X-drecton to the sphere poston n the reference coordnate r system, q ( B, C ), at B =, C j = ~36. j j 32

40 The nfluence of the ntal resettng can be numercally removed from R-test error profles by: r q( B 1, C j ) = Db ( B ) D ( C c j ) w q( B, C j δxˆ δyˆ ) CY CY (3-5) r The symbol q ( B, C ) s wthout the bar, ndcatng that the j nfluence of ntal resettng s removed. After the compensaton of ntal resettng, the trajectory would be shown as n Fg. 3-3(b), where the exstence of the center offset s easer to be ntutvely understood (the crcular trajectory s shfted by the dstance and the drecton equal to the center offset of C-axs). In Eq. (3-5), ˆ δ x CY and y CY δ ˆ represent the estmate of δx CY and δy CY. w They can be obtaned by best-fttng the raw measured trajectory, q ( B, C ), to the model descrbed n Eq. (3-4) for B = and C j = ~36. Ths can be done n an analogous manner as the algorthm to be presented n Secton 3.6. j c. Elmnaton of nfluence of offset errors of rotary axs: For example, the center offset of B-axs n the Z-drecton, denoted by δz BY (see Secton 2.3.1), s often caused by a calbraton error of the tool length (.e. the dstance from the spndle gauge lne to the sphere center). Ths should be regarded as a setup error, not the machne s nherent error. Smlarly, an offset error of the axs average lne of C-axs n X- and Y-drectons, denoted by δx BY and δy BY (note that δy BY equvalently represents the offset of C-axs average lne n the Y-drecton, and δx BY + δx CB represent the offset of C-axs average lne n the X-drecton at B = ), can be easly elmnated by properly tunng CNC control parameters, and thus are typcally regarded as setup errors. In our study, ther nfluence s numercally elmnated from measured R-test profles to more clearly present the machne s nherent error motons only. It must be emphaszed that ths operaton only elmnates the nfluence of average center offset,.e. locaton errors, and ther change wth the B-rotaton are shown. These parameters are also estmated by best-fttng the raw 33

41 w measured trajectory, q ( B, C ), to the machne s knematc model (for ts j detaled dentfcaton algorthm, see Secton 3.6). Denote the sensors nest s dsplacement profle after elmnatng the nfluence of estmated offset errors of C- and B-axes (.e. δ xˆ CY, δyˆ CY and z r δ ˆ BY ) by p ( B, C ). by: r r p( B, C j ) q( B, C j = 1 1 δxˆ ) δyˆ δzˆ CY CY BY j r 3 p( B, C ) R s gven j (3-6) The symbol, p, nstead of q, represents the sensors dsplacement where the nfluence of offset errors s removed. C Z Average crcle Measured poston Command poston Rotary table (a) Example 1: Wth squareness error of C- to X-axs C Z Average crcle Measured poston Command poston Rotary table (b) Example 2: Wth tlt error moton ( conng ) of C-axs Fg. 3-4 Average crcles wth R-test plot. d. Dsplayng average crcles: When the sphere s located away from the C-axs (.e. n Setups 1-a (outer 34

42 low) and 1-b (outer hgh) n Fg. 3-2(a)), measured trajectores are plotted wth average crcles, to more clearly observe average error motons at each B-angle. Two typcal examples of C-axs error motons are llustrated n Fgs. 3-4(a) and (b). For calculatng average crcles, the center and orentatons of a crcle are r best-ft to measured poston trajectory of the sensors nest, p ( B, C ) at each B, by the non-lnear least square method. j 3.5 Separaton of squareness error of lnear axes (1) Background and objectve All the prevous R-test studes [Wekert, 24, Brngmann, 26, Zargarbash, 29, Slamn, 21, Ibarak, 211] assumed error motons of lnear axes to be suffcently small, n order to calbrate error motons of rotary axes. When ths assumpton s not met, t may cause sgnfcant calbraton error. Ths secton proposes that a part of error motons of lnear axes can be separately dentfed by performng three R-test cycles llustrated n Fg. 3-2 (Setups 1-a,2-a, and 2-b). (2) Proposed analyss procedure In Setups 2-a (center low) and 2-b (center hgh), the R-test measurement cycle only requres C-axs rotaton at each B (=1,, N b ). Therefore, error motons of lnear axes do not affect the R-test measurement, except for the postonng error at the sphere s nomnal poston. When the sphere are located on the C-axs average lne, combnng Setups 2-a (center low) and 2-b (center hgh), the orentaton of lnes connectng these two setups represents the parallelsm error of the C-axs average lne to the Z-axs. On the other hand, when the sphere s located away from the C-axs (.e. Setup 1-a (outer low) or Setup 1-b (outer hgh)), the average orentaton of the measured trajectory at B = represents the squareness of the C-axs average lne to X- and Y-axes. Therefore, by combnng all measured profles (.e. Setups 1(outer) and Setups 2(center)), one can observe squareness errors of Z-axs to X- and Y-axes at B =. Analogous observaton apples at dfferent B. Ths s the basc dea 35

43 of the dscusson presented n ths secton. The procedure to separate squareness errors of lnear axes s presented as follows: a. Squareness of C-axs to X- and Y-axes of the B-coordnate system: In Setups 1-a (outer low), denote the measured sensor poston by r p 1a ( B, C j ). At each B, denote the unt normal vector of the average crcle of r p ( B, C ) (j=1,, N c ) by 1a j r 3 r n 1 a ( B ) R. Then, the orentaton of n 1a ( B ) from ts nomnal drecton represents the squareness error of the C-axs average lne and the X-axs (or Y-axs) average lne n the B-coordnate system (see Fg. r 3-1(c) as an llustratng example). The angular error of n ( B ) to the Y-axs of 1a the B-coordnate system around ts X-axs s denoted by B CY α ( B ) R, and that to the X-axs of the B-coordnate system around ts Y-axs s denoted by B β CX ( B ) R. Note that the B-coordnate system s defned as the coordnate system attached to the B-axs (see Secton (2) for more detals). The left-sde superscrpt B represents the quantty defned n the B-coordnate system. b. Parallelsm of C-axs to Z-axs of the B-coordnate system: Denote the measured sensors nest poston at B (=1,, N b ) and C j (j=1,, N c ) n Setups 2-a (center low) and 2-b (center hgh) respectvely by r r p ( B, C ) and p ( B, C ) n the reference coordnate system. At each B, 2a j 2b j r r denote the center of gravty of p ( B, C ) and p ( B, C ) (j=1,, N c ) by 2a j 2b j r 3 g 2 a ( B ) R and r g 3 ( R, respectvely. 2 b B ) r r Then, the unt orentaton vector of the lne connectng g ( B ) and 2a r r g ( B ) s denoted by n ( B ). The orentaton of n ( B ) from ts nomnal 2b 2a 2a drecton represents the parallelsm error of the C-axs average lne and the lne connectng actual hgher and lower sphere postons, whch can be seen as the Z-axs representatve lne n the B-coordnate system (see Fg. 3-11(c) as an r llustratng example). The angular error of n ( B ) to the Z-axs of the 2a 36

44 B-coordnate system around ts X-axs s denoted by B CZ α ( B ) R, and that around ts Y-axs s denoted by β ( B ) R. B CZ c. Squareness errors of Z-axs to X- and Y-axes of the B-coordnate system: By combnng (1) and (2), the squareness of Z-axs to X- (and Y-) axs of B B the B-coordnate system, denoted by β ( B ) (and α ( B ) ), can be calculated at each B (=1,, N b ) as follows (see Fg. 3-5): B B β ZX ZY ( B α ( B ZX B B ) = β ( B ) β ( B ) (3-7) CX CZ B B ) = α ( B ) α ( B ) (3-8) CY CZ B It must be emphaszed that β ( B ) represents the Z-X squareness error ZX n the B-coordnate system wth B, not the Z-X squareness error n the reference coordnate system. ZY B Z B Z B C (3)Squareness of Z-axs to X-axs (or Y-axs) of the B-coordnate system (1) Squareness of C-axs to X- axs (or Y-axs) of the B- coordnate system B X (2) Parallelsm of C-axs to Z-axs of the B-coordante sytem Table (angle of B-axs: B ) Fg. 3-5 Separaton of the squareness errors of Z-axs to X- and Y-axes of the B-coordnate system. ( B X and B Z represent the nomnal X-axs and Z-axs n the B-coordnate system; B Z represent the Z-axs of the B-coordnate system). d. Squareness errors of lnear axes: The squareness errors of lnear axes to be dentfed are squareness error of Z-axs to X-axs (denoted by β XZ ); squareness error of Z-axs to Y-axs (denoted byα YZ ); and squareness error of X-axs to Y-axs (denoted by γ YX ), all n the reference coordnate system (see Table 2-3). 37

45 It s generally not possble to convert squareness errors n the B-coordnate system to those n the reference coordnate system. However, under an assumpton that X-axs does not have sgnfcant angular errors and thus that squareness errors are the same at any X-postons n the measured volume, ther relatonshp can be knematcally formulated as follows: β ( B ) = β ZX cos(2b ) (3-9) B ZX α ( B ) = α ZY cos( B ) + γ XY sn( B ) (3-1) B ZY The squareness error of Z-axs to X-axs (.e. ˆβ XZ ) can be dentfed by ˆ ZX B best-fttng Eq. (3-9) to β ( B ) calculated n Eq. (3-7) by the least square method. Moreover, the squareness error of Z-axs to Y-axs (.e. calculated when B = from Eq. (3-1). After elmnatng the nfluence of from Eq. (3-1), ˆα YZ ) s ˆα YZ B ˆ γ YX s dentfed by best-fttng Eq. (3-1) to ˆ α ( B ) calculated n Eq. (3-8) by the least square method. The contrbuton of ths subsecton can be summarzed as follows: a. Z-X and Z-Y squareness errors can be estmated at each B-coordnate system. They can not be, however, straghtforwardly converted to squareness errors of lnear axes n the reference coordnate system. b. When the followng assumpton s met, squareness errors of lnear axes can be estmated n the reference coordnate system: squareness errors n the reference coordnate system are the same at any X postons. Ths assumpton can not be met when, for example, the X-axs has sgnfcant angular errors. c. When the assumptons above are met, the nfluence of squareness errors of lnear axes can be removed from R-test profle (see Secton 3.6.2). ZY 3.6 Identfcaton of poston-ndependent and poston-dependent geometrc errors of a rotary axs Objectve Whle the graphcal presentaton of R-test results, presented n Secton 3.4, s mportant to ntutvely understand error motons of rotary axes, t s also 38

46 mportant to numercally parameterze error motons as an error map of rotary axes, partcularly when ts numercal compensaton s ntended to apply. As was dscussed n Secton 2.3, an error map of rotary axes can be parameterzed as poston-dependent geometrc errors. We wll frst present an algorthm to dentfy locaton errors, or poston-ndependent geometrc errors, lsted n Table 2-3, from R-test results, n Secton Although the dentfcaton of locaton errors has been already presented by some researchers n the lterature [Brngmann, 26, Ibarak, 211], we start from ts bref revew as the bass. Then, n Secton 3.6.4, t wll be extended to poston-dependent geometrc errors. An algorthm to numercally parameterze poston-dependent geometrc errors of B-axs s proposed. Ths part s an orgnal contrbuton of ths thess Separaton of the nfluence of squareness errors of lnear axes The R-test measurement result of Setup 1-a (n Fg. 3-2(a)) s used for the dentfcaton. The nfluence of squareness errors of lnear axes on the TCP n the reference coordnate system, Secton 2.3.2: ˆ r s q 3 R, s derved from Eqs. (2-9) and (2-1) n r qˆ 1 = s r T t [ 1] T (3-11) r r y x Tt = Ty Tx Tt x Tt = Da ( α YZ ) Db ( β y Tx = Dc ( γ XY ) Dx ( X ) r Ty = D ( Y ) y XZ ) D ( Z z ) (3-12) The defnton of squareness errors of lnear axes s presented n Table 2-3 n Secton In r s qˆ, the ^ represents the dsplacement calculated wth the knematc model. The rght-sde superscrpt s denotes the nfluence of lnear axes squareness errors. The R-test measures the TCP dsplacement n the workpece coordnate system. The nfluence of squareness errors of lnear axes on the R-test measurement result, w ˆ s q ( B, C j 3 ) R, s gven by: 39

47 ) = r s qˆ Tr = ( Db ( B ) Dc ( C )) 1 w s r s q ˆ ( B, w 1 ˆ C j q 1 w ˆ s o q ( B, C j j 1 (3-13) w s w s o ) = qˆ ( B, C ) qˆ (, ) (3-14) The squareness errors of lnear axes estmated n Secton 3.5 (.e. β XZ, and γ YX α YZ, ) are numercally removed from the R-test measurement profle before the dentfcaton of geometrc errors of rotary axes Identfcaton of poston-ndependent geometrc errors It s mportant to note that R-test probes can only measure the dsplacement of sphere center from ts ntal postons (.e. the poston when B = C j = ). As was dscussed n Secton 3.4-(2), wth consderng the nfluence of ths ntal resettng, the sphere dsplacement measured by R-test probes s represented by Eq. (3-2) n Secton 3.4. Table 2-3 by: Denotes a set of locaton errors assocated wth the rotary table shown n T ω = [ δx BY, δy BY, δz BY, α BY, β BY, γ BY, δx CB, α CB] (3-15) The objectve of the algorthm s to dentfy ω. When the sphere w dsplacement, q ( B, C ), s measured by R-test probes, ω s dentfed by j solvng the followng problem: mn ˆ ω, j w w ( q( B, C ) qˆ j s ( B, C j )) w w o o ( qˆ( B, C j )) ( qˆ(, )) ω ω ω 2 (3-16) ˆ j w s where represents the 2-norm. q ( B, C ) represents the nfluence of squareness errors of lnear axes, dentfed n Eq. (3-14). The analytcal w formulaton of Jacoban matrces (.e. ( q ˆ( B, C j ))/ ω ) can be derved from the knematc model gven n Eq.(2-11), as s presented n [Ibarak, 211]. It can be also numercally computed [Brngmann, 26]. The problem (3-16) can be solved by the least square method. 4

48 Identfcaton of poston-dependent geometrc errors of a rotary axs As was descrbed n Secton 2.3, locaton errors only represent average poston and orentaton of a rotary axs. The poston and the orentaton of a rotary axs may vary wth ts rotaton. The objectve of ths subsecton s to present an algorthm to numercally parameterze poston-dependent geometrc errors assocate wth B-axs (see Table 2-4 n Secton ), such that an error map of B-axs can be obtaned to descrbe how poston and orentaton of B-axs changes wth the B-rotaton. Ths subsecton does not consder poston-dependent geometrc errors of C-axs, snce they can be drectly observed from R-test results presented n Secton 3.4. The nfluence of estmated poston-ndependent terms, ˆω, on the measured sphere dsplacement, ), ( ˆ j w C B q, s derved from Eqs. (2-9) and (2-1) n Secton 2.3.2: = = 1 ) ( 1 1 ), ( ˆ 1 q T q T C B q r w r r r w j w (3-17) c b b r w r T T T = (3-18) ) ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( B D z D y D x D D D D T b BY z BY y BY x BY c BY b BY a b r δ δ δ γ β α = (3-19) ) ( ) ˆ ( ) ˆ ( C D x D D T c CB x CB a c b δ α = (3-2) ), ( ˆ ), ( ˆ ), ( ˆ o o q C B q C B q w j w j w = (3-21) Denotes a set of the poston-dependent geometrc errors assocated wth B-axs to be dentfed shown n Table 2-4 by: T BY BY BY BY CB BY B B B B B z B y B x B )] ( ~ ), ( ~ ), ( ~ ), ( ~ ), ( ~ ), ( ~ [ ) ( γ β α δ δ δ ω = (3-22) The objectve of the algorthm s to dentfy ) ( B B ω for all gven B s. When the sphere dsplacement, ), ( j w C B q, s measured by R-test probes, ) ( B B ω s dentfed by solvng the followng problem wth the least square method: ( ) ( ) j B B w B j w j w j s w j w B B B q B C B q C B q C B q C B q B, 2 ) ( ˆ ) ( ) ( ), ˆ( ) ( ), ˆ( )), ( ˆ ), ( ˆ ), ( ( mn ω ω ω ω o o (3-23)

49 w Note that the Jacoban matrces (.e. ( q( B, C ))/ ω ( B ) j B ) n the formulaton above can be analytcally derved from Eq.(2-11) n an analogous manner as n Secton Expermental case studes Graphcal presentaton of R-test profles Expermental setup The objectve of the expermental case study to be presented n ths secton s to demonstrate the schemes proposed n ths chapter: 1) graphcal presentaton of R-test results (Secton 3.4); 2) separaton of squareness errors of lnear axes (Secton 3.5); and 3) dentfcaton of poston-dependent geometrc errors of a rotary axs (Secton 3.6). The R-test measurement was expermentally conducted on a commercal small-szed fve-axs machne tool of the confguraton shown n Fg. 2-5 (NMV15DCG by Mor Sek Co., Ltd.). Table 3-1 shows the man specfcatons of NMV15DCG. Fgure 3-5 shows the expermental setups: Setups 1-a (outer low) and 1-b (outer hgh) (see Fg. 3-2(a)). R-test measurement cycles are conducted n all of four setups n Fgs. 3-2(a) and (b). Nomnal sphere locatons n the workpece coordnate system are: Setup 1-a (outer low): w q 1a = [, -9., 4.6] (mm) Setup 1-b (outer hgh): w q 1b = [, -9., 14.6] (mm) Setup 2-a (center low): w q 2a = [,, 4.6] (mm) Setup 2-b (center hgh): w q 2b = [,, 14.6] (mm) In each setup, the R-test measurement cycle s conducted wth the followng command B and C angular postons: B = -75, -5,, 75 ( = 1,,7) C j =, 3,, 33 (j = 1,,12) Total 7 12 = 84 ponts are measured. The command trajectory of each axs (.e. X, Y, Z, B, and C-axs) for Setup 1-a (outer low) and Setup 1-b (outer hgh), s shown n Fg

50 Z Y X (a) Setup 1-a (outer low) n Fg. 3-2(a) Z Y X (b) Setup 1-b (outer hgh) n Fg. 3-2(a) Z Y X (c) Setup 1-b (outer hgh) at B = -75 Fg. 3-6 R-test expermentaton setups: Setup 1-a (outer low) and Setup 1-b (outer hgh) n Fg. 3-2(a). 43

51 Table 3-1 Specfcatons of NMV15DVG from Mor Sek [Mor Sek] X Y Z B C Stroke 42 mm 21 mm 4 mm -18~16 36 Drven system Ball screw and servo motor Drect Drve motor Sze of work table Φ25 mm Measurement result Fgure 3-8 shows raw sphere dsplacements measured by the R-test w sensors nest, q ( B, C ), n the workpece coordnate system (only the measured profle n Setup 1-a (outer low) s shown). j x mm y mm z mm b deg c deg Setup 1-a (outer low) Setup 1-b (outer hgh) ndex number Fg. 3-7 Commanded X, Y, Z, B, and C trajectores for the Setup 1-a (outer low) and Setup 1-b (outer hgh). ball dsplacement mm x y z data number Fg. 3-8 Raw sphere dsplacements measured by the R-test n the workpece coordnate system (Setup 1-a (outer low)). 44

52 B=-75 Table (a) At B = -75 B= Table (b) At B = º B=75 Table (c) At B = 75º Fg. 3-9 Sensor postons measured by R-test wth C-rotaton at B = -75º, º, and 75º(3D vew) n Setups 1-a (outer low) and 1-b (outer hgh). An error of measured poston ( marks) from ts command poston ( marks) s magnfed 1, tmes. 45

53 Table B = -75 Projecton onto XZ plane Projecton onto XY plane (a) At B = -75 B = -75 Table B = B = 75 Projecton onto XZ plane Projecton onto XY plane Squareness error of C-axs to X-axs ( B β CX (B )) (b) At B = + Table B = 75 Projecton onto XZ plane Projecton onto XY plane (c) At B = 75 Fg. 3-1 Sensor postons measured by R-test wth C-rotaton at B = -75º, º, and 75º (3D vew) n Setups 1-a (outer low) and 1-b (outer hgh), projected onto XZ plane and XY plane. 46

54 Table B = -75 Projecton onto XZ plane Projecton onto XY plane (a) At B = -75 Table B = Projecton onto XZ plane Projecton onto XY plane (b) At B = Parallelsm error of C-axs to Z-axs ( B β CZ (B )) + Table B = 75 Projecton onto XZ plane Projecton onto XY plane (c) At B = 75 Fg Sensor postons measured by R-test n Setups 2-a (center low) and 2-b (center hgh), projected onto XZ plane and XY plane. 47

55 The measured profle s converted to the reference coordnate system wth compensaton of ntal resettng, as well as numercal elmnaton of the nfluence of offset errors of rotary axs (.e. descrbed n Secton 3.3. ˆ δ x CY, y CY δ ˆ and δ zˆ BY ), as r Fgures 3-9(a) to (c) show measured p ( B, C ) wth full C-rotaton (C j =,, 33 ) at B = -75,, and 75 n Setups 1-a (outer low) and 1-b (outer hgh). Although an analogous profle s measured at total seven B angular postons (see Secton 3.7.1), only three of them are shown. In Fg. 3-9, an error r of the measured sensors nest poston ( marks), p ( B, C ), calculated by Eq. r (3-6) from ts command poston ( marks), denoted by q ( B, C ), gven by Eq. (3-1), s magnfed 1, tmes. Panted crcles represent average crcles presented n Secton 3.4. Table ndcates approxmate poston and orentaton of rotary table. Fgure 3-1 shows same measured sensors nest poston trajectores projected onto the XZ plane and XY plane. Fgure 3-11 shows measured sensors nest dsplacement profles n Setups 2-a (center low) and 2-b (center hgh) projected on the XZ plane and XY plane. j j j Observaton The graphcal presentaton of R-test measurement results presented n Secton allows us to make many ntutve observatons of error motons of rotary axes. Ths subsecton partcularly focuses on observng how error motons of the rotary table (C-axs) changes dependng on the swvelng axs (B-axs). For example: In Fg. 3-1(b) at B =, measured trajectores (both upper and lower trajectores) are nclned from ther command trajectores around Y-axs. Ths represents the squareness error of the C-axs average lne to the X-axs at B =. At dfferent B (see Fgs. 3-1(a) to (c)), ths squareness error slghtly changes, especally at B = 75. One of major contrbutors to these orentaton errors s the angular postonng error of B-axs. See Secton

56 and Secton for further dscusson. In Fg. 3-11(b) at B =, lnes connectng hgher and lower trajectores are tlted from the Z-axs around Y-axs. Ths represents the parallelsm error of the C-axs average lne to Z-axs at B = (see Secton 3.5 and Secton for further dscusson). The parallelsm error does not change sgnfcantly at dfferent B (see Fgs. 3-11(a) to (c)) Separaton of squareness error of lnear axes By the procedure proposed n Secton 3.5, squareness errors of lnear axes are dentfed by comparng R-test results n three setups (.e. Setups 1-a(outer low), 2-a(center low), and 2-b(center hgh) as shown n Fg. 3-2). Fgure 3-12(a) shows the squareness error of C-axs average lne to the B X-axs and Y-axs average lne of the B-coordnate system, α ( B ) and B β CX ( B ), calculated at each B (=1 7) from the orentaton of the average crcle of measured R-test profles n Setup 1-a (outer low) shown n Fg. 3-1 (see Secton 3.5(1)). Fgure 3-12(b) shows the orentaton error of C-axs average lne to the B B Z-axs average lne of the B-coordnate system, α ( B ) and β ( B ), calculated at each B (=1 7) from the orentaton of the lne connectng the center gravty of the measured R-test profles n Setup 2-a (center low) and that n Setup 2-b (center hgh) from ts nomnal drecton, as s shown n Fg (see Secton 3.5(2)). By combnng Fg. 3-12(a) and 3-12(b) (see Secton 3.5(3)), the squareness error of Z-axs to X-axs and Y-axs of the B-coordnate system B B (denoted by β ( B ) and α ( B ) (=1,, N b )) can be calculated, and s ZX shown n Fg. 3-12(c) (lnes wth and marks, respectvely). ZY CZ CY CZ 49

57 8 x 1-5 Orentaton error rad Squareness of C-axs to X-axs Squareness of C-axs to Y-axs B deg (a) Squareness of C-axs to X-axs and Y-axs of the B-coordnate system at each B, dentfed from the measured profle n Setup 1-a (outer low) Orentaton error rad 8 x Parallelsm of C-axs to Z-axs around Y-axs Parallelsm of C-axs to Z-axs around X-axs B deg (b) Parallelsm of C-axs to Z-axs of the B-coordnate system at each B, dentfed from measured profles n Setups 2-a (center low) and 2-b (center hgh) Orentaton error rad 8 x 1-5 Smulated nfluence 6 of α ZY and γ XY Smulated 4 Squareness of Z-axs to Y-axs ( B nfluence of β α ZY ) ZX 2-2 Squareness of Z-axs to X-axs ( B β ZX ) B deg (c) squareness of Z-axs to X- and Y-axes of the B-coordnate system calculated from (a) and (b) and the nfluence of squareness errors of lnear axes (.e. ˆα ZY, ˆβ ZX, and ˆ γ XY ) Fg Orentaton error of C-axs n the B-coordnate system from measured R-test profles n Fgs. 3-1 to 3-11, and separaton of the squareness errors of lnear axes. 5

58 Table B = -75 Projecton onto XZ plane Projecton onto XY plane (a) At B = -75 B = -75 Table B = B = 75 Projecton onto XZ plane Projecton onto XY plane (b) At B = Table B = 75 Projecton onto XZ plane Projecton onto XY plane (c) At B = 75 Fg Sensor postons measured wth R-test after elmnatng the nfluence of estmated squareness errors of lnear axes (.e. (Setups 1-a (outer low) and 1-b (outer hgh)). ˆα YZ, XZ ˆβ, and ˆ γ YX ) 51

59 The squareness errors of lnear axes (.e. α YZ, XZ β, and γ YX ) s estmated from Eqs. (3-9) and (3-1), as mentoned n Secton 3.5(4). From the measurement result shown n Fg. 3-12(c), the followng results are obtaned: ˆ α YZ = rad; ˆ β XZ = rad; and ˆ γ YX = rad. The nfluence of dentfed squareness errors (.e. ˆα YZ, ˆβ XZ, and ˆ γ YX ) on the squareness error of Z-axs to X- and Y-axes of the B-coordnate system, calculated by Eqs. (3-9) and (3-1), s also shown n Fg. 3-12(c) (lnes wth and marks, respectvely). B Fgure 3-12(c) shows that the measured profle of ˆ α ( B ) matches ZY well wth the smulated one wth ˆα YZ and ˆ γ YX. On the other hand, the B measured profle of ˆ β ( B ) does not match wth the smulated one wth ˆβ XZ. ZX Ths suggests that the Z-X squareness error n the reference coordnate system s not constant at dfferent X-postons (.e. the assumpton n Secton 3.5-(4) s not met). Ths may be caused by the ptch error moton of X-axs. As a result, we conclude that, the estmated Z-X squareness error, ˆβ XZ, contans large estmaton error (the Z-X squareness error can not be represented by a sngle poston-ndependent parameters) Identfcaton of geometrc errors assocated wth B-axs As was dscussed n Secton 3.6, f the machne meets the assumpton that squareness errors are the same at any X postons, squareness errors of lnear axes can be estmated as demonstrated above. In ths partcular expermental machne, ths assumpton was not satsfed, especally for X and Z axes. (1) Separaton of squareness errors of lnear axes By the algorthm proposed n Secton 3.6, error motons of B-axs observed by R-test are numercally parameterzed as poston-dependent geometrc errors. ˆ j w s Frst, the nfluence of the squareness errors of lnear axes, q ( B, C ), estmated by Eq. (3-14) n Secton s numercally elmnated from the R-test 52

60 measurement result of Setup 1-a (outer low), as mentoned n Secton 3.6. The sensor postons (Setups 1-a (outer low) and 1-b(outer hgh)) after elmnatng the nfluence of ˆα YZ, XZ w w ˆβ, and ˆ γ YX,.e. (, ) ˆ s q B C q ( B, C ), are shown n Fg. Orgnal R-test profles shown n Fg. 3-1 contan the nfluence of both rotary and lnear axes error motons. In Fg. 3-13, the nfluence of squareness errors of lnear axes s removed. It should be emphaszed that the nfluence of other error motons of lnear axes, e.g. straghtness errors or lnear postonng errors, stll remans. Snce the R-test measures only the relatve dsplacement of the rotary table (drven by rotary axes) to the spndle-sde sphere poston (drven by lnear axes), t s n prncple not possble to completely elmnate the nfluence of lnear axes. However, n practcal applcatons, squareness of lnear axes are often domnant error factors on the volumetrc error n the three-axs knematcs. Comparng Fg wth Fg. 3-1, we can fnd that both trajectores, wth and wthout the nfluence of squareness errors of lnear axes, do not dffer much (the maxmum dfference n measured sensor poston s about 1.5 μm). In ths partcular applcaton example, t suggests that the nfluence of squareness errors of lnear axes on R-test profles s not sgnfcant. As was dscussed n Secton 3.7.2, we found that the estmated Z-X squareness error, ˆβ XZ, had a large estmaton error. The analyss presented n ths subsecton nevertheless used ths estmate, wth a prmal nterest n demonstratng the proposed scheme presented n Secton j j (2) Identfcaton of locaton errors Locaton errors were estmated by Eq. (3-16) n Secton The dentfed locaton errors are lsted n Table 3-2. (3) Identfcaton of poston-dependent geometrc errors assocated wth B-axs Identfcaton of poston-dependent geometrc errors assocated wth B-axs was conducted by Eq. (3-22) n Secton The dentfed 53

61 poston-dependent geometrc errors of B-axs are shown n Fg Ths can be seen as an error map of B-axs. Table 3-2 Identfed locaton errors Symbol Value (μm) Symbol Value ( 1-5 rad) δx BY -7.8 α BY 1.8 δy BY 12.1 β BY 4.1 δz BY -38. γ BY.8 4 x 1-3 translatonal errors mm 2-2 ~ δ δx BY ((B)) δy BY ((B)) x BY B ~ δz (B) ( B BY ) ~ δy CB B B deg (a) ~ ~ δ x ( B ), δy ( B ), δ~ z ( B ) BY CB BY 4 x 1-5 angular errors rad 2-2 ~ α BY ((B) BY B ) ~ γ γ BY ( (B) BY B ) ~ ββ BY ((B) BY B ) B deg (b) ~ ~ α ( B ), β ( B ), ~ γ ( B ) BY BY Fg Identfed poston-dependent geometrc errors of the rotary table dependent on B-axs angular poston. BY 54

62 (4) Observatons a. Locaton errors (.e. poston-ndependent terms) as lsted n Table 3-2, suggest that statc center offsets of B- and C-axes average lnes from ther nomnal postons (.e. δx BY, δy BY, and δz BY), as well as constant angular postonng error,β BY (about rad ( 8 arcsec)), are domnant for ths expermental case. b. All of prevous R-test works revewed n Secton 1.2 focused only on the dentfcaton of locaton errors. As can be clearly seen by comparng Table 3-2 and Fg. 3-14, locaton errors (n Table 3-2) only represent the average of error motons observed n Fg For example, when tlt ~ BY B ~ BY B error motons of B-axs, parameterzed by α ( ) and γ ( ), vary sgnfcantly wth the B-rotaton as can be observed n Fg. 3-14(b), ther average values do not have much sgnfcance. Ths llustrates a major contrbuton of ths chapter. ~ c. Axal error moton of B-axs,.e. the varaton n observed δ y CB ( B ) wth the B-rotaton, s not sgnfcant (about 1μm). ~ ~ d. Radal error moton of B-axs,.e. the varaton δ x BY ( B ) and δ z BY ( B ) wth the B-rotaton, was observed. However, the radal error moton of B-axs (.e. run-out, see Fg. 2-7(b) n Secton ) s only about ±1 μm (peak-to-peak), whch s not sgnfcant compared to the measurement uncertanty of the R-test devce. ~ BY B ~ BY B e. Tlt error moton of B-axs,.e. α ( ) and γ ( ), was observed. Tlt error moton of B-axs, or conng (see Fgs. 2-7(c) n Secton ), s about ±3 1-5 rad (peak-to-peak). 3.8 Concluson Compared to ball bar measurements [Abbaszaheh-Mr, 22, Kakno, 1994, Tsutsum, 23], the R-test has a strong potental advantage n ts applcablty to hgh-effcent, fully-automated calbraton of error motons of rotary axes on fve-axs machne tools. 55

63 A graphcal presentaton method of R-test measurements was presented n ths chapter. Expermental results ntutvely clarfed how error motons of rotary table changes dependng on the B-angle. Ths chapter compares two R-test setups where the sphere s away from or on the nomnal C-axs average lne. By combnng both measurements, the drectonal relatonshp of C-, Z- and X- (or Y-) axes can be obtaned at each B-angle. Under a certan assumpton, squareness between X-axs, Y-axs, and Z-axs could be estmated, n addton to orentaton errors of the C-axs average lne at each B-angle. An algorthm to numercally parameterze poston-dependent geometrc errors of B-axs s proposed n ths chapter. The expermental case study shows that 1) statc center offsets of B- and C-axes average lnes from ther nomnal postons (.e. δx BY, δy BY, and δz BY) are the sgnfcant error factors, 2) however, angular postonng error of B-axs (.e. β BY (B)), as well as tlt error moton of B-axs, (represented by α BY (B) and γ BY (B)) are also observed n ths expermental case. 56

64 Chapter 4 Observaton of thermal nfluence on error motons of rotary axes by statc R-test 4.1 Introducton As was revewed n Secton 1.2, many researchers have recently reported a calbraton scheme of knematc errors n a fve-axs machne tool. However, an evaluaton method of thermal nfluence on the fve-axs knematcs s rarely found n the lterature. Among many error sources n machne tool knematcs, thermal errors can be one of domnant error factors under extended usage of the machne [Ramesh, 2, Schwenke, 28]. ISO 23-3 and ISO [ISO 23-3, 27, ISO , 27] descrbe tests to evaluate thermal dstortons on machne tools caused by rotatng spndle and recprocatng moton of lnear axes. For the applcaton to a fve-axs machne tool, these standards have the followng ssues: 1) No test s descrbed n these standards on the thermal nfluence on a rotary axs. 2) The tests descrbed n these standards, such as the measurement of thermal dstortons due to movng X-axs shown n Fg. 4-1 (descrbed n ISO 23-3 [ISO 23-3, 27]), only measure thermal nfluence on the TCP (tool center poston) and ts orentaton, and do not evaluate thermal nfluence on error motons of an axs. 3) In seral-lnk knematcs, error motons of one axs are often affected by those of the axs on whch t s mounted on. For example, n the tltng rotary table confguraton (see Fg. 2-5 n Secton 2.3), where a rotary table (C-axs) s nstalled on a swvelng axs (B-axs), error motons of C-axs are often nfluenced by the angular poston of B-axs, due to e.g. the gravty-nduced deformaton. The thermal deformaton may change ths dependency of C-axs error motons on the B-axs angular poston. No tests 57

65 descrbed n ISO 23-3 and can see such an nfluence. Fg. 4-1 The set-up for measurement of thermal dstortons due to movng X axs table of machnng centre. [ISO 23-3, 27] 4.2 Objectve and orgnal contrbuton of ths chapter The objectve of ths research s to propose a method to observe the nfluence of thermal dstortons on error motons of a rotary axs n fve-axs knematcs by R-test measurement. To ntutvely observe geometrc errors of rotary axes, a graphcal presentaton of dsplacement profles measured by R-test was proposed n Secton 3.4. An algorthm to dentfy not only locaton errors, but also poston-dependent component errors of rotary axes wth statc R-test, was proposed n Secton 3.6. The measurement and analyss scheme presented n ths chapter s a straghtforward applcaton of these approaches to thermal tests. 4.3 Test procedure (1) Recprocatng moton of a rotary axs In the proposed thermal test, thermal nfluence caused by a servo motor of a rotary axs wll be nvestgated. Frst, the rotary axs to perform recprocatng moton to generate the heat must be chosen. In our test, snce the deformaton of 58

66 the swvelng B-axs may sgnfcantly nfluence error motons of the rotary table, B-axs s naturally selected to be the recprocatng axs. The B-axs performs a recprocatng movement. Note that other axes (.e. lnear axes and C-axs) are stopped wthn the B-axs recprocatng moton. At the gven tme nterval, the recprocatng moton of B-axs s nterrupted and the followng R-test measurement cycle s conducted. (2) R-test measurement procedure At gven tme nterval, ths B-axs recprocatng moton s nterrupted, and the R-test measurement cycle wth Setup 1-a (outer low), as shown n Fg. 3-1 and Fg. 3-2(a), s conducted. See Secton for detals on the R-test measurement cycle. 4.4 Analyss of R-test profles Graphcal presentaton of R-test profle For more ntutve understandng of rotary axes error motons, the measured R-test profle s graphcally dsplayed by the procedure presented n Secton Error parameters of the rotary table to be dentfed The locaton and orentaton of axs of rotaton may vary due to ts rotaton (.e. poston-dependent geometrc errors descrbed n Secton ). Moreover, as was dscussed n [Srvastava, 1995], the thermal dstortons can be modeled by geometrc errors that vary wth the tme. The poston-dependent geometrc errors of B-axs, lsted n Table 4-1, represent poston and orentaton errors of the C-axs (rotary table) average lne at each B angle. In other words, they represent how the C-axs rotates at each B-angle. The algorthm therefore targets numercal parameterzaton of these parameters, as already descrbed n Secton 3.6. In Chapter 3, total four R-test cycles are conducted wth dfferent sphere locatons. In ths test, only one R-test cycle s performed at each measurement 59

67 nterval. Ths s smply because t s not possble to perform multple R-test cycles at a measurement nterval. As the result, the present scheme manly targets the observaton of the thermal nfluence on the B-axs error motons. Table 4-1 Descrpton of geometrc errors of a rotary table Symbol δx BY (B, t) δy CB (B, t) δz BY (B, t) α BY (B, t) β BY (B, t) γ BY (B, t) Descrpton Locaton changes of B-axs of rotaton n X-drecton dependng on B-angle and tme t. Locaton changes of the rotary table s axs of rotaton n Y-drecton dependng on B-angle and tme t. Locaton changes of B-axs of rotaton n Z-drecton dependng on B-angle and tme t. Orentaton changes of B-axs of rotaton around X-axs dependng on B-angle and tme t. Angular postonng error of B-axs of rotaton dependng on B-angle and tme t. Orentaton changes of B-axs of rotaton around Z-axs dependng on B-angle and tme t Calbraton procedure of geometrc errors of the rotary table From each R-test profle measured at the gven tme nterval, locaton errors and poston-dependent geometrc errors are numercally dentfed. Its detaled algorthm s presented n Secton Case study Expermental setups In ths experment, the R-test sensors nest s nstalled at w q = [-.448 mm, mm, mm] T n the workpece coordnate system (defned n Secton 2.3), as shown n Fg Fve thermocouple sensors (sheathed type) are attached on the locatons shown n Fg. 4-2 for roughly correlatng calbrated geometrc errors and thermal dstrbuton. The machne s cold-started. Then, the B-axs keeps swvelng between -9 and -6 by the angular velocty of 5, degree/mn. At every 15 mnutes, ths B-axs recprocatng moton s nterrupted, and the R-test measurement cycle s conducted wth the followng command B and C angular postons: 6

68 B = -6, -3,, 6 (=1,,5) C j =, 72,, 288 (j=1,,5) Total 5 5 = 25 ponts are measured. Fgure 4-3 shows command X, Y, Z, B, and C trajectores for an R-test measurement cycle. One R-test measurement cycle for 25 ponts takes wthn 2 mnutes ncludng the setup tme n ths expermental case. T2 T1 T3 T4 R-test T5 Fg. 4-2 Setup of R-test and thermocouple sensors (T1: thermocouple sensor for ambent temperature; T2: on the B-axs cover near B-axs center; T3: on the cover near brdge between B-shaft and C-table; T4: on C-table; T5: on R-test). x mm y mm z mm b deg c deg ndex number Fg. 4-3 Command X, Y, Z, B, and C postons for each R-test measurement cycle. 61

69 4.5.2 Measured temperatures Fgure 4-4 shows measured temperatures throughout ths experment, contnued for about 3 hours. Observatons can be made as follows: The temperature on the B-axs cover near B-axs center (T2) rose by 1.4 C. Note that the thermo couple sensor T2 s attached to the B-axs cover, not the B-axs motor drectly, smply because we are not allowed to dsmount the cover. The temperature of the B-axs motor tself was not measured. The temperature rse on the cover near the brdge between B-shaft and C-table (T3) was about.7 C. The temperature rse on the C-table (T4) was not observed. It suggests that the transfer of the heat caused by recprocatng moton of B-axs to the surface of rotary table was suffcent small. Throughout ths experment, the change n the ambent temperature (T1) and the temperature of R-test devce (T5) was wthn.5 C T1: Ambent temperature T2: Near center of B-shaft Temperature C T3: Near brdge between B-shaft and C table 27 T4: on C-table 26.5 T5: on R-test 26.5 h 1 h 1.5 h 2 h 2.5 h 3 h Tme hour Fg. 4-4 Measured temperature by thermal couple sensors. To see the causal connecton of recprocated motons and temperature rse, as well as error motons of rotary axes, we have conducted another experment n dfferent condtons of recprocatng B-axs moton. In ths test, the B-axs was swveled between -15 to 15 n 5, deg/mn. Compared to the present test (-9 to -6 n 5, deg/mn), the temperature rse of the B-shaft was sgnfcantly smaller. In the present test, the B-axs motor outputs sgnfcantly larger torque due to the gravty nfluence. By comparng these two tests, we 62

70 reasonably conclude that temperature rse observed n Fg. 4-4, as well as the change n error motons presented n the followng subsecton, s caused manly by the B-axs recprocatng moton Graphcal presentaton of R-test profle (1) Test results w The R-test profle measured at each test ntervals, q ( B, C ) ( = 1~5, j = 1~5), s transformed to the reference coordnate system, denoted by r p ( B, C ) as s presented n Secton 3.4. Sensor poston profles measured at, j 1, and 2 hours from the begnnng of the test (denoted by tme t = h, 1h, 2h) are shown n Fgs. 4-5(a) to (c) n the three-dmensonal vew. Although the R-test cycle was conducted at total 11 tmes n 3 hours, only three profles (t = h, 1h, 2h) are shown to smplfy the plot. Note that an error of measured postons ( ) from ts command poston ( ) s magnfed 8, tmes. Fgures 4-5(a), (b), and (c) present sensor poston profles measured wth C-rotaton (C j = to 288 ) at B = -6,, 6, respectvely. In each plot, three panted crcles represent the average crcle ft to measured sensor postons for t = h, 1h, and 2h. These average crcles are shown to help easer understandng of poston and orentaton of the C-axs average lne (see Fg. 3-4 n Secton 3.4). Table shows rough poston of the rotary table. Fgures 4-6(a) to (c) show ther projecton onto the XY plane and XZ plane. It s to be noted that the statc axs shft of B-axs n Z-drecton at the start of ths seres of measurements (.e. t = h) s numercally elmnated from the R-test profle n these plots, snce t s typcally caused by the mscalbraton of tool length and thus s not regarded as the machne s error motons (see Secton 3.4). The change wth tme n ths axs shft of C-table to the Z-drecton s shown. These plots clearly show the poston and the orentaton of C-axs average lne at each B, and how they changes wth the tme proceeds. j 63

71 B = -6 Table (a) B = -6 B = -6 B = 6 Table B = (b) B = Table B = 6 (c) B = 6 Fg. 4-5 Sensor postons measured by R-test n the reference coordnate system wth C-rotaton at B = -6,, 6. An error of measured sensor postons ( ) from ts command postons ( ) s magnfed 8, tmes. 64

72 Projected on XY plane (a) B = -6 Projected on XZ plane Projected on XY plane (b) B = Projected on XZ plane Projected on XY plane Projected on XZ plane (c) B = 6 Fg. 4-6 Sensor postons measured by R-test n the reference coordnate system wth C-rotaton at B = -6,, 6, projected onto XY plane and XZ plane. 65

73 (2) Observaton Observatons can be made as follows: a. In projecton on XY plane of Fgs. 4-6(a) to (c), measured sensor postons move to (-X, -Y) drecton as tme proceeds (about -2 μm to X-drecton and -6 μm to Y-drecton n 3 hours). It suggests that the temperature rse gradually shfted the poston of the axs average lne of C-axs. b. In projecton on XZ plane at B = (Fg. 4-6(b)), gradual shft of measured trajectores to the Z-drecton can be observed, although t s sgnfcantly smaller than the shft to the Y-drecton. In ths machne, the nfluence of heat by the B-axs moton manly affects the rotary table poston n the Y-drecton, rather than n X- or Z-drectons. c. In projecton on XZ plane of Fg. 4-6(b), measured sensor postons are tlted from the nomnal trajectory around the Y-axs. As already dscussed n Secton 3.4, ths suggests the squareness error of the C-axs average lne to the X-axs, whch can be caused by the angular postonng error of B-axs at B =. Ths orentaton error of measured profles s larger at B = 6 (n projecton on XZ plane n Fg. 4-6(c)). Ths represents larger B-axs angular postonng error at B = 6 relatve to that at B =, whch could be also observed from projecton on XY plane of Fg. 4-6(c). A slght change n ths angular postonng error wth the temperature rse can be observed, although t s not sgnfcantly large. In ths analyss, the nfluence of squareness errors of lnear axes s not removed, unlke the case presented n Sectons 3.3 to 3.6. Ths s because: 1) the nfluence of squareness errors of lnear axes s, n ths expermental machne, suffcently small compared to error motons of rotary axes, as shown n Fg ) It s not possble to evaluate the thermal nfluence on error motons of lnear axes, snce t s not possble to conduct multple R-test cycles wth dfferent sphere postons at each test nterval. Therefore, the observatons above assume that error motons of lnear axes are suffcently small, and thus errors observed n Fgs. 4-5 and 4-6 are caused by rotary axs error motons only. When 66

74 ths assumpton s not met, error profles are nfluenced by not only error motons of rotary axes, but also those of lnear axes. In the present thermal test, however, t can be reasonably assumed that the heat transfer to lnear axes s lmted (see Secton 4.5.2) and thus the thermal nfluence on ther error motons s suffcently small Calbraton of geometrc errors of the rotary table (1) Test results From the R-test results presented n Secton 4.5.3, the poston and the orentaton of C-axs average lne at each B can be calculated. Sx geometrc error parameters of the rotary table, shown n Table 4-1, can be parameterzed essentally from the poston and the orentaton of average crcles of measured sensor postons shown n Fg. 4-5 and 4-6. Fgures 4-7(a) to (f) show how calbrated geometrc error parameters change wth the tme proceeds. (2) Observaton The dentfcaton results shown n Fg. 4-7 tell that: a. The gradual shft of the rotary table n X-, and Y-drecton observed n the prevous subsecton s parameterzed by a gradual change n δx BY and δy CB n Fgs. 4-7(a) and (b). b. From the dentfed result δz BY shown n Fg. 4-7(c), a gradual shft of the Z-poston of B-axs centerlne s also observed (about -2 μm n 3 hours) as tme proceeds. c. Compared to the measurement uncertanty of the R-test devce (estmated to be about 2μm [Wekert, 24]), the nfluence of thermal dstorton of machne tool on the tlt error moton of B-axs (.e. α BY, β BY, and γ BY ) n Fgs. 4-7(d) to (f) s relatvely small, whch could also be observed from Fgs. 4-5 and 4-6. However, a slght change of the tlt error motons of B-axs was dentfed, partcularly on the tlt error moton around Z-axs (γ BY ). 67

75 δx BY mm B deg h.3 h.5 h.7 h 1 h 1.3 h 1.6 h 2 h 2.2 h 2.5 h 2.7 h (a) δx BY δy CB mm B deg h.3 h.5 h.7 h 1 h 1.3 h 1.6 h 2 h 2.2 h 2.5 h 2.7 h (b) δy CB δz BY mm B deg h.3 h.5 h.7 h 1 h 1.3 h 1.6 h 2 h 2.2 h 2.5 h 2.7 h (c) δz BY 68

76 α BY rad 4 x B deg h.3 h.5 h.7 h 1 h 1.3 h 1.6 h 2 h 2.2 h 2.5 h 2.7 h (d) α BY β BY rad 8 x B deg h.3 h.5 h.7 h 1 h 1.3 h 1.6 h 2 h 2.2 h 2.5 h 2.7 h (e) β BY γ BY rad 4 x B deg h.3 h.5 h.7 h 1 h 1.3 h 1.6 h 2 h 2.2 h 2.5 h 2.7 h (f) γ BY Fg. 4-7 Calbrated geometrc errors of the rotary table at each tme nterval (See Table 4-1 for defnton of error parameters). 69

77 4.6 Concluson By ntroducng an approach proposed n Secton 3.4 and Secton 3.6, calbraton method of thermally nduced geometrc errors of C-axs caused by recprocatng movements of B-axs was proposed n ths secton. Its expermental applcaton example was demonstrated. The present thermal test clarfes how error motons of the rotary table change wth the tme proceeds. From expermental results, the followng observatons are made: (1) Temperature rse of B-shaft manly caused a gradual poston shft of C-axs average lne. Such a shft may potentally cause sgnfcant geometrc errors of the fnshed workpece by fve-axs machnng, as wll be demonstrated for an example of the cone frustum machnng test n Chapter 6. (2) R-test results clarfed the orentaton error of C-axs average lne varyng wth B-rotaton due to recprocated moton of B-axs. A slght change on ths tlt error moton was observed, partcularly on the tlt error moton around the Z-axs, whch s potentally caused by the thermal nfluence of B-axs. (3) By comparng wth analogous test results under the condton where the B-axs motor generates smaller heat, t can be observed that the observed change n error motons s caused manly by the heat generated by the B-axs motor, although ths causal connecton s not proved n ths experment. 7

78 Chapter 5 Non-contact R-test for dynamc measurement on fve-axs machne tools 5.1 Introducton Chapter 3 and 4 presented the applcaton of the conventonal R-test devce to the measurement of statc error motons of rotary axes. Smlarly, as was revewed n Secton 1.2, most of the researches n the lterature focused on error calbraton on fve-axs machne tools n statc R-test measurement [Wekert, 24, Brngmann, 26 Ibarak, 211, Zargarbash, 29]. Note that statc measurement means the measurement under the condton that the machne tool keeps stll when loggng the measurement data. On the other hand, a dynamc measurement s defned as a measurement that s conducted when the machne tool s drven wth a velocty. All the conventonal R-test devces, ncludng commercally avalable ones [IBS, Fda], use contact-type lnear dsplacement sensors wth a flat-ended probe. When flat-ended probes are contacted wth the sphere, the three-dmensonal poston of the sphere center can be calculated by a smple formula [Wekert, 24, Brngmann, 26, Ibarak, 211] from measured dsplacements by usng pre-calbrated drecton vectors of lnear dsplacement sensors only (see Secton 2.2.2). However, the nfluence of frcton between probes and the sphere, or the dynamcs of a sprng supportng probes, could potentally mpose sgnfcant nfluence on the measured dsplacement, partcularly n dynamc measurements. Furthermore, the safety of measurement, avodng the crash caused by e.g. ms-programmng, can be a crtcal ssue wth the contact-type R-test. Therefore, a non-contact R-test has potentally sgnfcant advantages partcularly for dynamc measurement on fve-axs machne tool. 5.2 Objectve and orgnal contrbuton of ths chapter As was revewed n the prevous secton, all the conventonal R-test 71

79 devces [Wekert, 24, IBS, Fda, Ibarak, 29] use contact-type dsplacement sensors. The objectve of ths chapter s to construct a non-contact type R-test wth laser dsplacement sensors, partcularly for dynamc measurement on fve-axs machne tools. As a non-contact dsplacement sensor, we employ a laser dsplacement sensor for ts longer reference dstance and larger measurement range than other non-contact sensors, such as capactve and nductve dsplacement sensors. Potental nherent advantages of applyng non-contact dsplacement sensors to the R-test nclude: 1) The measurement s not affected at all by the frcton on the sphere surface and probes. 2) The measurement s not affected at all by the dynamcs of a supportng sprng n contact-type dsplacement sensors. 3) The measurement s safer due to longer workng dstance between the sphere and sensors. When a non-contact dsplacement sensor s used, or when the sensor does not touch the sphere wth a flat surface, e.g. sphere-ended probes, the offset of the sphere center from the lne representng the sensor s senstve drecton could ntroduce an error for calculatng the three-dmensonal dsplacement of the sphere when the conventonal algorthm for contact-type R-test s used (see Secton for further detals). A new algorthm for calculatng three-dmensonal dsplacement of sphere wth non-contact dsplacement sensors should be proposed. Another crtcal ssue wth the applcaton of a laser dsplacement sensor to the non-contact R-test s ts measurement uncertanty due to the nclnaton of the target surface. When the sensor s senstve drecton s off the center of the sphere, the target surface, where the laser spot hts, s nclned from the sensor s senstve drecton, whch often results n the measurement error. An approprate laser dsplacement sensor should be chosen, whose measurement uncertanty s less senstve to the nclnaton of the target surface. Furthermore, such a measurement error must be numercally compensated from measured 72

80 dsplacement profles. For ths end, the contrbuton of ths chapter can be summarzed as follows: 1) a profle measurement of a sphere by usng laser dsplacement sensors wth dfferent measurng prncples s conducted to nvestgate the measurement uncertanty due to the nclnaton of the target surface (Secton 5.3); 2) an algorthm to calculate three-dmensonal dsplacement of the sphere center for non-contact R-test s proposed (Secton 5.4); 3) a model to nterpolate the measurement error of the non-contact type R-test measurement s proposed. The compensaton wth the estmated measurement error s also conducted to mprove the measurement accuracy wth the non-contact R-test (Secton 5.5); 4) a prototype non-contact R-test s developed. Case studes, ncludng a statc measurement and a dynamc measurement by the developed non-contact R-test devce, are conducted to evaluate ts measurement performance by comparng the measurement results wth the contact-type R-test devce (Secton 5.6). 5.3 Selecton of laser dsplacement sensor for non-contact R-test Objectve Laser dsplacement sensors generally exhbt the best measurement performance when the measured surface s placed normal to the sensor s senstve drecton. Therefore, when measurng a sphere surface, the measurement uncertanty s supposedly mnmzed when the sensor s drected exactly to the sphere center. When the sphere center s shfted from there, the measured surface becomes tlted from the sensor s senstve drecton, and as a result, the measurement uncertanty s expected to ncrease. The measurement uncertanty mentoned above s expected to be sgnfcantly dependent on the measurng prncple. For the applcaton to a non-contact R-test, t s crucal to select an approprate type of laser dsplacement sensor that has the measurement uncertanty less senstve to the nclnaton of the target surface. Meanwhle, t s also mportant to have larger measurng range and smaller nose. Several laser dsplacement sensors wth dfferent measurng prncple are 73

81 avalable n today s market. Ths secton expermentally nvestgates the measurement uncertanty of four laser dsplacement sensors wth dfferent measurng prncple when they are appled to measure a sphere surface. For a non-contact R-test devce, t s favorable to select the dsplacement sensor wth the followng propertes: (1) Larger measurable area on a sphere surface can be obtaned. In ths chapter, measurable area represents the area perpendcular to the sensor s senstve drecton where vald reflected laser beam returns to the sensor and the measurement s possble. (2) The measurement range, defned as the workng dstance along the sensor s senstve drecton where the measurement s possble, s larger. (3) The measurement uncertanty caused by the nclnaton of the target surface, due to the curvature of sphere surface, s smaller. (4) The nose n the measured profle s smaller. The measurement nose s typcally caused by the speckle nose n the laser beam Laser dsplacement sensors used n ths study The nvestgated laser dsplacement sensors nclude spectral nterference type (SI-F1 by Keyence), specular reflecton type (LK-G1 by Keyence), dffuse reflecton type (LK-H52 by Keyence), and confocal type (LT-91MS by Keyence) [Keyence]. Table 5-1 shows man specfcatons of the nvestgated laser dsplacement sensors. The external vews and schematcs of the measurng prncple are shown n Fg Table 5-1 Man specfcatons of laser dsplacement sensors used n ths study [Keyence] Maker: Keyence LK-G1 LK-H52 SI-F1 LT-91MS Measurng prncple Specular reflecton type Dffuse reflecton type Spectral nterference type Confocal reflecton type Reference dstance 1 mm 5 mm 11.3 mm to 6 mm Measurement range ±1 mm ± 1 mm mm ±.3 mm Spot dameter (at reference Φ2 μm Φ5 μm Φ4 μm Φ2 μm dstance) 74

82 Laser head CCD 75 Dsplacement Specular reflecton component External vew Measurng prncple (a) Specular reflecton type (LK-G1) CCD Laser head Lens Dsplacement External vew Dffuse reflecton component Measurng prncple (b) Dffuse reflecton type (LK-H52) Head Laser head (Multwavelength) Interference Gratng CCD Dsplacement s analyzed from the spectrum. Reference Dsplacement External vew Measurng prncple (c) Spectral nterference type (SI-F1) 75

83 Laser head CCD Lens s moved to get a peak on CCD Objectve lens Measured dsplacement Dsplacement External vew Measurng prncple (d) Confocal type (LT-91MS) Fg. 5-1 Laser dsplacement sensors wth dfferent measurng prncples. [Keyence] The measurng prncples of these four laser dsplacement sensors are brefly ntroduced as follows (more detaled nformaton can be found n [Keyence]): (a) Specular reflecton type (LK-G1 by Keyence), and (b) Dffuse reflecton type (LK-H52 by Keyence) As shown n Fgs. 5-1(a) and (b), theses laser dsplacement sensors focus the laser beam on a target. The target reflects the beam back through the lens where t s focused on a lght-recevng element (CCD arrays). By detectng the dsplacement of the beam spot on the CCD arrays, the target s dsplacement (n the vertcal drecton n Fgs. 5-1(a) and (b)) can be determned. The surface-reflected laser from a target object conssts of a regular reflecton and a dffuse reflecton component. (a) and (b) accept regular reflecton component and dffuse reflecton component, respectvely. (c) Spectral nterference type (SI-F1 by Keyence) As shown n Fg. 5-1(c), a target-reflected laser beam wth wde wavelength band, nterferes wth the laser beam from a reference surface. The spectrum of the nterfered laser beam s detected by a lght-recevng element and analyzed to determne the dsplacement of the target. 76

84 (d) Confocal type (LT-91MS by Keyence) As shown n Fg. 5-1(d), a laser beam s focused on a target surface through an objectve lens that vbrates up and down at hgh speed by means of a tunng fork. The beam reflected off the target surface s converged on a pnhole and then enters a lght-recevng element. By measurng the exact poston of the objectve lens when the lght enters the recevng element, the target heght can be determned Expermental nvestgaton of measurement uncertanty of laser dsplacement sensors for profle measurement of sphere (1) Test objectve As mentoned n Secton 5.3.1, the measurement uncertanty s supposed to ncrease when the laser dsplacement sensor s senstve drecton s off the center of the target sphere. Therefore, n ths subsecton, a test to measure the profle of a sphere wth the laser dsplacement sensor s conducted. By conductng such a test, the measurable area, the measurement uncertanty due to the nclnaton of the target surface, and the nose when the laser scans the sphere surface (as mentoned n Secton 5.3.1), are nvestgated. (2) Test procedure Frst, the ceramc precson sphere wth the radus of 12.7 mm (other specfcatons are shown n Table 2-2), s attached to the machne spndle. One of four laser dsplacement sensors n Secton s fxed on the machne table vertcally as llustrated n Fg In each test, the sensor s set up such that the poston (X, Y) = (, ) corresponds to the pont where the sensor s senstve drecton approxmately ponts to the sphere center. Then, for the specular reflecton type laser dsplacement sensor LK-G1 for example (see Fg. 5-3), the machne s moved from X = -.6 mm to +.6 mm at Y = -.6 mm, where the laser dsplacement s contnuously logged. The same scannng operaton s repeated at Y = -.5, -.4,, +.6 mm. Smlar scannng test s conducted for each sensor, although the scannng drecton, the scannng 77

85 range, and the scannng path ptch dffer for each sensor. If the sensor s senstve drecton s exactly pontng to the sphere center at (X, Y) = (, ), the measured laser dsplacement should be the shortest (.e. maxmum value) there. Due to ts setup error, ths top locaton, as well as the orentaton of measured profle, may be shfted from the nomnal poston and orentaton. To cancel the nfluence of ths setup error, the deal sphere profle s best-ft to the measured profle. The dfference of measured profles from the best-ft deal sphere s then recalculated. The profle of the measured sphere surface s generated wth the nomnal X- and Y-poston of the sensor. Scannng path Z Sphere attached on the spndle Y X Laser dsplacement sensor Fg. 5-2 Schematcs of the profle measurement of a sphere. Z Y Scan the laser probe n horzontal plane Laser CCD X Scannng of sensor Sphere Error caused by tlt angle Fg. 5-3 Measurement uncertanty to the nclnaton of the target surface wth specular reflecton type laser dsplacement sensor (LK-G1). 78

86 (3) Analyss objectve and procedure As was mentoned before, the measurement uncertanty s supposedly mnmzed when the sensor s senstve drecton s exactly algned to the lne pontng to the sphere center. As the laser spot moves away from ths top poston on the sphere surface, the measurement uncertanty s expected to ncrease (see Fg Ths schematc dagram shows the specular reflecton type laser dsplacement sensor as an example, but smlar observaton can be made for other sensors). Eventually, as the dstance from ths top poston s larger than certan value, the reflected lght fals to return to the sensor and thus the measurement fals. Based on test results, we dvde the sphere surface dependng on the dstance from the lne pontng to the sphere center as follows: 1) The range where the measurement dsplacement s suffcent small,.e. the range that can be regarded as a flat surface (smaller than 1 μm n ths study); 2) The range where the profle measurement error s smaller than the defned tolerance; 3) The range where the profle measurement error s larger than the defned tolerance; 4) The range where the measurement of the sphere profle fals. Note that the profle measurement error s defned as the measured dsplacement wth respect to the nomnal sphere geometry. The tolerance s set to be 1 μm n ths secton. The geometrc naccuracy of the ceramc precson sphere s pre-calbrated to be suffcently small compared to the sensor s measurement uncertanty. The man objectve of experments n ths secton s to fnd the sze of each range (1~4) for each of four dfferent laser dsplacement sensors presented n Secton 5.2. Naturally, the sensor havng larger range s preferred for the applcaton to the non-contact R-test. (4) Test results 79

87 Fgures 5-4 show the profle measurement result of a sphere wth specular reflecton type laser dsplacement sensor (LK-G1). Fgure 5-4(a) shows the test setup. Fgure 5-4(b) shows measured raw profles of laser dsplacements plotted wth nomnal X and Y postons of the sensor. As presented n Secton (2), the nomnal sphere profle (also shown n Fg. 5-4(b)) s best-ft to measured profles. In Fg. 5-4(c), for each of scannng lne to the X-drecton, the dfference n the measured profle from the nomnal profle s plotted wth nomnal X and Y postons. Ths represents the measurement error, where the nfluence of the sensor s setup error s removed. Ths profle s called the profle measurement error. Fg. 5-4(d) shows the projecton of profle measurement error n Fg. 5-4(c) along X-drecton (all the scannng lnes are supermposed). Fgures 5-5 to 5-7 are the setup and the profle measurement result wth dffuse reflecton type laser dsplacement sensor (LK-H52), spectral nterference type laser dsplacement sensor (SI-F1), and confocal reflecton type laser dsplacement sensor (LT-91MS), respectvely. LK-G1 attached on the Z-axs Z X Y Sphere fxed on the table Z mm measured sphere profle X mm.5 nomnal sphere profle Y mm.5 1 (a) Setup (b) Profle measurement result 8 x profle measurement error mm X mm (c) Profle measurement error (d) Profle measurement error along X-axs Fg. 5-4 Profle measurement result of a sphere wth specular reflecton type laser dsplacement sensor (LK-G1). 8

88 nomnal sphere profle Z Y Z mm X measured sphere profle X mm -2 2 Y mm -2 (a) Setup (b) Profle measurement result.3.25 profle measurement error mm x mm (c) Profle measurement error (d) Profle measurement error along X-axs Fg. 5-5 Profle measurement result of a sphere wth dffuse reflecton type laser dsplacement sensor (LK-H52). (a) Setup Z Y X Z mm nomnal sphere profle measured sphere profle.2 X mm Y mm (b) Profle measurement result 1 x 1-3 profle measurement error mm x mm (c) Profle measurement error (d) Profle measurement error along X-axs Fg. 5-6 Profle measurement result of a sphere wth spectral nterference type laser dsplacement sensor (SI-F1). 81

89 Z Y X Z mm x X mm measured sphere profle -.5 nomnal sphere profle Y mm.5 (a) Setup (b) Profle measurement result (c) Profle measurement error (d) Profle measurement error along X-axs Fg. 5-7 Profle measurement result of a sphere wth confocal type laser dsplacement sensor (LT-91MS). (5) Observatons The followng observatons can be made for each laser dsplacement sensor: 1) Specular reflecton type (LK-G1, Fgs. 5-4) Snce the object s a sphere, the angle of reflecton would vary wth tlt angle of the normal drecton of measured surface wth respect to the sensor s senstve drecton, whch would cause measurement error (see Fgs. 5-4(c) and (d)). When the tlt angle of the normal drecton of measured surface wth respect to the senstve drecton of laser dsplacement sensor s reversed, the drecton of the measurement error s expected to be reversed. As can be observed n Fgs. 5-4(c) and (d), wthn the range ±.5 mm n the XY plane from the sphere top, the average error from the sphere s nomnal geometry s wthn ±3 μm approxmately. The hgher-frequency varaton n measured profles of the ampltude up to 2 μm s also observed over the entre range, whch s lkely caused by the speckle nose n laser beam. 82

90 2) Dffuse reflecton type (LK-H52, Fgs. 5-5) Dffuse reflecton type laser dsplacement sensor (LK-H52 from Keyence) succeeded to measure the wdest range of sphere surface (over ±2.5 mm n the XY plane) among the four sensors, whle hgher-frequency varaton s sgnfcantly larger (ts ampltude s up to 15 μm, as shown n Fgs. 5.5(c) and (d)) than other laser dsplacement sensors, whch could be caused by the speckle nose n laser beam. More studes on the measurement uncertanty of ths sensor for profle measurement can be found n [Kmura, 211]. 3) Spectral nterference type (SI-F1, Fgs. 5-6) Although the spectral nterference type laser dsplacement sensor has smaller measurement area (±.3 mm), the test result n Fgs. 5-6 (b) to (d) shows smallest errors over the range ±.3 mm (the mean error s wthn ±.2 μm). Its hgher-frequency component s also the lowest among the four sensors. 4) Confocal type (LT-91MS, Fgs. 5-7) For confocal type laser dsplacement sensor, both the profle measurement error (wthn 1μm n the range ±.5 mm) and hgher-frequency error (wthn ±.8 μm) are relatvely small among the four sensors. However, occasonal spkes, of the ampltude 6 μm at maxmum, are observed. Ths could be caused by the focusng error due to the laser s speckle nose. The summary of profle measurement results s shown n Table 5-2. Nose level s defned as the standard devaton (1σ) of hgher-frequency component n the profle measurement error. Based on measured profles, the ranges 1) to 3) presented n Secton (3) are evaluated for each sensor. In Table 5-2, profle measurable area shows the sze of each range n the XY plane wth the center at the sphere top. Specular reflecton type laser dsplacement sensor (LK-G1 by Keyence), whch has relatvely large measurement range, relatvely large measurable area, as well as relatvely good measurement performance (.e. measurement uncertanty and the nose level), s chosen to be the sensor to develop an R-test prototype. The cost of each sensor s also one of reasons for ths choce. The prce of each sensor s: LK-H52 < LK-G1 < LT-91MS < SI-F1. 83

91 Table 5-2 Profle measurement result of a ceramc sphere wth radus of 12.7 mm. Model Specular reflecton type (LK-G1) Dffuse reflecton type (LK-H52) Spectral nterference type (SI-F1) Confocal type (LT-91MS) Profle measurable area 1): ±.1 mm 2): ±.3 mm 3): ±.6 mm 1): ±.1 mm 2): ±1 mm 3): over ±2.5 mm 1): ±.1 mm 2): ±.3 mm 3): ±.3 mm 1): ±.1 mm 2): ±.3 mm 3): ±1.5 mm Nose level (1σ) 1μm 5μm.1μm.5μm 5.4 Constructon of algorthm to calculate three-dmensonal dsplacement of sphere center Dfference n algorthms for non-contact and contact-type R-tests For the conventonal R-test, a contact-type lnear dsplacement sensor wth flat-ended probes s pushed to beng contacted wth the sphere. Therefore, ts dsplacement would not be affected by center offset of the sphere n the plane perpendcular to the sensor s senstve drecton (see Fg. 5-8 (a)). As presented n Secton 2.2.2, the algorthm to calculate the sphere center dsplacement by the conventonal contact-type R-test s revewed as follows: Fgure 5-9 shows the measurement setup of conventonal contact-type R-test (re-posted from Fg. 2-2). A precson sphere s attached to the spndle of test machne. Three dsplacement sensors ( = 1, 2, 3) are nstalled on the table (Fg. 5-9 only shows one dsplacement sensor). The orgn of the coordnate system, O = (,, ) T, s defned at the sphere center n ts ntal poston. The unt drecton vector V = (u, v, w ) T ( = 1, 2, 3) represents the senstve drecton of the -th dsplacement sensor. The dsplacement of sphere center, O j = (O jx, O jy, O jz ) T (j = 1,..., N), can be calculated as follows (see Secton 2.2.2): 84

92 1 u1 u2 u3 jx jy jz 1 j 2 j 3 j v1 v2 v3 (5-1) [ O O O ] = [ d d d ] w 1 w 2 w 3 where d j ( = 1, 2, 3; j = 1,..., N) denotes the measured dsplacement of the -th dsplacement sensor when the sphere center s moved from O to O j. It suggests that when three dsplacement sensors touch the sphere from dfferent drectons, the three-dmensonal dsplacement of the sphere center can be calculated from pre-calbrated drecton vectors and measured dsplacements of three probes only. sphere poston after offset ntal sphere poston contact-type dsplacement sensor (a) By contact-type dsplacement sensor wth flat-end probe sphere poston after offset ntal sphere poston nfluence of sphere offset on measurement dsplacement non-contact type dsplacement sensor (b) By non-contact dsplacement sensor Fg. 5-8 Influence of sphere center s offset on the measured dsplacement wth contact-type dsplacement sensor and wth non-contact type dsplacement sensor. 85

93 O O j ntal sphere poston sphere poston after center offset d j V Fg. 5-9 Measurement setup of conventonal contact-type R-test. [Ibarak, 29] However, for the case of a non-contact R-test, the measured dsplacement of the probe changes when the sphere center s shfted perpendcular to the senstve drecton of laser dsplacement sensor (see Fg. 5-8 (b)). It suggests that an error would be ntroduced by the center offset of sphere when the conventonal algorthm for contact-type R-test s used. Note that, although Zargarbash and Mayer [Zargarbash, 29] have proposed a non-contact measurement devce named Cap-test wth capactance dsplacement sensors, the algorthm used for calculatng the three-dmensonal dsplacement of sphere seemed the same wth the algorthm used n conventonal R-test. To elmnate the nfluence of center shft llustrated n Fg. 5-8 (b), a new algorthm for calculatng three-dmensonal dsplacement of sphere wth non-contact dsplacement sensors should be proposed Algorthm to calculate three-dmensonal dsplacement of sphere center Fgure 5-1 shows the measurement prncple of a non-contact R-test. A precson sphere wth the radus R s attached to the spndle of test machne. Three laser dsplacement sensors ( = 1, 2, 3), whch are nstalled on the table, are roughly drected to the sphere center. The orgn of the coordnate system, O = (,, ) T, s defned at the sphere center n ts ntal poston; the orentaton of the coordnate system s defned based on the machne coordnate system [Schwenke, 28]. The unt drecton vector V = (u, v, w ) T ( = 1, 2, 3) 86

94 represents the senstve measurement drecton of the correspondng laser dsplacement sensor. The ntersecton pont of the lne representng -th sensor s senstve drecton and the sphere surface at the ntal poston s defned as P = (x, y, z ) T ( = 1, 2, 3). When the sphere center s postoned at O j = (O jx, O jy, O jz ) T (j = 1,..., N), the ntersecton moves to P j = (x j, y j, z j ) T ( = 1, 2, 3). d j ( = 1, 2, 3; j = 1,..., N) denotes the measured dsplacement of the -th laser dsplacement sensor when the sphere center s moved from O to O j. spndle sphere V P d j O Pj Oj laser dsplacement sensor 1 laser dsplacement sensor 3 laser dsplacement sensor 2 Fg. 5-1 Measurement setup of non-contact R-test wth three laser dsplacement sensors. Equatons (5-2) represent the relatonshp between the dsplacement of laser dsplacement sensors and the poston of the sphere center: Pj P j = P O j + 2 d j V = R 2 (5-2) The procedure to calculate the three-dmensonal poston of the sphere center, O j, wth non-contact R-test s shown as follows: (1) Calbraton of parameters of each sensor Smlarly as the conventonal contact-type R-test (see Secton 2.2.2), the drecton vector of each sensor, V ( = 1, 2, 3), must be pre-calbrated n the machne coordnate system. However, the estmaton of the sphere center, O j, from laser dsplacements, d j ( = 1, 2, 3), requres not only the orentaton of each sensor s senstve drecton, V, but also ntersecton ponts when the sphere 87

95 s at the ntal poston, represented by P ( = 1, 2, 3). Ths s an essental dfference n algorthms for contact-type and non-contact R-tests. The calbraton procedure of parameters mentoned above s shown as follows: (a) Perform the same calbraton cycle presented n Secton (2). Poston the sphere center at gven reference postons, O j (j = 1,..., N), wthn the measurement volume; (b) Assumng that the machne s postonng error at O j s suffcently small, calbrate P and V of each laser dsplacement sensor separately ( = 1, 2, 3), from O j ( j = 1,..., N) and measured laser dsplacements d j (j = 1,..., N) by solvng the followng mnmzaton problem: mn P, V j = 1,..., N ( P + dj V O j R) 1 2 (5-3) where equalty constrants P = R, V = 1 ( = 1, 2, 3) are appled. (2) Estmaton of sphere center dsplacement from laser dsplacements Once the drecton vector of each sensor, V ( = 1, 2, 3), as well as ts poston, represented by P ( = 1, 2, 3), s calbrated, the three-dmensonal dsplacement of sphere center, O j (j = 1,..., N 2 ), can be calculated from measured laser dsplacements, d j ( = 1, 2, 3), by solvng the followng mnmzaton problem: mn Oj ( P + dj V O j R) = 1,2,3 2 (5-4) Both Eq. (5-3) and Eq. (5-4) can be solved wth the nonlnear least square method [Matlab, 22] Expermental verfcaton of the proposed algorthm (1) Test objectve and procedure The objectve of ths subsecton s to expermentally nvestgate estmaton accuracy of the sphere dsplacement by the proposed algorthm n Secton The procedure of experments s lsted as follows: Identfcaton of sensor parameters: 88

96 (a-1) The spndle-sde sphere s postoned at total 9 postons gven n Secton (2). The nterval of each pont, l n Secton (2), s dfferent for each sensor (snce the measurable range of each sensor s dfferent). In Fg. 5-11, the center pont and 8 corner ponts are command postons. (a-2) The dsplacements of laser dsplacement sensors at 9 postons (the center poston and the 8 corner postons) are used to calbrate V and P ( = 1, 2, 3) wth Eq. (5-3). Estmaton of sphere postons: (b-1) Then, the spndle-sde sphere s postoned at total 27 ponts (9 ponts above are ncluded) wthn the same cubc volume as shown n Fg (b-2) By usng calbrated V and P ( = 1, 2, 3), the sphere center poston for each commanded poston s calculated from measured dsplacements, d j, by Eq. (5-4). (b-3) Estmated postons of sphere center are plotted by magnfyng the error vector from commanded postons. Note that, a ceramc precson sphere wth radus of 12.7 mm s used (see Table 2-2 of Secton for detals). (2) Test result Same tests are conducted wth four sensors presented n (1). The estmated postons of sphere center for each of four laser dsplacement sensors are shown n Fg The error scale s shown n each plot n Fg. 5-11, respectvely. The machne s postonng error s assumed to be suffcently small compared to measurement uncertanty. Then, errors of dentfed postons from command postons observed n Fg. 5-11(a) to (d) can be seen as estmaton errors of non-contact R-tests. Table 5-3 summarzes the estmaton error of sphere postons of the conducted statc R-test measurements. mean represents mean value of norm of measurement errors wthn the 27 postons and std represents standard devaton (1σ) of norm of measurement errors. The test s 89

97 conducted for 3 tmes. The repeatablty n Table 5-3 shows the maxmum devaton of norm of measurement error wthn any two tests. z mm Identfed poston 1μm y mm.3 command -.2 poston x mm (a) Wth specular reflecton type (LK-G1) z mm Identfed poston μm y mm 1.5 command -1 poston x mm -1.5 (b) Wth dffuse reflecton type (LK-H52) z mm Identfed poston μm y mm command poston x mm (c) Wth spectral nterference type (SI-F1) 9

98 .2.1 Identfed poston command poston z mm μm y mm x mm (d) Wth confocal type (LT-91MS) Fg Commanded and dentfed poston of sphere center n a statc R-test measurement. Smlar tests are conducted wth dfferent measurement volumes (e.g. X.24 mm Y.24mm Z.24 mm for LK-G1), although plots of measurement errors are not shown. All the test results are summarzed n Table 5-3. For some sensors, only a part of these measurement volumes are tested, due to the lmtaton n measurable area (see Secton (3)). For example, the measurement volume mm s tested only for LK-H52. Note that Fg only shows one of these tests for each sensor. Table 5-3 Profle measurement error (mean, std) and repeatablty of sphere dsplacement n a statc R-test measurement. Measurement volume (mm 3 ) LK-G1 (μm) LK-H52 (μm) SI-F1 (μm) LT-91 MS(μm) Measurement wthn.24 3 (2.4,.9) (.8,.3) (.8,.3) error wthn.4 3 (4.7, 2.3) (12.9, 7.1) (1.,.4) (mean, std) wthn 2 3 ( 14.3, 6.5 ) Repeatablty wthn wthn wthn

99 (3) Observaton From Table 5-3, spectral nterference type laser dsplacement sensor (SI-F1 by Keyence) and confocal type laser dsplacement sensor (LT-91MS by Keyence) exhbt smallest measurement errors (both n the mean and the standard devaton) among the tested laser dsplacement sensors, although the measurable volume by these two sensors s smaller (wthn.4 3 mm 3 for SI-F1, wthn.24 3 mm 3 for LT-91MS). Dffuse reflecton type laser dsplacement sensor (LK-H52 by Keyence) can measure the largest volume (over 2 3 mm 3 ), whle the measurement error s sgnfcantly larger both n the mean and the standard devaton than other types of laser dsplacement sensors. Specular reflecton type laser dsplacement sensor (LK-G1 by Keyence), has relatvely large measurable area as well as relatvely good estmaton performance compared to LK-H Compensaton scheme of the measurement error caused by laser dsplacement sensor Objectve As was presented n Secton 5.4.3, when specular reflecton type laser dsplacement sensor (LK-G1) s used, the estmaton error of sphere dsplacement s 4.7 μm n average wthn mm. Ths s sgnfcantly large compared to the target measurement accuracy. From test results shown n Secton (Fg. 5-11(a) for LK-G1), t s reasonably concluded that the measurement uncertanty to the nclnaton of the target surface s the crtcal error factor for ths estmaton error of sphere center, snce the proposed algorthm dd not consder ths nfluence. Moreover, Table 5-3 shows that the measurement results of sphere center wth the proposed algorthm exhbt good repeatablty compared to the machne s postonng repeatablty. For LK-G1, the repeatablty of the norm of the three-dmensonal measurement error s below 1 μm, as shown n Table 5-3. It suggests that the measurement accuracy of sphere center can be mproved, when the measurement error due to the curvature of sphere surface s 92

100 compensated. The objectve of ths secton s to propose ts compensaton scheme. The overall estmaton accuracy of the non-contact R-test wth the proposed compensaton scheme wll be expermentally nvestgated n Secton Interpolaton of the measurement error wth RBF Network A radal bass functon (RBF) network s an artfcal neural network that uses radal bass functons (such as the functon shown n Eq. (5-7)) as actvaton functons. The RBF Network s typcally used n such applcatons as functon approxmaton, tme seres predcton, and control, especally when the analytcal formulaton of the problem s hard. In ths secton, the RBF Network s used as a three-dmensonal look-up table to nterpolate the measurement error of non-contact R-test. General schematc of the RBF Network s shown n Fg In our applcaton, at each nomnal sphere center poston (denoted by x () the non-contact R-test, denoted by 3 R ), the measurement error by 3 b ( xˆ( )) R ( = 1,..,M), s the output of the RBF Network, as shown n Eq. (5-5). Note that 3 x ˆ( ) R represents the estmated sphere center poston of x () by the algorthm proposed n Secton 5.4.2, and s the nput to the RBF Network. In the RBF Network [Seshagr, 2], a weght matrx 3 w j ( xˆ( )) R (j = 1,,N) and radal bass functons 3 ϕ j ( xˆ( )) R are together used to descrbe the measurement error b ( xˆ( )) ( = 1,..,M) as n Eq. (5-6). The radal bass functon, as shown n Eq. (5-7), represents the three-dmensonal dsplacement of the sphere center poston x () from another poston, x (j) (j = 1,,N), n the measurement volume. However, snce the nomnal sphere center poston (.e. x ()) s not known, the estmated sphere center poston x ˆ( ) replaces the nomnal poston x (), assumng the estmated x ˆ( ) s close enough to the nomnal x (). b ( xˆ( )) = x ( ) xˆ( )( = 1,..., M ) (5-5) 93

101 N j = 1 ϕ ( xˆ( )) w ( xˆ( )) = b ( xˆ( ))( 1,..., M ) (5-6) j j = ϕ ( xˆ( )) = xˆ( j) xˆ( )( j 1,..., N) (5-7) j = ˆx(1) Inputs RBFs outputs x ˆ( ) ϕ j ( xˆ( )) b ( x ˆ( )) b 1 ( xˆ(1 ))... xˆ ( ) weghts w j ( xˆ( )) b... ( xˆ( )) Fg Interpolaton of measurement error wth RBF Network. The procedure to nterpolate the three-dmensonal measurement error of the sphere center wth the RBF Network s shown as follows: (1) Calbraton of weght matrces To buld the RBF Network, ts weght matrces, ( xˆ( )), must be dentfed. The calbraton procedure of weght matrces s shown as follows: (a) The same calbraton cycle s performed as presented n Secton 5.4. The sphere center s postoned at total N = 27 postons n a cube for calbraton. The machne s postonng error s assumed to be suffcently small compared to measurement uncertanty. The command sphere center postons are denoted by x (); (b) The sphere center postons are roughly estmated by the algorthm proposed n Secton 5.4.2, whch are denoted by x ˆ( ) ( = 1,,N). The radal bass functons are calculated wth Eq. (5-7), and the weght matrx s best-ft wth the least square method: N ( b ( xˆ( )) j ( xˆ( )) wj ( xˆ( )) ) 2 mn ϕ (j = 1,, N) (5-8) wj ( xˆ( )) = 1 (2) Estmaton of measurement error of sphere center poston When weght matrces, ( xˆ( )), are dentfed, the objectve of the RBF w j w j 94

102 Network s to estmate the estmated measurement error of sphere center poston b ( xˆ( )) (Eq.(5-5)) from roughly estmated sphere center poston x ˆ( ). It s gven by: N = x b ˆ ( xˆ( )) ϕ ( xˆ( )) w ( xˆ( )) ( = 1,, N) (5-9) j = 1 j j Fgure 5-13 shows the overall block dagram representaton of the sphere center calculaton algorthm for non-contact R-test, wth compensatng the measurement error of the laser dsplacement sensor. Fg Overall block dagram representaton of the sphere center calculaton algorthm for non-contact R-test. 5.6 Developed prototype of non-contact R-test devce A non-contact R-test prototype wth three specular reflecton type laser dsplacement sensors (LK-G1) was developed. Ths secton presents the developed prototype, as well as expermental nvestgaton of ts measurement accuracy of sphere dsplacement Developed prototype of non-contact R-test wth LK-G1 The developed R-test prototype s shown n Fg The specfcatons of the laser dsplacement sensor LK-G1 are shown n Table 5-1 of Secton The specfcatons of the ceramc sphere are shown n Table 5-4. Note that a 95

103 sphere wth larger dameter (Φ 5 mm) compared to the one tested n Secton (Fg. 5-11) and Secton (Fg. 5-4) (see Table 2-2 of Secton 2.2.2) s used to get a larger measurable area for each sensor, and consequently, larger measurable volume for sphere dsplacement. The laser dsplacement sensors LK-G1 are nstalled on a sensors nest, whose orentaton s tlted 45 from the horzontal plane, as shown n Fg Wth ths set-up, the rotaton range of the B-axs angle can reach ±9 n an R-test measurement cycle (see Secton 3.3.1). Sphere LK-G1 97 mm 45 (a) External vew (1) (b) External vew (2) Fg Non-contact R-test measurement devce wth three laser dsplacement sensors (LK-G1). Table 5-4 Specfcatons of the reference ball (from Kolb & Baumann). Dameter 5 mm +1.5μm at 2.6 Sphercty below.4 μm Materal ceramc Profle measurement of a sphere wth LK-G1 In Fg. 5-3 n Secton 5.3.4, a profle measurement of a sphere of the dameter 25.5 mm was presented. Snce a sphere of the dameter 5 mm s used for the non-contact R-test prototype, the same profle measurement s conducted for ths sphere. Ths profle measurement s requred to construct the RBF 96

104 Network presented n Secton For the procedure for a profle measurement of a sphere, a reader should refer to Secton The measurement result s shown n Fg When the measurement range s extended to ±1 mm from the sphere top, the average error from the sphere s nomnal geometry s about ±3 μm approxmately. Compared wth the test shown n Secton 5.3.3, the measurement range s proportonally extended as the dameter of the sphere s enlarged. It s also observed that the measurement uncertanty s sgnfcantly dependent on the nclnaton of the target surface. Y Z X Tested sensor (a) Expermental setup (b) Profle measurement result (c) Profle measurement error (d) Profle measurement error along Y-drecton Fg Profle measurement result of a sphere usng LK-G1 and the sphere n Table

105 5.6.3 Expermental verfcaton of the proposed algorthm The objectve of ths subsecton s to expermentally nvestgate the valdty of the algorthm to estmate the sphere center poston from laser dsplacements, wth the compensaton of measurement errors due to the sphere curvature, presented n Secton 5.5. The same test presented n Secton s conducted for the non-contact R-test prototype developed n the prevous subsecton. The spndle-sde sphere s postoned at total 27 postons wthn a cubc volume (1 1 1 mm) shown n Fg. 5-16(a) for parameter calbraton. Then, the spndle-sde sphere s postoned at total 1183 postons shown n Fg. 5-16(b) wthn the same volume to verfy the measurement accuracy wth the proposed algorthm shown n Fg After calbratng the parameters, the sphere center postons wth compensaton are estmated and shown n Fg. 5-16(b). Frst, when the conventonal algorthm for conventonal contact-type R-test presented n Secton s used, errors n the estmated sphere center poston from ts command poston (n X, Y, and Z drectons) s shown n Fg. 5-16(c). Smlarly as n Secton 5.4.3, when the machne s postonng error s assumed to be suffcent small, these errors can be seen as the estmaton error of the R-test measurement. Fgure 5-16(d) shows the three-dmensonal estmaton error of sphere center postons by the algorthm proposed n Secton wthout compensatng the measurement error of the laser dsplacement sensor tself; Fgure 5-16(e) shows the three-dmensonal estmaton error of sphere center postons by the algorthm proposed n Secton 5.5 (see Fg. 5-13) wth compensatng the measurement error of the laser dsplacement sensor. It clearly shows that, 1) when the proposed algorthm n Secton s appled wthout the compensaton, estmaton errors are only slghtly smaller than those wth the conventonal algorthm; there are stll large estmaton errors (wthn ±.2 mm); 2) the estmaton accuracy s sgnfcantly mproved when the compensaton presented n Secton 5.5 s appled (the estmaton error s summarzed n Table 5-5). In the tests above, both the calbraton test (n Fg. 5-16(a)) and the 98

106 verfcaton test (n Fg. 5-16(b)) were done wth the same setup of the non-contact R-test. To further nvestgates ts measurement accuracy under dfferent setups, the measurement shown n Fg. 5-16(b) s also conducted when the rotary table s postoned at B =, C = ; B = 9, C = ; and B =, C = 9, to evaluate the repeatablty and the effectveness of the algorthm proposed n Secton 5.5. The results are shown n Table 5-5. Note that the estmaton error s evaluated by (mean, std). See Secton for the defnton of (mean, std). Table 5-5 ndcates that the developed non-contact R-test devce has the measurement uncertanty about 1.5 μm n the mean, and about.8 μm n the standard devaton wthn the measured volume mm. It should be noted that the test results shown n Table 5-5 ncludes the postonng uncertanty of the machne tool as well. 1.5 Identfed poston z mm μm y mm.5 1 command poston.5 1 x mm (a) Command and estmated sphere center postons for parameter calbraton. (b) Command and estmated postons wth compensaton 99

107 (c) Error wth conventonal algorthm for contact-type R-test (for total 1183 ponts shown n (b)) (d) Error wthout compensaton (e) Error wth compensaton Fg Estmaton errors of sphere poston wthn mm. Table 5-5 Measurement error (mean, std) and repeatablty of sphere Measurement uncertanty dsplacement wth specular reflecton type laser dsplacement sensor (LK-G1). Measurement Sphere center estmaton Sphere center estmaton wth volume algorthm (μm) compensaton (μm) (mm 3 ) B=, C= B=, C=9 B=9, C= B=, C= B=, C=9 B=9, C= wthn 1 3 (6.5, 3.8) (6., 3.7) (6.3, 3.8) (1.4,.7) (1.6,.7) (1.7,.8) 5.7 Case studes The objectve of ths subsecton s to demonstrate the applcaton of the developed non-contact R-test devce to statc and dynamc measurements, as well as to present the expermental comparson of measured results by contact-type 1

108 and non-contact R-test devces. Present expermental case studes ncludes: 1) a statc measurement to calbrate an error map of a rotary axs, and 2) a dynamc measurement to observe dynamc errors n the synchronous moton between rotary axs and lnear axes Calbraton of an error map of a rotary axs n statc measurement (1) Test objectve and procedure The objectve of ths test s to expermentally nvestgate the measurement performance of the developed non-contact R-test n a statc R-test measurement cycle presented n Secton 3.3.1, as well as the expermental comparson wth the contact-type R-test. The statc R-test measurement cycle presented n Secton s expermentally conducted on the same fve-axs machne tool shown n Fg. 2-5 (n Secton ), wth the contact-type R-test and wth the non-contact R-test. The detaled expermental procedure can be found n Secton However, n ths subsecton, only the R-test measurement cycle of Setup 1-a (outer low), as shown n Fg. 3-1(a) of Secton 3.3.1, was conducted. Sphere locatons n the workpece coordnate system are: Setup 1-a (outer low) wth the contact-type R-test: w q ct = [.2, -83.9, 4.5] (mm) Setup 1-a (outer low) wth the non-contact R-test: w q nc = [.7, -8.1, -3.] (mm) The R-test measurement cycle s conducted wth the followng command B and C angular postons: B = -75, -5,, 75 (=1,,7) C j =, 3,, 33 (j=1,,12) The R-test measurement cycle was repeated for 3 tmes wth the contact-type R-test and wth the non-contact R-test, separately. The measurement result s plotted, as was proposed n Secton 3.4. Note that statc center offset of C-axs n X-, Y-, and Z-drecton (.e. δx BY, δy CB, δz BY ) s numercally elmnated, snce t s usually not the nherent error of B-axs (see Secton 3.4 for 11

109 more detals). The expermental setups (Setup 1-a(outer low) n Secton 3.3.2) wth the contact-type R-test and wth the non-contact R-test are shown n Fg (a) and (b), respectvely. R-test R-test (a) wth contact-type R-test (b) wth non-contact R-test Fg Expermental setups (Setup 1-a (outer low) n Secton 3.3.2) wth contact-type R-test and wth non-contact R-test. Table Fg Measured sensor postons n the statc R-test measurement cycle by contact-type and non-contact R-test devces. 12

110 Table (a) At B = 5 projected on XZ plane (b) At B = 5 projected on XY plane Table (c) At B = projected on XZ plane (d) At B = projected on XY plane Table (e) At B = -5 projected on XZ plane (f) At B = -5 projected on XY plane Fg Measured sensor postons n the statc R-test measurement cycle by contact-type and non-contact R-test devces, projected on XZ plane and XY plane. 13

111 δy CB mm 3 x non-contact type contact-type (a)δx BY (B) (c)δz BY (B) α BY rad B deg x 1-5 (b)δy CB (B) contact-type non-contact type B deg (d)α BY (B) β BY rad x contact-type 4 2 non-contact type B deg γ BY rad x contact-type -2 non-contact type B deg (e)β BY (B) (f)γ BY (B) Fg. 5-2 Calbrated geometrc errors of the rotary table. (2) Test results Fgure 5-18 shows the graphcal representaton of the measured postons of the R-test sensors nest n the reference coordnate system when B = (see Secton 3.4 for more detals on the representaton scheme). The measured profles at B = 5,, and -5 are projected onto (a) XZ plane and (b) XY 14

112 plane, as plotted n Fg The errors are magnfed 1, tmes (.e. wth an error scale of 1 μm/1 mm). Note that the dfference n the heght (.e. the dstance from the table surface) of two profles by contact-type and non-contact R-test devces n Fgs to 5-19 s due to the dfference n the Z-poston of the sphere n both measurements. Poston-dependent geometrc errors assocated wth the B-axs, or an error map of the rotary table, s calbrated wth the method proposed n Secton 3.6. The calbraton result s shown n Fg See Table 2-4 for the descrpton of each geometrc error parameter. Note that all the three measurement cycles wth the contact-type R-test and wth the non-contact R-test are plotted n Fg. 5-2, to evaluate the repeatablty of measurement. (3) Observaton From the results shown n Fgs to 5-2, the followng observatons could be made: a. Measured poston profles shown n Fg. 5-19, as well as geometrc error estmated n Fg. 5-2, by contact-type and non-contact R-test devces, show a good agreement wthn measurement uncertantes (and the machne s postonng uncertantes). b. Both contact-type and non-contact R-test devces show good repeatablty compared to the machne s postonng repeatablty. c. However, there are stll some dfference n measurement results, partcularly n theδz BY (B) and β BY (B), as shown n Fg. 5-2(c) and (e). The devaton n the δ z BY (B) s 3 μm approxmately,, partcularly at B = 25. the angular devaton n the β BY (B) s about ±2 1-5 rad, partcularly at B = ±75. Ths may be caused by the thermal nfluence on the machne s error motons, although ts exact cause s not fully nvestgated. 15

113 5.7.2 Dynamc measurement wth synchronous moton of rotary axs and lnear axes (1) Test objectve and procedure As mentoned n Secton 5.1, a dynamc measurement s defned as a measurement that s conducted when the machne tool s drven wth a velocty. The objectve of ths test s to expermentally nvestgate the measurement performance of the developed non-contact R-test n a dynamc R-test measurement, as well as the expermental comparson wth the contact-type R-test. The dynamc measurement was conducted wth the synchronous moton of rotary axs (C-axs) and lnear axes (X- and Y-axes). The same machne tool presented n Secton 2.2 was tested. The procedure s shown as follows: a. The C-axs s commanded to rotate wth a constant feedrate at B =. The spndle-sde sphere s commanded to follow the R-test sensors nest on the table, as shown n Fg b. The R-test contnuously measures the relatve dsplacement n the workpece coordnate system. At the same tme, X, Y, and C postons n the CNC system measured by the lnear encoder are logged. c. The dynamc measurement was conducted at two angular velocty of C-axs (358 degree/mn and 3,583 degree/mn n ths case study), to observe the nfluence of the feedrate on dynamc errors. Note that sphere locatons n the workpece coordnate system are: Setup 1-a (outer low) wth the contact-type R-test: w q ct = [.2, -83.9, 4.5] (mm) Setup 1-a (outer low) wth the non-contact R-test: w q nc = [.7, -8.1, -3.] (mm) whch are the same as the statc measurement n Secton

114 XY moton to follow C-rotaton C-rotaton Fg Synchronous moton of C-axs and XY-axes. (2) Test results As mentoned n Secton 2.2, R-test measures the dsplacement of sphere (.e. the TCP) n the workpece coordnate system, relatve to the R-test sensors nest attached on the work table. As shown n Fg. 5-22, the measured R-test dsplacements are decomposed nto the radal, tangental, and axal drectons. Note that, n radal and tangental drectons, the nfluence of the statc center offset of B-axs n X- and Y-drecton (.e. δx CY, δy CY) s numercally elmnated (n the same manner as n Secton 3.4). Fgure 5-23 shows measured dsplacement profles n (a) radal, (b) tangental, and (c) axal drectons, when the angular velocty of C-axs s 358 degree/mn (.e. the feedrate of the crcular nterpolaton wth lnear axes s about 5 mm/mn). Fgure 5-24 shows the measured dsplacement profles, when the angular velocty of C-axs s 3,583 degree/mn (.e. the feedrate of the crcular nterpolaton wth lnear axes s about 5, mm/mn). Tangental component of the measured synchronous error Y Spndle-sde sphere R-test sensors nest X Start C-rotaton Radal component of the measured synchronous error Fg Representaton of the measured synchronous errors wth synchronous moton of C-axs and X-, Y-axes. 17

115 1 R-test No dsplacement 1 R-test No dsplacement Y mm 5 Feedback 2μm/dv Y mm 5 Feedback 2μm/dv Start C-rotaton X mm -1 Start C-rotaton X mm Contact-type R-test Non-contact R-test (a) Radal drecton Y mm Feedback R-test 2μm/dv Start C-rotaton 1 R-test 5-5 No dsplacement -1 No dsplacement X mm X mm Y mm Feedback 2μm/dv Start Contact-type R-test Non-contact R-test (b) Tangental drecton C-rotaton Error n axal drecton mm 4 x R-test Feedback C deg Contact-type R-test Error n axal drecton mm x (c) Axal drecton R-test Feedback C deg Non-contact R-test Fg Measured synchronous errors wth synchronous moton of C-axs and X-, Y-axes (at C-axs angular velocty: 358 degree/mn). 18

116 1 R-test No dsplacement 1 No dsplacement 5 5 Y mm Feedback 2μm/dv Y mm Feedback 2μm/dv -5-1 Start C-rotaton X mm -5 R-test -1 Start X mm Contact-type R-test Non-contact R-test (a) Radal drecton C-rotaton 1 R-test Feedback 1 R-test 5 5 Y mm -5 2μm/dv Y mm -5 2μm/dv Feedback -1 No dsplacement Start C-rotaton -1 No dsplacement C-rotaton Start X mm X mm Contact-type R-test Non-contact R-test (b) Tangental drecton 4 x x 1-3 Error n axal drecton mm 2-2 R-test Feedback C deg Contact-type R-test Error n axal drecton mm (c) Axal drecton R-test Feedback C deg Non-contact R-test Fg Measured synchronous errors wth synchronous moton of C-axs and X-, Y-axes (at C-axs angular velocty: 3,583 degree/mn). 19

117 In Fg. 5-23(a) and Fg. 5-24(a), the radal component of the measured sphere dsplacement s magnfed 2,5 tmes and polar-plotted. R-test represents the radal-drecton trajectory measured by the contact-type (left) or the non-contact (rght) R-test. Feedback represents the same trajectory calculated from X, Y, and C-postons measured by lnear (rotary) encoders. The trajectory Feedback s dentcal n both left and rght plots. The start poston of the C-rotaton and the rotaton drecton s ponted out n the fgures. The crcle wth no dsplacement represents the level where there s no dsplacement. When the radus of the magnfed error plot of R-test s larger than the reference crcle representng zero error, t means the radus of the sphere center trajectory s larger than that of the R-test sensors nest trajectory on the table. In Fg. 5-23(b) and Fg. 5-24(b), the tangental component of the measured dsplacement s magnfed 2,5 tmes and polar-plotted. When the radus of the magnfed error plot of R-test s smaller than the reference crcle representng zero error, t means the spndle-sde sphere center s delayed relatve to the R-test sensors nest on the table. The axal component of the measured dsplacement s plotted along the C-rotaton angle, as shown n Fg. 5-23(c) and Fg. 5-24(c). The axal error n the postve drecton means the sphere center moves to + Z drecton relatve to the R-test sensors nest. (3) Observaton Many observatons can be made from Fgs and Fg. 5-24: a. In the tangental drecton (Fg. 5-23(b) and Fg. 5-24(b)), the measurement result wth the contact-type R-test shows a constant delay n the spndle-sde sphere poston relatve to the R-test sensors nest (about 3μm under 3,58 deg/mn). Moreover, the delay enlarges as the angular velocty of C-axs ncreases (about 6μm under 3,583 deg/mn). However, there s approxmately no constant delay at both veloctes of C-axs wth the non-contact R-test. Ths may show the measurement error by the contact-type R-test devce due to the 11

118 nfluence of the frcton or the dynamcs of sensors themselves. Its exact cause s, however, not clarfed at ths stage. To clarfy t, true synchronous error of C-axs to XY-axes should be measured for comparson by a more relable measurng nstrument of the traceable measurement uncertanty. It s not done at ths stage. b. In the tangental drecton (Fg. 5-23(b) and Fg. 5-24(b)), when the machne tool starts the synchronous moton, a spke-shaped error (about ± 5μm or ± rad under 3,583 deg/mn) could be observed both wth the contact-type R-test and the non-contact R-test. Ths error can be also observed n the feedback data. Ths error s caused by transent synchronous errors of C- and X-axes. c. In all plots (a) to (c) by the non-contact R-test, the nose (or hgh-frequency components of the measured profle) s larger (peak-to-peak 2 μm approxmately). It could be caused by the measurement uncertanty of laser dsplacement sensor due to the speckle nose n laser beam. In profle scannng of a sphere shown n Fg n Secton 5.6.2, smlar nose can be observed. d. Except for the nose, the radal trajectory (Fg. 5-23(a) and Fg. 5-24(a)) and the axal trajectory (Fg. 5-23(c) and Fg (c)) by contact-type and non-contact R-test devces show a good match. 5.8 Concluson (1) All the prevous studes on the R-test n the lterature used contact-type dsplacement sensors wth a flat-ended probe. Ths chapter presented the development of the non-contact R-test devce usng laser dsplacement sensors. A non-contact R-test devce was developed wth the specular reflecton type laser dsplacement sensor (LK-G1) n ths study. (2) The measurement accuracy of four laser dsplacement sensors wth dfferent measurng prncples for profle measurement of a sphere surface was expermentally nvestgated. The performance of the four laser dsplacement sensors n the applcaton to the non-contact R-test was 111

119 studed. (3) A new algorthm was proposed to estmate the three-dmensonal dsplacement of sphere center by usng a non-contact type R-test wth laser dsplacement sensors. It shows that the algorthm should consder the measurement uncertanty caused by the nclnaton of the target surface. (4) The measurng performance of the developed non-contact R-test was nvestgated compared wth the contact-type R-test n the applcaton to error calbraton of an error map of the rotary table n statc measurement, as well as a dynamc measurement of synchronzaton errors of rotary axes and lnear axes. a. Measurement results n the statc measurement wth contact-type and non-contact R-test devces show a good agreement. Both of contact-type and non-contact R-test devces exhbt good repeatablty. b. Both R-test devces exhbted slghtly dfferent result n the dynamc measurement, partcularly n the tangental drecton to the C-axs rotaton (about 3μm under C-axs velocty of 358 deg/mn, about 6μm under 3,583 deg/mn). Possbly, the dynamcs of the contact-type R-test may be an error factor, whle ts exact cause should be clarfed n the future. c. The prototype non-contact R-test devce developed n ths chapter s subjected to hgh-frequency nose of the ampltude about 2 μm due to the speckle nose n laser beam. When a laser dsplacement sensor of dfferent measurng prncple (e.g. the spectral nterference type laser dsplacement sensor, studed n ths chapter) s used, ths nose may be sgnfcantly reduced, although the measurable area may become smaller. 112

120 Chapter 6 Influence of geometrc errors of rotary axes on a machnng test of cone frustum by fve-axs machne tools 6.1 Introducton The prevous chapters (Chapter 3 to 5) focused on the error calbraton method of geometrc errors of rotary axes by R-test. One of nherent dffcultes wth fve-axs machnng s n that t s dffcult to understand how error motons of machne tools are coped as the geometrc error of the machned workpece. For example, n case of three-axs machnng wth X, Y, and Z axes, the squareness error of lnear axes s coped as the squareness error of two edges of the machned workpece. On the other hand, n fve-axs machnng, t s dffcult to ntutvely understand how the squareness error of a rotary axs to a lnear axs, one of locaton errors, s coped onto the machned workpece. It s even more dffcult to understand the nfluence of more complex error motons, parameterzed as poston-dependent geometrc errors n ths thess. As a result, t s dffcult for machne tool bulders or users to understand how mportant t s to calbrate such a complex error moton. If error motons of rotary axes calbrated n prevous chapters (Chapter 3 and 5) do not mpose sgnfcant nfluence on the machnng accuracy, t s of no mportance to calbrate them accurately. For the same reason, t s dffcult to dagnose the cause n machne tools for the geometrc naccuracy of the machned workpece. As a typcal example, a machnng test of cone frustum, descrbed n NAS (Natonal Aerospace Standard) 979 [NAS 979, 1969], s wdely accepted by machne tool bulders to evaluate the machnng performance of fve-axs machne tools. In ths test, even when the geometrc error of the machned test pece exceeds the acceptable tolerance, t s generally very dffcult for machne tool bulders to fnd ts causes, and then to 113

121 fnd where to mprove n the machne confguraton to acheve the target accuracy. Ths chapter dscusses the nfluence of varous error motons of rotary axes on a fve-axs machne tool on the machnng geometrc accuracy of cone frustum machned by ths test. From such an analyss, we can evaluate the mportance of each error moton wth respect to the nfluence on the machnng accuracy n the gven machnng applcaton. Furthermore, such an analyss can be the fundamental for error dagnoss from the geometrc naccuracy of the machned workpece. It must be emphaszed that the cone frustum machnng test s just an example of machnng applcatons. Ths chapter consders the cone frustum machnng test only, as an example of wdely accepted machnng tests n the machne tool ndustry. Analogous error senstvty analyss methodology to be presented n ths chapter can be straghtforwardly appled to any machnng applcatons n general. 6.2 Objectve and orgnal contrbuton of ths chapter NAS 979 [NAS 979, 1969] descrbes the evaluaton of machnng accuracy of a fve-axs machne tool by the machnng of a cone frustum, whch s wdely accepted to many machne tool bulders as a fnal performance test for fve-axs machne tools. Equvalent non-cuttng measurement methods usng a ball bar measurement have also been studed by Ihara et al. [Ihara, 25, Matano, 27], and ts ncluson n the revsed ISO [ISO , 211] s currently under dscusson n ISO/TC39/SC2. A crtcal ssue wth ths cone frustum test s that the nfluence of the machne s error sources on the geometrc accuracy of the machned workpece s very dffcult to understand for machne tool bulders. The nfluence of the machne s poston-ndependent geometrc errors (locaton errors) on the geometrc error of the machned cone frustum test pece was dscussed n [Uddn, 29]. Matsushta et al. [Matsushta, 28] and Yumza et al. [Yumza, 27] presented smlar analyss to dscuss how ther nfluences 114

122 are correlated to the locaton and the orentaton of test pece. As s clear from these error analyses, a part of poston-ndependent geometrc errors mposes a sgnfcant nfluence on the crcularty of the machned test pece. From our experences, however, on the latest commercal small-szed fve-axs machne tools, the crcularty error can be typcally as small as fve to ten mcrometers. It suggests that poston-ndependent geometrc errors on such a machne are tuned suffcently small. To further mprove the crcularty of the machned test pece on such a machne, more complex error motons of a rotary axs, such as the gravty deformaton, angular postonng error of a rotary axs, pure radal error motons or tlt error motons of a rotary axs, must be reduced. Such more complex error motons can be modeled as poston-dependent geometrc error, as presented n Secton 2.2. To our knowledge, no work n the lterature extended the analyss to poston-dependent geometrc error. The objectve of ths paper s to present a numercal analyss of the nfluence of major error motons on the crcularty error of the machned cone frustum test pece. Error motons that have relatvely larger nfluence on crcularty, and those that have neglgbly small nfluence, are found out. Based on the present dscusson, expermental case studes are presented to demonstrate the error dagnoss on a cone frustum machnng test. It must be emphaszed that the man contrbuton of ths chapter s on the proposal of the analyss methodology of the senstvty of poston-dependent geometrc errors on the machnng geometrc accuracy. It can be appled bascally to any machnng applcatons. 6.3 Setup of cone frustum machnng test Fgure 6-1 shows machnng confguraton and parameters of tlted cone frustum to be consdered n ths paper. D, φ, ψ are defned as dameter of tool path, tlted angle of cone frustum about Y-axs n the workpece coordnate system and half-apex angle of the cone frustum, respectvely. (C x, C y, C z ) s the center locaton of tool tp trajectory n the workpece coordnate system. The orgn of the workpece coordnate system s defned at the ntersecton of 115

123 nomnal B-axs and C-axs. For smplcty of computaton, ths secton smulates a tool center trajectory whch can be nterpreted as a geometrc profle of the machned workpece surface when the tool radus s zero. Table 6-1 shows the condtons for cone frustum machnng test used n smulatons presented n Sectons 6.4 and 6.5. The smulaton wll be conducted under two condtons: (a) φ=15, ψ=3 and (b) φ=75, ψ=3. The machne confguraton shown n Fg. 2-5 s assumed. The command trajectory of each axs n each case s shown n Fg The algorthm to calculate the command trajectory X (k), Y (k), Z (k), B (k), and C (k) (k = 1,,N) can be found n e.g. [Uddn, 29]. Z Y X Tool D Tool path along cone frustum sde surface Z Tool Y X Bottom Cone frustum Top surface ψ Tool tp locaton n CLdata Base cylnder Inclned jg D φ Center locaton of tool tp trajectory (C x, C y, C z ) Fg. 6-1 Setup for machnng test of cone frustum. Table 6-1 Test condtons for the machnng test of cone frustum Parameter Value Dameter of tool path, D (mm) Tlt angle, φ( ) case (a): 15 case (b): 75 Half-apex angle, ψ ( ) 3 Center locaton of tool path (-81.8,,189.3) (C x, C y, C z ) (mm) 116

124 (a) φ<ψ (φ=15, ψ=3 ) (b) φ>ψ (φ=75, ψ=3 ) Fg. 6-2 Command trajectory of each axs (assumng feedrate 1, mm/mn). It s to be noted that Ihara and Tanaka [Ihara, 25] showed that command trajectores for cone frustum machnng can be categorzed nto two groups. When φ<ψ, C-axs rotates for 36, whle B-axs rotates for 2φ, as s shown n Fg. 6-2 (a). However when φ>ψ, B-axs rotates for 2ψ, whle C-axs does not rotate for 36, as s shown n Fg. 6-2 (b). Although the followng secton shows only two smulaton results for representatve cases φ<ψ (φ=15, ψ=3 ) and φ>ψ (φ=75, ψ=3 ), the dscusson n [Ihara, 25] ndcates that the nfluence of each error moton on the contourng accuracy can be also qualtatvely categorzed nto ether case (φ<ψ or φ>ψ). 6.4 Influence of poston-ndependent geometrc errors of rotary axes (1) Analyss objectve and procedure Ths subsecton frst dscusses the quanttatve nfluence of poston-ndependent geometrc errors, or locaton errors shown n Table 2-3 n Secton , on the crcularty of the machned cone frustum test pece. Although such a senstvty analyss of poston-ndependent geometrc errors can be found n prevous studes n the lterature [Uddn, 29, Matsushta, 28, 117

125 Yumza, 27], ths subsecton brefly revews t, snce t s the fundamental for the analyss to be presented n Secton 6.5. Note that for the defntons and symbols of geometrc errors, ncludng poston-ndependent geometrc errors (n Table 2-3), and poston-dependent geometrc errors (n Table 2-4), a reader should refer to Secton When each poston-ndependent geometrc error of rotary axes n Table 2-3 s set ether.1mm for lnear errors or.1 for angular errors, the tool poston n the workpece coordnate system, w q 3 R, s calculated by usng the knematc model (wth Eq. (2-11) n Secton 2.3.2), for the gven command poston (X (k), Y (k), Z (k), B (k), and C (k)). By repeatng ths smulaton at each poston n command trajectores n Fg. 6-2, a contour error trajectory n the workpece coordnate system s computed. (2) Analyss result A smulated error trajectory for each poston-ndependent geometrc error s shown n Fg The crcularty error, or the crcular devaton G (defned n ISO 23-4 [ISO 23-4, 25]), defned as the dfference between maxmum and mnmum radal errors, s computed from the smulated error trajectory. Note that the center of smulated error trajectory s set to the MRS center (mnmum radal separaton center) [ISO 23-4, 25], where the smallest crcularty error s obtaned. The smulated senstvty of crcularty error to each poston-ndependent geometrc error s summarzed n Table 6-2. From the smulaton results shown n Fg. 6-3, t can be observed that: 1) the nfluence of δx BY, δx CB, and β BY on the geometry of the contour error trajectory s qualtatvely smlar, ether of whch affect the contour error trajectory n X-drecton (or n Y-drecton). Note that the ampltude of dstorton depends on the value of the assumed error; 2) the nfluence of δy BY, α CB, and α BY s qualtatvely smlar, ether of whch affects the contour error trajectory to 45 drecton to the X-axs; 3) both δz BY and γ BY do not affect the contour error trajectory of cone frustum at all. Therefore, t s generally not possble to conduct 118

126 error dagnoss of each poston-ndependent geometrc error of rotary axes by a sngle cone frustum machnng test. (a) φ<ψ (φ=15, ψ=3 ) (b) φ>ψ (φ=75, ψ=3 ) Fg. 6-3 Influence of each poston-ndependent geometrc error on smulated contour error profle. 119

127 Table 6-2 Crcularty error smulated wth each Contrbutor poston-ndependent geometrc error Crcularty error ( µm ) φ<ψ (φ=15, ψ=3 ) φ>ψ (φ=75, ψ=3 ) α BY β BY γ BY α CB δx BY δy BY δz BY δx CB Influence of poston-dependent geometrc errors of rotary axes Analyss objectve and basc methodology Ths subsecton further extends the analyss to more complex error motons of a rotary axs, descrbed by poston-dependent geometrc errors. The objectve of ths subsecton s to present the senstvty of the crcularty of cone frustum workpece to each error moton under two condtons (n Table 6-1). The basc analyss methodology s as follows: as an llustratng example, consder the problem to nvestgate the nfluence of B-axs angular postonng error on the crcularty of the machned cone frustum workpece (see Secton 6.5.2). A profle of B-axs angular postonng error, as a functon of the B angular poston (B ), can be arbtrary. It s not possble to study the nfluence of an nfnte number of B-axs angular postonng error profles. Instead, we assume a typcal functon for ths trajectory. For example, the B-axs angular postonng error s typcally a lnear functon of the B-angular poston, caused by e.g. homogenous expanson of a lnear encoder. The gradent of ths functon may dffer at dfferent regon, due to e.g. a calbraton error of a lnear encoder. By consderng typcal error profles found n actual machne tools, we model a profle of B-axs angular postonng error as the superposton of multple frst-order functons of B-angular poston. Ther gradent, ampltude and offset are parameters to be specfed. 12

128 Then, the nfluence of ths B-axs angular postonng error profle on the crcularty of the cone frustum workpece s smulated by usng the knematc model (wth Eq. (2-11) n Secton 2.3.2), for the gven command poston (X (k), Y (k), Z (k), B (k), and C (k)). The nfluence of the B-angular postonng error on the crcularty of the cone frustum workpece s studed by usng the Monte Carlo smulaton. In other words, parameters of the geometrc error (the gradent, the ampltude, and the offset of the profle n ths example) are randomly gven wth specfed mean and standard devaton, and the workpece crcularty s smulated n each case. The senstvty of nput geometrc errors to statstcal mean and standard devaton of smulated crcularty s studed. Such a statstcal analyss based on the Monte Carlo smulaton s common and well establshed n the measurement uncertanty analyss to evaluate the nfluence of each uncertanty contrbutor on the overall measurement uncertanty [JCGM 1, 28]. For example, n [Brngmann, 29], the nfluence of error motons of lnear axes on the estmaton of locaton errors was studed by usng the Monte Carlo smulaton. Smlarly as n ths subsecton, error motons of lnear axes are modeled by typcal functons. It can be seen analogous to the analyss procedure presented n ths chapter. Ths subsecton consders the followng error motons of rotary axes, whch are potentally common errors observed n commercal fve-axs machne tools Angular postonng error of B-axs As was dscussed n the prevous subsecton, we assume that the B-axs has an angular postonng error profle shown n Fg. 6-4(a) wth arbtrary a 1, a 2 and b. Note that a =mean (β BY (B )) and a 1 represents the gradent of the least square ft lne for β BY (B). Such an error profle can be decomposed nto three components as s shown n Fg. 6-4(b)-(d) (.e. β BY (B ) = β BY + β 1 BY(B ) + β 2 BY(B )). Each poston-dependent component s formulated as follows: a B B β (6-1) 1 1 max BY ( B ) = + a1 2 Bmax Bmn 121

129 a2 B b + a2 B ( Bmn, b) 2 2 b Bmn β BY ( B ) = (6-2) a2 B Bmax + a 2 B ( b, Bmax) 2 Bmax b where a 1 and a 2 are gven by N (.1,.33 )sgn, where N (μ, σ) represents a normally dstrbuted random number wth the mean value μ and the standard devaton (abbrevated by std) σ. sgn s ether +1 or -1 wth 5% possblty. B mn and B max are respectvely set to be mnmum or maxmum of rotaton angle of the B-axs. For example, for case (a) (φ=15, ψ=3 ), B mn and B max are set to be -45 and -15, respectvely, because the command B-angle s between -45 and -15 as shown n Fg 6-2(a). b s the rotaton angle where the gradent of modeled error profle (β 2 BY(B)) changes, whch s gven by b=u (B mn, B max ), where U (m, n) represents a unformly dstrbuted random number on the nterval (m, n). β 1 BY(B) refers to frst-order component of the consdered angular postonng error profle of B-axs, as shown n Fg. 6-4(c). β 2 BY(B) refers to the ramp-type component as shown n Fg. 6-4(d). The mean and the dstrbuton of nput random parameters (e.g. μ and σ for a 1 and a 2 ) should be selected accordng to typcal level of postonng accuracy n commercal machne tools. Although t s n practce dffcult for us to select typcal level of moton errors, t should be noted that the major contrbuton of the present analyss s to fnd out the senstvty, or the rato, of nput moton errors to the output crcularty, not absolute value of the crcularty. It must be emphaszed that nvestgatng the nfluence of a perodc error of hgher frequency (e.g. a perodc error caused by worm gear or a bearng) s not n man scope of ths secton, snce the nfluence of such an error can be understood relatvely easly. Ths secton studes the nfluence of lower-frequency error profles as represented n Fg. 6-4 as a potentally crtcal error factor n typcal fve-axs machnes. The nfluence of poston-ndependent component, namely β BY, was studed n Secton 6.3 and summarzed n Table 6-2. For β 1 BY(B), total 1, smulatons to compute geometrc error profle of the machned cone frustum workpece are performed wth randomly gven a 1. Smlarly, the nfluence of 122

130 β 2 BY(B) s studed wth 1, smulatons. The mean value and standard devaton ( 1σ ) of smulated crcularty errors s calculated. Table 6-3 shows the senstvty of the crcularty of cone frustum workpece to angular postonng errors of B-axs. For φ<ψ (φ=15, ψ=3 ), t can be observed that the nfluence of β 1 BY(B) s relatvely small, snce such a monotonously ncreasng (decreasng) postonng error mostly changes the radus and the center poston of error trajectory, and thus has small nfluence on the crcularty. On the other hand, the crcularty s sgnfcantly more senstve to β 2 BY(B) Axal error moton of B-axs An axal poston error of B-axs, typcally caused by geometrc profle error of a bearng, s analogously modeled by δy 1 BY(B) and δy 2 BY(B) as n Eq. (6-1) and Eq. (6-2), whle a 1 and a 2 are gven by N (1 µm, 3.3 µm)sgn. The results are also shown n Table 6-3. Table 6-3 Influence of angular postonng error of B-axs, axal error moton of B-axs and lnear error moton of B-axs to Z- drecton Contrbutor Angular postonng error of B-axs Axal error moton of B-axs Lnear error moton of B-axs to Z- drecton Parameter Crcularty error ( µm ) settng φ=15, ψ=3 φ=75, ψ=3 (a 1 and a 2 ) mean std mean std β 1 BY(B) N (.1, β 2 BY(B).33 )sgn δy 1 BY(B) δy 2 BY(B) N (1 µm, µm)sgn δz 1 BY(B) N (1 µm, 3.3 δz 2 BY(B) µm)sgn 123

131 βby(b) a 2 a a 1 Βmn b Βmax Β (a) Total angular postonng error profle of B-axs ( β BY (B) ) β BY a Βmn Βmax Β (b) Poston-ndependent component ( β BY(B) ) β 1 BY(B) a 1 Βmn Βmax Β (c) Frst-order component ( β 1 BY(B) ) β 2 BY(B) a2 Βmn b Βmax Β (d) Ramp-type component ( β 2 BY(B) ) Fg. 6-4 Angular postonng error profle of B-axs consdered n senstvty study. 124

132 6.5.4 Lnear error moton of B-axs to Z-drecton A lnear error moton of B-axs to Z-drecton, typcally caused by the gravty-nduced deformaton of a rotary unt, s modeled as δz 1 BY(B) and δz 2 BY(B) n the same way wth δy BY (B) n Secton Snce lnear error moton of B-axs to Z-drecton ( δz BY (B) ) s at the non-senstve drecton, the nfluence of δz BY (B) s neglgbly small for both setups, as the smulaton results shown n Table Angular postonng error of C- axs The angular postonng error of C-axs γ CB (C) s assumed to be perodc for 36 rotaton of C-axs. Therefore, an angular postonng error proportonal to C angular poston, wth a dfferent gradent before and after the gven c, s consdered. Analogous to error shown n Fg. 6-4 (d), t s modeled as follows: a2 C c + a2 C ( Cmn, c) 2 2 c Cmn γ CB( C ) = (6-3) a2 C Cmax + a 2 C ( c, Cmax) 2 Cmax c where c s gven by c=u (C mn, C max ), a 2 s gven by a 2 =N (.1,.33 )sgn. C mn and C max are respectvely set to be mnmum or maxmum of rotaton angle C. Smlar smulaton process s conducted as n Secton 6.5.2, and results are shown n Table 6-4. Table 6-4 Influence of angular postonng error, axal error moton of C-axs Contrbutor Angular postonng error of C-axs Axal error moton of C-axs γ 2 CB(C) δz 2 CB(C) Parameter Crcularty error ( µm ) settng φ=15, ψ=3 φ=75, ψ=3 (a 2 ) mean std mean std N (.1,.33 )sgn N (1 µm, 3.3 µm)sgn

133 6.5.6 Axal error moton of C-axs An axal error moton of C-axs, typcally caused by geometrc profle error of bearng, s modeled by δz 2 CB(C) as n Eq. (6-3), whle a 2 s gven by a 2 =N (1 µm, 3.3 µm)sgn. The smulaton results are shown n Table Perodc pure radal error moton of B- and C-axs Fgure 2-7(b) n Secton (4) shows perodc pure radal error moton of C-axs, or run-out of C-axs, whch s modeled as follows: δx δy CB CB ( C ) = a cos( C ( C ) = a sn( C ζ ) + ζ ) Perodc pure radal error moton of B-axs s modeled as follows: (6-4) δx δz BY BY ( B ) = a cos( B ( B ) = a sn( B ζ ) + ζ ) (6-5) where a 2 and ζ are gven by: a 2 =N(1 μm, 3.3 μm), ζ= U(, 36 ). The smulaton results are shown n Table 6-5. From the smulaton results, the nfluence of pure radal error moton of B- and C-axs on crcularty s suffcently small compared to gven errors. Table 6-5 Influence of perodc pure radal error moton and concal tlt error moton of B- and C-axs Contrbutor Perodc pure radal error moton of C-axs Perodc pure radal error moton of B-axs Perodc concal tlt error moton of C-axs Perodc concal tlt error moton of B-axs δx CB (C) and δy CB (C) δx BY (B) and δz BY (B) α CB (C) and β CB (C) α BY (B) and γ BY (B) Parameter Crcularty error ( µm ) settng φ=15, ψ=3 φ=75, ψ=3 (a 2 ) mean std mean std N (1 µm, 3.3 µm) N (.1,.33 )

134 6.5.8 Perodc concal tlt error moton of B- and C-axs As s shown n Fg. 2-7(c) n Secton (4), perodc concal tlt error moton of C-axs s modeled as follows: α β CB CB ( C ) = a cos( C ( C ) = a sn( C ζ ) + ζ ) (6-6) Smlarly, perodc concal tlt error moton of B-axs s modeled as follows: α β BY BY ( B ) = a cos( B ( B ) = a sn( B ζ ) + ζ ) (6-7) where a 2 and ζ are gven by: a 2 =N(.1,.33 ), ζ= U(, 36 ). The smulaton results are shown n Table 6-5. The nfluence of perodc concal tlt error moton of C-axs on crcularty s suffcently small compared to gven errors. However, perodc concal tlt error moton of B-axs have sgnfcantly larger nfluence, especally when φ>ψ (φ=75, ψ=3 ) Change n poston and orentaton of C-axs centerlne dependng on B-axs rotaton Due to the gravty-nduced deformaton of B-axs, axs average lne of C-axs could be tlted or drfted n radal drecton when B rotates. The change n poston and orentaton of C-axs centerlne dependng on B-axs rotaton are respectvely modeled as follows: δx δy CB CB ( B ) = a 1 ( B ) = a 1 B cos( ζ )( B B sn( ζ )( B max max B B B B mn mn mn mn ) ) (6-8) α β CB CB ( B ) = a 1 ( B ) = a 1 B cos( ζ )( B B sn( ζ )( B max max B B B B mn mn mn mn ) (6-9) ) where a 1 s gven by a 1 =N(1 μm, 3.3 μm) for Eq. (6-8), and a 1 =N(.1,.33 ) for Eq. (6-9), respectvely. ζ s gven by ζ= U(, 36 ). Ths model 127

135 assumes error motons of C-axs ncreased proportonally as B rotates from B mn to B max, as s shown n Fg Note that B mn and B max are respectvely set to be mnmum and maxmum of absolute value of B-axs rotaton angle. In our smulaton, for case (a) ( φ=15, ψ=3 ), B mn and B max are set to be 15 and 45, respectvely. The smulaton results are shown n Table 6-6. δycb(b) a B= B max ζ B=B δx CB(B) B= B mn Fg. 6-5 Modelng of change n poston of C-axs dependng on B-axs rotaton ( δx CB (B) and δy CB (B) ). Table 6-6 Influence of change n poston and orentaton of C-axs dependng on B-axs rotaton Contrbutor Change n poston of C-axs dependng on B-axs Change n orentaton of C-axs dependng on B-axs δx CB (B) and δy CB (B) α CB (B) and β CB (B) Parameter Crcularty error ( µm ) settng φ=15, ψ=3 φ=75, ψ=3 (a 2 ) mean std mean sd N (1 µm, 3.3 µm) N (.1,.33 ) Perodc pure radal error moton and tlt error moton of C-axs dependng on B-axs rotaton The smulaton results conducted n Sectons and show that perodc pure radal error moton and tlt error moton of C-axs have neglgbly small nfluence on crcularty. Ths secton consders the case where these error motons of C-axs ncrease as B rotates from B mn to B max, as s shown n Fg. 128

136 6-6. Ths error, caused typcally by the gravty-nduced deformaton, s often the case n practce (see Secton 6.6). Perodc pure radal error moton of C-axs dependng on B-axs rotaton s modeled as follows: δx δy CB CB ( C, B ) = a 2 ( C, B ) = a 2 B B cos( C + ζ )( B B max B B sn( C + ζ )( B B where a 2 and ζ are gven by a 2 =N(1 μm, 3.3 μm) and ζ= U(, 36 ). max mn mn mn mn ) ) (6-1) δycb(c,b) ζ a δx CB(C,B) B= B mn B=B B= B max Fg. 6-6 Modelng of radal error moton of C-axs dependng on B-axs rotaton ( δx CB (B,C) and δy CB (B,C) ). Table 6-7 Influence of perodc pure radal error moton and tlt error moton of C-axs dependng on B-axs rotaton Contrbutor Perodc pure radal error moton of C-axs dependng on B-axs Perodc tlt error moton of C-axs dependng on B-axs δx CB (B,C) and δy CB (B,C) α CB (B,C) and β CB (B,C) Parameter Crcularty error ( µm ) settng φ=15, ψ=3 φ=75, ψ=3 (a 2 ) mean standard mean standard devaton devaton N (1 µm, µm) N (.1,.33 ) Perodc tlt error moton of C-axs dependng on B-axs rotaton, s smulated n the same way as n Eq. (6-1), whle a 2 s gven by N(.1, 129

137 .33 ). The smulaton results are shown n Table 6-7. Compared wth Sectons and 6.5.8, crcularty error of cone frustum s more senstve to perodc pure radal error moton and tlt error moton of C-axs when they ncrease wth B rotaton Summary of analyss Tables 6-3 to 6-7 show the senstvty of the crcularty of cone frustum test pece, to poston-dependent geometrc errors, representng typcal low-frequency error motons of a rotary axs. Many observatons can be made from these smulaton results. For example, some poston-dependent geometrc errors such as lnear error moton of B-axs to Z drecton, perodc pure radal error moton and concal tlt error moton of C-axs, have a neglgbly small nfluence on crcularty of cone frustum. However, perodc pure radal error moton and tlt error moton of C-axs enlarged wth B rotaton could be crtcal factors for cone frustum machnng test. It has been shown n Secton 6.3 or [Matsushta, 28] that a center offset of a rotary axs, one of poston-ndependent geometrc errors, has a sgnfcant nfluence on the crcularty error of machned workpece. In typcal machne setup by an operator, the locaton of C-axs rotaton center lne s measured wth B =. When the C-axs rotaton center lne s moved as the B-axs rotates from B =, typcally due to the gravty nfluence, the present smulaton shows that such an error may cause a large crcularty error. Analogous observaton can be made for pure radal and tlt error motons of C-axs. 6.6 Expermental case study Objectve Based on the error analyss presented n prevous sectons, ths secton demonstrates an expermental case study to fnd out a crtcal error factor n a cone frustum machnng test. 13

138 DBB Fg. 6-7 Setup of the ball bar measurement. Z Y X + C B + Fg. 6-8 Expermental fve-axs machne tool. Instead of actual machnng test, a contourng error profle was measured wth the same CL (cutter locaton) trajectory as n a cone frustum machnng test by a ball bar measurement [Ihara, 25]. Fg. 6-7 shows the setup of ball bar measurement. The confguraton of the expermental fve-axs machne tool s shown n Fg Note that the expermental machne tool studed here s dfferent from the one used n Chapter 3 to 5. Test condtons are summarzed n Table 6-1. Only the case wth the tlt angle φ=15 and the half-apex angle ψ=3 was tested. The ball bar nomnal length s 15 mm, and the feedrate n the workpece coordnate system s 1, mm/mn. Fg. 6-9 shows contour error profles for the cone frustum CL trajectory measured by ball bar measurement. 131