CIRP Annals - Manufacturing Technology 68 (2019) 511–514
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Improved gear metrology based on the calibration and compensation of rotary table error motions Qichang Wang a,*, Yue Peng a, Ann-Kathrin Wiemann b, Felix Balzer c, Martin Stein b, Norbert Steffens c, Gert Goch (1)a a
University of North Carolina at Charlotte, USA Physikalisch–Technische Bundesanstalt, Germany c Hexagon Manufacturing Intelligence, Germany b
A R T I C L E I N F O
A B S T R A C T
Keywords: Gear Compensation Rotary table
Geometric measurements of cylindrical gears are mainly conducted by either applying the generation principle or by coordinate measurements of points on a flank surface. In both cases, a rotary table (RT) is highly involved in the inspection process. The RT error motions and the misalignments of the gear axis with respect to the table’s effective rotary axis affects the measurement results, whose impacts increase with the gear dimensions. This paper presents the influence of such errors on gear measurements using both the classical line oriented and the areal inspection methods. A compensation method to remove these influences in gear evaluations is proposed using a RT, which was calibrated with a ball plate artifact. The methods are validated by measuring and evaluating a large gear artifact applying multiple strategies with and without compensation. © 2019 Published by Elsevier Ltd on behalf of CIRP.
1. Introduction Geometric measurements of cylindrical gears are mainly performed by two methods: (i) generative measurement following the generation principle (GP), (ii) coordinate measurements of points on flank surfaces. The first method requires simultaneous linear motion of the probe tip and rotary motion of the gear flank. Accordingly, rotary tables (RTs) are mandatory components of the applied multi-axis measuring devices able to perform GP, including gear measuring instruments (GMIs) and 4-axis coordinate measuring machines (CMMs). The second method uses three-axes linear motions of the probe tip, so that it is mainly applied on the CMMs. RTs have three translational error motions dx , dy and dz , two tilt error motions ex and ey , and one angular positioning deviation ez (Fig. 1). In gear measurement, the error motions introduce extra deviations to the measured points on gear flanks, which should be separated from these gear’s geometric deviations. Therefore, gear
Fig. 1. Schematic of six error motions of rotary table.
* Corresponding author. E-mail address:
[email protected] (Q. Wang). https://doi.org/10.1016/j.cirp.2019.04.078 0007-8506/© 2019 Published by Elsevier Ltd on behalf of CIRP.
measurement and evaluation could be improved by calibrating and compensating the error motions of the RT. In area-oriented gear measurement (multiple profile and helix lines covering a gear flank) [1], RTs mainly work in four modes: (i) The RT is disabled and kept stationary at an arbitrary angle (usually zero). The error motions at this angle introduce a homogeneous transformation to the entire gear. (ii) The RT positions the targeted flanks to a proper angle, so that all flanks can be accessed by a single probe, which greatly simplifies the probe configuration. It remains stationary during the measurement of one flank, while the probe moves in two-dimensions (for a profile line) or three-dimensions (for a helix line) to capture the points. (iii) The RT positions each measured profile/helix line of a flank to the initial angular position, so that all lines can be accessed by a single probe. The RT also remains stationary during probing of the targeted line. (iv) For profile measurement, the RT rotates simultaneously with the probe moving in the line of action direction, performing the generative measurement. For helix measurement, this motion is done by rotating the RT while moving the probe in the axial direction. The error motions of the RT are functions of the rotation angle. Therefore, points on the same flank deviate uniformly in case (i) and (ii), while in case (iv) every point is affected differently by RT error motions. Moreover, the range of the rotation angle needed to cover an entire flank increases with the helix angle and the face width, which might result in increased relative deviations from point to point. As illustrated in Fig. 2, the magenta dots represent case (i), which is used as a reference to compare with the other cases. The blue stars show the coordinates that are captured in machine
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When measuring a profile following the generation principle (case iv), the sensor captures the deviations in the linear motion along the line of action (in which the probe moves). 1 ð3Þ PMCS;i ¼ T RTCS; MCS þ R1 RTCS;MCS T LCS; RTCS þ RLCS;RTCS P LCS;i T LCS; RTCS and RLCS; RTCS are the transformations from RTCS to the line coordinate system (LCS). PLCS;i defined by Eq. (4) is used as the basic model in LCS for the probe moves in Y axis. The line of action could be in an arbitrary direction as long as it is tangent to the base circle in the measured transverse plane. PLCS;i ¼ ½ rb Fig. 2. Cases of gear measurement using a RT by measuring profile (a–c) or helix lines (d–f); magenta dots in all figures represents case (i); (a, d): Case (ii); (b, e): Case (iii); (c, f): Case (iv).
coordinate system (MCS) in cases (ii)–(iv). ui is the angle that the RT rotated to position the flank, line, or point to measure. Multiple ui are marked in Fig. 2 to show the RT motions in each case. 2. Methodology 2.1. Mathematical model Case (i)–(iii) discussed in Section 1 could be represented with the metrology loop as in Fig. 3. The point coordinates in MCS (PMCS ), as a vector, is described by Eq. (1) (i = 1, 2, . . . , N). n h io 1 1 1 PMCS;i ¼ T RTCS;MCS þ R1 RTCS;MCS T RTErr;RTCS;i þ RRTErr;RTCS;i Ru;ECS;i T WCS;uCS þ RWCS;uCS P WCS;i
ð1Þ
Fig. 3. Metrology loop of gear measurement with RT.
T RTCS;MCS and RRTCS;MCS are the translational and rotational transformations from the MCS to the RT coordinate system (RTCS). T RTErr;RTCS and RRTErr;RTCS are the error motions of the RT measured in the RTCS. Ru; RTCS is the commanded RT rotation, which is performed in the errors-integrated RT coordinate system (ECS). T WCS;uCS and RWCS;uCS are the transformations defining the workpiece coordinate system (WCS), which are constructed using the datum features of the gear (such as the center bore). PWCS; i gives the point coordinates in WCS, which include the gear’s deviations and should be used for gear evaluations. i indices the points and N is the total number of measured points. 3 dlot cosbb sin js;i þ Ls;i 7 6 rb cos js;i þ Ls;i þ js;i þ rb 6 " #7 6 7 d cosbb ¼6 7 cos js;i þ Ls;i 7 6 rb sin js;i þ Ls;i js;i þ lot 4 5 rb 2
PWCS; i
ð2Þ
zs;i þ dlot sinbb
In Eq. (2), dlot is the plumb line distance from the measured point i to the helical gear flank in the surface normal direction [2]. rb , bb , at are the basic parameters to define a gear, which are base radius, helix angle at base radius, and transverse pressure angle, respectively. js;i is the roll angle to describe the flank in the profile direction. Ls;i is the rotational alignment where the involute starts z tanb on the base radius, and Ls;i ¼ L0 þ s;i rb b . zs;i characterizes the flank in the axial direction.
yi
zc T ¼ rb
rb jc;i þ dy;i
zc;i
T
ð4Þ
Plane z ¼ zc;i is the transverse plane in which the inspected profile and in which the probe is commanded to move. jc;i is the roll angle of the point intended to be measured. dy;i is the deviation captured by the sensor at the roll angle jc;i , i.e. at this commanded position. Each measured point is found by intersecting the line of the probe’s motion and the gear flank represented in the MCS, which means equating Eqs. (1) and (3). Then, the coordinates of the surface points can be reconstructed from the deviations data. If a helix line is measured by the generation principle, the probe moves along a line as described by Eq. (5). PLCS ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r2b þ rb jc;i
2
T dy
zc;i
ð5Þ
2.2. Compensation of the RT error motions Before measuring a workpiece, the RTCS is either determined using a reference sphere (usually for CMMs) or it is well known with respect to the MCS (usually for GMIs). The WCS is then constructed using datum features. Therefore, T RTCS;MCS , RRTCS;MCS , T WCS;uCS , and RWCS;uCS are known while establishing the CSs. The commanded RT rotation, Ru; RTCS;i , is calculated by the instrument controller, depending on the measuring mode. The CMM used in the experiments described here offers the angle of the RT at each measured point. The error motions can be identified with a circular ball plate artifact, which is centered on the RT (Fig. 8(a)). In each cycle of repeated multi-step measurement, all spheres are measured. Error motions are separated from the deviations of the CMM and artifact [3,4]. If the error motions are periodic, they can be compensated numerically to improve gear measurement results. Values at an arbitrary rotary angle can be interpolated using sampled data. In such a way, T RTErr;RTCS;i and RRTErr;RTCS;i are obtained. Applying these transformations to the captured points, the point coordinates in the workpiece coordinate system with RT error motions compensated are obtained by Eq. (6). PWCS; i ¼
RWCS; uCS Ru;ECS RRTErr;RTCS RRTCS;MCS P MCS;i T RTCS;MCS T RTErr;RTCS T WCS;uCS
ð6Þ 3. Numerical simulation The influences of error motions and the mathematical models are verified by simulation with a designed gear, RT error motions, and CSs definitions. The parameters of the gear are the same as the artefact used in the experiments in Section 4. The starting angle of the involute on the transverse planez ¼ 0 is 0 (Fig. 4(m)). The measured error motions of the RT in Section 4.1 (see Fig. 9) were integrated in the simulation. If PWCS; i constitutes a perfect involute surface known by design, which means the deviations of the gear itself are zero, the evaluation results purely show the influences of the error motions. Fig. 4 illustrates the influence of each error motion when performing profile measurements in case (i) and (iii). Each figure is an areal deviation map [1] when a single error motion is applied.
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Germany [5]. It is designed as a segment of a complete gear. Three teeth have different helix angles and hands of helices: (i) 10 /right (ii) 0 /spur (iii) 20 /left. The gear standard sits on a base frame via a kinematic coupling. A total mass of 455.7 kg creates a significant asymmetric load on the RT when the gear shaft is centered. Therefore, a 400 kg cylindrical segment counterweight is placed opposite to the gear standard for balancing purpose. An aluminum positioning accessory realizes the accurate positioning of counterweight and base frame. The gear standard is measured by a 4-axis CMM, which consists of a Leitz PMM-G 50.40.20 and a Zollern aerostatic RT. This ;1000 mm RT has a maximum load of 3.0 tons. 4.2. Design of experiments
Fig. 4. Effect of one error motion on the flank deviation map of case (i) (a–f) and case (iii) (g–l): (a) dx , (b) dy , (c) dz , (d) ex , (e) ey , (f) ez , (g) dx , (h) dy , (i) dz , (j) ex ,
(k) ey , (l) ez .
A deviation map displays the differences from the measured points to the reference geometry (designed gear) with respect to the extensions in profile (U) and helix (V) directions. Thedlot axis is the plumb line distance.dlot;i ¼ 0 means that the measured flank shows no deviation. The benefits of using areal deviation maps are that a 3D view of the flank’s deviations is displayed, and the evaluation parameters represent the entire flank. From Fig. 4, each error motion introduces a different deviation pattern to the flank deviations. The effects vary from case to case as well. Fig. 5 further shows the total effects of the error motions in the two different cases. The effects also depend on the position of the flank. The errors are effectively compensated in simulation.
The 3rd tooth was selected since it had the largest helix angle, thus using a largest range of the RT. For both profile and helix measurements, four experiments were conducted using four modes of RT operation including the reference group: (i) positioning the flank at u = 0 (reference) and, (ii) at u = 270 , (iii) RT positioning of each profile or helix line, (iv) generative measurements of profile or helix lines. In all four experiments, 41 profile lines or 11 helix lines on the right flank were measured for area-based evaluation. The conventional CMM software offers line-oriented evaluations for each measured line. Fig. 7 shows the thereby evaluated slope deviation at each line. Each experiment (Exp.1 to Exp.4 correspondingto the 4 experiments) was repeated 3 times (T1 to T3). The results showed reasonable repeatability. Each point on a line in Fig. 7(a) represents one profile slope deviation, evaluated using the profile at thecorrespondingline index. Firstly, Fig. 7(a) shows that the profile slope deviation changes if measured at different axial positions in different modes, which introduces uncertainty to the evaluation results. For the profile at the middle of the flank (line # 21), the largest difference caused by different measuring modes is 1.975 (mm). For the helix at the line #6, the largest difference caused by different measuring modes is 3.2 m m.
Fig. 5. Deviation maps of case (i) (a–b) and case (iii) (c–d) as caused by actual RT error motions: (a) profile measurements, (b) helix measurements, (c) profile measurements, (d) helix measurements.
4. Experimental verification
Fig. 7. Profile or helix slope deviation evaluated at all measured lines in 4 experiments: (a) profile experiments; (b) helix experiments; (c) areal deviation map of experiment 1.
4.1. Experimental setup A large gear calibration artifact (Fig. 6) for universal use on GMIs and CMMs was built at Physikalisch-Technische Bundesanstalt (PTB),
One advantage of areal evaluation is that the parameters are evaluated based on the entire flank rather than on a local line. The areal deviation map [1,6] of test 1 in experiment 1 (Exp. 1, T1) is presented in Fig. 7(c). The highlighted point in (a) shows the profile slope deviation of the highlighted line in (c). Please note that (a) is measured in the transverse plane while the deviation map in (c) is measured in the surface normal direction (dlot as explained in Section 3). Table 1 presents the areal evaluation results of the 4 experiments. The largest difference is between the twist deviation of experiment 2 and 4, which is 2.4 m m. 4.3. RT calibration
Fig. 6. Experimental setup: (a) Gear artifact (segment of a complete gear) with three teeth having different helix angles and hands of helices (b) Right flank of tooth #3 with 41 profile lines and 11 helix lines.
In many configuration of large 4-axis CMMs, the RTs are fixed on the granite frames, which permanently reduce the measuring
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Table 1 Areal deviation parameters of test 1 in 4 experiments. Dev. Parameters (mm) A
f Ha (profile slope dev.) A f Hb (helix slope dev.) A f Ca (profile crowning dev.) A f Cb (helix crowning dev.) A f S (twist dev.)
Exp1
Exp2
Exp3
Exp4
8.5
9.5
8.2
8.1
8.9
8.6
8.5
10.4
0.1
0.1
0.1
0.2
0.4
0.7
0.5
1.3
0.2
1.7
1.2
0.7
volumes. In the CMM used here, the RT is placed on a 0.5 m high steel frame, which can be lifted and transported outside of the measuring volume. However, not enough measuring volume remains on top of the gear standard in Fig. 6 to add a circular ball plate, which can be used for in-situ RT calibration at the active loading condition. Instead, the error motions in this setup were estimated by calibrating the RT under no load, 1-ton and 2-tons of symmetric loads. 1-ton steel disks were used to create symmetric loads. The circular ball plate was centered and fixed on top of the disks (see Fig. 8).
Fig. 8. Experimental setups to identify the error motions of RT under symmetric loads: (a) non-load (b) 2 tons.
In all 3 load variation experiments, the table was rotated by four cycles in both the counter-clock-wise (CCW) and clock-wise (CW) rotating directions in steps of 10 . At each step, 5 points were probed on each of the six tungsten carbide spheres. The separated error motions of RT showed high repeatability in all cases (4 repetitions each). As shown in Fig. 9, the RT had similar error motions in three levels of symmetric loads. The periods of all six error motions were 2p. The RT also had comparable error motions in both rotating directions, at all three levels of loads. Therefore, it is reasonable to use the separated error motions at no load or 1-ton symmetric load for gear measurement compensation. The RT was calibrated again under no load condition in steps of 2 . Increasing the resolution improved the approximation of error motions at an arbitrary angle by interpolation. Moreover, the increased rotational resolution helped to catch key features including valleys and peaks. For example, it detected the valley A in the dy plot of Fig. 9, while the 10 per step cases missed it. For dy, it is noticeable that the results of no load, CCW, 2 case deviated from other results. It may originate from thermal drift during long data acquisition time, necessary for the much smaller rotary step. Therefore, only the results of CW, 2 case were used to compensate the error motions in both rotating directions.
Table 2 Areal deviation parameters of test 1 in 4 experiments, RT error motions compensated. Dev. Parameters (mm)
Exp1
Exp2
Exp3
Exp4
A f Ha (profile slope dev.) A f Hb (helix slope dev.) A f Ca (profile crowning dev.) A f Cb (helix crowning dev.) A f S (twist dev.)
8.1
9.0
7.9
7.7
9.6
9.3
9.0
11.0
0.1
0.1
0.1
0.2
0.4
0.7
0.5
1.3
2.0
1.7
1.8
0.7
4.4. Compensation The compensation procedure consists of four steps: (i) export the CSs and calculate the transformation matrices; (ii) export coordinates of the measured points; (iii) recalculate the translational error motions at each angle to the height z2 of the RTCS, because the reference point for the calibrated RT error motions is at height z1 ; (iv) calculate point coordinates in WCS using Eq. (6), and evaluate the flank with the calculated points. dx;z2 dy;z2 ¼ ðz2 z1 Þey þ dx;z1 ðz2 z1 Þex þ dy;z1 ð7Þ In step (ii), the CMM used in the experiments exports coordinates in the WCS, without knowing the actual RT error motions. Therefore, the coordinates are transformed back to the MCS first. Table 2 shows the results of the compensation for all 4 experiments. A large source of uncertainty noticed in the experiments is the repeatability of the location of the WCS. The compensation procedure will be improved, and more experiments will be conducted to investigate this issue in the future. 5. Conclusions This paper combined the rotary table calibration technique and areal evaluation technique to improve gear metrology result on CMMs, providing new possibilities for future applications in gear industry. RT error motions could add complex deviations to the gear flank measurement. According to the simulation results, translations in micrometer level and rotations in arcsec level could cause deviations of the evaluated parameters in micrometer level for the given gear. There are other influencing factors in the measurement procedure: the eccentricity and misalignments of the gear flanks with respect to the datum features; the CMM deviations; probe radius correction errors; and deformations and scale reading errors caused by temperature change; difference between the axis of rotation determined by the RT calibration and the axis of rotation determined when building the RTCS. Acknowledgement The authors gratefully acknowledge for the financial supports from the Center for Precision Metrology at UNC Charlotte, USA and Hexagon Manufacturing Intelligence, Germany. The authors also thank Mr. Achim Wedmann for his help in setting up experiments. References
Fig. 9. Six error motions of RT under three levels of symmetric loads.
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