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Self-Calibration Method for a Ball Plate Artefact on a CMM
CIRP Annals - Manufacturing Technology 65 (2016) 503–506
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CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Self-Calibration Method for a Ball Plate Artefact on a CMM Anke Guenther a,*, Dirk Sto¨bener b, Gert Goch (1)c a b c
Reishauer AG, Wallisellen, Switzerland Bremen Institute for Metrology, Automation and Quality Science (BIMAQ), University of Bremen, Germany Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Metrology Calibration Gear artefact
The design and the application of a circular ball plate artefact, introduced a couple of years ago, appeared to be very simple. But at the end, a measurement result and a measurement uncertainty can be determined only via NMI calibrated artefacts. Recently, PTB and NMIJ published self-calibration methods for pitch artefacts, one based on the rosette-method. This paper describes the testing and verification of this calibration method, applied to a gear artefact. The method was extended to calibrate not only the pitch position of the balls, but also their radial and height position on the circular ball plate (gear artefact). The concept of this advanced method, experiences regarding its application and test results will be presented. ß 2016
1. Introduction and state-of-the-art Improvements of the gear quality for power transmission systems can be achieved by modifications of the manufacturing processes, the materials and the gear shape. They lead to new requirements for the gear quality inspection, where new probing and evaluation methods with higher resolution and decreasing uncertainty have to be developed [1]. As the functionality of a gear for an automotive power train is significantly influenced by deviations of a few micrometres, the uncertainty of the quality inspection process should be less than 2 mm (preferably substantially less) for profile, helix and pitch deviations [2]. The quality inspection of these deviation parameters is generally carried out using gear measuring instruments (GMI) or coordinate measuring machines (CMM) [2]. The accuracy of these measuring devices is checked and calibrated by measurements of specific gear artefacts, whose uncertainties significantly influence the uncertainty of the measuring process for the gear. Therefore, artefacts for the various measurement tasks (profile, helix and pitch deviations) with the smallest possible uncertainties are required. Commercially available artefacts only partially comply with this requirement [1]. Several authors reported about the use of a circular ball plate as an artefact for gear measurements. Kondo and Mizutani used a ball artefact to calibrate spur gear profile measurements. It consists of two balls mounted onto a metal base plate [3]. Komori et al. presented a magnetically self-aligned multiball pitch artefact [4] and Kondo et al. designed a multiball artefact with ceramic balls, which are attached onto a ceramic cylinder by spring forces [5]. Gu¨nther et al. presented a circular ball plate as bevel gear measurement standard [6]. Generally, these artefacts consist of spheres mounted onto a stable plate in a circular pattern (Fig. 1) * Corresponding author. Tel.: +41 78 9259988. E-mail address: [email protected] (A. Guenther). http://dx.doi.org/10.1016/j.cirp.2016.04.080 0007-8506/ß 2016
and this pattern offers the opportunity to apply reversal or multiple-orientation (self-calibration) techniques for the determination of the artefacts’ parameters and their uncertainties.
Fig. 1. Investigated ball plate as an artefact for gear measurements.
The self-calibration techniques are time consuming, but they enable a separation of the systematic deviations caused by the probing and the positioning system from the deviations of the measured object by multiple measurements [7]. Mark used this approach for the determination of the pitch deviations of a master gear [8] and Osawa et al. presented an application for the calibration of cylindrical workpieces [9]. The National Metrology Institutes of Japan (NMIJ) [10] and Germany (PTB) [11] derived two different methods for the self-calibration of standards for pitch measurements. Both methods use multiple-orientation techniques in combination with a pitch artefact. The PTB approach is based on the rosette-method published by Noch and Steiner [12] and achieves a pitch uncertainty for the tested artefact of 0.4 mm. If a ball plate artefact shall be used for the physical representation of topography points in bevel gear metrology [6]
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A. Guenther et al. / CIRP Annals - Manufacturing Technology 65 (2016) 503–506
or for profile lines in cylindrical gear metrology [3], the knowledge about the relative angular positions (azimuth) of the spheres (as determined in [10–12]) is not sufficient. Deviations of involute profile lines or topographic point grids can be calculated only from the full 3-D-positions of the spheres. In order to achieve uncertainties below 2 mm for the various gear deviation parameters, the uncertainties of the artefact’s ball positions should be below 1 mm for all directions. These small uncertainty values cannot be reached by standard CMM measurements. Therefore self-calibration methods have to be applied, which separate the ball position deviations from the systematic deviations of the measuring system. Hence, the ball position uncertainty budget is independent from these systematic deviations and mainly influenced by non-systematic measurement deviations, which are estimated according to [11]. This paper presents the extension of the rosette-method to the determination of the full 3-D-position parameters of the balls in order to provide a method for the characterisation of ball plates as artefacts for bevel and cylindrical gears. It presents first application results and discusses the influences of position deviations on the determination of gear parameters. 2. Measuring procedure and evaluation method 2.1. Multiple-orientation evaluation algorithm The arrangement of the ball plate (relative positions of calibration spheres to each other) is given and constant. A simple measurement of ball positions would include several deviations of the measurement device (MD). In order to reduce or eliminate these unavoidable deviations, multiple reversal measurements of the ball plate have to be applied (rosette-method). The method is based on the assumption that the measured deviation for each ball’s angular position fp is represented by the superposition (see Eq. (1)) of the deviations of the ball centre positions Afp, the angle deviation of the positioning system for each plate orientation Bfp, and the pitch deviation of the rotary table encoder scale of the CMM Cfp. The sum of deviations is completed by non-systematic error contributions e.
All measurements are evaluated at the same WCS. For several reasons, the evaluation method requires the definition of the WCS by approximating a plane and a circle to all ball centre points at the first angular position of the rotary table. The direction of the z-axis of the WCS was defined by the normal direction of this plane, whereas the centre of the approximated circle determines the origin of the WCS. 2.3. New aspects for the use of the multiple-orientation method In contrast to the published application [11,13], the rosettemethod is not only applied to determine the pitch deviations of the spheres but also their relative radial position fr and height fz deviations. This is possible under the assumption that the observed deviations (from their mean value) of the radial and height positions are (like the pitch deviations) a sum of the deviations caused by the ball plate itself, the positioning deviation for each plate orientation and the deviations induced by the rotary table. Because of the special WCS-definition (see above), the sums of these position deviations in the matrix all become zero, too. Therefore, the use of the method can be extended to the computation of the inter-sphere position deviations in radial and z-direction. For this calculation, 2 further matrices will be filled with z-coordinates or radial distances instead of angle deviations. The separation of ball plate related shares (Af) and MD related shares (Bf and Cf) occurs in the same way as for the pitch deviation calculation in Eq. (1) (see Eqs. (2) and (3)). f z ¼ A fz þ B fz þ C fz
(2)
f r ¼ A fr þ B fr þ C fr
(3)
Due to the ‘closure principle’, the pitch deviations of each contributing influence (Afp, Bfp, Cfp) have to sum up to zero if all measuring positions are considered [8,11]. The separation of the different deviation contributions (Afp, Bfp, Cfp) is prepared by storing the measured pitch values of the ball centre points in a 2-dimensional matrix in a specific manner. The position for each value within this matrix is defined by its ball number and the angular step number (angular position) of the rotary table. This special arrangement of the measured values allows summarizing rows, columns and diagonals of the matrix [11]. Separately, each sum delivers only the deviation contribution of one influencing quantity, because the other systematic and non-systematic contributions sum up to zero. Further details about this rosette-method can be studied in [13].
The other new application aspect of the method is the selfcalibration of the mean radius of the ball circle, which is necessary for the full 3-D-position determination. Its value is calculated during method application and its uncertainty can directly be derived from the variance of the measured values, because the systematic deviations of the measuring system eliminate each other due to the multiple-orientation measurements. In z-direction no self-calibration is obtained, because all ball plate orientations vary in the x–y-plane. Hence, systematic deviations of the measuring system are not negligible for the mean height position and the height deviation of each ball. But, the WCS for each application case (bevel and spur gear calibration) is always derived from the ball centre positions, which leads to a mean height position very close to zero. Therefore, only the small individual height deviations of the ball centres from the x–y-plane at z = 0 are of interest for the determination of the gear deviation parameters and their uncertainties. The influence of systematic z-deviations of the probing system on these height deviations can be assessed as negligible and the deviation uncertainty is defined by the variance of the position measurements (comparable to the radial position uncertainty). The final knowledge of the complete 3D-positions of all calibrations spheres on the ball plate enables calculating the expected pitch deviations AND grid deviation patterns. This applies if the ball plate is used as a bevel gear artefact, whereas profile line deviations are calculated in case of cylindrical gear measurement.
2.2. Measuring procedure
3. Evaluation results
The investigations were carried out with a ball plate consisting of 12 ceramic balls glued onto a cylindrical plate (see Fig. 1). The balls are regular calibration spheres with an overall form deviation of less than 75 nm, which is required to achieve a good repeatability for the definition of a workpiece coordinate system (WCS) and the measurement results as well. The ball plate was clamped on the rotary table of a bridge-CMM (Leitz PMM-C) which measured 29 points on the upper hemisphere of each ball. A least squares algorithm determined the centre coordinates and the radius of each ball. The used evaluation method (see Section 2.1) requires the measurement of all 12 balls in 12 different angular positions of the rotary table (angular step: 3608/12 = 308).
Section 2 describes the separation of the deviations assigned to the ball plate from those of the measurement system. In order to verify this separation ability, measurements with and without rotary table compensation have been executed. The evaluation clearly showed the same results for the ball plate pitch deviations, A fp, but significantly differing values for contributions Bfp and Cfp, both belonging to the measurement system. Several executions (about 30) of the rosette-method yielded stable results for all evaluated ball plate characteristics. Fig. 2 presents the calculated relative angle deviations (Afp) of the individual balls of the ball plate obtained by 3 measurement runs (12 12 balls, run No. 167-169). The Afp-values (pitch) are
f p ¼ A fp þ B fp þ C fp þ e
(1)
A. Guenther et al. / CIRP Annals - Manufacturing Technology 65 (2016) 503–506
Fig. 2. Calculated pitch deviations of the ball centres Afp, referred to the ball circle radius R of the ball plate (R = 119.0184 mm) and z = 0 mm.
very stable over all runs. Their uncertainty is estimated as U(Afp) = 0.74 mm (k = 2). It is mainly determined by the variance of the measured data [11]. The positioning deviation (Bfp) and the pitch of the table encoder (Cfp) display the same stability, but the range of deviation values is much smaller (less than 10% of the ball centre position deviations). Fig. 3 shows the calculated individual, relative z-deviations of the balls from the mean z-reference values of all 3 rosetteevaluations. The values are very stable, too, and the uncertainty of the z-position deviations of the balls is determined to
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runs. The evaluated radial deviations vary by less than 0.5 mm for all runs, whereas the deviations for the individual balls show a range of almost 30 mm. The uncertainty of the radial deviations of the balls is estimated to U(Afr) = 0.76 mm (k = 2). The last necessary information for a complete characterization of the ball plate is the variation of the individual ball radii. These have been calculated to 0.8 mm from their mean value of 9.9973 mm. After calculating all x-, y-, and z-coordinates of all ball centre points based on the data of Figs. 2–4, it is possible to determine the reference values for gear parameter deviations (e.g. pitch or pressure angle) for bevel and cylindrical gear calibrations. For a GMI or CMM bevel gear calibration, the nominal ball positions are converted into target grid-point positions according to [6]. Additionally, target deviation parameters are derived from the position deviations. In order to measure a sphere instead of a bevel gear flank, an appropriate parameter set of gear parameters needs to be chosen for the instrument under evaluation. A perfect instrument should obtain the target deviation parameters for the grid points. Different deviation values indicate calibration deviations and the differences between targeted and obtained values have to be used as corrections for further gear measurements. The uncertainties of the ball positions are propagated according to [14] into the uncertainties of the grid points, which determine the uncertainties for the gear parameter deviations. For a cylindrical gear calibration, the ball positions are not converted into a target grid. Instead, the equator line of the ball is assumed as a part of an involute flank and the corresponding target deviations and their uncertainties are calculated from the ball position parameters comparable to the bevel gear procedure [3]. Fig. 5 shows the influences of all 4 position deviations on the expected pitch deviation of the ball plate, measured at the mean ball circle radius. It is clearly visible that the angular position deviation of the balls represents the dominating effect. The deviations of the z- and radial positions as well as the ball radius variation influence much less (cosine effect of the sphere form).
Fig. 3. Calculated individual z-deviations of the ball centres Afz from the plane at z = 0 mm (reference height of the WCS).
U(Afz) = 0.64 mm (k = 2). The contributions Bfz, and Cfz show very small variations and the values are much smaller (less than 3%). Fig. 4 illustrates the calculated individual radial deviations of the balls from their mean radial position for the three measurement Fig. 5. Estimated influence of the individual position deviations (see Figs. 2–4) and of the relative sphere radius deviation (from mean value r = 9.9973 mm) regarding the expected pitch deviation.
Fig. 4. Calculated individual radial position deviations of the ball centre points Afr from the mean radial distance R (ball circle radius).
The considered ranges of position deviations on the abscissa of Fig. 5 cover exactly the calculated values from the rosette-method. A similar analysis of the influencing parameters applied for the pressure angle deviation fHa reveals that the radial position of the individual balls is the dominant parameter. The influence of the other 3 components is negligible. In order to measure a sphere instead of a real involute flank, an appropriate set of gear parameters needs to be chosen. Fig. 6 illustrates this situation for an intentionally enlarged involute gear flank (moduli m = 9.92 mm, tooth number z = 24, pressure angle a = 21.18, base radius rb = 114.02 mm, pitch circle rt = 119.09 mm). The distribution of the measuring points at the ball surface is proportional to the roll length. All discussed measurements included the systematic probing of the balls with 29 measured points each. Therefore, one
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A. Guenther et al. / CIRP Annals - Manufacturing Technology 65 (2016) 503–506
Fig. 6. One individual sphere of the ball plate, shown together with an ideal involute curve of a theoretical gear (enlarged displayed).
rosette-measurement (12 12 ball measurements) required about 3 h, which is a long time span and can cause time effects. Further measurements with a reduced number of measured points showed less influence than expected on the stability of results (less than 0.2%) and seemed to be much more useful. For example, 5 measured points per sphere shortened the time for one run down to 40 min. 4. Comparison with pitch measurements In order to use the ball plate as a bevel gear artefact, first a nominal data set needs to be created based on the ball plate data [6]. Afterwards, a standard bevel gear program measures the ball plate instead of a real bevel gear. Fig. 7 compares a bevel gear pitch measurement result (mean values of cumulative pitch deviations of left and right flanks) with the pitch deviations of the ball plate calibration in Fig. 2. The gear measurement was performed with a CMM and a portable rotary table. The calibration of the ball plate was executed using a CMM of the same type but with an integrated rotary table. The differences (j1.5 mmj) can be explained by the pitch deviations of the rotary table of the second CMM and the estimated uncertainty of the ball positions.
Therefore, it is strongly recommended to determine the general tilt of the entire point cloud (x, y, z-coordinates without attention of their rotary axis position) and to correct it first. This effect results from a perfect but tilted circle of points, which shows an elliptical character in the radial values instead of a circle. Additionally, the mean radial distance of all points will be smaller than the theoretical one. The rosette-method cannot separate this kind of effect from object and system deviations. GMIs (not portal CMMs) are not able to perform measurements on all balls of the artefact in one rotary position. Therefore, an additional GMI-eligible WCS needs to be derived from the reference elements of the artefact. This WCS is determined additionally at the first rotational position at the beginning of the rosette-method measurements. The finally evaluated x-, y-, z-coordinates of the ball centre points have to be transformed into this additional coordinate system before calculating the expected deviations. 6. Conclusions and outlook With sufficient stable information about the individual spheres it is now possible to calculate expected deviations of involute profile lines with respect to the ‘‘camel-type’’ characteristic [3], used in cylindrical gear metrology. Additionally, deviations of bevel gear grid patterns can be calculated and used as set-values for calibration measurements. This allows the operator to use artefacts with elementary, high accurate geometries, relatively simple to manufacture and to calibrate them with less efforts by themselves. The achieved uncertainties for the ball positions are smaller than the target value of 1 mm. Therefore the approach is usable for the calibration of a ball plate as a gear standard. Additionally, the experiments will be repeated with an independent indexing table in order to confirm that a similar concept is independent of the CMM’s rotary table coordinate system and the degree of its compensation. Acknowledgements The authors gratefully acknowledge the financial support by the German Federal Ministry for Economic Affairs and Energy for the project ‘‘EVeQT’’ – Project No. 0325490A. The authors thank Hexagon Metrology GmbH, Wetzlar, Germany, for providing a CMM for the execution of the experiments. References
Fig. 7. Comparison of the pitch deviation values, obtained by a standard bevel gear measurement, and the rosette-method result for angle deviations.
5. Further information derived by the evaluation method Numerical tests with simulated eccentricity and wobble effects of the ball circle relative to its reference elements showed that all resulting deviations of ball positions from their theoretically perfect positions can be determined in full 3-D-coordinates. A remaining eccentricity of the rotary axis towards the WCS indicates appropriate values of the calculated Cfp and Cfr contributions of the system. But a wobble between these axes shows corresponding contributions at Cfz and transfers additional contributions to the artefact deviations in Afp. That means this effect of the rotation system cannot be perfectly separated.
[1] Goch G (2003) Gear Metrology. Annals of the CIRP 52(2):659–695. [2] Komori M, Takeoka F, Kubo A, Okamoto K, Osawa S, Sato O, Takatsuji T (2009) Evaluation Method of Lead Measurement Accuracy of Gears Using a Wedge Artefact. Measurement Science and Technology 20:025109. [3] Kondo K, Mizutani H (2002) Measurement Uncertainty of Tooth Profile by Master Balls. VDI-Berichte, International Conference on Gears, Mu¨nchen, 797–810. [4] Komori M, Takeoka F, Kiten T, Kondo Y, Osawa S, Sato O, Takatsuji T, Takeda R (2015) Magnetically Self-aligned Multiball Pitch Artifact Using Geometrically Simple Features. Precision Engineering 40:160–171. [5] Kondo Y, Sasajima K, Osawa S, Sato O, Komori M (2009) Traceability Strategy for Gear-Pitch-Measuring Instruments: Development and Calibration of a Multiball Artefact. Measurement Science and Technology 20(6):065101. [6] Gu¨nther A, Kniel K, Ha¨rtig F, Lindner I (2013) Introduction of a New Bevel Gear Measurement Standard. CIRP Annals – Manufacturing Technology 62(1):515–518. [7] Evans CJ, Hocken RJ, Estler WT (1996) Self-Calibration: Reversal Redundancy, Error Separation, and ‘Absolute Testing’. CIRP Annals 45(2):617–634. [8] Mark WD (1998) Method for Precision Calibration of Rotary Scale Errors and Precision Determination of Gear Tooth Index Errors. Mechanical Systems and Signal Processing 12(6):723–752. [9] Osawa S, Busch K, Franke M, Schwenke H (2005) Multiple Orientation Technique for the Calibration of Cylindrical Workpieces on CMMs. Precision Engineering 29:56–64. [10] Kondo Y, Osawa S, Sato O, Komori M, Takatsuji T (2012) Evaluation of Instruments for Pitch Measurement Using a Sphere Artifact. Precision Engineering 36:604–611. [11] Kniel K, Ha¨rtig F, Osawa S, Sato O (2009) Two Highly Accurate Methods for Pitch Calibration. Measurement Science and Technology 20(11):115110. [12] Noch R, Steiner O (1966) Die Bestimmung von Kreisteilungsfehlern nach einem Rosettenverfahren. Zeitschrift fu¨r Instrumentenkunde 74(10):308–316. [13] Ha¨rtig F, Kniel K (2007) Fehlertrennverfahren-ein Weg zu hochgenauen Verzahnungsmessungen, GETPRO-Kongress in Wu¨rzburg, Germany1–9. [14] GUM JCGM 100:2008, Guide to the Expression of Uncertainty in Measurement. www.bipm.org.
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Self-Calibration Method for a Ball Plate Artefact on a CMM Anke Guenther a,*, Dirk Sto¨bener b, Gert Goch (1)c a b c
Reishauer AG, Wallisellen, Switzerland Bremen Institute for Metrology, Automation and Quality Science (BIMAQ), University of Bremen, Germany Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Metrology Calibration Gear artefact
The design and the application of a circular ball plate artefact, introduced a couple of years ago, appeared to be very simple. But at the end, a measurement result and a measurement uncertainty can be determined only via NMI calibrated artefacts. Recently, PTB and NMIJ published self-calibration methods for pitch artefacts, one based on the rosette-method. This paper describes the testing and verification of this calibration method, applied to a gear artefact. The method was extended to calibrate not only the pitch position of the balls, but also their radial and height position on the circular ball plate (gear artefact). The concept of this advanced method, experiences regarding its application and test results will be presented. ß 2016
1. Introduction and state-of-the-art Improvements of the gear quality for power transmission systems can be achieved by modifications of the manufacturing processes, the materials and the gear shape. They lead to new requirements for the gear quality inspection, where new probing and evaluation methods with higher resolution and decreasing uncertainty have to be developed [1]. As the functionality of a gear for an automotive power train is significantly influenced by deviations of a few micrometres, the uncertainty of the quality inspection process should be less than 2 mm (preferably substantially less) for profile, helix and pitch deviations [2]. The quality inspection of these deviation parameters is generally carried out using gear measuring instruments (GMI) or coordinate measuring machines (CMM) [2]. The accuracy of these measuring devices is checked and calibrated by measurements of specific gear artefacts, whose uncertainties significantly influence the uncertainty of the measuring process for the gear. Therefore, artefacts for the various measurement tasks (profile, helix and pitch deviations) with the smallest possible uncertainties are required. Commercially available artefacts only partially comply with this requirement [1]. Several authors reported about the use of a circular ball plate as an artefact for gear measurements. Kondo and Mizutani used a ball artefact to calibrate spur gear profile measurements. It consists of two balls mounted onto a metal base plate [3]. Komori et al. presented a magnetically self-aligned multiball pitch artefact [4] and Kondo et al. designed a multiball artefact with ceramic balls, which are attached onto a ceramic cylinder by spring forces [5]. Gu¨nther et al. presented a circular ball plate as bevel gear measurement standard [6]. Generally, these artefacts consist of spheres mounted onto a stable plate in a circular pattern (Fig. 1) * Corresponding author. Tel.: +41 78 9259988. E-mail address: [email protected] (A. Guenther). http://dx.doi.org/10.1016/j.cirp.2016.04.080 0007-8506/ß 2016
and this pattern offers the opportunity to apply reversal or multiple-orientation (self-calibration) techniques for the determination of the artefacts’ parameters and their uncertainties.
Fig. 1. Investigated ball plate as an artefact for gear measurements.
The self-calibration techniques are time consuming, but they enable a separation of the systematic deviations caused by the probing and the positioning system from the deviations of the measured object by multiple measurements [7]. Mark used this approach for the determination of the pitch deviations of a master gear [8] and Osawa et al. presented an application for the calibration of cylindrical workpieces [9]. The National Metrology Institutes of Japan (NMIJ) [10] and Germany (PTB) [11] derived two different methods for the self-calibration of standards for pitch measurements. Both methods use multiple-orientation techniques in combination with a pitch artefact. The PTB approach is based on the rosette-method published by Noch and Steiner [12] and achieves a pitch uncertainty for the tested artefact of 0.4 mm. If a ball plate artefact shall be used for the physical representation of topography points in bevel gear metrology [6]
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A. Guenther et al. / CIRP Annals - Manufacturing Technology 65 (2016) 503–506
or for profile lines in cylindrical gear metrology [3], the knowledge about the relative angular positions (azimuth) of the spheres (as determined in [10–12]) is not sufficient. Deviations of involute profile lines or topographic point grids can be calculated only from the full 3-D-positions of the spheres. In order to achieve uncertainties below 2 mm for the various gear deviation parameters, the uncertainties of the artefact’s ball positions should be below 1 mm for all directions. These small uncertainty values cannot be reached by standard CMM measurements. Therefore self-calibration methods have to be applied, which separate the ball position deviations from the systematic deviations of the measuring system. Hence, the ball position uncertainty budget is independent from these systematic deviations and mainly influenced by non-systematic measurement deviations, which are estimated according to [11]. This paper presents the extension of the rosette-method to the determination of the full 3-D-position parameters of the balls in order to provide a method for the characterisation of ball plates as artefacts for bevel and cylindrical gears. It presents first application results and discusses the influences of position deviations on the determination of gear parameters. 2. Measuring procedure and evaluation method 2.1. Multiple-orientation evaluation algorithm The arrangement of the ball plate (relative positions of calibration spheres to each other) is given and constant. A simple measurement of ball positions would include several deviations of the measurement device (MD). In order to reduce or eliminate these unavoidable deviations, multiple reversal measurements of the ball plate have to be applied (rosette-method). The method is based on the assumption that the measured deviation for each ball’s angular position fp is represented by the superposition (see Eq. (1)) of the deviations of the ball centre positions Afp, the angle deviation of the positioning system for each plate orientation Bfp, and the pitch deviation of the rotary table encoder scale of the CMM Cfp. The sum of deviations is completed by non-systematic error contributions e.
All measurements are evaluated at the same WCS. For several reasons, the evaluation method requires the definition of the WCS by approximating a plane and a circle to all ball centre points at the first angular position of the rotary table. The direction of the z-axis of the WCS was defined by the normal direction of this plane, whereas the centre of the approximated circle determines the origin of the WCS. 2.3. New aspects for the use of the multiple-orientation method In contrast to the published application [11,13], the rosettemethod is not only applied to determine the pitch deviations of the spheres but also their relative radial position fr and height fz deviations. This is possible under the assumption that the observed deviations (from their mean value) of the radial and height positions are (like the pitch deviations) a sum of the deviations caused by the ball plate itself, the positioning deviation for each plate orientation and the deviations induced by the rotary table. Because of the special WCS-definition (see above), the sums of these position deviations in the matrix all become zero, too. Therefore, the use of the method can be extended to the computation of the inter-sphere position deviations in radial and z-direction. For this calculation, 2 further matrices will be filled with z-coordinates or radial distances instead of angle deviations. The separation of ball plate related shares (Af) and MD related shares (Bf and Cf) occurs in the same way as for the pitch deviation calculation in Eq. (1) (see Eqs. (2) and (3)). f z ¼ A fz þ B fz þ C fz
(2)
f r ¼ A fr þ B fr þ C fr
(3)
Due to the ‘closure principle’, the pitch deviations of each contributing influence (Afp, Bfp, Cfp) have to sum up to zero if all measuring positions are considered [8,11]. The separation of the different deviation contributions (Afp, Bfp, Cfp) is prepared by storing the measured pitch values of the ball centre points in a 2-dimensional matrix in a specific manner. The position for each value within this matrix is defined by its ball number and the angular step number (angular position) of the rotary table. This special arrangement of the measured values allows summarizing rows, columns and diagonals of the matrix [11]. Separately, each sum delivers only the deviation contribution of one influencing quantity, because the other systematic and non-systematic contributions sum up to zero. Further details about this rosette-method can be studied in [13].
The other new application aspect of the method is the selfcalibration of the mean radius of the ball circle, which is necessary for the full 3-D-position determination. Its value is calculated during method application and its uncertainty can directly be derived from the variance of the measured values, because the systematic deviations of the measuring system eliminate each other due to the multiple-orientation measurements. In z-direction no self-calibration is obtained, because all ball plate orientations vary in the x–y-plane. Hence, systematic deviations of the measuring system are not negligible for the mean height position and the height deviation of each ball. But, the WCS for each application case (bevel and spur gear calibration) is always derived from the ball centre positions, which leads to a mean height position very close to zero. Therefore, only the small individual height deviations of the ball centres from the x–y-plane at z = 0 are of interest for the determination of the gear deviation parameters and their uncertainties. The influence of systematic z-deviations of the probing system on these height deviations can be assessed as negligible and the deviation uncertainty is defined by the variance of the position measurements (comparable to the radial position uncertainty). The final knowledge of the complete 3D-positions of all calibrations spheres on the ball plate enables calculating the expected pitch deviations AND grid deviation patterns. This applies if the ball plate is used as a bevel gear artefact, whereas profile line deviations are calculated in case of cylindrical gear measurement.
2.2. Measuring procedure
3. Evaluation results
The investigations were carried out with a ball plate consisting of 12 ceramic balls glued onto a cylindrical plate (see Fig. 1). The balls are regular calibration spheres with an overall form deviation of less than 75 nm, which is required to achieve a good repeatability for the definition of a workpiece coordinate system (WCS) and the measurement results as well. The ball plate was clamped on the rotary table of a bridge-CMM (Leitz PMM-C) which measured 29 points on the upper hemisphere of each ball. A least squares algorithm determined the centre coordinates and the radius of each ball. The used evaluation method (see Section 2.1) requires the measurement of all 12 balls in 12 different angular positions of the rotary table (angular step: 3608/12 = 308).
Section 2 describes the separation of the deviations assigned to the ball plate from those of the measurement system. In order to verify this separation ability, measurements with and without rotary table compensation have been executed. The evaluation clearly showed the same results for the ball plate pitch deviations, A fp, but significantly differing values for contributions Bfp and Cfp, both belonging to the measurement system. Several executions (about 30) of the rosette-method yielded stable results for all evaluated ball plate characteristics. Fig. 2 presents the calculated relative angle deviations (Afp) of the individual balls of the ball plate obtained by 3 measurement runs (12 12 balls, run No. 167-169). The Afp-values (pitch) are
f p ¼ A fp þ B fp þ C fp þ e
(1)
A. Guenther et al. / CIRP Annals - Manufacturing Technology 65 (2016) 503–506
Fig. 2. Calculated pitch deviations of the ball centres Afp, referred to the ball circle radius R of the ball plate (R = 119.0184 mm) and z = 0 mm.
very stable over all runs. Their uncertainty is estimated as U(Afp) = 0.74 mm (k = 2). It is mainly determined by the variance of the measured data [11]. The positioning deviation (Bfp) and the pitch of the table encoder (Cfp) display the same stability, but the range of deviation values is much smaller (less than 10% of the ball centre position deviations). Fig. 3 shows the calculated individual, relative z-deviations of the balls from the mean z-reference values of all 3 rosetteevaluations. The values are very stable, too, and the uncertainty of the z-position deviations of the balls is determined to
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runs. The evaluated radial deviations vary by less than 0.5 mm for all runs, whereas the deviations for the individual balls show a range of almost 30 mm. The uncertainty of the radial deviations of the balls is estimated to U(Afr) = 0.76 mm (k = 2). The last necessary information for a complete characterization of the ball plate is the variation of the individual ball radii. These have been calculated to 0.8 mm from their mean value of 9.9973 mm. After calculating all x-, y-, and z-coordinates of all ball centre points based on the data of Figs. 2–4, it is possible to determine the reference values for gear parameter deviations (e.g. pitch or pressure angle) for bevel and cylindrical gear calibrations. For a GMI or CMM bevel gear calibration, the nominal ball positions are converted into target grid-point positions according to [6]. Additionally, target deviation parameters are derived from the position deviations. In order to measure a sphere instead of a bevel gear flank, an appropriate parameter set of gear parameters needs to be chosen for the instrument under evaluation. A perfect instrument should obtain the target deviation parameters for the grid points. Different deviation values indicate calibration deviations and the differences between targeted and obtained values have to be used as corrections for further gear measurements. The uncertainties of the ball positions are propagated according to [14] into the uncertainties of the grid points, which determine the uncertainties for the gear parameter deviations. For a cylindrical gear calibration, the ball positions are not converted into a target grid. Instead, the equator line of the ball is assumed as a part of an involute flank and the corresponding target deviations and their uncertainties are calculated from the ball position parameters comparable to the bevel gear procedure [3]. Fig. 5 shows the influences of all 4 position deviations on the expected pitch deviation of the ball plate, measured at the mean ball circle radius. It is clearly visible that the angular position deviation of the balls represents the dominating effect. The deviations of the z- and radial positions as well as the ball radius variation influence much less (cosine effect of the sphere form).
Fig. 3. Calculated individual z-deviations of the ball centres Afz from the plane at z = 0 mm (reference height of the WCS).
U(Afz) = 0.64 mm (k = 2). The contributions Bfz, and Cfz show very small variations and the values are much smaller (less than 3%). Fig. 4 illustrates the calculated individual radial deviations of the balls from their mean radial position for the three measurement Fig. 5. Estimated influence of the individual position deviations (see Figs. 2–4) and of the relative sphere radius deviation (from mean value r = 9.9973 mm) regarding the expected pitch deviation.
Fig. 4. Calculated individual radial position deviations of the ball centre points Afr from the mean radial distance R (ball circle radius).
The considered ranges of position deviations on the abscissa of Fig. 5 cover exactly the calculated values from the rosette-method. A similar analysis of the influencing parameters applied for the pressure angle deviation fHa reveals that the radial position of the individual balls is the dominant parameter. The influence of the other 3 components is negligible. In order to measure a sphere instead of a real involute flank, an appropriate set of gear parameters needs to be chosen. Fig. 6 illustrates this situation for an intentionally enlarged involute gear flank (moduli m = 9.92 mm, tooth number z = 24, pressure angle a = 21.18, base radius rb = 114.02 mm, pitch circle rt = 119.09 mm). The distribution of the measuring points at the ball surface is proportional to the roll length. All discussed measurements included the systematic probing of the balls with 29 measured points each. Therefore, one
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Fig. 6. One individual sphere of the ball plate, shown together with an ideal involute curve of a theoretical gear (enlarged displayed).
rosette-measurement (12 12 ball measurements) required about 3 h, which is a long time span and can cause time effects. Further measurements with a reduced number of measured points showed less influence than expected on the stability of results (less than 0.2%) and seemed to be much more useful. For example, 5 measured points per sphere shortened the time for one run down to 40 min. 4. Comparison with pitch measurements In order to use the ball plate as a bevel gear artefact, first a nominal data set needs to be created based on the ball plate data [6]. Afterwards, a standard bevel gear program measures the ball plate instead of a real bevel gear. Fig. 7 compares a bevel gear pitch measurement result (mean values of cumulative pitch deviations of left and right flanks) with the pitch deviations of the ball plate calibration in Fig. 2. The gear measurement was performed with a CMM and a portable rotary table. The calibration of the ball plate was executed using a CMM of the same type but with an integrated rotary table. The differences (j1.5 mmj) can be explained by the pitch deviations of the rotary table of the second CMM and the estimated uncertainty of the ball positions.
Therefore, it is strongly recommended to determine the general tilt of the entire point cloud (x, y, z-coordinates without attention of their rotary axis position) and to correct it first. This effect results from a perfect but tilted circle of points, which shows an elliptical character in the radial values instead of a circle. Additionally, the mean radial distance of all points will be smaller than the theoretical one. The rosette-method cannot separate this kind of effect from object and system deviations. GMIs (not portal CMMs) are not able to perform measurements on all balls of the artefact in one rotary position. Therefore, an additional GMI-eligible WCS needs to be derived from the reference elements of the artefact. This WCS is determined additionally at the first rotational position at the beginning of the rosette-method measurements. The finally evaluated x-, y-, z-coordinates of the ball centre points have to be transformed into this additional coordinate system before calculating the expected deviations. 6. Conclusions and outlook With sufficient stable information about the individual spheres it is now possible to calculate expected deviations of involute profile lines with respect to the ‘‘camel-type’’ characteristic [3], used in cylindrical gear metrology. Additionally, deviations of bevel gear grid patterns can be calculated and used as set-values for calibration measurements. This allows the operator to use artefacts with elementary, high accurate geometries, relatively simple to manufacture and to calibrate them with less efforts by themselves. The achieved uncertainties for the ball positions are smaller than the target value of 1 mm. Therefore the approach is usable for the calibration of a ball plate as a gear standard. Additionally, the experiments will be repeated with an independent indexing table in order to confirm that a similar concept is independent of the CMM’s rotary table coordinate system and the degree of its compensation. Acknowledgements The authors gratefully acknowledge the financial support by the German Federal Ministry for Economic Affairs and Energy for the project ‘‘EVeQT’’ – Project No. 0325490A. The authors thank Hexagon Metrology GmbH, Wetzlar, Germany, for providing a CMM for the execution of the experiments. References
Fig. 7. Comparison of the pitch deviation values, obtained by a standard bevel gear measurement, and the rosette-method result for angle deviations.
5. Further information derived by the evaluation method Numerical tests with simulated eccentricity and wobble effects of the ball circle relative to its reference elements showed that all resulting deviations of ball positions from their theoretically perfect positions can be determined in full 3-D-coordinates. A remaining eccentricity of the rotary axis towards the WCS indicates appropriate values of the calculated Cfp and Cfr contributions of the system. But a wobble between these axes shows corresponding contributions at Cfz and transfers additional contributions to the artefact deviations in Afp. That means this effect of the rotation system cannot be perfectly separated.
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