Variable speed compensation method of errors of probes for CNC machine tools

Variable speed compensation method of errors of probes for CNC machine tools

Accepted Manuscript Title: Variable speed compensation method of errors of probes for CNC machine tools Authors: Adam Wozniak, Michał Jankowski PII: D...

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Accepted Manuscript Title: Variable speed compensation method of errors of probes for CNC machine tools Authors: Adam Wozniak, Michał Jankowski PII: DOI: Reference:

S0141-6359(16)30434-2 http://dx.doi.org/doi:10.1016/j.precisioneng.2017.03.001 PRE 6546

To appear in:

Precision Engineering

Received date: Revised date: Accepted date:

22-12-2016 6-2-2017 6-3-2017

Please cite this article as: Wozniak Adam, Jankowski Michał.Variable speed compensation method of errors of probes for CNC machine tools.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2017.03.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Variable speed compensation method of errors of probes for CNC machine tools (Variable speed compensation method) Adam Wozniak1, Michał Jankowski2 1,2

Warsaw University of Technology, Faculty of Mechatronics, Institute of Metrology

and Biomedical Engineering. Poland, św. A. Boboli 8 st., 02-525 Warszawa. E-mail: 1 [email protected], 2 [email protected] Corresponding author: Adam Wozniak. Warsaw University of Technology, Faculty of Mechatronics, Institute of Metrology and Biomedical Engineering. Poland, św. A. Boboli 8 st., 02-525 Warszawa. E-mail: [email protected]. Tel.: +48 22 849 03 95 Highlights    

A new method of errors compensation for CNC machine tool probes was proposed; The method does not require any additional hardware; It was verified both in laboratory and on CNC machine tool; Probes’ errors are reduced 10 times.

Abstract: Systematic errors of kinematic touch-trigger probes for CNC machine tools may exceed errors of the machine tool itself. As a result, the machining accuracy is strongly dependent on the probe’s accuracy. Numerical correction of probes’ systematic errors can be used. However, it requires executing calculations by the CNC machine tool controller. To avoid this troublesome requirement, a new method of errors compensation is proposed. In this approach, a modification of the probe’s pre-travel in a given direction is achieved by modification of measurement speed in this direction. Because all measurement speeds can be calculated offline, the controller does not have to do any calculations. The proposed method has been tested for sample kinematic probes and the error reduction was at least 10-fold. Keywords: machine tool; ; ; , touch-trigger probe, errors compensation, on-machine measurement

1. Introduction Touch-trigger probes for CNC machine tools, mounted in place of the cutter or the turning tool, are used for automatic workpiece set-up, coordinate measurement of machined part and, less usually, for determination of the machine tool’s kinematic errors [1 – 9]. That is why probe’s errors influence the accuracy of machining. In the case of the most popular kinematic probes, probe error can exceed 10 µm [10 – 16] and be larger than machine tool’s errors. To increase the probing accuracy various measures can be implemented. The most obvious of them is to equip the machine tool with the more precise probe, e.g. the strain gauge one instead of kinematic one. Precise probes are, however, expensive and not every machine tools user can qualify the added cost. Another way of improving on-machine measurements accuracy is to apply a numerical correction to probe’s systematic errors. This method was successfully applied to probes dedicated to coordinate measuring machines (CMMs) [17 – 19] and can be equally effectively applied to kinematic probes for CNC machine tools [13, 20] because their accuracy is more influenced by the systematic errors than by the random ones [15, 16]. However, this solution requires uploading an error map to the machine tool’s controller and executing calculations by this controller. It means that the correction is strongly controller hardware- and software-dependent. To overcome this drawback a new method of probes’ systematic errors compensation was developed. 2. Systematic errors of touch-trigger probes The systematic errors of the probe can be easily described by triggering radius characteristics and variation value. The triggering radius can be defined as follows [15]: if position of the stylus tip corresponding to triggering of the probe’s transducer during measurement in direction i is indicated with TGi and centre of best-fitted element 2

determined for triggering points TG for various directions is given by OS, the triggering radius in measurement direction i, indicated with ri, equals to a distance between points OS and TGi. The triggering radius variation Vr is a difference between maximum and minimum values of triggering radius of the probe. The triggering radius variation corresponds with the form deviation of the test sphere obtained during measurement performed with a use of the probe, so it describes practical performance of the probe well. However, theoretical modelling of the probe’s behaviour is easier using another parameter – pre-travel w [16 – 19, 21, 22]. This parameter can be defined as follows: if the neutral position of the stylus tip is given by N, the pre-travel in direction i, indicated with wi, is equal to distance between points N and TGi. The neutral position N, relative to the probe’s body, may be different for each measurement, for example because of probe’s hysteresis. However, it’s usually assumed that it’s constant – changes are assumed to be negligible. Graphical interpretation of these two parameters is shown in Fig. 1. Assuming that the points OS and N lie in the same place, ri = wi. This assumption is rarely met in practice, but if the distance between abovementioned points is small enough, it can be reasonably assumed that ri ≈ wi. For the purpose of this research, such assumption was done. Results presented in the following parts of the paper indicate that this approach was correct

Each direction i can be defined by values of two angles, α and β. α is an

angle between projection of the direction i on a plane perpendicular to the probe’s axis and chosen line on this plane, while β is an angle between direction i and the plane perpendicular to the probe’s axis. Previous researches showed that the pre-travel in a given direction is proportional to the measurement speed [23] and equals [24]: 3

𝑤(𝛼, 𝛽) = 𝑤𝑇 (𝛼, 𝛽) + 𝑤𝐼 (𝛼, 𝛽) = 𝑤𝑇 (𝛼, 𝛽) + 𝑣(𝛼, 𝛽)𝜏, (1) where: wT – pre-travel component related to the probe’s transducer, wI – pretravel component related to the measurement speed, v – measurement speed, τ – delay between the triggering of the transducer and the change of the probe’s controller output. If the measurement speed is the same in all measurement directions, the pretravel component related to the measurement speed is also equal in all the measurement directions, as it is shown in Fig. 2. As a result, the effective probing radius (stylus tip radius minus pre-travel, that is the distance by which a measured machine tool position has to be moved in order to get coordinates of the measured point on the surface of the workpiece) is different for different directions, while the machine tool controller usually uses only one value of the effective probing radiusThe idea of the proposed probe error compensation method is based on programming a variable measurement speed, so that the pre-travel component related to the measurement speed is also variable, but the overall pre-travel is equal in every direction. Then, the probing error is only affected by random errors of the probe. Constant pre-travel value can be easily included in a single effective probing radius value used by the machine controller. 3. The errors compensation method principle In order to compensate the variability of probe’s pre-travel component related to the probe’s transducer, the pre-travel component related to the measurement speed has to decrease with an increase of the transducer-related pre-travel’s component according to the equation: 𝑤𝐼 (𝛼, 𝛽) = 𝑤 ̿ − 𝑤𝑇 (𝛼, 𝛽), (2) 4

where 𝑤 ̿ is a mean value of pre-travel for all directions. If this condition is met, the pre-travel characteristics looks as in the Fig. 3 and the probe’s systematic errors are compensated. To achieve this goal, measurement speed values should be calculated as follows: 𝑣1 (𝛼, 𝛽) = 𝑣0 (𝛼, 𝛽) +

̿̿̿̿−𝑤 𝑤0 0 (𝛼,𝛽) 𝜏

,

(3) where: v1 – measurement speed applied to compensate the probe errors, v0 – measurement speed before applying error compensation, w0 – overall pre-travel before applying error compensation. As mentioned above, pre-travel is a parameter appropriate to probe behaviour modelling, but, in practice, the triggering radius is more important. Due to this it may be necessary to replace pre-travel values in equation 3 by the triggering radius values. In that case one iteration of probe’s errors mapping and measurement speed values’ calculation can be insufficient. If so, the measurement speed determination should be performed several times, each time using the previously calculated v1 measurement speed values as v0 valuesApplication of the proposed method does not change the mean pre-travel value – the mean pre-travel value is a target pre-travel value, the value to be obtained after compensation. That’s why also mean measurement speed doesn’t change – there are measurement directions for which the speed is increased, but there are also directions in which the speed is decreased. Consequently, measurement time can be longer when the proposed method is used. For example, assuming that the measurement speed is set 10 mm from the measured surface, and that from this point the measurement time is calculated, measurement of 2 points, each with the speed of 50 mm/min, takes 24.0 s, while measurement of 1 point with the speed of 30 mm/min and 5

1 point with the speed of 70 mm/s takes 28.6 s. That’s why the measurement speed should be set as close to the measured surface as possible. Fortunately, for majority of the measurement directions speed decrease or increase is casual. 4. Experimental verification of the proposed method The proposed method was experimentally verified in two steps: in laboratory, to check, if the developed procedure can reduce the triggering radius variation, and on the machine tool, to verify, if obtained probe errors elimination is significant in a real onmachine measurement environment. The most popular method of probe error determination is an on-machine measurement of a master artefact [10 – 14, 25 - 28]. The main disadvantage of this method is that machine tool errors influence the results. In the first step, in order to avoid this influence, experimental verification of the proposed method was performed with a test setup that is a practical implementation of the moving master artefact method which was described in [15]. The scheme of the setup is shown in Fig. 4, while photo of a mechanical part of this setup is presented in Fig. 5. The tested probe (1) is fixed so that its stylus tip (2) is in the centre of the master artefact (3) – an inner hemisphere artefact or a ring gauge. The master artefact is mounted on the 3-axial piezoelectric stage (4) which position is set by the control unit. The control unit contains the probe’s interface, so it receives the probe’s triggering signal. To test the probe’s triggering radius in a given direction, the master artefact is displaced in this direction by the piezostage and its coordinates corresponding to the triggering point of the probe are read. The expanded (for coverage factor k = 2) uncertainty of determination of triggering radius variation U(Vr) = 0.6 µm.

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Application of the proposed probes’ errors compensation method does not only require obtaining the probe’s errors characteristics, but also knowledge of the delay time τ. A method of determination of this value was described in [24] and it is based on a mean triggering radius measurements for various measurement speeds. Because: ∆𝑟̿ = ∆𝑣 ∙ 𝜏, (4) where Δ𝑟̿ is an increase of the mean measured triggering radius, and Δv – an increase of measurement speed, the τ value can be easily calculated. In order to verify the proposed error compensation method, it was applied to two 3-point kinematic probes: Renishaw OMP40-2 and Renishaw OMP60. Both of these probes worked with the same OMI wireless interface and τ values, obtained as described above, were: 12.68 ms for the OMP40-2 probe and 25.35 ms for the OMP60 probe. An initial measurement speed of 50 mm/min was set for both of the probes and both of the probes were equipped with a 50 mm long stylus. Before applying the error compensation method, the triggering radius variation of the OMP40-2, determined from 10 measurements, was equal to 11.0 µm. The obtained triggering radius characteristics of this probe for different values of β angle are presented in Fig. 6 (all presented values are mean values from 10 measurements). As it can be seen, the characteristics have a 3-lobed shape. This shape is clearly visible for small β angle values and actually non-existent for big β angle values: for β = 0° triggering radius variation Vr = 8.6 µm, while at β = 80° triggering radius variation Vr = 2.3 µm. The three-lobed shape of the triggering radius characteristics is responsible for most of triggering radius variation – the triggering radius variation in the plane 7

perpendicular to the probe’s axis (for

β = 0°) is only slightly smaller than in 3D. These

results are consistent with the existing knowledge about the kinematic probes [10 – 18, 21 – 23] – the peaks of triggering radius characteristics correspond to directions parallel to the probe’s transducer’s 3-armed moving part. After applying the error compensation method, the measurement speed varied from 26.46 mm/min to 69.87 mm/min and the triggering radius variation, determined from 10 measurements, decreased to 1.0 µm, which is the value of the probes random errors – there is no point in further reduction of systematic errors below this value. The obtained triggering radius characteristics for different values of β angle are shown in Fig. 7 (all presented values are mean values from 10 measurements). As can be seen, the spatial triggering radius characteristic is quasi-spherical. It means that the systematic errors of the probe were successfully eliminated. When the OMP60 probe is considered, the triggering radius variation, determined from 10 measurements before the error compensation was equal to 16.8 µm. The obtained triggering radius characteristics of this probe – before applying the error compensation – for different values of β angle are shown in Fig. 8 (all presented values are mean values from 10 measurements). The same as in the case of the OMP40-2 probe, the triggering radius characteristics have the 3-lobed shape that is typical for the 3-point kinematic probes. This shape disappeared after applying the developed error compensation method. The measurement speed varied from 31.24 mm/min to 66.18 mm/min and the triggering radius variation, determined from 10 measurements, dropped to 1.0 µm. The improved, quasi-circular triggering radius characteristics for different values of β angle

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are shown in Fig. 9 (all presented values are mean values from 10 measurements). As can be seen, as in the case of the OMP40-2 probe, the systematic errors are eliminated. Tests performed on the laboratory setup confirmed that the proposed method is capable of reducing the probes systematic errors more than 10 times. To verify if application of this method would reduce errors of a measurement system consisting of a probe and a machine tool, on-machine measurements of a test sphere were performed. The third probe, an OMP40-2 (another one than the OMP40-2 used in in-laboratory tests), equipped with the 100 mm long stylus and the Haas VF8 machine tool were used. To determine the delay time τ for this probe, 15 on-machine measurements of a nominally 52 mm gauge ring were performed: 5 with the constant speed of 50 mm/min, 5 with the constant speed of 100 mm/min and 5 with the constant speed of 150 mm/min. Each time the ring was measured in 36 points. Using obtained data, the delay time τ = 14.5 ms was calculated. To determine the systematic errors of the used probe, the test sphere was measured 10 times with the constant speed 100 mm/min, each time in 109 points. Obtained form deviations which are equal to the probe systematic errors are shown in Fig. 10 (shown values are mean values from 10 measurements). Assuming that form error of the test sphere is negligible, the systematic component of probing error of used OMP40-2 probe with a 100 mm stylus, on Haas VF8 machine is equal to the measured form error at 21 µm – this is measured form error, when mean values from 10 measurements are taken into account. Because of random errors, form error in a single measurement can be greater. The maximum obtained value was 23 µm. After the abovementioned measurements were completed, a new set of measurement speed values was calculated (minimal value was 43.594 mm/min and 9

maximal value was 131.277 mm/min) and 10 measurements with these new speed values were performed. Obtained results are shown in Fig. 11 (shown values are mean values from 10 measurements). Measured form deviation was equal to 2 µm, when data from 10 measurements were taken into account. Maximal value of form error obtained in a single measurement was 6 µm. This shows that also on machine tool the proposed method can reduce systematic errors of the probe beyond the level of probe random errors. Nevertheless the requirement of multi-iteration procedure was taken into consideration, in the real measurement environment the efficient systematic errors reduction was obtained in the first iteration of the procedure. To determine if the measurement time increased significantly, the calculation of the relative time increase ΔtR (counting only the measurement-speed dependent time) was performed as follows: speed-dependent component of measurement time for a single point tS is equal to: 𝑆

𝑡𝑆 = 𝑣 (5) where S is an probe’s approach distance – the distance from a point in which the measurement speed is set to the measured point. Measurement time for multiple points tM is a sum of measurement times for single points: 𝑆

𝑡𝑀 = ∑𝑛𝑖=1 𝑣(𝛼 ,𝛽 ) 𝑖

𝑖

(6) where n is a number of measurements.

10

To determine the relative measurement time increase due to applying the error compensation method ΔtR, the measurement time after applying the method is divided by the measurement time before applying the method and 1 is subtracted: ∆𝑡𝑅 =

𝑆 𝑣1 (𝛼𝑖 ,𝛽𝑖 ) 𝑆 ∑𝑛 𝑖=1𝑣0 (𝛼 ,𝛽 ) 𝑖 𝑖

∑𝑛 𝑖=1

−1=

1 𝑣1 (𝛼𝑖 ,𝛽𝑖 ) 𝑛 𝑣0 (𝛼𝑖 ,𝛽𝑖 )

∑𝑛 𝑖=1

− 1,

(7) Using values applied on the machine tool (109 measurements, calculated measurement speeds), ΔtR = 4.6%. Measurement time increase, calculated from the given equation (7), depends on the measurement program obviously. If measurement directions with higher measurement speeds are used more frequently than measurement directions with lower measurement speeds, then even a measurement time reduction can be achieved. If the position of the measured part is fixed, then it can be achieved by angular positioning of the probe. It is recommended to determine the uncompensated error of the probe for every measurement direction which is used in the measurement program, but it is not necessary – new measurement speed can be determined basing on values in other directions, e.g. using linear interpolation. In both cases all calculations should be performed offline, during preparation of the measurement program. As it was said before, the assumption that ri = wi was done. Obtained reduction of the probes’ systematic errors indicates that it was justified. 5. Conclusions Three-point kinematic probes usually have triggering radius variation larger than 10 µm and their errors are an important source of on-machine measurement errors. 11

Because dominant part of the probes’ errors are systematic errors, the numerical error correction is possible, however, its application requires executing calculations by the CNC machine tool’s controller. In case of measurements along the single axis of the machine, calculations are simple: it’s sufficient to add or subtract the known systematic error’s value from the X, Y or Z coordinate reading. However, in case of measurements performed not along a single axis, calculations are getting more complicated, because correction has to be applied separately in 2 or 3 axes. It can be achieved by uploading to the machine tool’s controller 3 error correction’s values for each measurement directions (one for each coordinate) or by uploading one systematic error’s value for each direction and performing calculations using trigonometric functions. To avoid the necessity of such calculations, the new method of probe error compensation was developed. By setting proper measurement speeds, varying for different measurement directions, errors of the probe can be significantly reduced. The experiments conducted for exemplary 3-point kinematic probes showed that the probes’ triggering radius variation can be reduced from 16.8 µm down to 1 µm and that probing error on the machine tool can be reduced from 23 µm down to 6 µm while reducing its systematic component to 2 µm. The advantage of the proposed method is that all calculations can be done offline to evaluate the new speed values in the machine G-code, before any on-machine measurement starts. Instead of separately calculating three corrected values (X, Y and Z), machine tool’s controller has to change a single value of the measurement speed. Application of the developed method would make it possible for users of the kinematic probes to perform on-machine measurements with the accuracy comparable with that achieved through applying more sophisticated and more expensive strain gauge probes. 12

The same goal can be equally well obtained using numerical error correction, but the proposed method is an alternative which – in some cases – is easier in application. Acknowledgements The research has been partially funded project PBS2/B6/16/2013 of The National Centre for Research and Development of Poland. References [1]

Erkan T, Mayer JRR, Dupont Y. Volumetric distortion assessment of a five-axis machine by probing a 3D reconfigurable uncalibrated master ball artefact. Prec Eng 2011;35(1);116-125

[2]

Mayer JRR. Five-axis machine tool calibration by probing a scale enriched reconfigurable uncalibrated master balls artifact. CIRP Ann – Manuf Techn 2012;61;515-518

[3]

Ibaraki S, Iritani T, Matsushita T. Error map construction for rotary axes on fiveaxis machine tools by on-the-machine measurement using a touch-trigger probe. Int J Mach Tool Manu 2013;68;21-29

[4]

Ibaraki S, Ota Y. Error calibration for five-axis machine tools by on-the-machine measurement using a touch-trigger probe. International Journal of Automation Technology 2013;8(1);20-27

[5]

Alami Mchichi N, Mayer JRR., Axis location errors and error motions calibration for a five-axis machine tool using the SAMBA method. Procedia CIRP 2014;14;305-310

[6]

Guiassa R, Mayer JRR., Balazinski M., Engin S., Delorme F-E. Closed door machining error compensation of complex surfaces using the cutting compliance

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coefficient and on-machine measurement for a milling process. Int J Comp Integ M 2014;27(11);1022-1030 [7]

Jiang Z, Bao S, Zhou X, Tang X, Zheng S. Identification of location errors by a touch-trigger probe on five-axis machine tools with a tilting head. Int J Adv Manuf Tech 2015;81(1);149-158

[8]

Mayer JRR, Rahman MM, Łoś A. An uncalibrated cylindrical indigenous artefact for measuring inter-axis errors of a five-axis machine tool. CIRP Ann - Manuf Techn 2015;64(1);487-490

[9]

Rahman MM, Mayer R. Calibration performance investigation of an uncalibrated indigenous artefact probing for five-axis machine tool. J Mach Eng 2016;16(1);33-42

[10] Cho MW, Seo TI. Inspection planning strategy for the on-machine measurement process based on CAD/CAM/CAI integration. Int J Adv Manuf Tech 2002;19(8);607-617 [11] Cho MW, Seo TI. Machining error compensation using radial basis function network based on CAD/CAM/CAI integration concept. Int J Prod Res 2002;40(9);2159-2174 [12] Cho M., Seo TI, Kwon HD. Integrated error compensation method using OMM system for profile milling operation. J Mater Process Technol 2003;136(1-3);8899 [13] Choi JP, Min BK, Lee SJ. Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system. J Mater Process Technol 2004;155-156; 2056-2064

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[14] Zeleny J, Janda M. Automatic on-machine measurement of complex parts. Modern Machinery Science Journal 2009;(1);92-95 [15] Jankowski M, Woźniak A, Byszewski M. Machine tool probes testing using a moving inner hemispherical master artifact. Prec Eng 2014;38(2);421-427 [16] Jankowski M, Woźniak A., Mechanical model of errors of probes for numerical controlled machine tools. Measurement 2016;77; 317-326 [17] Estler WT, Phillips SD, Borchardt B, Hopp T, Levenson M, Eberhardt K, McClain M, Shen Y, Zhang X. Practical aspects of touch-trigger probe error compensation. Prec Eng 1997;21(1);1-17 [18] Estler WT, Phillips SD, Borchardt B, Hopp T, Witzgall C, Levenson M, Eberhardt K, McClain M, Shen Y, Zhang X. Error compensation for CMM touch trigger probes. Prec Eng 1996;19(2-3);84 – 96 [19] Krawczyk M, Gąska A, Sładek J. Determination of the uncertainty of the measurements performed by coordinate measuring machines. tm-Technisches Messen 2015; 82(6);329-338 [20] Qian XM, Ye WH, Chen XM. On-machine measurement for touch-trigger probes and its error compensation. Key Eng Mater 2008;375-376;558-563 [21] Woźniak A, Dobosz M. Metrological feasibilities of CMM touch trigger probes. Part I: 3D theoretical model of probe pretravel. Measurement 2003;34(4);273-286 [22] Woźniak A, Dobosz M. Influence of measured objects parameters on CMM touch trigger probe accuracy of probing. Prec Eng 2005;29(3);290-297 [23] Bohan Z, Feng G, Yan L. Study on Pre-travel Behaviour of Touch Trigger Probe under Actual Measuring Conditions. Procedia CIRP 27, 13th CIRP conference on Computer Aided Tolerancing, 2015;53-58 15

[24] Woźniak A, Jankowski M. Wireless communication influence on CNC machine tool probe metrological parameters. Int J Adv Manuf Tech 2016;82(1);535-542 [25] ISO 230-10:2011: Test code for machine tools -- Part 10: Determination of the measuring performance of probing systems of numerically controlled machine tools [26] Semotiuk L, Józwik J, Kuric I. Measurement uncertainty analysis of different CNC machine tools measurement systems. Adv Sci Technol Res J 2013;7(19);4147 [27] Verma MR, Chatzwagiannis E, Jones D, Maropoulos PG. Comparison of the measurement performance of high precision multi-axis metal cutting machine tools. Procedia CIRP 2014;25;138-145 [28] Rahman MM, Mayer JRR. Measurement Accuracy Investigation of Touch Trigger Probe with Five-Axis Machine Tools. Arch Mech Eng 2016;63(4); 495510.

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Figure captions: Fig. 1. Graphical interpretation of probe’s triggering radius and pre-travel Fig. 2. Probe’s pre-travel components and overall pre-travel for constant measurement speed: a schematic view from the side (left) and from the top (right) Fig. 3. Probe’s pre-travel components and overall pre-travel after application of the variable measurement speed errors compensation method: a schematic view from the side (left) and from the top (right) Fig. 4. Scheme of the test setup implementing the moving master artefact method: 1 – tested probe, 2 – stylus tip, 3 – master artefact, 4 – 3-axial piezostage Fig. 5. Photo of a mechanical part of the test setup implementing the moving master artefact method: 1 – tested probe, 2 – stylus tip, 3 – master artefact, 4 – 3-axial piezostage Fig. 6. OMP40-2 triggering radius characteristics for different values of β angle, before error compensation Fig. 7. OMP40-2 triggering radius characteristics for different values of β angle, after applying error compensation Fig. 8. OMP60 triggering radius characteristics for different values of β angle, before error compensation Fig. 9. OMP60 triggering radius characteristics for different values of β angle, after applying error compensation Fig. 10. Measured form deviations of the test sphere, equal to the systematic errors of the OMP40-2 used on the machine tool, for different values of β angle

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Fig. 11. Measured form deviations of the test sphere, equal to the systematic errors of the OMP40-2 used on the machine tool, for different values of β angle, after applying error compensation

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TGi+1

TGi+2

TGi

N

TGi+3

ri wi TGi+4

OS

. Fig. 1.

19

Z

Y

overall w ≠ const.

wI(α,β) wT(α,β)

TG(α,β)

α overall w ≠ const. wI(α,β) wT(α,β)

X

N

TG(α,β)

β

.

X

N

Fig. 2.

Z

Y overall w = const.

TG(α,β) wT(α,β) α

overall w =const. wI(α,β) wT(α,β)

N

TG(α,β)

β

.

N

X

Fig. 3.

20

wI(α,β) X

1

control 2

part

3

4

Fig. 4.

1

2

3 4

Fig. 5.

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α,° 0 β,° 0 10 20 30 40 50 60 70 80

30

30 300

60 20 10

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Fig. 6.

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0 10 20 30 40 50 60 70 80

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+5

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30 30

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