project square final - bipm · file: project_square_final-an.doc / issue 28. 02. 2005 slovenský...

File: Project_SQUARE_Final-an.doc / Issue 28. 02. 2005

Slovenský metrologický ústavSlovak Institute of MetrologySlowakisches Institut für Metrologie

EUROMET Supplementary Comparison #570

Comparison of squarenessmeasurements

Final Report


Jiri Mokros, SMU, Bratislava, February 2005


Content

1 Introduction.............................................................................12 Organisation ............................................................................13 Participants..............................................................................24 Time schedule..........................................................................35 Description of the Standards......................................................36 Measurement instructions..........................................................36.1 Definitions .............................................................................37 Measurement (calibration) results ..............................................57.1 Supplementary data ................................................................68 Measurement uncertainty ..........................................................69 Measuring devices and procedures..............................................79.1 SMU ......................................................................................79.2 METAS...................................................................................79.3 PTB .......................................................................................89.4 GUM.................................................................................... 109.5 MIRS (SMIS) ........................................................................ 119.6 MIKES ................................................................................. 129.7 IPQ ..................................................................................... 139.8 BNM-LNE.............................................................................. 139.9 SP....................................................................................... 149.10 OMH.................................................................................... 159.11 CMI .................................................................................... 169.12 NMI VSL............................................................................... 1610 Results.................................................................................. 1710.1 Granite square...................................................................... 1810.1.1 Straightness............................................................................................... 1810.1.2 Squareness ................................................................................................ 2810.2 Cylinder Square .................................................................... 3110.2.1 Straightness............................................................................................... 3110.2.2 Squareness ................................................................................................ 4511 Conclusion............................................................................. 50

Final Report of EUROMET Comparison of squareness measurements No: 570 1

1 Introduction

The comparison of squareness measurement was aimed to compare and verify thedeclared calibration measurement capabilities of participating laboratories and toinvestigate the effect of systematic influences in the measurement process and theirelimination.As regards the technical parameters, the standards which were circulated representthe standards currently used in the metrological praxis. It makes possible to comparethe standard devices in the real conditions.The standards had to be calibrated by the measurement process currently used inthe participant’s laboratory (i.e. in the horizontal or vertical position of the square).

This comparison could help us to gain an important information revealingimperfections of measurement process related to measurement systems of individualparticipating NMIs. Such an information could be in turn used for the upgrade ofmeasurement system or measurement procedure. For the sensing of real profilemust be performed prior to the determination of the angle deviation between twoarms of square, ISO 1101 was not applied. The straightness deviation and angleshave been calculated separately.

2 Organisation

This comparison will be submitted as the EUROMET supplementary comparison inthe framework of the Mutual Recognition Arrangement (MRA) of the MetreConvention and shall support confidence in calibration and measurement certificatesissued by the participating national metrology institutes (NMI).The comparison was organised according to the rules set up by the BIPM1.

1 T.J. Quinn, Guidelines for CIPM key comparisons (Appendix F to the MRA, 1. March 1999, BIPM,

Paris)


3 Participants

NMI Address Name E-mail Telephone FAX

Slovak Institute of Metrology SMUKarloveská 63,SK-842 55 BratislavaSlovakia

Jirko Mokroš [email protected] +421 2 60294 253 +421 2 65429 592

Swiss Federal Office ofMetrology and Accreditation METAS

Lindenweg 50,CH-3003 Bern-WabernSwitzerland

Ruedi Thalmann [email protected] +41 31 323 33 85 +41 31 323 32 10

Reinhard Probst [email protected] +49 531 592 5220 +49 531 592 5205Physikalisch-TechnischeBundesanstalt PTB

Bundesallee 100,D-38116 BraunschweigGermany Otto Jusko [email protected] +49 531 592 5310 +49 531 592 5305

Czech metrological institute CMIV Botanice 4, Praha 5, CZ-15072Czech republic

Vít Zelený [email protected] +420 257 288 387 +420 257 328 077

Central Office of Measures GUMul. Elektoralna 2,00-950 Warszawa,P-10, Poland

BarbaraSmereczynska [email protected] +48 22 620 54 38 +48 22 620 8378

National Office of Measures OMHNemetvolgyi út 37-39,H-1124 BudapestHungary

Edit Banreti [email protected] +36 1 4585 800+36 1 4585 944 +36 1 214 3157

Instituto Português da Qualidade IPQRua António Gião, 2,2829-513 CAPARICAPortugal

Maria FernandaSaraiva [email protected] +351 21 2948160 +351 21 2948188

University of Maribor,Faculty of MechanicalEngineering,Laboratory for ProductionMeasurement

MIRS(SMIS)

Smetanova 17,2000 MariborSlovenia

Bojan Ačko [email protected] +386 2 220 7581 +386 2 220 7990

MIKES Konepajametrologia MIKESMetallimiehenkuja 6FIN-02150 ESPOOFinland

Heikki Lehto [email protected] +358 9 4565350 +358 9 460627

BNM - Laboratoire Nationald'Essais

BNM-LNE

1, rue G. Boissier75724 PARIS Cedx 15France

Georges Vailleau [email protected] +33 1 40 43 37 00 +33 1 40 43 37 37

NMi VSL B.V. P.O. Box 654, 2600 AR,Delft, The Netherlands Rob Bergmans [email protected] +31 15 269 15 00 -31 15 261 29 71

Swedish National Testing andResearch Institute SP

Department of Metrology,Brinellgatan 4, SE-501 15BORÅS, Sweden

Stefan Källberg [email protected] +46 33 16 56 26 +46 33 10 69 73

mailto:[email protected]














4 Time schedule

The time schedule was currently modified few times, reflecting the requirements ofsome participants. In the interim the standards were delivered back to the SMU in June2001, in order they be equipped with a new ATA-carnet. The real time schedule isshown in the following table.

Laboratory Country DateSMU SK September – October 2000METAS, part 1 CH October – December 2000PTB DE December 2000 – February 2001GUM PL January – March 2001MIRS (SMIS) SI April 2001MIKES FI May – June 2001SMU (no measurement) SK June 2001IPQ PT July – August 2001METAS, part 2 CH August 2001BNM-LNE FR September – October 2001SP SE October – November 2001OMH HU November – December 2001CMI CZ December 2001 – February 2002NMi VSL NL February – May 2002SMU SK June – November 2002

One abnormality has happened during the circulation – in the period between themeasurements in CMI and in NMi VSL, the local damage (approximately 1,5 x 1 mm) toone edge of the granite square has been observed. Nevertheless, this damageseemed to have no effect to the measured parameters.

5 Description of the Standards

Two standards were calibrated:- granite squareness standard of rectangular shape (500x300x70) mm with four

marked functional surfaces, weight 26 kg,- cylindrical squareness standard of steel with 102 mm diameter and 401,5 mm height

with marked positions for the profile lines, weight 25 kg.

6 Measurement instructions

6.1 DefinitionsZero point of the coordinate system – the intersection of the functional planes and themeasurement plane (see Fig. 1).


Local deviation from straightness – the distance between the measured point and theLS regression-line fitted through the measured profile in the measured plane; thepositive value corresponded to the orientation outside from the material of square (seeFig. 1).

Angle between fitting lines (in the case of granite standard) – interior angle γLS

between the LS regression-lines fitted through the measured profiles AB and AC (seeFig. 1). The fitting line of the profile AC could be replaced by the envelope regressionline (interior angle γB) – see Fig. 1.Angle between fitting line (in the case of the cylindrical standard) – the angles areunderstood as the interior angles between the corresponding LS regression-lines fittedthrough the measured profiles at 0°, 90°, 180°, 270° and the envelope plane of thebasis (see Fig. 2).

Fig. 1Specification of the granite square


Fig. 2Specification of the cylindrical square

The measured profiles (in the case of the granite standard) are defined in thelongitudinal axis in the middle of each functional plane.The measured profiles (in the case of cylindrical squareness standard) – the generatrixprofiles at 0°, 90°, 180°, 270° (marked on the "TOP" plane) around the circumferenceof a cylinder.The starting point of measurement – 5 mm from the zero point (defined above). In thecase of cylindrical squareness standard, four zero points are given by the inter-sectionof four generatrix profiles with BASIS plane. The density of measuring points of theprofile shall be 0,5 mm (in extra cases should be allowed the integer multiple of0,5 mm, max. 2 mm).Angles of the squares were measured using the technique currently applied by theparticipant. This method was described in details by each participant in the annex A2"Measurement Report".The squares were measured in the position currently used in the laboratory –horizontally or vertically.

7 Measurement (calibration) results

Due to the properties of measurement devices, two following kinds of definition of thesquare angle are being applied: „envelope – LS regression“ and „LS regression –LS regression“. Of totally 12 NMIs involved, 4 NMIs are using the first definition


(envelope – LS regression) and 9 NMIs are using the second one (LS regression –LS regression), while one NMI carried out the measurements applying both definitions.As for the final evaluation, it is necessary to choose a single definition from those twomentioned above (with the transformation between "envelope - LS regression"). Thepractical determination of the LS regression line is a simple mathematical operation,which doesn‘t differ from the theoretical definition. The practical determination of theenvelope line is significantly more complicated, since it is determined by the contactpoints of the functional plane and the basis of the measurement device. It is difficult todescribe this contact exactly, because of the flatness deviations of the basis andelastic deformations of the parts adjacent to the contact points.

The following parameters had to be calibrated:

- granite squareness standard of rectangular shape: interior angle γB between twolines AB and AC (envelope - LS regression) fitted through the measured profiles,

and / or

- granite squareness standard of rectangular shape: interior angle γLS between twoLS regression-lines AB and AC fitted through the measured profiles,

- cylindrical squareness standard: interior angles γ0, γ90, γ180, γ270 between the LSregression-line fitted through the measurement profiles at 0°, 90°, 180°, 270° andthe envelope plane of the basis (resting on a surface plate),

- local LS straightness deviation for all measured profiles (2 and 4) of both standards(results had to be reported in electronic format only).

7.1 Supplementary data

- radius of the probe tip,

- ambient temperature and its time drift during the measurement period,

- description of the standard device on which the calibration has been performed,

- description of the measurement methods and the data evaluation,

- the method of calculation of the combined standard uncertainty uC (k = 1) related tothe angle between fitting lines and uncertainty of local deviation from straightness,

- measurement uncertainty budget.

8 Measurement uncertainty

The combined standard uncertainty uC (k = 1) of all measurement results had to beestimated according to the ISO Guide for the Expression of Uncertainty inMeasurement.


The participants were asked to report their measurement uncertainty budget in theannex A2 "Measurement Report".

9 Measuring devices and procedures

The measuring devices used by the participants are shortly described below. Therequired form of data reporting was designed in order to reveal possible error sourcesof individual NMIs. Despite of this procedure is more demanding for the evaluationthan simple reporting of squareness and straightness deviation according to ISO 1101,the content of information provided by the participant was noticeably larger. That iswhy all participants had to report the measurement data corresponding to localstraightness deviation and angles between LS lines.

9.1 SMUDescription of measuring deviceThe measuring device NME 90° (with 1300 mm straightness column, resting on asurface plate and air bearing carriage) compares form and angle position of verticalarm of measured rectangular standard with form and position of measuring column.Air bearing carriage bears two inductive sensors, which read a profile line of themeasured square. The square standard under test is placed on a granite base plateso its horizontal arm is connected with this plate (envelope plane). The angle ofsquare is defined by the fitting line (evaluated from individual measured points onvertical arm) and by the horizontal plane, given by the granite base plate of device.Such a determination of square was chosen by the device producers, because thisway is usually used in industry.

Procedure of measurementFor the measurement of angle standard the well-known method of error separationtechnique (reversal technique) by means of “self-calibration” is used. This methodallows the evaluation of the profile of square vertical arm without beforehandinformation about the profile of the measuring column. Process of the measurementconsists of two steps – measurement of the square standard in 0° position and in180° position.

Procedure of result calculation:Measurement profiles are transferred to an Excel worksheet. The slope of theprofiles is calculated by linear regression, resulting in the angle of the squarenessstandard. Deviation of each measured point from the LS regression line is the localstraightness deviation.

9.2 METASDescription of measuring device


Horizontal Measurements: Numerically controlled 800 mm straightness measuringdevice (STRAIGHTLine®) with air bearing carriage, combined with numericallycontrolled air bearing rotary table with Heidenhain RON 905 encoder.Vertical Measurements: Numerically controlled 1000 mm straightness column(SQUAREMaster®) with air bearing carriage, resting on a surface plate.Procedure of measurementHorizontal Measurements: Straightness measurement instrument is calibrated withreversal technique using a 1000 mm ceramic straight edge, resulting in errorcorrection file.The granite squareness standard is resting horizontally on the rotary table at the 0°position, while probing line AB with the straightness measuring instrument.Subsequent rotation of granite square by 90°, then probing line AC.Vertical Measurements: The squareness standard (granite square or cylinder) isresting vertically on a surface plate. The device under test is rotated around itsvertical axis to apply reversal error separation.

Procedure of result calculation:Measurement profiles are transferred to an Excel worksheet. The slope of theprofiles is calculated by linear regression, resulting in the angle of the squarenessstandard.

9.3 PTBFor granite squareAir bearing linear guideway, PTB construction, photoelectric linear encoderHEIDENHAIN LIDA 185, resolution 0,05 µm, measurement range L=1000 mm,indexing table ULTRADEX for 90° turn, reference: 90° angle block STARRETT, two-axis electronic autocollimator Möller-Wedel ELCOMAT 2000, resolution 0,01",inductive displacement transducer Mahr Millitron 1204 with probe 1303, resolution0,01µm, measurement range ±3µm.Description of measuring deviceMeasurement of straightness deviation on line A – B from B to A, 90° turn of theobject by use of indexing table,Measurement of 90° deviation by use of angle block and autocollimator,Measurement of straightness deviation on Line A – C from A to CProcedure of measurement controlled by computerProcedure of result calculation:Calculation of the straightness deviation from the regression fitting lines,Calculation of the 90° angle deviation from the slope difference of the regressionfitting lines,


Correction of the straightness deviation of the guideway by reversal technique andof the angle deviation of the 90° angle block from calibration.Software used: Microsoft EXEL 97


For cylinder squareDescription of measuring deviceThe measuring device used for calibration of cylindrical square standard wasdeveloped by PTB, based on the form tester MFU 8 (Mahr). With this instrument themeasurands diameter, roundness, straightness, parallelism and squareness ofcylindrical and spherical objects can be calibrated in one single set-up.Procedure of measurementThe straightness deviations of the z-guide amount to 0,15 µm over 400 mm. Theangle between C-axis and z-guide can be adjusted within about ± 2 µm/ 400 mm(1´´). A complete reversal technique has been used to correct both, the straightnessdeviations and the orientation of the z-axis.

9.4 GUMDescription of measuring deviceCoordinate measuring machine SIP CMM5.Procedure of measurementGranite square. The angle "A" between corresponding LS fitting lines was measuredin four positions in XY measuring plane of the machine in order to eliminatesquareness and straightness deviation of the machine axes. In every position theangle "A" was measured five times. The LS fitting lines were measured in two ways.Firstly by probing points using special program for line measurement. Secondly byprobing points using scanning mode. The density of measuring points on each linewas about 0,5 mm in both used methods of measurement. These two methods ofmeasurement were used in order to compare obtained results.The cylindrical square was measured in vertical position in ZX measuring plane ofthe machine. It was placed on the granite surface plate which was set on measuringtable of the machine. This surface plate established a measuring basis. Thecorresponding LS fitting lines were measured using two methods of measurement(in the same way as during measurement of granite square). Firstly by probingpoints using special program for line measurement. Secondly by probing pointsusing scanning mode. The density of measuring points on each line was about 0,5mm in both used methods of measurement. The angles between the respective LSfitting lines (0°, 90°, 180°, 270°) and envelope plane of the basis were measured fivetimes. The reference system was established according to the axis of the cylindricalsquare. These two methods of measurement were used for comparison of theresults.

Procedure of result calculation:Granite square:In each measuring position the value of the angle "A" was calculated as a meanvalue obtained from five measurements. The calculations were carried out for thetwo methods of measurement. The angle "A" between the respective LS fitting lines


was calculated as a mean value obtained from four positions of the granite squareon the measuring table of the machine. Calculations for the two methods ofmeasurement were done separately.As regards the local straightness deviation the corresponding LS fitting lines wereestimated from all measured points probed only in scanning mode because thesoftware of the machine does not show the values of points using special programfor line measurement. Then the distances between the respective points andcorresponding points lying on these lines were calculated. As a result the worstdeviations were chosen from all measured lines for each respective LS fitting line.Cylindrical square:The angles between the respective LS fitting lines (0°, 90°, 180°, 270°) and thebasis were calculated as a mean value obtained from all measurements made forthe each line. The calculations were carried out for the two methods ofmeasurement separately.As regards the local straightness deviation of the respective LS fitting line thecalculations were carried out in the same way as in case of granite square. As aresult the worst deviations from the respective LS fitting lines were chosen from allmeasured lines.

9.5 MIRS (SMIS)Description of measuring device:Coordinate measuring machine CMM Zeiss UMC 850.Procedure of measurement: - Square:The granite square was measured in horizontal position (plane XY of CMM). It wasput on the CMM table. Line AC was positioned in Y axis direction and served as abasis element for the coordinate system transformation (from CMM to the square).The square was positioned very precisely along Y axis (physically).The angle wasmeasured in positions A (probing direction for line AB: +Y) and B (probing directionfor line AB: -Y). The line AC was in both cases probed in –X direction. When thesquare was turned from position A to B, the line AC remained in the same position.The result was calculated as a mean value of both measured angles. The wholemeasurement was repeated 4 times and the mean value was calculated.Straightness was measured separately. Lines AC and AB were probed in twoopposite directions (+ X and – X). The beginning and ending points were set asrequired by the instructions. Density of points was 2 mm because the results weretransferred manually to PC and the number was anyway quite high. The result wascalculated as a mean value of point coordinates in position A (probing direction +X)and in position B (probing direction -X).

- Cylinder:The cylinder was measured on the CMM in vertical position. It vas put on a veryprecise Heidenhain granite plate (in fact this is a stand used for measurements with


incremental precise probes). This plane served as an envelope plane of the cylinderbasis. The envelope plane served as a basis element for the coordinate systemtransformation (from CMM to the cylinder). The position of the cylinder regardingmeasurement lines was first positioned as position A.

All lines were measured in this position and angles were calculated (γ0°A, γ90°A, γ180°A,γ270°A). After that the cylinder was turned into position B.

The lines were measured again and the angles (γ0°B, γ90°B, γ180°B, γ270°B) werecalculated. The result was calculated as a mean value of both measured angles (Aand B).Straightness of lines was measured separately. The cylinder was put in horizontalposition along Y axis and each line was measured in two positions(position A –probing direction –X, position B – probing direction +X). The beginning and endingpoints were set as required by the instructions. Density of points was 2 mm from thesame reason as by the square. The result was calculated as a mean value of pointcoordinates in position A and in position B.

9.6 MIKESDescription of measuring device:The straightness was measured by using the vertical movement of Talyrond 262cylindricity measuring machine made by Rank Taylor Hobson Ltd. The verticalmovement “L” is 510 mm and the horizontal movement is 200 mm. The straightnessdeviation of the vertical movement is 630 nm and the repeatability of the movementis 65 nm. The machine is mostly used for roundness and cylindricity measurements.The squareness was measured with squareness measuring machine made byMikes. Main parts of the machine are: the surface plate 1600 x 800 mm (made byMikes), the granite square 1000 x 600 x 100 mm (made by Planolith), the unit formovements “L” 1000 mm and “y” 250 mm (made by IF Werner GmbH) and severalinductive sensors (made by Tesa Sa). The flatness of the surface plate is 2 µm(limited area), The flatness of the granite square is on 1000x100 mm surface 3,7 µmand on 600x100 surface 2,4 µm, but the straightness on the used area is (of surface1000x100 mm) is 0,5 µm. The deviation of the angle between surface plate andgranite square is 1,5” (on the used area). The machine is used for calibration ofsquares.

Procedure of measurement:The straightness was measured by comparison using the straightness of theTalyrond 262 vertical movement as reference.The “squareness” was measured by repeating the measurements when thestandard has been turned 180° with the squareness measuring machine.

Procedure of result calculation:


The mean of three straightness results has first be corrected with the actual valuesof the vertical movement and then fixed to the 0,5 mm division. From this values theLS-regression line has been calculated and used as reference to the given values.The squareness has been calculated using a mean of five measurements for eachmeasured surface (granite square lines A-B and C-D and from the cylindricalstandard the lines 0°, 90°, 180° and 270°). From the lines only the direction of theLS-regression line is used. The results of this calculations has been corrected withthe difference of “diameter”.

9.7 IPQThe description of measurement system is not described here, because this NMIasked to be excluded from the project.

9.8 BNM-LNEDescription of measuring device:3D Measuring machine SIP CMM5, equipped with measuring probe head.The machine is traceable through laser interferometer for X and Y axis.Perpendicularity of X axis versus Y axis is checked with a ball plate and results aretaken into account in the software.Procedure of measurement (for granite square):The probed points are situated on a line which is situated in the middle of each face.The square have been measured in 4 positions.Procedure of measurement (for cylindrical square):391 points on each generatrix line and 24 on the face.The cylinder have been measured in 2 positions, first with the axis in X machinedirection and second with the axis in Y machine direction.

Procedure of result calculation:For granit squareThe result is the mean value of the four measurements.Angle are given by the angle between the LS linesFor cylindrical squareAngle are the mean value of the 2 measurementsStraightnessNo error separation technique have been employed


9.9 SPDescription of measuring device:Straightness measurements: A form-measuring instrument, type Tr 260,manufactured by Rank Taylor-Hobson. The pillar is 500 mm (max L) and on highresolution the pick-up’s measuring range is ±200 µm with resolution 12 nm.Straight angle measurements: Comparison against a reference angle standardusing a height-measuring instrument in combination with a dial-gauge. In this casethe height-measuring instrument is used only as a vertical column holding the dial-gauge, which measures the deviations in interest.

Procedure of measurement (for granite square):Cylindrical straightness measurements: An error separation technique is used withTr 260; each line along the cylinder’s surface is measured twice with the cylinderrotated 180 degrees between each measurement. By combining the twomeasurements, the straightness deviation of the instrument is removed.Straightness measurements on granite square: Using Tr260, the measured profilesare corrected by subtracting the instruments form error (obtained from calibrationwith a 500 mm cylinder standard and error separation).Deviation from straight angle: Comparison with reference angle standard in twoheights, 10 mm and 390 mm. Several measurements have been made, with theobject and reference placed in different positions on the plane table. A typicalmeasurement series consists of three repeated measurements on the referencesquare, three measurements on the object and finally back to the reference again.In order to reduce effects from local surface variations, a mean value of the dial-gauge’s readings over a 1 mm distance was used.

Procedure of result calculation:Cylindrical straightness measurements: The instrument’s straightness error isremoved by, for each line (0°, 90° etc.), taking the sum of two unfiltered straightnesscurves measured in different directions (rotated 180° in between) and then dividethis sum by 2. Since the measuring points are separated by only approximately 0,15mm, the reported straightness profiles have been obtained taking the values nearestevery whole 0,5 mm. Finally the profiles have been aligned using a least square fit.Straightness measurements on granite square: The measured profiles have beencorrected by subtracting the instrument’s deviation from straightness in theappropriate measurement range (5 mm to 295 mm or 5 mm to 495 mm). Themeasuring points are separated by only approximately 0,10 mm – 0,20 mm so thereported straightness profiles have been obtained taking the values nearest everywhole 0,5 mm point. Finally the profiles have been aligned using a least square fit.Deviation from straight angle: The following method has been used for both thecylinder and the granite square.The calibration certificate for the reference square (400x300x55 mm) expresses thedeviation from straight angle as the angle between “envelope – LS regression”.When the reference square is standing on a plane, this angle correspond to certain


deviations (expressed in µm) from a perfect straight angle, depending on the heightabove the table and the straightness profile of the reference square. Since themeasurements are performed by comparing measured deviations (using a “height-measuring instrument” in combination with a dial-gauge) at 10 mm and 390 mm,basically only these two height are of interest.Below, 10 mm and 390 mm are referring to the measured deviations in theseheights:X (µm) = Object (390 mm – 10 mm) – Reference (390 mm – 10 mm)X is then translated to the object’s “envelope - LS regression”-angle by taking intoaccount the local straightness profiles of both the object and the reference.

9.10 OMHDescription of measuring device:Three coordinate measuring machine SIP CMM5

Procedure of measurement:No special corrections for the machine parameters were applied (as it is measuredfor normal customers).Granite:The planes involving AC and AB were probed. The intersection line of the 2 planeswas used for spatial alignment, the normal vector of the plane AB for the planaralignment and the intersection between the mentioned line and the upper surfacefor 0 setting.The granite was measured in the position as indicated on Fig. 1 of the writtenprocedure (AC in x direction, AB in y).There was not enough time to measure the standards in reversed positions but thestraightness error of the machine is within the value of max.0,4 µm measured bylaserinterferometer and the perpendicular error of the x-y axis is less than 0,3”according to previous measurements.These values are taken into account in the uncertainty determination.The measurement was repeated 5 times. The mean value of 5 is given as the result.Cylinder:The plane was probed and its normal vector was defined as the element for spatialalignment (x axis) and for x=0. The cylinder was taken from 6 points and used for 0setting on y and z.The cylinder gauge was measured first in the position where 90-0-270 could beprobed and after with rotating the cylinder by 180° in the position where 270-180-90could be measured. With this method a rough estimation could be taken (for 90 and180) if the deviation comes from the straightness of the machine or of the cylinder.The measurements were repeated 5 times.


Procedure of result calculation:The final results were calculated as the mean of 5 measurements. To determine therepeatibility of the measuring process, the standard deviation was calculated andused.No special correction was taken like in the normal procedure for customers.The regression line and the individual straightness deviation were calculated by theConcerto software.

9.11 CMIDescription of measuring device:Three coordinate measuring machine SIP CMM5

Procedure of measurement:

Granit square: 5 times 2 measurements 180° rotated to eliminate systematic errors

Steel cylinder: 3 times 4 measurements 90° rotated to eliminate systematic errors

Procedure of result calculation:Straightness: pointwise average from all pairAngle: average from all pair

9.12 NMI VSLDescription of measuring device:Coordinate Measuring Machine Zeiss UC550, Measuring volume 1200x550x450mm

Procedure of measurement:Generally speaking, our CMC calibration (quality system) does not include theoption of a local straightness analysis for customers’ calibrations. Instead of themeasured profile or its residuals to the LS-line, our software returns one value forthe global straightness only.Therefore, the profile/residuals presented with our work were sampledindependently from the normal calibration procedure in order to still allowcomparison.Granite square: The two relevant sides of the granite square are probed with theordinary ball probe of the CMM. The calibration is performed 12 times applying thereversal method, eliminating the systematic deviation of the CMM. The


measurement and its results are completely done with the software of the CMM.First the basis line (short arm) is probed and a LS-line fitted to it. Then the long armis probed relative to the LS-line of the short arm. The software then automaticallyreturns the global straightness of the long arm and the angle between the LS-linesthrough the long arm and the basis line.In addition and for verification purposes only, the electronic level is applied todetermine the straightness of the long arm.Cylindrical square: An electronic level with a pitch of 20mm is attached to the CMMas a probe. It is applied to measure the straightness of the cylinder. At eachmeasurement point, the slope of the level is recorded. The distance between themeasurement points equals the pitch of the level, 20 mm. The profile is thensequentially reconstructed by multiplying the slope with the pitch and adding it to theprevious profile height.The angle of squareness is determined by the angle between the LS-line throughthe straightness profile and the LS-plane through the basis plane of the cylinder,see 3.1.2 for details. The basis plane of the cylinder is measured with an ordinaryCMM ball probe and the CMM software fits an LS-plane to the basis plane. Relativeto this LS-plane, the first and the last point of the profile are probed with the stylus.Applying the reversal method and changing the position and orientation of thesquare a total of 8 times, each profile is measured 4 times. The systematic deviationof the CMM is thus eliminated.

Procedure of result calculation:Granit square: The basis line (short arm) is probed at 15 points with 5 mm spacing.The CMM software then fits a LS-line through these points and adjusts itscoordinate system so that this line becomes the new x-axis. Then the long arm isprobed at 24 points with 5 mm spacing, relative to the new coordinate system. Thesoftware of the CMM determines then the global straightness of the sides as well asthe angle through the least-squares lines through the two profiles, see also “5.Comments”. This measurement is repeated 12 times at different positions andorientations of the square, using the reversal method.Cylindrical square: Applying the reversal method and changing the position andorientation of the square a total of 8 times, each profile line is measured 4 times.

10 Results

In the following analyses the individual NMIs were given in the order corresponding tothe real time schedule. IPG asked to exclude their results from the measurementanalysis.After the first evaluation at the pilot laboratory, unexpectedly large deviationscorresponding to the angles and the shape of profile lines were found. Because of this,all participants were asked to check the provided data. After collecting the correcteddata, they were analysed again.


10.1 Granite square

10.1.1 StraightnessSome NMIs provided data with random disturbances, as e.g. bounce of measuring tip,dirt etc. These were for calculation of reference profile eliminated if the local deviationexceeded twice the value corresponding to the dispersion of remaining points aroundthe LS regression line.As the consequence of the surface structure of the granite square, the P-V value ofshort-period waviness is approximately 12 µm. For this reason, this component waseliminated from the measured profiles by the filtration using the drifting arithmeticmean with length of 5 mm. These modified data were used just for the calculation ofthe most conceivable middle profile (reference profile).Since not all NMIs measured with step 0,5 mm, the linear interpolation of the measuredpoints with increment of 0,5 mm has been carried out (with the exception of theNMi VSL).At the PTB, the local straightness was defined by the following way: the zero point ofthe coordinate system was located to the centre of profile line and in the case of profileAB the orientation of the L coordinate was opposite to other participants (direction BA).Exact position of the zero point with respect to its definition according to chapter 6 wasnot given. Since the length measured at PTB was smaller than its original length, thecoordinator after agreement PTB transformed the first measured point to theintersection of the profile with the bevel edge.By means of graphical comparison of the shape of profile lines there was found, thatthe „L“ coordinate stated at MIKES was in fact shorter by approximately 15 mm. Forthis reason, the multiplication coefficient k = 1,040107 was applied to all points of theprofile AC. In the case of profile line AB, the „L“ coordinate was correct.The weighted mean profile (reference profile) was calculated, applying the weightingcoefficients

( ) 2−= NMI

cuw (1)

The profile of NMi VSL was not contributing to the weighted mean, since it did notcorrespond to the conditions according to Chapter 6 (distances between adjacentmeasured points were 20 mm) and the profile AC was not measured at all (the profileof NMi VSL for the granite square was supplied as a bonus for verification only).Uncertainties of reference profiles:

mean of uC = 0,03 µm (581 values) max. of uC = 0,09 µmFrom the individual NMi profiles, without filtration, modified by interpolation only - step0,5 mm (only in PTB by means of transformation and in MIKES by correction of lengthgauge), there were calculated the AB and AC profiles deviations from the referenceprofileSince the profile lines provided by GUM seem to be too far from the weighted meanand their shape is very different from the mean profile, there is possible to handle withthem as they are out of tolerance (application of the Grubbs test for local straightnessis due to the input data too complicated). Similarly, the profiles measured at NMi VSL


were not included in the calculation of weighted mean, from the reason mentionedabove.For this reason was the weighted mean determined again, but after the exclusion ofthe mentioned NMIs.The profiles measured by the individual NMIs, reference (weighted mean) profile anddeviations from the reference profile are shown in the following graphs. The scale if itpossible is uniformed.Profiles, measured by the individual NMIs:


Reference profile:

Deviations from the reference profile for profile AB:


Analogically, for the profile AC:


Comments to the graphs:The scale was chosen uniform. In order to consider the stability of devices, evaluatedwere all (incl. repeated) measurements of NMIs. In the case of METAS (themeasurements of cylinder square were not completed due to the problems withmeasuring device, in the second stage the repeating of measurements of bothstandards) the differences fit into uC. In the case of SMU, the differences were causedby the replacement of axial inductive probe by the lever type probe in 2002.Kinematics properties of axial probe combined with elasticity of their holders causedthe additive deviations of indicated profile. The measurements (SMU 1 and METAS 1from year 2000) were not included to calculation of the reference profile.

10.1.2 SquarenessMeasuring facilities of individual NMIs are based on two used definitions of regressionprofile lines („envelope – LS regression“ and „LS regression – LS regression“). For thesake of uniformity, the angle between envelope line and LS regression line of line ACwas determined. The value of this angle has been estimated from the resultingreference profile-graphs of AC profile:

γLS - γB = -0,07”.The measurements (SMU 1 and METAS 1 from year 2000) were not included tocalculation of the reference profile. Applying this correction, all results weretransformed to the system „LS regression – LS regression“ and the mean value ofangle was calculated, using the weighting coefficient „w“ the weighted mean value of


angle and deviations of γLS from it. Since the values of γLS were differing significantly,the test of statistic consistence of data by means of Birge ratio (see Annex A) wascarried out. The calculated value RB = 1,39 is nearly equal to the critical value (forcoverage factor of k = 2) RB

crit = 1,34. Therefore the Grubbs test has been applied (seeAnnex A) both to the values γLS and to the differences from the weighted mean. On thebasis of such analysis there is obvious, that values of three NMI MIKES are possible tobe considered as outliers. After the exclusion of results of these NMI, the weightedmean was calculated again. The change of weighted mean (0,03”) was after exclusionof these NMIs negligible. Results of this analysis are contained in the following table:


Angle between:LS regression-fitting line and basis (γB)or LS regression-fitting line (γLS) (”):

γLS − γB = -0,07 differencefrom NMI: after normalisation: Meas. Device γ - (weighted mean)

Grubbs testfor:

γBasis γLSAngle

uc

γB (")

γLS (")

uc (")

straight.uc

(µm)Angle Straightness diff I

γLS

diff IIγLS

γLS diff I

SMU 2000 -1,60 0,20 -1,60 -1,53 0,20 0,10 Square-device Square-device -0,12 -0,08 -0,509 -0,323METAS horiz.Dec. 2000

89° 59’ 58.44” 0.12 -1,56 0,12 0,08 Rotary Table linear guideway -0,15 -0,11 -0,592 -0,406

PTB -1,55 0,07 -1,55 0,07 0,08 90° Angle block linear guideway -0,14 -0,10 -0,565 -0,380GUM 89,99956° 0,24 -1,58 0,24 0,34 CMM CMM -0,18 -0,14 -0,656 -0,471MIRS 89° 59´ 58,4” 0,45 -1,60 0,45 0,30 CMM CMM -0,19 -0,15 -0,699 -0,514MIKES -0,35 0,3 -0,35 -0,28 0,30 0,31 Square-device Talyrond 262 1,13 1,17 2,853 3,039METAS horiz.Aug. 2001

89° 59’ 58.66” 0.12 -1,34 0,12 0,08 Rotary Table linear guideway 0,07 0,11 0,000 0,186

METAS vert.Aug. 2001

89° 59’ 58.73” 0.13 -1,27 -1,20 0,13 0,12 Square-device Square-device 0,21 0,25 0,377 0,562

BNM-LNE 89° 59’ 58.52’’ 0,4 -1,48 0,40 0,60 CMM CMM -0,07 -0,03 -0,376 -0,191SP -1,48 0,31 -1,48 -1,41 0,31 0,25 Ref. Square RTH Tr260 0,00 0,04 -0,188 -0,003OMH 89,99975° 0,5 -0,90 0,50 0,50 CMM CMM 0,51 0,55 1,185 1,370CMI 89,99959° 0.00017° -1,48 0,61 0,28 CMM CMM -0,07 -0,03 -0,366 -0,180NMi VSL 89° 59’ 58.48” 0,67 -1,52 0,67 0,13 CMM Bridge 20mm -0,11 -0,07 -0,484 -0,299SMU 2002 -1,40 0,20 -1,40 -1,33 0,20 0,10 Square-device Square-device 0,08 0,11 0,020 0,205

Birge ratio RB = 1,39 1,38mean (excl. SMU 2000, METAS 2000) -1,31 RB (crit.) = 1,34 Gp 5% 2,507

weighted mean I (excl. SMU 2000, METAS 2000) -1,41 Gp 1% 2,755weighted mean II (excl. SMU 2000, METAS 2000, MIKES) -1,44

uint 0,04uxt 0,06


Angle deviations of individual NMIs from the weighted mean are shown in the followinggraph:

Comments to the graph:Angle deviations of first 5 NMIs compared to the values of remaining participantsindicate the suspect of change of measurand itself, i.e. the change of angle ofmeasured artefact (casual change of profile as the consequence of the contact with thebase plate of measuring facilities). Nevertheless, after the analysis of profile lines ofboth arms at the beginning and at the end of comparison no significant change couldbe revealed, which could support this hypothesis. Analysing the data of individualNMIs, for MIRS there must be taken into account that angles and local straightnessdeviation are independent (they were measured separately).

10.2 Cylinder Square

10.2.1 StraightnessSimilarly as in the case of granite square, some NMIs provided data incl. the randomdisturbations, as e.g. bounces of measuring tip, dirt etc. These were excluded from thecalculation of reference profile if the local deviation exceeded twice the valuecorresponding to the dispersion of remaining points around the LS regression line.As the consequence of the artefact surface finishing, the P-V value of short-periodwaviness of generatrixes of the cylinder is approximately 1,5 µm. For this reason, theshort-periodicity component was eliminated by means of drifting arithmetic mean with


length of 5 mm. These modified data were used just for the calculation of the mostconceivable middle profile (reference profile).Since not all NMIs measured with step 0,5 mm, the linear interpolation of the measuredpoints with increment of 0,5 mm has been carried out (with the exception of theNMi VSL).From the comparison of graphs there was found, that „L“ coordinate MIKES wasshorter by approximately 15 mm. For this reason, the coefficient k = 1,040107 hasbeen applied for all the measuring points.Similarly as in the case of granite square, applying the weighting „w“ coefficient theweighted mean profiles corresponding to 0°, 90°, 180° and 270° were calculated. Dueto the two-fold measurement at the SMU (SMU 1, SMU 2) and large shape differenceof measured results obtained (GUM, MIRS, CMI), four data sets (SMU 1, GUM, MIRS,CMI) were excluded from the reference profile calculation.After the calculation of the weighted mean, the results of GUM, MIRS and CMI aredeemed to be outlying. Likewise as in the case of granite square, the application ofGrubbs test is too complicated due to the input data. Moreover, from the course oflines is obvious, that at GUM the sign for the line 180° has been changed, at CMI thesign of „L“ coordinate was changed for all lines. Since all three mentioned NMIschecked their values after the information of the pilot and they confirmed data providedbefore, the evaluation was done without change of originally sent data set.The profiles, measured by the individual NMIs, reference (weighted mean) profile anddeviations from the reference profile, are shown in the following graphs. The scale if itpossible is uniformed.Uncertainties of reference profiles:

mean of uC = 0,02 µm (792 values) max. of uC = 0,10 µm


Profile, measured by the individual NMIs.Profile 0°:


Profile 90°:


Profile 180°:


Profile 270°:


Reference profile:

Deviations from the reference profile for profile 0°:


10.2.2 SquarenessCorrespondingly as for the granite square, the mean value of angles of LS regressionlines with envelope basis plane was calculated, using the “w” coefficient and thedeviations of γ from it.

Since the values of γ were too differing each other, the test of statistic consistence ofdata by means of Birge ratio (see Annex A) was carried out. The calculated valueRB = (4,30; 1,10; 4,26; 1,10) is greater than critical value (for coverage factor of k = 2)RB

crit = 1,38. Therefore the Grubbs test was applied (see Annex A) both for the valuesof γ and for the differences from the weighted mean. From the comparison of resultsof both tests there is obvious, that values of angles provided by GUM and MIRS canbe considered as outlying. After the exclusion of results of SMU 2000 (SMU 1), GUMand MIRS, the weighted mean value was calculated repeatedly. Results of thisanalysis are given in the following table:


Angle between:LS regression-fitting line and basis

from NMI: after normalisations: Meas. Device

NMI γ0° γ90° γ180° γ270° Angle

uc γ0° γ90° γ180° γ270°

uc (")

straightuc

(µm)Angle Straightness

SMU 2000 2,08 -0,1 -2,81 -0,35 0,2 2,08 -0,1 -2,81 -0,35 0,2 0,1 Square-device Square-devicePTB 2,8 -0,2 -3,2 -0,1 0,5 2,8 -0,2 -3,2 -0,1 0,5 0,05 MFU 8 - PTB MFU 8 - PTBGUM 90,00095o 90,0004° 89,99950o 90,00025o 0,56 3,42 1,44 -1,8 0,9 0,56 0,48 CMM CMMMIRS 89° 59´ 56,4” 90° 0´ 0” 90° 0´ 2,9" 89° 59´ 59,3” 0,45 -3,6 0 2,9 -0,7 0,45 0,3 CMM CMMMIKES 2,4 0,5 -2,2 -0,4 0,29 2,4 0,5 -2,2 -0,4 0,29 0,12 Square-device Talyrond 262METAS vert. 90° 0’ 2.55” 90° 0’ 0” 89° 59’ 57.02” 89° 59’ 59.62” 0,22 2,55 0 -2,98 -0,38 0,22 0,1 Square-device Square-deviceBNM-LNE 90° 00' 02,0" 90° 00' 00,4" 89° 59' 57,4" 89° 59' 59,4" 1 2 0,4 -2,6 -0,6 1 0,6 CMM CMMSP 2,44 -0,09 -3,31 -1,13 0,39 2,44 -0,09 -3,31 -1,13 0,39 0,13 Ref. Square RTH Tr260OMH 89,99989° 90,00001° 89,99969° 90,00004° 0,8 -0,396 0,036 -1,116 0,144 0,8 0,6 CMM CMMCMI 90,00057° 90,00015° 89,99957° 89,99999° 0,00014° 2,052 0,54 -1,548 -0,036 0,504 0,29 CMM CMMNMi VSL 90° 00’ 02.46” 89° 59’ 59.99” 89° 59’ 57.35” 90° 00’ 00.07” 0,75 2,46 -0,01 -2,65 0,07 0,75 0,07 CMM Bridge 20mm & levelSMU 2002 2,35 -0,13 -3,25 -0,60 0,2 2,35 -0,13 -3,25 -0,60 0,20 0,1 Square-device Square-device

mean 1,87 0,22 -2,23 -0,29weighted mean I (excl. SMU2000) 2,06 0,06 -2,56 -0,41

weighted mean II (excl. SMU2000, GUM, MIRS) 2,368 0,051 -2,863 -0,467uint 0,10uext 0.41 0.11 0.41 0.11


difference: γ - weighted mean I Grubbs test difference II: γ - weighted mean II

NMI dif γ0° dif γ90° dif γ180° dif γ270° for

γLS 0

forγLS 90

forγLS 180

forγLS 270

for diff0°

for diff90°

for diff180°

for diff270°

diffγ0°

diffγ90°

diffγ180°

diffγ270°

SMU 2000 0,021 -0,201 -0,324 0,079 c -0,750 -0,371 -0,131 0,011 -0,421 -0,189 0,155 -0,288 -0,151 0,053 0,117PTB 0,741 -0,301 -0,714 0,329 0,501 -0,966 -0,599 0,355 0,390 -0,631 -0,418 0,642 0,432 -0,251 -0,337 0,367GUM 1,356 1,385 0,760 1,311 0,827 2,563 0,220 2,298 0,713 2,906 0,445 2,556 1,052 1,389 1,063 1,367MIRS -5,659 -0,101 5,386 -0,271 -2,865 -0,535 2,968 -0,811 -2,978 -0,211 3,153 -0,527 -5,968 -0,051 5,763 -0,233MIKES 0,341 0,399 0,286 0,029 0,291 0,541 -0,014 -0,228 0,179 0,838 0,168 0,057 0,032 0,449 0,663 0,067METAS vert. 0,491 -0,101 -0,494 0,049 0,370 -0,535 -0,470 -0,189 0,258 -0,211 -0,289 0,096 0,182 -0,051 -0,117 0,087BNM-LNE -0,059 0,299 -0,114 -0,171 0,080 0,325 -0,248 -0,616 -0,031 0,628 -0,066 -0,332 -0,368 0,349 0,263 -0,133SP 0,376 -0,145 -0,750 -0,719 0,312 -0,729 -0,663 -1,646 0,198 -0,304 -0,439 -1,401 0,072 -0,141 -0,447 -0,663OMH -2,455 -0,065 1,370 0,573 -1,180 -0,458 0,620 0,829 -1,292 -0,135 0,802 1,118 -2,764 -0,015 1,747 0,611CMI -0,012 0,485 1,012 0,375 0,108 0,627 0,367 0,479 -0,006 1,018 0,593 0,732 -0,316 0,489 1,315 0,431NMi VSL 0,401 -0,111 -0,164 0,499 0,322 -0,557 -0,277 0,685 0,211 -0,232 -0,096 0,973 0,092 -0,061 0,213 0,537SMU 2002 0,287 -0,229 -0,766 -0,166 0,262 -0,811 -0,630 -0,607 0,151 -0,480 -0,449 -0,323 -0,022 -0,180 -0,390 -0,128

s = 1,901 0,465 1,710 0,515 1,90 0,477 1,709 0,513Birge ratio RB = 4,30 1,10 4,26 1,10 Gp 5% 2,412

RB (crit.) = 1,38 Gp 1% 2,636


Angle deviations of individual NMIs from the weighted mean are in the following graphs, the scale is uniform:


Comment to graphs:Both NMIs are using CMM. It would be interesting to analyse the measurementprocedure with respect of the equal error compensation of CMM for the X and Y axes.Analysing the results of individual NMIs, it should be kept in mind, that at MIRS theangles and straightness deviation were measured separately. The graphs indicate anunderestimation of uncertainties by some NMIs, which caused a deviation of thereference value.

11 Conclusion

In order to compare the individual deviations mutually (25 profiles for the granit squareand 44 profiles for the cylinder) the graphical illustration of “standard deviations“ andboth extreme values (max. and min.) of deviations was created.

Granit square, table of angels and profiles:

NMI diff.γLS

Grubbstest

diff. γLS

sdAB

MinAB

MaxAB

sdAC

MinAC

MaxAC

SMU 2000 -0,08 -0,33 0,18 -0,48 0,48 0,07 -0,21 0,27METAS horizontal 1 -0,11 -0,41 0,09 -0,29 0,35 0,06 -0,28 0,22PTB -0,10 -0,39 0,13 -0,52 0,54 0,10 -0,32 0,41GUM -0,14 -0,48 0,44 -1,62 1,11 0,47 -1,25 1,01MIRS -0,15 -0,52 0,21 -1,35 0,73 0,23 -0,58 0,86MIKES 1,17 3,09 0,24 -0,89 1,00 0,22 -0,80 0,53METAS horizontal 2 0,11 0,19 0,07 -0,32 0,27 0,05 -0,23 0,19METAS vertical 0,25 0,57 0,15 -0,62 0,53BNM-LNE -0,03 -0,19 0,20 -0,69 0,74 0,16 -0,54 0,49SP 0,04 0,00 0,20 -0,80 0,68 0,18 -0,55 0,58OMH 0,55 1,39 0,25 -0,79 0,70 0,18 -0,50 0,50CMI -0,03 -0,18 0,24 -0,80 0,83 0,18 -0,84 0,60Nmi VSL -0,07 -0,30SMU 2002 0,11 0,21 0,11 -0,42 0,33 0,06 -0,21 0,21

Gp 5%= 2,51


Granit square, graphs of profiles:


Granit square, graphs of angels:

Cylinder square, table of profiles:

NMI sd0°

Min0°

Max0°

sd90°

Min90°

Max90°

sd180°

Min180°

Max180°

sd270°

Min270°

Max270°

SMU 2000 0,09 -0,13 0,48 0,10 -0,15 0,50 0,16 -0,24 0,70 0,15 -0,23 0,55PTB 0,05 -0,17 0,15 0,02 -0,18 0,06 0,03 -0,23 0,07 0,02 -0,18 0,06GUM 0,39 -1,80 0,71 0,40 -1,66 0,91 1,15 -1,77 2,66 0,80 -1,33 2,27SMIS 0,86 -2,51 1,65 0,35 -1,00 0,80 0,70 -1,41 1,53 0,79 -1,67 1,56MIKES 0,05 -0,17 0,12 0,05 -0,17 0,10 0,06 -0,17 0,16 0,04 -0,20 0,11METAS vertical 0,02 -0,12 0,05 0,02 -0,09 0,15 0,02 -0,13 0,06 0,02 -0,07 0,05BNM-LNE 0,10 -0,31 0,34 0,10 -0,40 0,26 0,11 -0,45 0,38 0,13 -0,43 0,33SP 0,04 -0,14 0,09 0,04 -0,11 0,09 0,03 -0,10 0,07 0,04 -0,13 0,11OMH 0,17 -0,51 0,40 0,09 -0,28 0,30 0,17 -0,43 0,54 0,11 -0,41 0,27CMI 0,33 -0,88 0,94 0,30 -0,87 0,96 0,34 -1,12 0,91 0,21 -0,61 0,61SMU 2002 0,05 -0,08 0,16 0,04 -0,08 0,15 0,06 -0,10 0,20 0,04 -0,08 0,21


Cylinder square, graphs of profiles:


Cylinder square, table of angles:

NMI diff IIγ0°

diff IIγ90°

diff IIγ180°

diff IIγ270°

Grubbstest

diff. 0°

Grubbstest

diff. 90°

Grubbstestdiff.180°

Grubbstest

diff. 270°

SMU 2000 -0,29 -0,15 0,05 0,12 0,01 -0,42 -0,19 0,15PTB 0,43 -0,25 -0,34 0,37 0,39 -0,63 -0,42 0,64GUM 1,05 1,39 1,06 1,37 0,71 2,91 0,45 2,56MIRS -5,97 -0,05 5,76 -0,23 -2,98 -0,21 3,15 -0,53MIKES 0,03 0,45 0,66 0,07 0,18 0,84 0,17 0,06METAS vertical 0,18 -0,05 -0,12 0,09 0,26 -0,21 -0,29 0,10BNM-LNE -0,37 0,35 0,26 -0,13 -0,03 0,63 -0,07 -0,33SP 0,07 -0,14 -0,45 -0,66 0,20 -0,30 -0,44 -1,40OMH -2,76 -0,02 1,75 0,61 -1,29 -0,14 0,80 1,12CMI -0,32 0,49 1,32 0,43 -0,01 1,02 0,59 0,73NMi VSL 0,09 -0,06 0,21 0,54 0,21 -0,23 -0,10 0,97SMU 2002 -0,02 -0,18 -0,39 -0,13 0,15 -0,48 -0,45 -0,32

Gp 5% = 2,41

Cylinder square, graphs of angles:


This regional supplementary comparison was the first comparison in this field. It hasprovided the independent information about metrological properties of measuringequipment and method comparing to participated NMIs.Comparison results of squareness were in the good conformity with probable values ofstandards for most of participated laboratories. Some NMIs probably did not include allthe possible error sources into the calculation and therefore their uncertainties weretoo small in relation to the deviations from the reference.Shape of profile lines provided by some NMIs was significantly different from theweighted mean. The reason could be in the insufficient compensation of the errors ofguiding probing part in the measuring procedure.This comparison provided the information about state of metrological servicesprovision in the field of big squares measurement. For the sake of effectiveexploatation of the energy demanded by this comparison, it has to be followed by theanalysis of influencing factors in each participating NMI and revision of this category inAppendix C MRA.Coordinator is thankful to all participants for the good cooperation.


ANNEX A

A.1 Consistency of results and outliers

Some reported measurement result seemed to be inconsistent with other results, andcould change the reference values.

A.1 Arithmetic mean

The arithmetic mean reference value xref was calculated by the average of allmeasurement values xi :

∑=

=n

iiref x

nx

1

1 (1)

The arithmetic mean does not take into account the uncertainty of the individual resultscontributing to the reference value. For a relatively small number of participants,results with large deviations, but still not to be considered as outliers, can stronglyinfluence the mean.The standard uncertainty u(xref) of the arithmetic mean can either be determined byapplication of the error propagation law, i.e. by taking into account the uncertaintiesu(xi) of the individual results [Eq. (2)], or by the spread of the results, i.e. by thestandard deviation divided by the square root of the number n of results contributing tothe mean [Eq. (A3)].

nu

xun

xu rmsn

iiref == ∑

=1

2 )(1)( (2)

or

∑=

−−

=n

irefiref xx

nnxu

1

2

11 )(

)()( (3)

A.2 Statistical consistency

The statistical consistency of a comparison can be investigated by the so-called Birgeratio2 RB , which compares the observed spread of the results with the spreadexpected from the individual reported uncertainties.

2 Statistical Analysis of Interlaboratory Comparisons, EUROMET workshop held at

NPL on 11.-12. November 1999,http://www.npl.co.uk/ssfm/download/documents/sss_m_00_173.pdf

http://www.npl.co.uk/ssfm/download/documents/sss_m_00_173.pdf


The application of least squares algorithms and the χ2-test leads to the Birge ratio

in

extB u

uR = (4)

where uin, the internal standard deviation, is given by the reported uncertainties2/1

1

2 )(−

=

−

= ∑

n

iiin xuu (5)

and the external standard deviation uext is the standard deviation of the spread of theresults xi, weighted by the associated uncertainties u(xi):

2/1

2

22

)()1())((

−−

=∑

∑−

−

i

wiiext xun

xxxuu (6)

wx is the weighted mean given by

∑

∑

=

−

=

− ⋅= n

ii

n

iii

wxu

xxux

1

2

1

2

)(

)((7)

The Birge ratio has an expectation value of RB = 1. For a coverage factor of k = 2, thedata in a comparison are consistent provided that

)1/(81 −+= nRcritB (8)

who n = number of participating laboratoriesA value RB > RB

crit may be interpreted such that the laboratories have underestimatedtheir uncertainties.

A.3 Weighted mean

The weighted mean reference value xref was calculated by the mean of allmeasurement values xi weighted by the inverse square of the standard uncertaintiesu(xi) associated with the measurements.

∑

∑

=

−

=

− ⋅= n

ii

n

iii

refxu

xxux

1

2

1

2

)(

)((9)

The weighted mean approach requires the individual uncertainties from thelaboratories can be estimated according to a common approach (which should be thecase, since all participants were requested to estimate the uncertainties according tothe ISO Guide). If this is not the case, a single "wrong" value with a stronglyunderestimated (too small) uncertainty could strongly influence or even fullydetermine the weighted mean. On the other hand, a high quality measurement with


overestimated uncertainty would contribute to the reference value only to a smallextent.The standard uncertainty u(xref) of the reference value is calculated either byappropriately combining the individual uncertainties [Eq. (10)], or by the spread of theresults [Eq. (11)], which is identical to the internal and external standard deviationgiven in Eq. (5, 6)

21

1

2/

)()(−

=

−

= ∑

n

iiref xuxu (10)

or

∑∑

−

−

−

−=

)()1(

))(()( 2

22

i

refiiref xun

xxxuxu (11)

It has to be noted that Eqs. (3) and (11) do not result from the law of error propagationand are certainly not in accordance to the GUM. In statistically consistent cases, thesestandard deviations should be approximately equal to the standard uncertaintiesevaluated according to Eqs. (2) and (10), respectively, resulting in a Birge ratio ofapproximately 1 (see section A.1 Statistical consistency).

A.4 Grubbs' test

The Grubbs’ test according to the ISO 5725-2: 20003 was applied for one outlyingobservation.

The test applied on the smallest and largest value of the set of data, xi for i = 1,2, …p.The Grubbs’ statistic is calculated as

( )s

xxG p

p

−= or

( )s

xxG p1−

= where (12)

∑=

=p

iix

px

1

1 and ( )∑=

−−

=p

ii xx

ps

1

2

11 (13)

If the test statistic is less than or equal to 5% of critical value and less than or equal toits 1% of critical value, the item tested is called a straggler. If the test is greater than its1 % of critical value, the item is called a statistical outlier.

3 ISO 5725-2 : Accuracy (trueness and precision) of measurement methods and results – part 2 : Basicmethod for the determination of repeatability and reproductibility of a standard measurement method.


Outliers do not participate in the calculation of the means.

project square final - bipm · file: project_square_final-an.doc / issue 28. 02. 2005 slovenský...

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