Detecting 6 DoF geometrical errors of rotary tables

Journal Pre-proofs Detecting 6 DoF geometrical errors of rotary tables Karin Kniel, Matthias Franke, Frank Härtig, Frank Keller, Martin Stein PII: DOI: Reference:

S0263-2241(19)31230-8 https://doi.org/10.1016/j.measurement.2019.107366 MEASUR 107366

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Measurement

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15 July 2019 29 November 2019 3 December 2019

Please cite this article as: K. Kniel, M. Franke, F. Härtig, F. Keller, M. Stein, Detecting 6 DoF geometrical errors of rotary tables, Measurement (2019), doi: https://doi.org/10.1016/j.measurement.2019.107366

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Detecting 6 DoF geometrical errors of rotary tables Abbreviated title: 6 DoF error detection for rotary tables Karin Kniel, Matthias Franke, Frank Härtig, Frank Keller, Martin Stein Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Abstract An error separation method called three-rosette method allows all errors in rotary tables integrated into coordinate measuring machines (CMMs) to be determined by means of an uncalibrated circular ball plate. The calibration procedure takes advantage of the fact that error contributions can be separated and eliminated by means of multiple measurements of each ball center in welldefined measurement positions. The skilled evaluation of these results allows all six degrees of freedom (DoF) of a rotary table to be detected: the angular positioning error motion, the axial error motion and two error components for describing both the tilt error motion and the radial error motion. The method is of practical and economic relevance, as ball plates can be manufactured at low cost and require no further calibration. A detailed measurement and evaluation strategy as well as example results are presented. Keywords: Coordinate metrology, error separation method, three-rosette method, rotary table, error model, calibration procedure 1. Introduction Errors in machine guideways have a significant influence on the quality of results, and thus on uncertainty. In the wake of the first industrial revolution, when steam engines allowed the use of work tools, tolerances of workpieces decreased continuously. Thus, it was necessary for the mechanical accuracy of the machine tools (and thus of the measuring machines used for quality control) to become more accurate, which increased the cost of production processes. Since computerized numerical control (CNC) machines began to dominate the market, cost reduction has played a major role. An economic way of enhancing the accuracy of measuring machines is to numerically correct the guideway deviations [1, 2]. In this context, some of the most effective measurement procedures of error detection are error separation methods, which allow systematic errors to be minimized by means of multiple well-defined measurements of an object in normal and sometimes also reversal positions. By averaging the results, error contributions can be separated and eliminated. The most important prerequisite for applying error separation methods to error detection in guideways is that the axis deviations must be reproducible, which allows the systematic behavior of the guideway motion to be systematic. In addition to their reasonable costs and easy and robust handling, error separation methods allow the use of uncalibrated measurement objects such as hole plates

and ball plates [2, 3]. A pointwise evaluation of all measurements performed yields systematic errors for the measuring systems investigated as well as geometrical deviations for the measurement object used. The following section describes how the three-rosette method as an example for an error separation method can be applied to determine errors in rotary table guideways integrated into coordinate measuring machines (CMMs) with cartesian axes [4]. The first-ever detailed description of this measurement and evaluation strategy is provided in this paper which has been developed at the Physikalisch-Technische Bundesanstalt (PTB). Partial applications of the procedure were described previously to pitch deviations on gears [5] and to circular ball plates [6]. Moreover, in the recent past applications to large rotary tables used in gantry-type CMMs have been investigated in [7] and [8]. The latter paper also quantifies the impact of rotary table error motions on measurements of large involute gears. 2. Application of the three-rosette method to rotary tables 2.1 Principle of the method Rotary tables are based on rotational guideways whose errors of motion can be described by six degrees of freedom (see Figure 1): one axial error motion (ctz), one angular positioning error motion (crz), two components for the radial error motion (ctx and cty) and two components for the tilt error motion (crx and cry). The notation chosen for the errors of motion is in keeping with guideline VDI/VDE 2617 [9].

Figure 1. Error model of rotary table kinematics. Motion of rotary tables are describable by six degrees of freedom: one axial error motion (ctz), one angular positioning error motion (crz), two components for the radial error motion (ctx and cty) and two components for the tilt error motion (crx and cry) The error separation method described is called “three-rosette method”, which is a special implementation of the so-called closure technique for rotational measurement applications [10]. For determining the above mentioned six geometrical error contributions of the rotary table the measuring set-up consist of the three independent rosettes: (1) an uncalibrated circular ball plate, (2) a Cartesian CMM (moving along a circle) and (3) a rotary table.

The circular ball plate is mounted on the rotary table of the CMM that has a tactile probe [6] (see Figure 2). The balls are arranged in a circle at equally distributed intervals on the plate. Usual tolerances need to be kept for the plate and its clamping; however, the form of the balls must be as accurate as possible, since the probing points of the CMM on the balls can vary slightly. A special measurement strategy (see Chapter 2.2) referred to as the three-rosette method makes it possible to determine all six error contributions of the rotary table as well as the respective error contributions of the CMM and the ball plate itself. This method can be successfully applied even if only the stability of the plate and the reproducibility of the rotary table and the CMM are ensured.

Figure 2. Setup of a ball plate. The Ball plate is mounted on a rotary table of a Cartesian CMM that has a tactile probe. 2.2 Measurement strategy In Figure 3, the measurement scheme for six balls on the ball plate and six angular positions (𝑛 = 6) of the rotary table are shown as an example. In each row of the scheme, the rotary table stays in the same angular position 𝑖 = 1,…,𝑛 while in each column, the same ball 𝑗 = 1,…,𝑛 on the plate is measured. In the diagonals from top right to bottom left and from bottom left to top right, the CMM assumes the same measuring position. The average of the corresponding measured positions in a row contains the influences 𝑅𝑇𝑖 as the deviations of the rotary table in angular position 𝑖, the mean value 𝑃 of all deviations of the ball positions on the plate and the mean value 𝑀 of all deviations of the measuring positions of the CMM. The systematic contributions in 𝑃 and 𝑀 are the same in all rows and are eliminated by subtracting the total mean. This yields the single influences 𝑅𝑇𝑖 of the rotary table subtracted by the mean value 𝑅𝑇 over all rotary positions (= offset elimination).

Figure 3. Measurement scheme for determining errors in a rotary table using a ball plate with 6 balls. Black spheres represent the location of the first ball, crossed spheres represent the ball measured in the current angular position, and white spheres indicate the position of the remaining balls. 𝑅𝑇𝑖 are the deviations of the rotary table in angular position 𝑖, 𝑃 is the mean value of the deviations of the ball plate and 𝑀 is the mean value of the deviations of the CMM The measurement scheme requires each ball to be measured once in each angular position of the rotary table. Each ball measurement corresponds to the movement of the linear axes and the rotary table at a certain position. Depending on whether the measuring scheme is processed line by line, column by column or diagonally, different motion sequences result for the linear axes and the rotary table. The angular positions of the rotary table and the measuring positions of the CMM are programmed in an inner and an outer loop. In the inner loop, all balls are measured several times in different positions, while in the outer loop, only one rotation over the whole measurement sequence is performed. The order selected for the measuring positions influences the measuring deviations that occur on the rotary table, the CMM and the ball plate due to the factors such as thermal drift effects during the measurements. Therefore, the best measurement strategy based on Figure 3 depends on the given situation. Three cases must be distinguished as follows: 1) Investigation of the component deviations in accordance with VDI/VDE 2617 Part 4 [9] for the acceptance and reverification testing of CMM with an additional axis of rotation:  The scheme of Figure 3 is processed row by row.  The CMM and the ball plate are operated in the inner loop and the rotary table in the outer loop. While the CMM measures all balls on the plate, the rotary table does not move. Then, the rotary table takes the next angular position in the outer loop and the CMM begins to measure the

balls again. At the end of the calibration procedure, the rotary table turns 360° only once and the CMM 𝑛 times. 2) Investigation of the mean systematic error behavior of the rotary table for example, with the objective of correction:  Scheme of Figure 3 is processed along the diagonal from top right to bottom left.  The rotary table and the ball plate are operated in the inner loop and the CMM in the outer loop. In this case, the CMM maintains its measuring position while the rotary table positions the balls. After one revolution, the CMM moves to the next measuring position. In total, the CMM turns 360° only once, but the rotary table turns 𝑛 times. 3) Investigation of the mean systematic deviation behavior of both the rotary table and the CMM with the same emphasis:  Scheme of Figure 3 is processed column by column.  The CMM and the rotary table are operated in the inner loop and the ball plate in the outer loop. At the end, in all three cases, 𝑛2 ball center positions are available in terms of the respective coordinates 𝑥𝑖𝑗, 𝑦𝑖𝑗 and 𝑧𝑖𝑗 for each ball 𝑗 in each position 𝑖. Applying Case 2, for example, the following single measuring steps must be completed: a) Aligning the ball plate in such a way that Ball 1 points approximately in the direction of x-axis b) Determining the rotary axis within the CMM measuring volume c) Defining a local coordinate system in such a way that the z-axis is determined by the specified rotary axis and the x-axis passes through the center of Ball 1 which must be fixed for all measurements d) Obtaining the output results that contain the coordinates of the ball center positions and the angle of the rotary table in the coordinate system from Step c) In Table 1 the target positions for Step d) are listed if the rotary table and the CMM rotate counterclockwise (mathematically positive) and the balls are also numbered counterclockwise (mathematically positive). The ball plate used has 6 balls and a ball center circle radius of 182.5 mm. Me CM RT Ball 𝒙𝒊𝒋 as. M Pos. No. in No. Pos. 𝒊 𝒋 mm 1 182. 1 1 5 2 = 182. 2 6 5 1 3 182. 3 5 5

𝒚𝒊𝒋 𝒛𝒊𝒋 Me CM RT Ball 𝒙𝒊𝒋 𝒚𝒊𝒋 𝒊 as. M Pos. No. in in in in in ° No. Pos. 𝒊 𝒋 mm mm mm mm 91.2 158. 0 0 0 7 2 1 5 05 8 = 91.2 158. 0 0 60 3 6 5 05 2 9 91.2 158. 0 0 120 4 5 5 05

𝒛𝒊𝒋 𝒊 in in ° mm 0

60

0

120

0

180

Me CM RT Ball 𝒙𝒊𝒋 𝒚𝒊𝒋 as. M Pos. No. in in No. Pos. 𝒊 𝒋 mm mm 4 182. 4 4 0 5 5 182. 5 3 0 5 6 182. 6 2 0 5 13 158. 3 1 91.2 05 5 14 158. 4 6 91.2 05 5 15 158. 5 5 91.2 05 5 = 16 3 158. 6 4 91.2 05 5 17 158. 1 3 91.2 05 5 18 158. 2 2 91.2 05 5 25 5 1 91.2 158. 5 05 26 6 6 91.2 158. 5 05 = 27 5 1 5 91.2 158. 5 05 28 2 4 91.2 158. 5 05

𝒛𝒊𝒋 Me CM RT Ball 𝒙𝒊𝒋 𝒚𝒊𝒋 𝒊 as. M Pos. No. in in in in ° No. Pos. 𝒊 𝒋 mm mm mm 10 91.2 158. 0 180 5 4 5 05 11 91.2 158. 0 240 6 3 5 05 12 91.2 158. 0 300 1 2 5 05 19 0 120 4 1 182. 0 5 20 0 180 5 6 182. 0 5 21 0 240 6 5 182. 0 5 = 22 4 0 300 1 4 182. 0 5 23 0 0 2 3 182. 0 5 24 0 60 3 2 182. 0 5 31 91.2 0 240 6 1 158. 5 05 32 91.2 0 300 1 6 158. 5 05 = 33 6 91.2 0 0 2 5 158. 5 05 34 91.2 0 60 3 4 158. 5 05

𝒛𝒊𝒋 𝒊 in in ° mm 0

240

0

300

0

0

0

180

0

240

0

300

0

0

0

60

0

180

0

300

0

0

0

60

0

120

Me CM RT Ball 𝒙𝒊𝒋 𝒚𝒊𝒋 as. M Pos. No. in in No. Pos. 𝒊 𝒋 mm mm 29 3 3 91.2 158. 5 05 30 4 2 91.2 158. 5 05

𝒛𝒊𝒋 Me CM RT Ball 𝒙𝒊𝒋 𝒚𝒊𝒋 𝒊 as. M Pos. No. in in in in ° No. Pos. 𝒊 𝒋 mm mm mm 35 91.2 0 120 4 3 158. 5 05 36 91.2 0 180 5 2 158. 5 05

𝒛𝒊𝒋 𝒊 in in ° mm 0

180

0

240

Table 1. Target ball center coordinates (𝑥𝑖𝑗, 𝑦𝑖𝑗, 𝑧𝑖𝑗) and rotary table (RT) angle 𝑖. It is based on Figure 3 and is in accordance with Case 2 Investigations at PTB have shown that 20 probing points distributed over 5 horizontal circles, with 4 points each provide stable results. The upper horizontal circle was chosen 1 mm below the pole of the sphere, the lower line circle 1 mm below the equator. There was no probing point directly at the sphere's pole. In addition, it is recommended to check the repeatability of the ball center positions before applying the procedure. The variation gives an indication of the accuracy and reliability of the process. During the actual application of the procedure repetitive measurements should be avoided, as this extends the measurement time and, for example, drift effects would affect the accuracy of the process considerable. 2.3 Evaluation strategy The evaluation of the error contributions is based on the center point coordinates of the ball coordinates 𝑥𝑖𝑗, 𝑦𝑖𝑗 and 𝑧𝑖𝑗, which are evaluated by the CMM software. Furthermore, the geometrical deviations of the ball plate and the CMM are not explicitly calculated first and then subtracted to obtain the error contributions of the rotary table, instead, the error contributions of the rotary table are determined directly. The symbols and formulas presume that the rotation axis c is in the direction of the z-axis of the CMM. The origin of the coordinates is the intersection point of the rotation axis with the plane fitted into the ball center points. In the first angular position, the first ball is located on the x-axis. It is not necessary for these conditions to be met with more than a usual level of accuracy. If the rotary table is aligned with other axes of the CMM, the formulas must be converted analogously. The individual errors in each angle position 𝑖 can be described according to the following equations.  The axial error motion in the z-direction 𝒄𝒕𝒛𝒊 is determined by equation (1).

𝑛

𝑐𝑡𝑧𝑖 = 1/𝑛

∑𝑧

𝑛

𝑖,𝑗

― 1/𝑛2

𝑗=1

𝑛

∑ ∑𝑧

𝑖,𝑗

(1)

𝑖 = 1𝑗 = 1

 The radial error motion is described by its two components in the x- and y-direction 𝒄𝒕𝒙𝒊 and 𝒄𝒕𝒚𝒊 in accordance with equation (2) and (3). 𝑐𝑡𝑥𝑖 = 𝑥𝑖 ― 𝐴 cos (2𝜋(𝑖 ― 1)/𝑛) + 𝐵 𝑠𝑖𝑛(2𝜋(𝑖 ― 1)/𝑛) (2) 𝑐𝑡𝑦𝑖 = 𝑦𝑖 ― 𝐴 𝑠𝑖𝑛(2𝜋(𝑖 ― 1)/𝑛) ― 𝐵 𝑐𝑜𝑠(2𝜋(𝑖 ― 1)/𝑛)

(3)

The sub-totals of the x- and y-values 𝑥𝑖 and 𝑦𝑖 are calculated by equation (4) and (5) 𝑛

𝑥𝑖 = 1/𝑛

∑𝑥

𝑛

𝑖,𝑗

∑ ∑𝑥

― 1/𝑛2

𝑗=1

𝑖,𝑗

(4)

𝑖 = 1𝑗 = 1

𝑛

𝑦𝑖 = 1/𝑛

𝑛

∑𝑦

𝑖,𝑗

𝑛

― 1/𝑛2

𝑗=1

𝑛

∑ ∑𝑦

𝑖,𝑗

(5)

𝑖 = 1𝑗 = 1

and correction coefficients 𝐴 and 𝐵 in order to eliminate the eccentricity between the rotational axis of the rotary table and the center of the ball plate by means of equation (6). 𝑛

𝐴 = 1/𝑛

∑(𝑥 𝑐𝑜𝑠(2𝜋(𝑖 ― 1)/𝑛) + 𝑦 𝑠𝑖𝑛(2𝜋(𝑖 ― 1)/𝑛)) 𝑖

𝑖

(6)

𝑖=1 𝑛

𝐵 = 1/𝑛

∑( ―𝑥 𝑠𝑖𝑛(2𝜋(𝑖 ― 1)/𝑛) + 𝑦 𝑐𝑜𝑠(2𝜋(𝑖 ― 1)/𝑛)) 𝑖

𝑖

(7)

𝑖=1

Comment: The three translational error contributions 𝑐𝑡𝑥𝑖, 𝑐𝑡𝑦𝑖 and 𝑐𝑡𝑧𝑖 are derived from the calculation of the center of gravity of the ball centers and a correction for its eccentricity. This applies to measuring arrangements with cyclic permutation even for a CMM with strong nonlinear systematic deviations. The correction of the radial error motion is a sine function with the same amplitude for the two radial components in 𝑥 and 𝑦 and with a phase lead of 90 ° in 𝑥. Hence, the corrections cannot be determined independently for the two components. A simple circle best-fit without attention to the point sequence on the circle is impermissible. This also applies to the correction of the plate inclination for determining of the tilt error contributions.  The angular positioning error motion 𝒄𝒓𝒛𝒊 is calculated in accordance with to equation (8) 𝑛

𝑐𝑟𝑧𝑖 = 𝑟𝑧𝑖 ― 1/𝑛

∑𝑟𝑧

𝑖=1

𝑖

(8)

in which 𝑟𝑧𝑖 as the sub-total of rotation angle around the z-axis, is calculated by means of equation (9), 𝑛

𝑟𝑧𝑖 = 1/𝑛𝑅

∑( ―𝑥

𝑖,𝑗sin (2𝜋((𝑖

― 1)/𝑛 + (𝑗 ― 1)/𝑛))

𝑗=1

(9)

+ 𝑦𝑖,𝑗cos (2𝜋((𝑖 ― 1)/𝑛 + (𝑗 ― 1)/𝑛))) with the average radius of the ball center circle 𝑅 in equation (10). 𝑅 𝑛

= 1/𝑛2

𝑛

∑∑

(10 )

𝑖 = 1𝑗 = 1

(𝑥𝑖,𝑗cos (2𝜋((𝑖 ― 1)/𝑛 + (𝑗 ― 1)/𝑛)) + 𝑦𝑖,𝑗sin (2𝜋((𝑖 ― 1)/𝑛 + (𝑗 ― 1)/𝑛)))  The tilt error motion is described by its two rotational components around the x- and y-axis 𝒄𝒓𝒙𝒊 and 𝒄𝒓𝒚𝒊 in accordance with equation (11) and (12). 𝑛

𝑐𝑟𝑥𝑖 = 𝑟𝑥𝑖 ― 1/𝑛

∑𝑟𝑥 ― 𝐶cos (2𝜋(𝑖 ― 1)/𝑛) + 𝐷sin (2𝜋(𝑖 ― 1)/𝑛) 𝑖

(11)

𝑖=1 𝑛

𝑐𝑟𝑦𝑖 = 𝑟𝑦𝑖 ― 1/𝑛

∑𝑟𝑦 ― 𝐶sin (2𝜋(𝑖 ― 1)/𝑛) ― 𝐷cos (2𝜋(𝑖 ― 1)/𝑛) 𝑖

(12)

𝑖=1

The sub-totals of rotation angles around the x- and y-axes 𝑟𝑥𝑖 and 𝑟𝑦𝑖 are calculated by equation (13) and (14), 𝑛

𝑟𝑥𝑖 = 2/𝑛𝑅

∑𝑧

𝑖,𝑗sin (2𝜋((𝑖

― 1)/𝑛 + (𝑗 ― 1)/𝑛))

(13)

𝑗=1 𝑛

𝑟𝑦𝑖 = ―2/𝑛𝑅

∑𝑧

𝑖,𝑗cos (2𝜋((𝑖

― 1)/𝑛 + (𝑗 ― 1)/𝑛))

(14)

𝑗=1

while the correction coefficients for the plate inclination 𝐶 and 𝐷 are calculated by means of equation (15) and (16). 𝑛

𝐶 = 1/𝑛

∑(𝑟𝑥 cos (2𝜋(𝑖 ― 1)/𝑛) + 𝑟𝑦 sin (2𝜋(𝑖 ― 1)/𝑛)) 𝑖

𝑖

(15)

𝑖=1 𝑛𝑤

𝐷 = 1/𝑛

∑( ―𝑟𝑥 sin (2𝜋(𝑖 ― 1)/𝑛) + 𝑟𝑦 cos (2𝜋(𝑖 ― 1)/𝑛)) 𝑖

𝑖

(16)

𝑖=1

Comment: The correction of the tilt errors 𝑐𝑟𝑥𝑖 and 𝑐𝑟𝑦𝑖 is a sine function with the same amplitude for the two radial components in 𝑥 and 𝑦 and with a phase lead of 90 ° in 𝑥. Hence, the corrections must not be

determined independently for the two components. Moreover, a simple circle best-fit is impermissible if it does not include the point sequence on the circle. 3. Verification measurements For the verification of the three-rosette-method a rotary table was selected that had been integrated into a commercial CMM (see Figure 2). The method was applied by means of a ball plate consisting of 12 balls measured in accordance with the strategy Case 2 from Chapter 2.3 (investigation of the mean systematic deviation behavior of the rotary table), which is visualized in Figure 4. The evaluation of the measurement values is based on the mathematical descriptions in Chapter 2.4, here, the algorithms were implemented as a MATLAB program to enable an automated application. As input parameters, the xyz-coordinates of the ball centers as well as the angular positions of the rotary table were determined and transferred to the MATLAB program in accordance with Table 1 (actual instead of nominal values).

Figure 4. Measurement strategy applied in accordance with the description in Chapter 2.3. The two different coordinate systems are marked with (xCMM, yCMM, zCMM) for the coordinate measuring machine and (XRT, YRT, ZRT) for the rotary table The three-rosette-method was applied under different conditions in order to ensure the correct elimination of the errors in the ball plate as well as the guideway errors of the linear axes of the CMM. First (i), the procedure was performed under the best possible conditions with all online error compensations for the rotary table and with the linear axes enabled in the firmware of the CMM. In the second run (ii), the ball plate was rotated by 90 ° on the rotary table before starting the procedure so that the rotary table errors yielded would be the same as those before. Then (iii), the correction of the linear axes of the CMM was switched off, again, this was done to reproduce the same errors. Finally (iv), the corrections to the linear axes and for the axial error motion to

the rotary table were switched off. It was expected that the values for the latter error will correspond to the correction table determined by the manufacturer. In Figure 5, the results are presented in six diagrams, with one diagram assigned to each geometrical deviation of the rotary table ctx, cty, ctz, crx, cry and crz. In each diagram, the deviations are plotted against the angular positions in four curves according to runs above described (i), (ii), (iii) and (iv). For the angular positioning error crz, two additional curves are depicted because for this error contribution the CMM manufacturer offers a correction; here, this correction is used for a comparison with the results of the three-rosette-method. In this case, curves (iii) and (iv) are not nearly identical but describe the geometrical deviation measured by means of the three-rosette-method after switching off the manufacturer correction. Of the two additional curves, (v) shows the correction values of the manufacturer and (vi) shows the difference between (iv) and (v). The ordinates of the diagrams are all in the range of  0.06 µm and  0.3 μrad, respectively. The fact that the deviations are within this very small range is already evidence of for the very small geometrical deviations of this rotary table. However, the curves show that even these small residual errors can be determined in such a way that they can be easily reproduced, with a mean variation in the range of 0.02 µm and 0.2 µrad with the three-rosette method, regardless of the ball plate position on the rotary table and the correction of the linear CMM axes. The diagram for crz proves that the results of the threerosette-method achieved deviation values at the common grid points that were similar to those of the manufacturer correction, even though the manufacturer correction has a much larger number of supporting points. To obtain a larger number of supporting points with the three-rosette-method, it is necessary to increase the number of balls on the ball plate. The differences in the crz deviations at the twelve supporting points between the rosette-method and the manufacturer correction show the same curve progression as the deviations of the runs (i), (ii) and (iii) with a maximum difference of 0.4 µrad. This indicates that these residual deviations are still present even after the manufacturer correction and can be revealed by means of the three-rosette-method.

Figure 5. Results for the six error contributions of the rotary table investigated. One diagram for each error motion of the rotary table (RT) ctx, cty, ctz, crx, cry and crz.

5. Conclusion Errors in rotary tables integrated into CMMs can be determined by means of an error separation method using the three-rosette method and an uncalibrated circular ball plate. This robust and inexpensive calibration procedure that is described here, is based on the performance of 𝑛2 measurements of the plate equipped with 𝑛 balls. The underlying measurement strategy sets the relative positions of the single balls, the rotary table angular positions and the CMM positions in a new relationship to each other for each measurement, which allows the individual error components to be separated. The measurement strategy changes slightly depending on whether the main focus is on the errors from the ball plate, from the rotary table or from the CMM. The process was tested on a CMM with an integrated rotary table using a 12-ball plate. The measurement results were suitable for the determination of all six

errors of the rotary table with high accuracy. Especially, the angular positioning error crz proves that the results of the three-rosette-method achieved deviation values that were similar to those of the manufacturer correction. This serves as a verification of the method. As expected, the high accuracy classes of the measuring equipment used, do not affect the reliability and measurement uncertainty of the method. PTB is currently developing reduced error separation methods, with almostconstant measurement uncertainty. The already developed approaches for pitch measurements on gears [11] will be applied to the method for rotary tables. Thus, in future, the number of measurements required (𝑛2) should decrease significantly, which will streamline the process even further. Funding: The basics of this work were supported by the Deutsche Forschungsgemeinschaft (DFG) within the project “Rückführung von Koordinatenmeßgeräten durch Abschätzung der zu erwartenden Meßabweichungen durch Simulation" (1994 - 1996). References [1] Evans R.J., Hocken R., Estler W.T.: Self-Calibration: Reversal, Redundancy, Error Separation and Absolute Testing. Annals of the CIRP, 1996, 45(2), 617-634. [2] Kunzmann H., Trapet E., Wäldele F.: A Uniform Concept for Calibration, Acceptance Test, and Periodic Inspection of Coordinate Measuring Machines Using Reference Objects. Annals of the CIRP, 1990, 39(1), 561564. [3] Schellekens P., Spaan H., Soons J., Trapet E., Loock V., Dooms J., de Ruiter H., Maisch M.: Development of Methods for Numerical Correction of Machine Tools. Proceedings of the 7th International Precision Engineering Seminar, 1993, Springer Verlag, Kobe (Japan), 213-223. [4] Trapet E.: Rückführung von Koordinatenmessgeräten durch Abschätzung der zu erwartenden Messabweichungen durch Simulation. Bericht zum DFG-Forschungsvorhaben, 1996. [5] Kniel K., Härtig F., Osawa S., Sato O.: Two Highly Accurate Methods for Pitch Calibration. Measurement Science and Technology, 2009, 20(11), 115-110. [6] Guenther A., Stöbener D., Goch G.: Self-Calibration Method for a Ball Plate Artefact on a CMM. Annals of the CIRP, 2016, 65, 503–506 [7] Wang Q., Miller J., von Freyberg A., Steffens N., Fischer A., Goch G.: Error mapping of rotary tables in 4-axis measuring devices using a ball plate artifact. CIRP Annals, 67(1), 2018, 559-562. [8] Wang Q., Peng Y., Wiemann A., Balzer F., Stein M., Steffens N., Goch G.: Improved gear metrology based on the calibration and compensation of rotary table error motions. CIRP Annals, 68 (1), 2019, 1, 511-514.

[9] VDI/VDE 2617 series: Accuracy of coordinate measuring machines Characteristics and their checking. [10] Noch R., Steiner O.: Die Bestimmung von Kreisteilungsfehlern nach einem Rosettenverfahren. Zeitschrift für Instrumentenkunde 74, 1966, 307316. [11] Keller F., Stein M., Kniel K.: Reduced error separating method for pitch calibration on gears. Advanced Mathematical and Computational Tools in Metrology and Testing XI, Series on advances in mathematics for applied sciences, 89, 2018, 220-228. Conflict of Interest and Authorship Conformation Form Please check the following as appropriate:

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Highlights of the article „Detecting 6 DoF geometrical errors of rotary tables“ MEAS-D-19-02199 • • • •

Error detection in rotary tables integrated into coordinate measuring machines Application of an error separation method called three-rosette method Measurement of an uncalibrated circular ball plate in multiple positions Verification with a 12-ball plate by determining all six error contributions