Brigid Mullany, Ph.D University of North Carolina, Charlotte

Transcription

1 Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte October 2007

2 Table of Contents Page Introducton 3 Standards relevant to the study 4 ISO 230-2: ISO 230-2: ISO 230-2: VDI / DQG ANSI B JIS Parameters to be reported. 12 Identcal and smlar parameters ISO 230-2:2006 & VDI / DQG ISO 230-2:2006 & ISO 230-2: ISO 230-2:1988 & VDI / DQG VDI / DQG 3441 & JIS B ISO 230-2:2006 & JIS B Numercal analyss Appendx A UNC Charlotte Mullany 2

3 Introducton Machne tool postonal accuracy and repeatablty are core descrptors of a machne tool and ndcate the machne s expected level of performance. Whle a number of standards and gudelnes exst outlnng how to evaluate machne tool postonal accuracy and repeatablty, they dffer n ther analyss procedures and n key parameter defnton. As a result the values reported for postonal accuracy and repeatablty for any one machne can vary dependng on whch standard was used. As all standards are equally vald t s benefcal to be aware how the standards dffer from each other and how the dfferent calculated values compare to each other. Ths document ams to do ths. The ams of ths report can be explctly broken down nto the followng: o Lst commonly used nternatonal or natonal standards that are related to machne tool postonal accuracy and repeatablty. o Determne whch standards are equvalent to each other and subsequently solate the core standards. o Compare the key parameters from each standard and dentfy dentcal and conceptually smlar parameters. o Perform numercal analyss to evaluate how smlar parameters compare to each other under dfferent condtons and determne f converson factors exst allowng parameters from dfferent standards to be drectly compared. Credt to revewers: The author would lke to thank the followng for ther valuable comments and nsghts: o Alkan Donmez, NIST, MD. o Wolfgang Knapp, Engneerng Offce Dr. W. Knapp, Swtzerland. o Scott Smth and Bob Hocken, UNC Charlotte, NC. UNC Charlotte Mullany 3

4 Standards relevant to the study The core standards ncluded n ths study are lsted n table 1. These standards prmarly deal wth determnng the postonal repeatablty and accuracy of a machne tool. Standards specfcally focused on the geometrc accuracy of the machne tool are not ncluded n the study. Table 1: Core standards under nvestgaton. Name Ttle Comment ISO 230-2:2006 Determnaton of accuracy and repeatablty of postonng numercally controlled machne axes ISO 230-2:1997 Determnaton of accuracy and repeatablty of postonng numercally controlled machne axes ISO 230-2:1988 Determnaton of accuracy and repeatablty of postonng of numercally controlled machne tools England BS ISO (1999) German VDI/DGQ 3441 DIN ISO (2000) USA ASME B5.54(2005) Chna GB/T (2000) Determnaton of accuracy and repeatablty of postonng numercally controlled machne axes Statstcal testng of the operatonal and postonal accuracy of machne tools Determnaton of accuracy and repeatablty of postonng numercally controlled machne axes Methods for performance evaluaton of computer numercally controlled machnng centers Determnaton of accuracy and repeatablty of postonng numercally controlled machne axes Japan JIS B 6192:1999 Determnaton of accuracy and repeatablty of postonng numercally controlled machne axes JIS B Test code for performance and accuracy of numercally controlled machne tools Ths replaced the ISO 230-2:1997 verson of the standard. Detals of the changes are gven later. Ths was replaced by ISO 230-2:1997. Dfferences between the two standards are gven later. Equvalent to ISO 230-2:1997 Equvalent to ISO 230-2:1997 Data analyss for machne tool accuracy s equvalent to ISO 230-2:1997 Equvalent to ISO 230-2:1997 Equvalent to ISO 230-2:1997 Ths standard s wthdrawn however detals on the analyss technques wll be gven. UNC Charlotte Mullany 4

5 Key Standards The followng standards wll be looked at n detal; o ISO 230-2:2006 o ISO 230-2:1997 o ISO 230-2:1988 o VDI/DQG 3441 o ANSI B5:54 o JIS B Whle detals of each standard are not gven, specfcs wth respect to the hstory of the standard are provded,.e. whch standard t replaced, whch standards replaced t, whch standards are based on the standard etc. NOTE: Ths document s not a substtute for readng the ndvdual standards. The actual standards should be used when undertakng any of the outlned tests or n determnng the postonal accuracy of a machne tool. UNC Charlotte Mullany 5

6 ISO 230-2: 2006 Test Code for Machne tools - Part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled axes. Older Versons : o ISO 230-2:1997 o ISO 230-2:1988 Other nternatonal standards based on ISO 230-2:2006 standard - NONE Scope of the standard: To specfy the methods of testng and evaluatng the accuracy and repeatablty of postonng of NC machne tools and components by drect measurement of ndependent axes on the machne. Used for type testng, acceptance testng, comparson testng, perodc verfcaton, machne compensaton. Dfferences between ISO 230-2:2006 and ISO 230-2:1997: o A measurement uncertanty statement s added to the 2006 verson and now the measurement uncertanty should be ncluded when reportng the key parameters. An annex s provded wth the standard detalng how to determne the measurement uncertanty. UNC Charlotte Mullany 6

7 ISO 230-2: 1997 Test Code for Machne tools - Part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled axes. Older Versons : o ISO 230-2:1988 Other nternatonal standards based on the ISO 230-2:1997 standard o GB/T (2000) (Chna) Test code for machne tools-part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled axes. Equvalent to ISO 230-2:1997 o JIS B 6192 (1999) (Japan) Test code for machne tools-part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled axes. Equvalent to ISO 230-2:1997 o BS ISO (1999) (England) Test code for machne tools-part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled axes. Equvalent to ISO 230-2:1997 o DIN ISO (2000) (Germany) Test code for machne tools-part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled axes. Equvalent to ISO 230-2:1997 Scope of the standard: To specfy the methods of testng and evaluatng the accuracy and repeatablty of postonng of NC machne tools and components by drect measurement of ndependent axes on the machne. Used for type testng, acceptance testng, comparson testng, perodc verfcaton, machne compensaton. Dfferences between ISO 230-2:1997 and ISO 230-2:1988: 1. The 1997 verson changed some of the termnology. The term standard uncertanty s used nstead of standard devaton to avod makng assumptons wth respect to the dstrbuton of the measured data. 2. The 1997 verson uses an expanded uncertanty coverage factor of two (k=2) nstead of three (k=3). 3. Calculaton of the bdrectonal systematc postonal devaton of an axs, E, s added to correlate to the Accuracy term n the ANSI B5.54 (1992). 4. Calculaton of the range of the bdrectonal postonal devaton range, M s added and t s equvalent to the Postonal Devaton term, P a, as descrbed n the VDI/DGQ UNC Charlotte Mullany 7

8 ISO 230-2:1988 Acceptance Code for Machne tools - Part 2: Determnaton of accuracy and repeatablty of postonng numercally controlled machne tools. Older Versons : NONE Other nternatonal standards based on the ISO 230-2:1988 standard: -NONE Scope of the standard: To specfy the methods of testng and evaluatng the accuracy and repeatablty of postonng of NC machne tools and components by drect measurement of ndependent axes on the machne. UNC Charlotte Mullany 8

9 VDI/ DQG 3441 Statstcal testng of the operatonal and postonal accuracy of machne tools Translated from the German ssue 3/1977 Older Versons: -NONE Other nternatonal standards based on the VDI/DGQ 3441 standard -NONE Scope of the gudelnes: The gudelne descrbes how statstcal methods can be appled to machnes that are and are not ted to a partcular part to determne operatonal or postonal accuracy of a machne. The standard s n two sectons, one secton focuses on operatonal uncertanty, ths s measured by determnng how accurately a machne can manufacture a defned workpece. The second secton detals how the postonal accuracy of the machne can be determned from drect length measurement,.e. under unloaded condtons. It s ths latter secton of the gudelnes that s referred to n ths document. UNC Charlotte Mullany 9

10 ASME B Methods for performance evaluaton of computer numercally controlled machnng centers Older Versons: ASME B Other nternatonal standards based on ASME B standard - NONE Other standards that reference the ASME B standard - NONE Dfferences between ASME B and ASME B : 1. Changes were made to brng consstency to termnology used between ths standard and ASME B methods for performance evaluaton of computer numercally controlled lathes and turnng machnes. Scope of the standard: Ths standard s very comprehensve and ncludes methodologes to specfy machne tool geometrc parameters, postonal accuracy and repeatablty. It also ncludes nformaton wth respect to envronmental condtons and thermal uncertantes. The secton on postonal accuracy and repeatablty s very smlar n approach to ISO 230-2:2006 however t ncludes a secton on perodc error (short wavelength perodc errors). UNC Charlotte Mullany 10

11 JIS B Test code for performance and accuracy of numercally controlled machne tools Ths standard was wthdrawn n 1987 Accordng to the JSA webpage t was replaced by JIS B The JIS B 6201 was frst mplemented n Ths standard has been revsed and reaffrmed several tmes n the past. The last revson was n 1993 and ths has been reaffrmed n 1998 and The JIS B standard does not explctly outlne tests to measure the machne tool postonal accuracy and repeatablty, perhaps earler versons dd. The JIS B whch was establshed n 1999 does however outlne how to measure machne tool postonal accuracy and the analyss secton of the standard s as per 1SO 230-2:1997. Scope of the standard: The standard outlnes a method for determnng the postonal accuracy and repeatablty of a machne tool UNC Charlotte Mullany 11

12 Key Parameters recommended for reportng by the dfferent standards. Table 2: Parameters recommended for reportng. ISO 230-2: 2006 A 06 & uncertanty (k=2) A 06 and A 06 E 06 & uncertanty (k=2) E 06 and E 06 M 06 & uncertanty (k=2) R 06 & uncertanty (k=2) R 06 and R 06 B 06 & uncertanty (k=2) ISO 230-2: 1997 ISO 230-2: 1988 VDI/DQG 3441 ANSI B JIS B A 06 A 88 P a Postonng A 06 accuracy test (P s ) A 06 and R 88 P smax A 06 and Repeatablty A 06 A 06 test (R s ) E 06 R 88 and P E s 06 Lost moton R 88 test (U s ) E 06 and B 88 U max E 06 and E 06 Least nput E 06 ncrement test M 06 U M 06 R 06 R 06 R 06 and R 06 R 06 and R 06 B 06 B 06 B 06 B 06 B 06 P 06 Appendx A gves a full lst of the nomenclature for each of the standards. Note 1: Whle the same parameter notaton s used n ISO 230-2:1988, 1997 and 2006 the actual mathematcal equatons may vary therefore a two dgt subscrpt (.e. 06 or 88 ) s used to denote the year of the standard beng referred to. The equatons and notaton used n ISO 230-2:1997 and ASME B54.5 are dentcal to those used n ISO 230-2:2006 and therefore the 06 subscrpt s used when referrng to ISO 230-2:1997 and ASME B54.4 parameters. Note 2: As no offcal abbrevatons are gven n the JIS 6330 Standard for the dfferent parameters, names have been assgned n ths report,.e. P s, U s and R s. UNC Charlotte Mullany 12

13 Comparson between the ISO 230-2:2006 and the VDI/DQG 3441 Table 3: Identcal and smlar parameters n the ISO 230-2:2006 and VDI/DQG ISO 230-2:2006 VDI/DQG 3441 Identcal or smlar Mean b-drectonal postonal devaton of an axs, M 06 M 06 = max x [ ] mn[ ] Mean Reversal value of an axs B 06 B 06 Where; B 1 m = B m = 1 = x x x Reversal value of an axs B 06 = max [ ] B B-drectonal accuracy of postonng of an axs, A 06 A 06 = max mn x [ x + 2s ;x + 2s ] [ 2 2 ] s ;x Undrectonal repeatablty of postonng of an axs, R 06 or R 06 R 06 R 06 = max = max [ 4s ] [ 4s ] s Postonal devaton, P a = x max x mn P a Mean Reversal Error, U 1 m U = U m = 1 Where; U = x x Max reversal error at a poston U max = max[ U ] Postonal Uncertanty, P 1 P = x + ( U + P s ) 2 Max 1 x 2 ( U + ) P s Mn Max Postonal Scatter, P s max P s max = P s max = max[6s ] Repeatablty of an Axs, R 06 No equvalent R max[ R ] 06 = parameter See Appendx A for detals on x, x, s and s Identcal - The dfference between the maxmum and mnmum averaged postonal devaton over the forward and reverse drectons. Smlar - Average reversal error. Due to slght dfferences n the equatons the values may vary. Ths s especally true f the averaged forward and reverse postonal errors lnes ntersect each other as n fgure 4. Identcal - Maxmum reversal error. Smlar - Maxmum range of values based on mean postonal errors, correspondng standard devatons and reversal errors along the axs. As the postonal uncertanty, P uses three tmes the standard devaton n ts calculaton and b-drectonal accuracy, A 06, only uses twce the standard devaton, P s expected to be bgger than A. Smlar - Indcates the maxmum spread of data ponts that occurred at an ndvdual target poston. P smax wll always be larger than ether R 06 or R 06 as P smax uses three tmes the standard devaton n ts calculaton whle R 06, only uses twce the standard devaton, Note that P smax s related to the averaged devaton over the forward and reverse drectons. If B 06 and U are zero then 2/3P smax should be smlar to R 06. Otherwse t s expected that U Ps max should be smlar to R 06. UNC Charlotte Mullany 13

14 Comparson between ISO 230-2:2006 and ISO 230-2:1988 Table 4: Identcal and smlar parameters n the ISO 230-2:2006 and ISO 230-2:1988. ISO 230-2:2006 ISO 230-2:1988 Comparson Mean Reversal value of an axs B 06 1 m B = B m = 1 B-drectonal accuracy of postonng of an axs, A 06 A = max 06 mn x [ x + 2s ;x + 2s ] [ 2 2 ] s ;x Undrectonal repeatablty of postonng of an axs, R 06 or R 06 R = max 06 R = max 06 [ 4s ] [ 4s ] Repeatablty of postonng of an axs, R 06 [ ] R max R 06 = Where; max [ 2s + 2s + B ; R ; R ] R = s Mean Reversal value of an axs B 88 1 n B = B n = 1 B-drectonal accuracy of postonng of an axs, A 88 A = max 88 mn [ x + 3s ;x + 3s ] [ x 3s ;x 3s ] Undrectonal repeatablty of postonng of an axs, R 88 or R 88 R = 6s ] 88 R = max[ 6s ] 88 Repeatablty of postonng of an axs, R 88 [ ] R max R 88 = Where; R max [ 3s + 3s + B ; R ; R ] = Identcal - Averaged reversal error Smlar - but ISO 230-2:1988 uses three standard devatons n ts calculatons whle the 2006 verson uses two standard devatons. A 88 s expected to be greater than A 06 Smlar- drect conversons between the two standards exst, R 06 = 2/3 R 88 R 06 = 2/3 R 88 Lkewse R 88 = 3/2 R 06 R 88 = 3/2 R 06 Smlar- Based on same concept, however ISO 230-2:1988 uses three standard devatons n ts calculaton as opposed to two standard devatons used n ISO 230-2:2006. R 88 wll be bgger than R 06. See Appendx A for detals on x, x, s and s UNC Charlotte Mullany 14

15 Comparson between ISO 230-2:1988 and VDI/DQG 3441 Table 5: Identcal and smlar parameters n the ISO 230-2:1988 and VDI/DQG ISO 230-2:1988 VDI/DQG 3441 Identcal or smlar Mean Reversal value of an axs B 88 B 88 Where; 1 n = B n = 1 B = x x B-drectonal accuracy of postonng of an axs, A 88 A 88 = max mn [ x + 3s ;x + 3s ] [ x 3s ;x 3s ] Undrectonal repeatablty of postonng of an axs, R 88 or R 88 R 88 R 88 = max = max [ 6s ] [ 6s ] Repeatablty of postonng of an axs, R 88 [ ] R max R 88 = Where; R [ 3s + 3s + B ; R ; R ] = max Mean Reversal Error, U 1 m U = U m = 1 Where; U = x x Postonal Uncertanty, P 1 P = x + ( U + P s ) 2 1 x ( U + P s ) 2 Max Mn Max Postonal Scatter, P s max P s max = P s max = max[6s ] No equvalent parameter Smlar - Average reversal error. Due to slght dfferences n the equatons the values may vary. Ths s especally true f the averaged forward and reverse postonal errors lnes ntersect each other as n fgure 4. Smlar - Maxmum range of values based on mean postonal error and correspondng standard devatons about each target poston. As the postonal uncertanty, P uses the averaged standard devaton over the forward and reverse drectons t s expected to be slghtly smaller than the b-drectonal accuracy, A 88. Smlar - Indcates the maxmum spread of data ponts that occurred at an ndvdual target poston. As P smax s based on the averaged standard devaton t s expected to be slghtly smaller than R 88 or R 88 If B 88 and U are zero then P smax should be smlar to R 88. Otherwse t s expected that U + P s max should be smlar to R 88 See Appendx A for detals on x, x, s and s UNC Charlotte Mullany 15

16 Comparson between VDI/DQG 3441 and JIS B Table 6: Identcal and smlar parameters n the VDI/DQG 3441 and JIS VDI/DQG 3441 JIS B Identcal or smlar Max reversal error at a poston U = max[ U max Max Postonal Scatter, P s max P s max = P s max = max[6s ] ] Lost Moton test U JIS = max [ U ] Repeatablty test, R s 1 R JIS = ± max[ R ] 2 JIS Where; = max[ R JIS x ] mn[ x ] Identcal, however U s s averaged over 7 measurements at each pont and not 5 as per the VDI/DQG Smlar: P smax s 6 tmes the largest averaged standard devaton of data ponts at a target pont, whle R s s ½ the maxmum range of data ponts measured at a target pont. R s wll be much smaller than P smax. Note the number of requred measurement ponts for the JIS B6330 s less than that requred for the VDI/DQG Comparson between ISO 230-2:2006 and JIS B Table 7: Identcal and smlar parameters n the ISO 230-2:2006 and JIS B ISO 230-2:2006 JIS B Reversal value of an axs B 06 = max [ ] B Lost Moton test U JIS = max [ U ] Identcal, however U s s averaged over 7 measurements at each target pont and not 5 as per the ISO Undrectonal repeatablty of postonng of an axs, R 06 or R 06 R 06 R 06 = 4s = 4s Repeatablty test, R s 1 R JIS = ± max[ R ] 2 JIS Where; = max[ R JIS x ] mn[ x ] Smlar: R 06 s 4 tmes the largest standard devaton of data ponts at a target pont, whle R s s ½ the maxmum range of data ponts measured at a target pont. R s wll be much smaller than R 06. B-drectonal systematc postonal devaton of an axs, E 06 E 06 mn x = max[ x ; x ] [ ; ] x Postonal accuracy PJIS = max x [ ] mn[ ] x Smlar: Both terms are smlar n concept, but as there are sgnfcant dfferences between the standards regardng the number of data ponts requred the two parameters may vary substantally. Note the number of requred measurement ponts for the JIS B6330 s less than that requred for the ISO 230-2:2006. UNC Charlotte Mullany 16

17 Numercal Evaluaton Identcal parameters from the dfferent standards need no further explanaton. Lkewse for parameters that have no comparable parameter n other standards. However t s worth examnng the relatonshp between smlar parameters and determnng f gudelnes can be wrtten that wll allow for translaton between the conceptually smlar parameters defned by the dfferent standards. For example, can the ISO 230-2:2006 bdrectonal accuracy of an axs, A, be converted drectly n the VDI/DQG 3441 postonal uncertanty, P? Numercal analyss was undertaken to determne the ratos between the conceptually smlar parameters and the expected range of ±percentage errors assocated wth each rato. Methodology A Monte Carlo approach s taken whereby several hundred sets of measurement ponts are generated usng the Gaussan random numbers generator functon n Matlab. Two dfferent approaches were taken when generatng the measurement ponts. In both cases the generated measurement ponts were analyzed as per the varous standards and the core parameters compared. Note on Gaussan assumpton: In general machne tools are not Gaussan n nature. Most errors are systematc. The frst approach taken (detals to follow) assumes a purely Gaussan dstrbuton of the measurement ponts, ths s somewhat lmted n ts valdty. The second approach (agan detals follow) whle stll usng Gaussan dstrbutons has a taken some, but not all, of the expected systematc errors nto consderaton. Nether approach can model machne tools accurately and t should be realzed that defntve (100% accurate) conversons between standards are not possble. That sad an apprecaton of how the dfferent parameters compare to each other s benefcal. It s also worth observng how the startng model assumptons affect the magntude of the expected ratos. 1. Gaussan Random Numbers For a chosen set of eleven target postons (relatve poston of each target pont to each other s not mportant) ten data ponts (fve n the forward drecton and fve n the reverse) were randomly generated n Matlab usng the Gaussan random number generator NORMRND. Ths functon requres a standard devaton and mean value. The fgure 1 below detals a typcal set of data ponts created when the mean and standard devaton were taken to be 0.5μm and 5 μm respectvely. Analyss: Usng the same mean and standard devaton fve thousand sets of measurement ponts were generated.. For each of the fve thousand sets the VDI/DQG 3441, ISO 230-2:1988, ISO 230-2:2006 and JIS6330 parameters were calculated. Whle a standard devaton value of 5μm maybe consdered somewhat on the large sde, t s worth notng that alterng the standard devaton used n the model does not sgnfcantly affect the resultng ratos or ther respectve standard devatons. The ratos between several comparable parameters are lsted n the frst column of table 10. The values lsted are the ratos averaged over fve thousand runs whle the number n parenthess s the standard devaton over the fve thousand runs. UNC Charlotte Mullany 17

18 x = Averaged postonal devaton at a target poston n the reverse drecton x = Averaged postonal devaton at a target poston n the forward drecton Fgure 1: Typcal data set generated wth a standard devaton of 5μm and a mean value of 0.5μm. 2. Gaussan Random Numbers wth Addtonal Constrants Ths approach also uses the Matlab NORMRND Gaussan random number generator to generate artfcal measurement ponts, however there are more constrants wth respect to the relatve locaton of the data ponts to the adacent target postons and the measurement ponts n the reverse drecton. Fgure 3 llustrates a typcal set of generated measurement ponts. The methodology can be descrbed n four steps, please also refer to fgure 2. The constrants used are based on ISO 230-2:2006 parameters, see table 8 for descrptons of the parameters used and fgure 3 for graphcal llustraton. 1. x, the frst target poston s randomly selected wthn a predefned range. 2. The mean value of postonal error, x at each of the subsequent target locatons n the forward drectons s randomly generated by a Gaussan functon wth the mean value taken as the poston of the target pont generated n step 1 and the devaton taken as a typcal undrectonal systematc postonal dvded by four, E / 4 or E / 4. E s the range n whch the averaged target poston error les and thus t devaton can be taken as E /4 (k=2). 3. Gaussan functons were used to generate the locaton of the mean postonal devaton values at each target poston n the reverse drecton. The value s determned by the subtractng a number generated usng the mean reversal value, B, as the mean and the UNC Charlotte Mullany 18

19 B /24 as the standard devaton. Takng ths approach the mean postonal errors n the forward and reverse drectons wll not ntersect. Ths ssue s addressed n secton Fve data ponts are randomly created for each of the mean postonal devatons n the forward and reverse drectons. For the Gaussan functon the mean value s taken to be the relevant x or x value and the devaton s a typcal R / 4 value. Step 1: x 1 Step 2: generate 10 other x Postonal error, mcrons 0 P1 P2 P3 Target locaton,p,mm Postonal error, mcrons Mean = P1, Stdev= E /4 0 P1 P2 P3 Target locaton,p,mm Locaton P1 s randomly pcked Ponts P2, P3 etc are generated usng a Gaussan dstrbuton random number generator n Matlab wth the mean and standard devaton set as llustrated above Step 3: generate 11 x ponts Postonal error, mcrons Mean = B, Stdev=Fn(B ) 0 P1 P2 P3 Target locaton,p,mm Step 4: generate data ponts at all target postons Mean = x, Stdev= R /4 Postonal error, mcrons 0 P1 P2 P3 Target locaton,p,mm Fgure 2: Steps nvolved n generatng the random numbers. Table 8: Parameters used n generatng the measurement pont sets and ther meanng. Parameter Meanng E It gves nformaton wth respect to the range of the averaged postonal devatons ( x or x ) calculated along the axs n ether the forward or the reverse drecton. It contans no nformaton wth respect to the spread of data ponts at each measurement poston. See table 7 for ts numercal defnton. B B, the reversal value at a target poston, P, s the dfference between the averaged postonal devaton n the forward x and the reverse x drectons. The mean reversal value recorded along the axs s taken as the average of all the reversal values, B, along an axs. See table 3 for ts numercal defnton. R R, the undrectonal repeatablty of postonng at a poston n the forward drecton, s related to the maxmum spread of measured postonal errors at a target poston, P, the spread s taken as 4 s (coverage k=2), where s s the standard devaton of the measurement ponts at a target poston. UNC Charlotte Mullany 19

20 x = Averaged postonal devaton at a target poston n the reverse drecton B E R x = Averaged postonal devaton at a target poston n the forward drecton Fgure 3: A typcal set of data ponts generated by the second method. Analyss: For ths approach there are effectvely three test parameters ( E, B, R ). Three test sets were run whereby upper and lower lmts were gven to each of the three parameters. The three testng bands were: 0.1 μm to 5 μm, 5 μm to 10 μm and lastly 10 μm to 30 μm. Wthn each test set the values were systematcally vared (eght dfferent combnatons) and fve hundred smulatons run for each combnaton, see table 9. Table 9: Combnatons tested for the frst bandwdth, 0.1μm to 5μm. Run 1 Run 2 Run 3 Run 4 Run 5 Run 6 Run 7 Run 8 E B R The key ISO 230-2:1988, ISO 230-2:2006, VDI/DQG 3441 and JIS 6330 parameters were calculated and the rato between conceptually smlar parameters as outlned n tables 3 to 7 were determned. The averaged rato and the averaged standard devaton (n parenthess) wthn each of the three test sets are lsted n the last three columns of table 10. UNC Charlotte Mullany 20

21 Table 10: Summary of the parameter ratos (and standard devatons) calculated from the Monte Carlo smulatons. Rato Gaussan 0.1 μm 5μm 5 μm 10μm 10 μm 30μm A 06 /P 0.8 (0.06) 0.9 (0.03) 0.86 (0.05) 0.87 (0.05) A 88 /A (0.04) 1.19 (0.03) 1.22 (0.04) 1.22 (0.04) A 88 /A (0.04) 1.29 (0.05) 1.33 (0.06) 1.32 (0.06) A 88 /P 1.16 (0.1) 1.06 (0.05) 1.05 (0.06) 1.05 (0.06) R 06 /P smax 0.75 (0.1) 0.76 (0.1) 0.76 (0.1) 0.76 (0.1) R 88 /P smax 1.13 (0.15) 1.14 (0.16) 1.14 (0.16) 1.14 (0.16) R 06 /(U+2/3P smax ) 1.14 (0.11) 1.06 (0.06) 1.03 (0.05) 1.04 (0.05) R 88 /(U+P smax ) 1.16 (0.11) 1.06 (0.06) 1.03 (0.05) 1.03 (0.05) R 88 /R (0.03) 1.26 (0.02) 1.27 (0.06) 1.28 (0.06) Rs/R (0.05) 0.35 (0.05) 0.35 (0.05) 0.35 (0.05) Rs/R (0.03) 0.26 (0.03) 0.26 (0.03) 0.26 (0.03) Rs/P smax 0.23 (0.03) 0.23 (0.03) 0.23 (0.03) 0.23 (0.03) Ps/E 0.66 (0.21) 0.73 (0.13) 0.76 (0.12) 0.75 (0.13) See Notes 1& 2 on page 12 and table 11. Table 11: Rato Key. Rato Detal A 06 /P B drectonal Accuracy (ISO 230-2:2006) / Poston Uncertanty (VDI/DQG 3441) A 88 /A 06 B drectonal Accuracy (ISO 230-2:1988) / B drectonal Accuracy (ISO 230-2:2006) A 88 /A 06 Undrectonal Accuracy (ISO 230-2:1988) / Undrectonal Accuracy (ISO 230-2:2006) A 88 /P B drectonal Accuracy (ISO 230-2:1988) / Poston Uncertanty (VDI/DQG 3441) R 06 /P smax Undrectonal Repeatablty (ISO 230-2:2006)/ Postonal Scatter (VDI/DQG 3441) R 88 /P smax Undrectonal Repeatablty (ISO 230-2:1988)/ Postonal Scatter (VDI/DQG 3441) R 06 /(U+2/3P smax ) Undrectonal Repeatablty (ISO 230-2:2006)/ (Reversal error + ⅔Postonal Scatter (VDI/DQG 3441) ) R 88 /(U+P smax ) Undrectonal Repeatablty (ISO 230-2:1988)/ (Reversal error + Postonal Scatter (VDI/DQG 3441) ) R 88 /R 06 Repeatablty (ISO 230-2:1988) / Repeatablty (ISO 230-2:2006) R s /R 06 Repeatablty (JIS B )/ Undrectonal Repeatablty (ISO 230-2:2006) R s /R 88 Repeatablty (JIS B )/ Undrectonal Repeatablty (ISO 230-2:1988) R s /P smax Repeatablty (JIS B )/ Postonal Scatter (VDI/DQG 3441) P s /E Postonal Accuarcy (JIS B )/ B-drectonal Postonal Devaton (ISO 230-2:2006) 2.1. Intersectng and A lmtaton of the prevous method used to generated measurement ponts s that the forward and reverse drectons wll not ntersect, a phenomena that may well occur n realty. The program was modfed so that the forward and reverse drectons were forced to ntersect. Ths was acheved by removng steps 2 and 3 outlned n secton 2 and specfyng the locaton of the averaged postonal error n each drecton, see fgure 4 for an example of a typcal output. R and E values were used to determne the spread of the data ponts at a target poston and the range of postonal errors n ether the forward or reverse drectons. The program was run 500 UNC Charlotte Mullany 21

22 tmes and the results are presented n Table 12. Agan the averaged rato and the averaged standard devaton (n parenthess) are gven for the three test bands. x = Averaged postonal devaton at a target poston n the reverse drecton x = Averaged postonal devaton at a target poston n the forward drecton Fgure 4: Randomly generated data ponts where and are forced to ntersect. Table 12: Summary of the parameter ratos (and standard devatons) calculated from ntersectng and. Rato 0.1 μm 5μm 5 μm 10μm 10 μm 30μm A 06 /P 0.85 (0.04) 0.82 (0.05) 0.84 (0.05) A 88 /A (0.03) 1.29 (0.04) 1.29 (0.04) A 88 /A (0.04) 1.29 (0.04) 1.29 (0.04) A 88 /P 1.08 (0.06) 1.07 (0.07) 1.08 (0.07) R 06 /P smax 0.76 (0.1) 0.76 (0.1) 0.76 (0.11) R 88 /P smax 1.14 (0.16) 1.14 (0.15) 1.14 (0.16) R 06 /(U+2/3P smax ) 1.26 (0.09) 1.17 (0.1) 1.19 (0.1) R 88 (U+P smax ) 1.23 (0.09) 1.12 (0.09) 1.08 (0.07) R 88 /R (0.04) 1.30 (0.05) 1.29 (0.05) R s /R (0.05) 0.35 (0.05) 0.35 (0.05) R s /R (0.03) 0.26 (0.03) 0.26 (0.03) R s /P smax 0.23 (0.03) 0.23 (0.03) 0.23 (0.03) See Notes 1& 2 on page 12 and table 11. UNC Charlotte Mullany 22

23 To provde the reader wth a hgher degree of transparency, table 13 detals the possble percentage varatons n the rato value assocated wth ± three standard devatons (k=3). In all cases there are sgnfcant percentage varatons possble. And n certan cases, such as comparng JIS 6330 parameters to ISO or VDI/DQG 3441 parameters, or comparng ISO repeatablty values to VDI/DQG 3441 postonal scatter values, the percentage varatons assocated wth the ratos are very large,.e. > ±34%. In these cases attemptng a drecton converson from one standard to another wth any degree of certanty s not advsed. Table 13: Ratos and ther percentage varatons. Rato Gaussan Rato +/- % error 0.1 μm 5 μm Rato +/- % error Intersectng 0.1 μm 5 μm Rato +/- % error A 06 /P 0.8 ±14.1% 0.9 ±10.0% 0.85 ±14.1% A 88 /A ±8.3% 1.19 ±7.6% 1.27 ±7.1% A 88 /A ±8.3% 1.29 ±11.6% 1.27 ±9.4% A 88 /P 1.16±25.9% 1.06 ±14.2% 1.08 ±16.7% R 06 /P smax 0.75 ±40.0% 0.76 ±39.5% 0.76 ±39.5% R 88 /P smax 1.13 ±39.8% 1.14 ±42.1% 1.14 ±42.9% R 06 /(U+2/3P smax ) 1.14 ±28.9% 1.06 ±17.0% 1.26 ±21.4% R 88 /(U+P smax ) 1.16 ±28.4% 1.06 ±17.0% 1.23 ±22% R 88 /R ±6.1% 1.26 ±4.8% 1.28 ±9.4% R s /R ±42.9% 0.35 ±42.9% 0.34 ±44.1% R s /R ±34.6% 0.26 ±34.6% 0.26 ±34.6% R s /P smax 0.23 ±39.1% 0.26 ±39.1% 0.23 ±39.1% P s /E 0.66 ±95% 0.73 ±53.4% See Note 1 on page 12 and table 11. Verfcaton of the Monte Carlo predcted ratos. To verfy the Monte Carlo analyss ratos two sets of data ponts were consdered. The frst set of ponts taken were from the worked example n the ISO 230-2:2006, the second set of measurement data ponts were taken from an actual test carred out on the Monarch mllng machne at UNC Charlotte. Both sets of data ponts underwent VDI/DQG 3441, ISO 230-2:2006 and ISO 230-2:1988 analyss to calculate the key parameters defned by each standard. The ratos between the conceptually smlar parameters were determned and compared to the range of rato values obtaned from the Monte Carlo analyss. The results are presented n table 14. The Predcted Rato Ranges reported n the fnal column of table 14 are the hghest and lowest possble ratos as gven by method 2 n the 0.1μm to 5μm test band (based on k=3). ISO ratos n column 2 refers to the ratos obtaned based on the data ponts lsted n ISO 230-2:2006, smlarly the Monarch ratos were determned from the data ponts taken off the Monarch machne at UNC Charlotte. UNC Charlotte Mullany 23

24 Table 14: Actual ratos compared to predcted ratos. Rato ISO Ratos Monarch Ratos Predcted Rato Ranges A 06 /P A 88 /A A 88 /P R 06 /P smax R 88 /P smax R 06 /(U+2/3P smax ) R 88 /(U+P smax ) R 88 /R See Notes 1 & 2 on page 12 and table 11. Conclusons Analyss of the dfferent standards solated parameters that were dentcal to each other and those that were smlar n concept but mathematcally dfferent, see tables 3 to 7. A Monte Carlo approach was taken to determne the relatonshp between conceptually smlar parameters. Two dfferent approaches were taken; purely Gaussan and Gaussan combned wth some systematc errors. Twelve relatonshp pars were consdered, however as the percentage varaton assocated wth the ratos was qute large n some cases (over 40%) only three conversons ratos could be consdered (A 88 /P, A 88 /A 06 and R 88 /R 06 ). These rato values stll had assocated ± percentage varatons up 14% and thus great cauton should be taken f consderng an attempt to convertng from one standard to another. Ultmately when seekng to compare values obtaned from the dfferent standards there s no substtute for actually conductng the analyss on actual measured data. UNC Charlotte Mullany 24

25 Appendx A Nomenclature used by the varous standards ISO 230-2:2006 & 1997 A & A Undrectonal Accuracy of postonng of an Axs A Bdrectonal Accuracy of postonng of an Axs E & E Undrectonal systematc postonal devaton of and axs E Bdrectonal systematc postonal devaton of an axs M Mean bdrectonal postonal devaton of an axs, M B Reversal value of an axs R & R Undrectonal repeatablty of postonng R B-drectonal repeatablty of postonng of an axs x & x Mean undrectonal postonal devaton at a target poston x Mean bdrectonal postonal devaton at a poston s & s Estmator for the undrectonal axs repeatablty of postonng at a target pont ISO 230-2:1988 A Accuracy of an axs W and W Range of postonal devaton B Mean reversal error B Reversal value of an axs R & R Undrectonal repeatablty of postonng R B-drectonal repeatablty of postonng of an axs x & x Mean undrectonal postonal devaton at a target poston s & s Estmator for the undrectonal axs repeatablty of postonng at a target pont VDI/DQG 3441 P smax Maxmum postonal scatter U max Maxmum reversal error at a poston U Mean reversal error P a Postonal devaton P Postonal uncertanty x & x Mean value of ndvdual values at a poston System Devaton from the desred value at a target poston x = s & s Standard devaton at a target poston n one drecton Mean standard devaton at a target poston s UNC Charlotte Mullany 25

26 ASME B A & A Undrectonal Accuracy of postonng of an Axs A Bdrectonal Accuracy of postonng of an Axs E & E Undrectonal systematc postonal devaton of and axs E Bdrectonal systematc postonal devaton of an axs M Mean bdrectonal postonal devaton of an axs, M B Reversal value of an axs R & R Undrectonal repeatablty of postonng R B-drectonal repeatablty of postonng of an axs P Perodc errors x & x Mean undrectonal postonal devaton at a target poston x Mean bdrectonal postonal devaton at a poston s & s Estmator for the undrectonal axs repeatablty of postonng at a target pont JIS Postonng Accuracy test Repeatablty test Lost moton test Least nput ncrement-feed P s R s U s L s UNC Charlotte Mullany 26