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Calibration and uncertainty evaluation of single pitch deviation by multiple-measurement technique
Precision Engineering 34 (2010) 156–163
Contents lists available at ScienceDirect
Precision Engineering journal homepage: www.elsevier.com/locate/precision
Calibration and uncertainty evaluation of single pitch deviation by multiple-measurement technique Osamu Sato a,∗ , Sonko Osawa a , Yohan Kondo b , Masaharu Komori c , Toshiyuki Takatsuji a a b c
AIST, AIST Tsukuba Central 3, 1-1-1 Umezono, Tsukuba, Ibaraki, Japan Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, Japan
a r t i c l e
i n f o
Article history: Received 21 November 2008 Received in revised form 20 April 2009 Accepted 20 May 2009 Available online 18 July 2009 Keywords: Gear metrology Pitch deviation Multiple-measurement technique Coordinate measuring machine Analysis of variance
a b s t r a c t Gears are key elements of power transmission systems, and the inspection of their pitch deviation is one of the most important tests on gears. The specifications of gears are assessed using gear measuring instruments (GMIs) or coordinate measuring machines (CMMs), and the results of the measurements must be validated under an appropriate traceability system. In the traceability system, calibrated gauges whose measuring uncertainties are estimated are necessary. In the case of pitch deviation measurement, special artefacts or gears manufactured with high dimensional accuracy are used as reference gauges. In this paper, authors propose calibration and uncertainty evaluation methods for the single pitch deviation of gears measured using CMMs. First, the evaluation of single pitch deviation using a multiple-measurement technique and the estimation of its uncertainty based on the analysis of variance are formulated. Second, a technique for reducing the measurement trials based on the symmetry of the measurement is discussed. Finally, the proposed calibration method is validated through experiments. © 2009 Elsevier Inc. All rights reserved.
1. Introduction Gears are key elements of power transmission systems in many applications. Therefore, inspections of the specifications of gears are necessary to ensure the performance of systems in which gears are included. In particular, the evaluation of pitch deviation is one of the most important tests on gears [1]. The specifications of gears are assessed using gear measuring instruments (GMIs), coordinate measuring machines (CMMs), or other specialized instruments. Because of the small tolerances in gear manufacturing, the inspection of gears requires high reliability in spite of the geometrical complexity of measurands on gears [1,2]. The results of the measurements must be validated under a rigorous traceability system [1,3]. In the traceability system, it is necessary that the measuring devices are verified using calibrated reference gauges and that the uncertainty of the measurement performed using the device is estimated appropriately [4]. In the case of pitch deviation measurement, special artefacts [5] or gears manufactured with high dimensional accuracy are used as calibrated standards [1]. To improve the performance of the inspection of gears, standards used in the verification of measuring devices should be calibrated with a small uncertainty.
In this study, authors propose calibration and uncertainty evaluation methods for the single pitch deviation of gears measured using CMMs. First, the evaluation procedure of single pitch deviation by a multiple-measurement technique using CMMs is described. Second, the estimation method of the pitch deviation measurement is formulated on the basis of the analysis of variance (ANOVA) [6]. Also, a technique for reducing the number of measurement trials based on ANOVA is proposed. Finally, a calibration is performed using the proposed method and its reliability is validated through a comparative experiment. 2. Measurement of single pitch deviation using CMMs The single pitch is determined by the coordinates of the points at the intersections of the reference circle with the tooth flanks. The single pitch deviation of a gear is the difference between the actual pitch and the corresponding nominal pitch on the reference circle. The sequence of the pitch measurement depends on the instruments used, e.g., GMIs or CMMs, and their equipments, e.g., with or without rotary tables [1]. In this section, a procedure for computing the single pitch deviation of a gear-shaped artefact measured using a CMM without a rotary table is described. 2.1. Calculation procedure
∗ Corresponding author. Fax: +81 29 861 4042. E-mail address: [email protected] (O. Sato). 0141-6359/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2009.05.009
Fig. 1 illustrates the scheme used to compute the coordinates of the points to be used for the pitch calculation. When the probe of
O. Sato et al. / Precision Engineering 34 (2010) 156–163
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the pitch calculation, p Ci, are computed by the coordinates of the starting point of the involute curve and the nominal diameter of reference circle. Usually, these procedures performed automatically by the software installed in the CMM. When measuring a gear whose diameter of reference circle is d and whose number of teeth is z, the ith single pitch deviation, Pi , is formulated as follows: Pi =
d d − i−1 − , 2 i z
(2)
where i is the phase angle of p Ci calculated as follows: i = arctan
Fig. 1. Scheme for computing the coordinates for pitch calculation.
the CMM touches the flank of the ith tooth, the coordinates of the T centre of the probe, c Ci = (c xi, c yi) , and the probing direction unit vector, ni , are read. Here, the coordinates of the corrected measured T
point, f Ci = (f xi, f yi) , are computed as follows: f
Ci = c Ci + rP ni ,
(1)
where rP is the calibrated radius of the probe tip. Because of the positioning error of the CMM, the corrected measured point is not exactly on the reference circle. Thus, the coordinates of the points T to be used for the pitch calculation, p Ci = (p xi, p yi) , are calculated f from Ci in accordance with the profile of the teeth. In case of the measurement on involute gear, for example, the starting point of the involute curve is determined by the coordinates of the corrected measured point, f Ci, and by the length between the centre of the gear and f Ci. Therefore, the coordinates of the points to be used for
p yi p xi
.
(3)
2.2. Multiple-measurement technique The results of measurements using CMMs include systematic errors derived from the parametric errors of CMMs, the directional sensitivity characteristics of the CMMs’ probing systems [7] and other factors related to the measured artefacts. To eliminate the systematic errors attributed to CMMs and their probing systems, the use of a reversal- or multiple-measurement technique is effective [8]. A multiple-measurement technique is suitable for reducing the systematic errors that occur in the measurement of rotationally symmetric artefacts such as cylinders or gears. Fig. 2 shows the principle of the elimination of systematic errors. Fig. 3 shows a schematic overview of the multiple-measurement of gear pitch. In the measurement, a gear is fixed in a position on a CMM, and each pitch of the gear is measured in a coordinate system
Fig. 2. Principle of systematic errors elimination by a multiple-measurement technique [8].
158
O. Sato et al. / Precision Engineering 34 (2010) 156–163
Fig. 3. Multiple-measurement of gear pitch.
on the gear. Then, the gear is rotated about the axis of it on the CMM, and the coordinate system is also transformed rotationally about the axis of the gear. After the transformation, each pitch is measured in the same way as in the previous step. The angle of rotation applied to the transformation is typically 360◦ /z, where z is the number of teeth. To confirm the effectiveness of the multiple-measurement technique, the measurement of the single pitch deviation of an involute spur gear, whose number of teeth is 36 and the diameter of the reference circle is 90 mm, was performed. Fig. 4 shows the result of an experiment on the measurement of single pitch deviation at each rotation step. The measurements at each step are collected and averaged. Fig. 4 shows the result of applying the multiple-measurement technique to gear measurement. To examine the efficacy of the technique, a comparative experiment was performed. In the comparison, authors changed the enable/disable status of the geometrical error compensation function of the CMM used in the experiments. Fig. 5(a) and (b) shows the measurement results obtained under the other condition. Upon comparing both results, there is little difference between the calculated single pitch deviations after averaging, regardless of the status of the error compensation function of the CMM. This suggests that the measurement deviation caused by the geometrical error of the CMM can be corrected by using the multiple-measurement technique. 3. Uncertainty evaluation
Fig. 4. Results of an experiment on single pitch measurement using a CMM. The geometrical error compensation function of the CMM was enabled. (a) Deviation of each single pitch measured in each rotation. (b) Averaged single pitch deviation.
steps. This means that the gear is measured in m positions. At each step, the gear is measured n times. Thus, the ith single pitch deviation is measured m × n times, and the following measure data set is obtained:
⎡1 1
2 Pi 1
Pi
3 Pi 1 3 Pi 2 3 Pi 3
In the estimation of the measurement uncertainties, each cause of uncertainty is evaluated and taken into account to obtain the total uncertainty [9]. However, it is difficult to enumerate the whole uncertainty causes when assessing the measurement result obtained using a CMM because of the complexity of the model of uncertainty propagation [10]. Therefore, it is proposed that the uncertainty of measurement is evaluated by ANOVA based on a small number of contributions to uncertainty. In this article, ANOVA is performed on the results based on two isolated uncertainty contributions:
⎢ 1 Pi 2 Pi ⎢2 2 ⎢1 ⎢ 3 Pi 23 Pi j Pi = ⎢ k ⎢. .. .. ⎢. . . ⎣.
1. the effect of the random errors of the CMM: u∗,rand and 2. the effect of the systematic errors of the CMM: u∗,sys .
Pi =
3.1. Uncertainty estimation based on ANOVA To formulate the uncertainty estimation by ANOVA, the measurement of the single pitch deviation of a gear whose number of teeth is z is considered. The measurements are performed in m
1 Pi n
2 Pi n
3 Pi n
... ... ... ..
.
...
m Pi 1 m Pi 2 m Pi 3
.. .
⎤
⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(4)
m Pi n
The average single pitch deviation after multiple-measurement, Pi , and that at the jth step, j Pi, are described as follows: 1 j Pi, k m·n m
n
j=1 k=1
j Pi
=
n 1 j
k
n
(5)
Pi.
k=1
The sum of the squared deviations of within class variation, SPi ,rand , and that of between class variation, SPi ,sys are derived as
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159
Here, the expectation of variance of within class variation, E(VPi ,rand ), and that of between class variation, E(VPi ,sys ), are obtained as follows: E(VPi ,rand ) = VPi ,rand , E(VPi ,sys ) = VPi ,rand + nVPi ,sys ,
(9)
therefore, the estimated standard deviation caused by the effect of the random errors of the CMM, ˆ Pi ,rand , and that by the effect of the systematic errors of the CMM, ˆ Pi ,sys , are expressed as follows:
V , Pi ,rand
ˆ Pi ,rand =
VPi ,sys − VPi ,rand
ˆ Pi ,sys =
n
(10) .
In the measurement, the ith pitch is measured n times in m positions. Thus, the contributions to uncertainty, uPi ,rand and uPi ,sys , are estimated as follows: ˆ P ,rand , uPi ,rand = √ i m×n ˆ Pi ,sys uPi ,sys = √ . m
(11)
The uncertainty of the measurement, UPi ,95 , is calculated as follows: UPi ,95 = k95
u2P ,rand + u2P ,sys ,
(12)
i
i
where k95 is the coverage factor (95%). 3.2. Effective data handling
Fig. 5. Results of an experiment on single pitch measurement using a CMM. The geometrical error compensation function of the CMM was disabled. Nevertheless, the result of each measurement step had greater variance than that observed using the CMM whose geometrical compensation function was enabled. However, both calculated single pitch deviations after averaging were about the same values. (a) Deviation of each single pitch measured in each rotation. (b) Averaged single pitch deviation.
follows: SPi ,rand =
n m
j (k Pi
j=1 k=1 n m
SPi ,sys =
− j Pi)
2
,
Pi ,rand = m(n − 1), Pi ,sys = m − 1.
z−1 P1 3 z P2 3
z P1 2
⎢ 2 P2 1 P2 ⎢1 2 ⎢3 ⎢ 1 P3 22 P3 j Pi = ⎢ k ⎢. .. ⎢. . ⎣. z Pz 1
2 P1 z
...
3 P2 z
⎤
.. .
⎥ ⎥ ⎥ . . . 4z P3 ⎥ . ⎥ ⎥ .. .. ⎥ . . ⎦
z−2 Pz 3
...
1 P3 3
z−1 Pz 2
...
(13)
1 Pz z j
(7)
Therefore, the unbiased variance of within class variation, VPi ,rand , and that of between class variation, VPi ,sys , are derived as follows:
VPi ,sys
P1
The measured value of the single pitch deviation, k Pi, consists of
The degree of freedom of within class variation, Pi ,rand , and that of between class variation, Pi ,sys , are calculated as follows:
Pi ,rand SPi ,sys = . Pi ,sys
1
(6)
j=1 k=1
VPi ,rand =
⎡1
2
(j Pi − Pi ) .
SPi ,rand
Strictly, each measurement uncertainty of single pitch deviation, UPi ,95 should be estimated individually. However, it is reasonable to expect that every uncertainty of single pitch deviation on a gear has the same value when the artefact to be measured and the measurement strategy performed on the artefact have rotational symmetry. This suggests that it is only necessary to evaluate the measurement uncertainty of single pitch deviation. Additionally, it suggests that it is possible to reduce the number of measurement trials by optimizing the use of the symmetry. Consider the example of the measurement of a gear whose number of teeth is z. The measurements are performed in z steps. In each step, the gear is measured once. Thus, the ith single pitch deviation is measured z times, and the following measure data set is obtained:
, (8)
its best estimate, Pi , the error derived from the random errors of j the CMM in each position, k r, and that derived from the systematic j
j
errors of the CMM, k s. Thus, k Pi is described as follows:
⎡
P1 + 11 r + 11 s
⎢ P + 2r + 2s ⎢ 2 1 1 ⎢ j ⎢ P3 + 31 r + 31 s Pi = k ⎢ ⎢ .. ⎣. Pz + z1 r + z1 s
P1 + z2 r + z2 s
P1 + z−1 r + z−1 s 3 3
...
P2 + 12 r + 12 s
P2 + z3 r + z3 s
...
P3 + 22 r + 22 s
P3 + 13 r + 13 s
...
. . .
. . .
..
Pz + z−1 r + z−1 s 2 2
Pz + z−2 r + z−2 s 3 3
...
j Since the expectation of k r
.
P1 + 2z r + 2z s
⎤
⎥ ⎥ P3 + 4z r + 4z s ⎥ . (14) ⎥ ⎥ . . ⎦ . P2 + 3z r + 3z s ⎥
Pz + 1z r + 1z s
in each position is zero, and since the summation of the single pitch deviation in each measurement step
160
O. Sato et al. / Precision Engineering 34 (2010) 156–163 j
must be zero, it is derived that the summation of k s over j is zero. Therefore, Pi is evaluated as follows: 1 j Pi. k z z
Pi =
(15)
j=1 j r k
j
j
and k s are extracted from k Pi by the subtraction of each Pi as follows:
⎡1
j R k
r + 11 s + 21 s + 31 s
1 2r 1 3r 1
⎢ ⎢ =⎢ ⎢. ⎣ ..
z r 1
+ z1 s
z r 2 1r 2 2r 2
.. .
z−1 r + z−1 s 3 3 z z r + s 3 3 1r + 1s 3 3
+ z2 s + 12 s + 22 s
z−1 r 2
.. .
+ z−1 s 2
z−2 r 3
+ z−2 s 3
⎤
. . . 2z r + 2z s . . . 3z r + 3z s ⎥ ⎥ . . . 4z r + 4z s ⎥ . ⎥ . .. ⎦ . .. . . . 1z r + 1z s
(16)
j
By shifting each element of k R in the jth column by j − 1 rows, and transposing the data set, the residuals of single pitch deviation are given as follows:
⎡1
j R k
⎢ ⎢
=⎢ ⎢
r + 11 s + 12 s + 13 s
1 1r 2 1r 3
⎣ ...
1r z
+ 1z s
2r 1 2r 2 2r 3
+ 21 s + 22 s + 23 s
2r z
+ 2z s
.. .
3r 1 3r 2 3r 3
+ 31 s + 32 s + 33 s
3r z
+ 3z s
.. .
... ... ... .. . ...
⎤
z r 1 z r 2 z r 3
+ z1 s + z2 s ⎥ ⎥ + z3 s ⎥ .
z zr
+ zz s
.. .
(17)
⎥ ⎦
The estimated standard deviations which are the causes of uncerj tainty, ˆ rand and ˆ sys , are calculated from k R in accordance with the procedure described above. In the actual measurement, the ith pitch is measured once in z positions. Thus, the contributions to uncertainty, uP,rand and uP,sys , are estimated as follows: uP,rand = √ rand , z×1 (18) sys uP,geo = √ . z Compared with the contributions to uncertainty evaluated with Eq. (11), uP,rand becomes large due to the reduction of the number of the measurement in one position. The uncertainty of the measurement, UP,95 , is calculated as follows: UP,95 = k95
u2P,rand + u2P,sys .
(19)
3.3. Example of uncertainty evaluation
Fig. 6. Residuals of single pitch deviation for uncertainty evaluation. (a) Residuals from the measurement data. (b) Residuals after index adjustment.
PMM 866 in a good environment. Each scale installed in each axis of the CMM was calibrated. Fig. 8 shows an overview of the measurement. The artefact is an involute spur gear whose number of teeth is 36 and whose diameter of reference circle is 90 mm. The artefact
To illustrate the detail of the data processing method formulated above, an uncertainty evaluation is demonstrated. Here, the case shown in Fig. 5 is used as the example. Fig. 5 shows the set of j
the measured data, k Pi, and the series of their best estimates, Pi , is shown in Fig. 5. j R k
j
is extracted from k Pi by the subtraction of each Pi , which is j
shown in Fig. 6. By shifting and transposing the k R, the residuals of single pitch deviation,
j R, k
are obtained as shown in Fig. 6. Fig. 7 j
shows the uncertainty of the measurement, UP,95 , derived from k R . In this case, uP,rand and uP,sys are computed as 0.02 and 0.17 m. And UP,95 is estimated to be 0.34 m (k95 = 2). 4. Calibration experiment 4.1. Overview of the calibration To verify the reliability of the proposed approach, a calibration experiment was performed. In the experiment, a gear manufactured with high dimensional accuracy was measured on a Leitz
Fig. 7. Calibration values and their uncertainty of single pitch deviation.
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was set upright on the moving table of the CMM. The workpiece coordinate system was constructed using the datum cylinder and the datum plane shown in Fig. 8. First, the single pitch deviations of the artefact were measured in the coordinate system. Next, the artefact was rotated 10 degrees around the central axis. After the rotation, each datum was measured and the workpiece coordinate system was reconstructed. The measurement in the next step was executed in the new coordinate system. 4.2. Results of the calibration In the calibration, single pitch deviations on both left and right flanks were measured. Fig. 9 shows the results of the measurement at every step. In accordance with the proposed method, the uncertainties of the calibration were calculated from the measurement results. Fig. 10 shows the results of the calibration. The values of the calibration uncertainty were 0.15 m for the left flank and 0.16 m for the right flank (see Table 1). 4.3. Comparison with results using other measuring devices 4.3.1. Comparison with other CMMs First, authors compared the calibration results to those obtained using two other CMMs, a highly accurate CMM (CMM-I) and a midlevel CMM (CMM-II). Both CMMs were correctly calibrated. The
Fig. 8. Overview of the calibration of single pitch measurement using the CMM. (a) Artefact on the CMM. (b) Workpiece coordinate system in the calibration.
Fig. 9. Results of the multiple-measurement. (a) Left flank. (b) Right flank.
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O. Sato et al. / Precision Engineering 34 (2010) 156–163
Fig. 10. Results of the calibration. The measurement uncertainties of single pitch deviation are 0.15 m and 0.16 m. (a) Left flank. (b) Right flank. Table 1 Results of the calibration. Single pitch deviation (Pi ) and its uncertainty.
Max. Pi [m] Min. Pi [m] uP,rand [m] uP,sys [m] UP,95 [m]
Left flank
Right flank
0.57 −0.69
0.69 −0.68
0.02 0.08 0.15
0.02 0.08 0.16
same artefact was measured using both CMMs and the uncertainty was estimated using the same protocol. Fig. 11 and Table 2 show the comparison results; the measurement result obtained using each CMM is about the same. This means that the multiple-measurement
Fig. 11. Comparison with other CMMs. (a) Left flank. (b) Right flank.
technique works effectively for the measurement of single pitch deviation. Furthermore, taking into consideration that the measurement results obtained using CMMs include their uncertainties, the measurement results obtained using CMM-I and CMM-II lie inside the confidence interval of the calibration value which authors derived. This demonstrates the adequacy of our proposed uncertainty evaluation procedure. 4.3.2. Comparison with other GMIs Next, authors compared the calibration results with those obtained using two GMIs manufactured by different companies (GMI-I and GMI-II). The same artefact was measured using both GMIs, however, the measurement and uncertainty evaluation protocol were different from those used with the CMMs. Fig. 12 and Table 3 show the comparison results. Comparing the results shown in Fig. 12 with those shown in Fig. 11, there seems to be some differ-
Table 2 Comparison with other CMMs. Single pitch deviation (Pi ) and its uncertainty. Left flank
Max. Pi [m] Min. Pi [m] uP,rand [m] uP,sys [m] UP,95 [m]
Table 3 Comparison with other GMIs. Single pitch deviation (Pi ) and its uncertainty.
Right flank
NMIJ
CMM-I
CMM-II
NMIJ
CMM-I
CMM-II
0.57 −0.69
0.62 −0.79
0.62 −0.63
0.69 −0.68
0.44 −0.58
0.46 −0.63
0.02 0.08 0.15
0.03 0.07 0.15
0.08 0.53 1.06
0.02 0.08 0.16
0.03 0.07 0.15
0.07 0.50 1.01
Left flank
Max. Pi [m] Min. Pi [m] UP,95 [m]
Right flank
NMIJ
GMI-I
GMI-II
NMIJ
GMI-I
GMI-II
0.57 −0.69
0.84 −0.93
0.60 −0.60
0.69 −0.68
1.01 −1.46
0.60 −0.80
0.15
1.00
1.00
0.16
1.00
1.00
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• applying the multiple-measurement technique to eliminate the systematic errors attributed to CMMs, • using ANOVA to estimate the calibration uncertainty, and • optimizing the uncertainty evaluation by considering the symmetry in the calibration. First, the performance of the multiple-measurement technique was demonstrated through the experiments. Regardless of the geometrical errors of the measuring devices, the same calibration values of single pitch deviation were derived from the measurement results obtained using CMMs. Next, the uncertainty of the measurement was rigorously formulated on the basis of ANOVA. As factors contributing to the uncertainty, two effects, the random errors and the systematic errors of the CMM, were considered. Also, the strategy of the number of measurement trials reduction by optimizing the use of the symmetry was proposed and demonstrated. Finally, the reliability of the proposed method was verified by comparing the calibration results to the measurement results obtained using other instruments. Throughout the comparative experiment, it was confirmed that the calibration result obtained by the proposed method corresponded to the measurement results obtained using other instruments. Acknowledgment The authors would like to thank Mr. Takeshi Hagino and Dr. Makoto Abbe (Mitutoyo Corporation) and Mr. Kinya Fujiwara (Honda R&D Co., Ltd.) for the providing the measurement data. References
Fig. 12. Comparison with other GMIs. (a) Left flank. (b) Right flank.
ence between the measurement results obtained using GMIs and CMMs. Taking into account the fact that the measurement results obtained using GMIs include their uncertainties, which are estimated 1.00 m, it is considered within the allowance. 5. Conclusion In this article, authors have described a calibration procedure for single pitch measurement. The features of this method include
[1] Goch G. Gear metrology. CIRP Ann 2003;52(2):659–95. [2] Dantan JY, Bruyere J, Baudouin B, Mathieu L. Geometrical specification model for gear—expression, metrology and analysis. CIRP Ann 2007;56(1):517–20. [3] Trapet E, Savio E, Chiffre De. New advances in traceability of CMMs for almost the entire range of industrial dimensional metrology needs. CIRP Ann 2004;53(1):433–8. [4] ISO 14253-1:1998 Geometrical Product Specifications (GPS)—inspection by measurement of workpieces and measuring equipment. Part 1. Decision rules for proving conformance or non-conformance with specifications. [5] Sammartinia MP, Chiffre LD. Development and validation of a new reference cylindrical gear for pitch measurement. Prec Eng 2000;24(4):302–9. [6] Winer BJ, Brown DRJP, Michels KM. Statistical principles in experimental design. Boston, MA: McGraw-Hill; 1993. [7] Weckenmann A, Estler T, Peggs G, McMurtry D. Probing systems in dimensional metrology. CIRP Ann 2004;53(2):657–84. [8] Osawa S, Busch K, Franke M, Schwenke H. Multiple orientation technique for the calibration of cylindrical workpieces on CMMs. Prec Eng 2005;29(1): 56–64. [9] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP and OIML. Guide to the expression of uncertainty in measurement. Geneva: ISO; 1995. [10] Wilhelm RG, Hocken R, Schwenke H. Task specific uncertainty in coordinate measurement. CIRP Ann 2001;50(2):553–63.
Contents lists available at ScienceDirect
Precision Engineering journal homepage: www.elsevier.com/locate/precision
Calibration and uncertainty evaluation of single pitch deviation by multiple-measurement technique Osamu Sato a,∗ , Sonko Osawa a , Yohan Kondo b , Masaharu Komori c , Toshiyuki Takatsuji a a b c
AIST, AIST Tsukuba Central 3, 1-1-1 Umezono, Tsukuba, Ibaraki, Japan Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, Japan
a r t i c l e
i n f o
Article history: Received 21 November 2008 Received in revised form 20 April 2009 Accepted 20 May 2009 Available online 18 July 2009 Keywords: Gear metrology Pitch deviation Multiple-measurement technique Coordinate measuring machine Analysis of variance
a b s t r a c t Gears are key elements of power transmission systems, and the inspection of their pitch deviation is one of the most important tests on gears. The specifications of gears are assessed using gear measuring instruments (GMIs) or coordinate measuring machines (CMMs), and the results of the measurements must be validated under an appropriate traceability system. In the traceability system, calibrated gauges whose measuring uncertainties are estimated are necessary. In the case of pitch deviation measurement, special artefacts or gears manufactured with high dimensional accuracy are used as reference gauges. In this paper, authors propose calibration and uncertainty evaluation methods for the single pitch deviation of gears measured using CMMs. First, the evaluation of single pitch deviation using a multiple-measurement technique and the estimation of its uncertainty based on the analysis of variance are formulated. Second, a technique for reducing the measurement trials based on the symmetry of the measurement is discussed. Finally, the proposed calibration method is validated through experiments. © 2009 Elsevier Inc. All rights reserved.
1. Introduction Gears are key elements of power transmission systems in many applications. Therefore, inspections of the specifications of gears are necessary to ensure the performance of systems in which gears are included. In particular, the evaluation of pitch deviation is one of the most important tests on gears [1]. The specifications of gears are assessed using gear measuring instruments (GMIs), coordinate measuring machines (CMMs), or other specialized instruments. Because of the small tolerances in gear manufacturing, the inspection of gears requires high reliability in spite of the geometrical complexity of measurands on gears [1,2]. The results of the measurements must be validated under a rigorous traceability system [1,3]. In the traceability system, it is necessary that the measuring devices are verified using calibrated reference gauges and that the uncertainty of the measurement performed using the device is estimated appropriately [4]. In the case of pitch deviation measurement, special artefacts [5] or gears manufactured with high dimensional accuracy are used as calibrated standards [1]. To improve the performance of the inspection of gears, standards used in the verification of measuring devices should be calibrated with a small uncertainty.
In this study, authors propose calibration and uncertainty evaluation methods for the single pitch deviation of gears measured using CMMs. First, the evaluation procedure of single pitch deviation by a multiple-measurement technique using CMMs is described. Second, the estimation method of the pitch deviation measurement is formulated on the basis of the analysis of variance (ANOVA) [6]. Also, a technique for reducing the number of measurement trials based on ANOVA is proposed. Finally, a calibration is performed using the proposed method and its reliability is validated through a comparative experiment. 2. Measurement of single pitch deviation using CMMs The single pitch is determined by the coordinates of the points at the intersections of the reference circle with the tooth flanks. The single pitch deviation of a gear is the difference between the actual pitch and the corresponding nominal pitch on the reference circle. The sequence of the pitch measurement depends on the instruments used, e.g., GMIs or CMMs, and their equipments, e.g., with or without rotary tables [1]. In this section, a procedure for computing the single pitch deviation of a gear-shaped artefact measured using a CMM without a rotary table is described. 2.1. Calculation procedure
∗ Corresponding author. Fax: +81 29 861 4042. E-mail address: [email protected] (O. Sato). 0141-6359/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2009.05.009
Fig. 1 illustrates the scheme used to compute the coordinates of the points to be used for the pitch calculation. When the probe of
O. Sato et al. / Precision Engineering 34 (2010) 156–163
157
the pitch calculation, p Ci, are computed by the coordinates of the starting point of the involute curve and the nominal diameter of reference circle. Usually, these procedures performed automatically by the software installed in the CMM. When measuring a gear whose diameter of reference circle is d and whose number of teeth is z, the ith single pitch deviation, Pi , is formulated as follows: Pi =
d d − i−1 − , 2 i z
(2)
where i is the phase angle of p Ci calculated as follows: i = arctan
Fig. 1. Scheme for computing the coordinates for pitch calculation.
the CMM touches the flank of the ith tooth, the coordinates of the T centre of the probe, c Ci = (c xi, c yi) , and the probing direction unit vector, ni , are read. Here, the coordinates of the corrected measured T
point, f Ci = (f xi, f yi) , are computed as follows: f
Ci = c Ci + rP ni ,
(1)
where rP is the calibrated radius of the probe tip. Because of the positioning error of the CMM, the corrected measured point is not exactly on the reference circle. Thus, the coordinates of the points T to be used for the pitch calculation, p Ci = (p xi, p yi) , are calculated f from Ci in accordance with the profile of the teeth. In case of the measurement on involute gear, for example, the starting point of the involute curve is determined by the coordinates of the corrected measured point, f Ci, and by the length between the centre of the gear and f Ci. Therefore, the coordinates of the points to be used for
p yi p xi
.
(3)
2.2. Multiple-measurement technique The results of measurements using CMMs include systematic errors derived from the parametric errors of CMMs, the directional sensitivity characteristics of the CMMs’ probing systems [7] and other factors related to the measured artefacts. To eliminate the systematic errors attributed to CMMs and their probing systems, the use of a reversal- or multiple-measurement technique is effective [8]. A multiple-measurement technique is suitable for reducing the systematic errors that occur in the measurement of rotationally symmetric artefacts such as cylinders or gears. Fig. 2 shows the principle of the elimination of systematic errors. Fig. 3 shows a schematic overview of the multiple-measurement of gear pitch. In the measurement, a gear is fixed in a position on a CMM, and each pitch of the gear is measured in a coordinate system
Fig. 2. Principle of systematic errors elimination by a multiple-measurement technique [8].
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Fig. 3. Multiple-measurement of gear pitch.
on the gear. Then, the gear is rotated about the axis of it on the CMM, and the coordinate system is also transformed rotationally about the axis of the gear. After the transformation, each pitch is measured in the same way as in the previous step. The angle of rotation applied to the transformation is typically 360◦ /z, where z is the number of teeth. To confirm the effectiveness of the multiple-measurement technique, the measurement of the single pitch deviation of an involute spur gear, whose number of teeth is 36 and the diameter of the reference circle is 90 mm, was performed. Fig. 4 shows the result of an experiment on the measurement of single pitch deviation at each rotation step. The measurements at each step are collected and averaged. Fig. 4 shows the result of applying the multiple-measurement technique to gear measurement. To examine the efficacy of the technique, a comparative experiment was performed. In the comparison, authors changed the enable/disable status of the geometrical error compensation function of the CMM used in the experiments. Fig. 5(a) and (b) shows the measurement results obtained under the other condition. Upon comparing both results, there is little difference between the calculated single pitch deviations after averaging, regardless of the status of the error compensation function of the CMM. This suggests that the measurement deviation caused by the geometrical error of the CMM can be corrected by using the multiple-measurement technique. 3. Uncertainty evaluation
Fig. 4. Results of an experiment on single pitch measurement using a CMM. The geometrical error compensation function of the CMM was enabled. (a) Deviation of each single pitch measured in each rotation. (b) Averaged single pitch deviation.
steps. This means that the gear is measured in m positions. At each step, the gear is measured n times. Thus, the ith single pitch deviation is measured m × n times, and the following measure data set is obtained:
⎡1 1
2 Pi 1
Pi
3 Pi 1 3 Pi 2 3 Pi 3
In the estimation of the measurement uncertainties, each cause of uncertainty is evaluated and taken into account to obtain the total uncertainty [9]. However, it is difficult to enumerate the whole uncertainty causes when assessing the measurement result obtained using a CMM because of the complexity of the model of uncertainty propagation [10]. Therefore, it is proposed that the uncertainty of measurement is evaluated by ANOVA based on a small number of contributions to uncertainty. In this article, ANOVA is performed on the results based on two isolated uncertainty contributions:
⎢ 1 Pi 2 Pi ⎢2 2 ⎢1 ⎢ 3 Pi 23 Pi j Pi = ⎢ k ⎢. .. .. ⎢. . . ⎣.
1. the effect of the random errors of the CMM: u∗,rand and 2. the effect of the systematic errors of the CMM: u∗,sys .
Pi =
3.1. Uncertainty estimation based on ANOVA To formulate the uncertainty estimation by ANOVA, the measurement of the single pitch deviation of a gear whose number of teeth is z is considered. The measurements are performed in m
1 Pi n
2 Pi n
3 Pi n
... ... ... ..
.
...
m Pi 1 m Pi 2 m Pi 3
.. .
⎤
⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(4)
m Pi n
The average single pitch deviation after multiple-measurement, Pi , and that at the jth step, j Pi, are described as follows: 1 j Pi, k m·n m
n
j=1 k=1
j Pi
=
n 1 j
k
n
(5)
Pi.
k=1
The sum of the squared deviations of within class variation, SPi ,rand , and that of between class variation, SPi ,sys are derived as
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Here, the expectation of variance of within class variation, E(VPi ,rand ), and that of between class variation, E(VPi ,sys ), are obtained as follows: E(VPi ,rand ) = VPi ,rand , E(VPi ,sys ) = VPi ,rand + nVPi ,sys ,
(9)
therefore, the estimated standard deviation caused by the effect of the random errors of the CMM, ˆ Pi ,rand , and that by the effect of the systematic errors of the CMM, ˆ Pi ,sys , are expressed as follows:
V , Pi ,rand
ˆ Pi ,rand =
VPi ,sys − VPi ,rand
ˆ Pi ,sys =
n
(10) .
In the measurement, the ith pitch is measured n times in m positions. Thus, the contributions to uncertainty, uPi ,rand and uPi ,sys , are estimated as follows: ˆ P ,rand , uPi ,rand = √ i m×n ˆ Pi ,sys uPi ,sys = √ . m
(11)
The uncertainty of the measurement, UPi ,95 , is calculated as follows: UPi ,95 = k95
u2P ,rand + u2P ,sys ,
(12)
i
i
where k95 is the coverage factor (95%). 3.2. Effective data handling
Fig. 5. Results of an experiment on single pitch measurement using a CMM. The geometrical error compensation function of the CMM was disabled. Nevertheless, the result of each measurement step had greater variance than that observed using the CMM whose geometrical compensation function was enabled. However, both calculated single pitch deviations after averaging were about the same values. (a) Deviation of each single pitch measured in each rotation. (b) Averaged single pitch deviation.
follows: SPi ,rand =
n m
j (k Pi
j=1 k=1 n m
SPi ,sys =
− j Pi)
2
,
Pi ,rand = m(n − 1), Pi ,sys = m − 1.
z−1 P1 3 z P2 3
z P1 2
⎢ 2 P2 1 P2 ⎢1 2 ⎢3 ⎢ 1 P3 22 P3 j Pi = ⎢ k ⎢. .. ⎢. . ⎣. z Pz 1
2 P1 z
...
3 P2 z
⎤
.. .
⎥ ⎥ ⎥ . . . 4z P3 ⎥ . ⎥ ⎥ .. .. ⎥ . . ⎦
z−2 Pz 3
...
1 P3 3
z−1 Pz 2
...
(13)
1 Pz z j
(7)
Therefore, the unbiased variance of within class variation, VPi ,rand , and that of between class variation, VPi ,sys , are derived as follows:
VPi ,sys
P1
The measured value of the single pitch deviation, k Pi, consists of
The degree of freedom of within class variation, Pi ,rand , and that of between class variation, Pi ,sys , are calculated as follows:
Pi ,rand SPi ,sys = . Pi ,sys
1
(6)
j=1 k=1
VPi ,rand =
⎡1
2
(j Pi − Pi ) .
SPi ,rand
Strictly, each measurement uncertainty of single pitch deviation, UPi ,95 should be estimated individually. However, it is reasonable to expect that every uncertainty of single pitch deviation on a gear has the same value when the artefact to be measured and the measurement strategy performed on the artefact have rotational symmetry. This suggests that it is only necessary to evaluate the measurement uncertainty of single pitch deviation. Additionally, it suggests that it is possible to reduce the number of measurement trials by optimizing the use of the symmetry. Consider the example of the measurement of a gear whose number of teeth is z. The measurements are performed in z steps. In each step, the gear is measured once. Thus, the ith single pitch deviation is measured z times, and the following measure data set is obtained:
, (8)
its best estimate, Pi , the error derived from the random errors of j the CMM in each position, k r, and that derived from the systematic j
j
errors of the CMM, k s. Thus, k Pi is described as follows:
⎡
P1 + 11 r + 11 s
⎢ P + 2r + 2s ⎢ 2 1 1 ⎢ j ⎢ P3 + 31 r + 31 s Pi = k ⎢ ⎢ .. ⎣. Pz + z1 r + z1 s
P1 + z2 r + z2 s
P1 + z−1 r + z−1 s 3 3
...
P2 + 12 r + 12 s
P2 + z3 r + z3 s
...
P3 + 22 r + 22 s
P3 + 13 r + 13 s
...
. . .
. . .
..
Pz + z−1 r + z−1 s 2 2
Pz + z−2 r + z−2 s 3 3
...
j Since the expectation of k r
.
P1 + 2z r + 2z s
⎤
⎥ ⎥ P3 + 4z r + 4z s ⎥ . (14) ⎥ ⎥ . . ⎦ . P2 + 3z r + 3z s ⎥
Pz + 1z r + 1z s
in each position is zero, and since the summation of the single pitch deviation in each measurement step
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O. Sato et al. / Precision Engineering 34 (2010) 156–163 j
must be zero, it is derived that the summation of k s over j is zero. Therefore, Pi is evaluated as follows: 1 j Pi. k z z
Pi =
(15)
j=1 j r k
j
j
and k s are extracted from k Pi by the subtraction of each Pi as follows:
⎡1
j R k
r + 11 s + 21 s + 31 s
1 2r 1 3r 1
⎢ ⎢ =⎢ ⎢. ⎣ ..
z r 1
+ z1 s
z r 2 1r 2 2r 2
.. .
z−1 r + z−1 s 3 3 z z r + s 3 3 1r + 1s 3 3
+ z2 s + 12 s + 22 s
z−1 r 2
.. .
+ z−1 s 2
z−2 r 3
+ z−2 s 3
⎤
. . . 2z r + 2z s . . . 3z r + 3z s ⎥ ⎥ . . . 4z r + 4z s ⎥ . ⎥ . .. ⎦ . .. . . . 1z r + 1z s
(16)
j
By shifting each element of k R in the jth column by j − 1 rows, and transposing the data set, the residuals of single pitch deviation are given as follows:
⎡1
j R k
⎢ ⎢
=⎢ ⎢
r + 11 s + 12 s + 13 s
1 1r 2 1r 3
⎣ ...
1r z
+ 1z s
2r 1 2r 2 2r 3
+ 21 s + 22 s + 23 s
2r z
+ 2z s
.. .
3r 1 3r 2 3r 3
+ 31 s + 32 s + 33 s
3r z
+ 3z s
.. .
... ... ... .. . ...
⎤
z r 1 z r 2 z r 3
+ z1 s + z2 s ⎥ ⎥ + z3 s ⎥ .
z zr
+ zz s
.. .
(17)
⎥ ⎦
The estimated standard deviations which are the causes of uncerj tainty, ˆ rand and ˆ sys , are calculated from k R in accordance with the procedure described above. In the actual measurement, the ith pitch is measured once in z positions. Thus, the contributions to uncertainty, uP,rand and uP,sys , are estimated as follows: uP,rand = √ rand , z×1 (18) sys uP,geo = √ . z Compared with the contributions to uncertainty evaluated with Eq. (11), uP,rand becomes large due to the reduction of the number of the measurement in one position. The uncertainty of the measurement, UP,95 , is calculated as follows: UP,95 = k95
u2P,rand + u2P,sys .
(19)
3.3. Example of uncertainty evaluation
Fig. 6. Residuals of single pitch deviation for uncertainty evaluation. (a) Residuals from the measurement data. (b) Residuals after index adjustment.
PMM 866 in a good environment. Each scale installed in each axis of the CMM was calibrated. Fig. 8 shows an overview of the measurement. The artefact is an involute spur gear whose number of teeth is 36 and whose diameter of reference circle is 90 mm. The artefact
To illustrate the detail of the data processing method formulated above, an uncertainty evaluation is demonstrated. Here, the case shown in Fig. 5 is used as the example. Fig. 5 shows the set of j
the measured data, k Pi, and the series of their best estimates, Pi , is shown in Fig. 5. j R k
j
is extracted from k Pi by the subtraction of each Pi , which is j
shown in Fig. 6. By shifting and transposing the k R, the residuals of single pitch deviation,
j R, k
are obtained as shown in Fig. 6. Fig. 7 j
shows the uncertainty of the measurement, UP,95 , derived from k R . In this case, uP,rand and uP,sys are computed as 0.02 and 0.17 m. And UP,95 is estimated to be 0.34 m (k95 = 2). 4. Calibration experiment 4.1. Overview of the calibration To verify the reliability of the proposed approach, a calibration experiment was performed. In the experiment, a gear manufactured with high dimensional accuracy was measured on a Leitz
Fig. 7. Calibration values and their uncertainty of single pitch deviation.
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was set upright on the moving table of the CMM. The workpiece coordinate system was constructed using the datum cylinder and the datum plane shown in Fig. 8. First, the single pitch deviations of the artefact were measured in the coordinate system. Next, the artefact was rotated 10 degrees around the central axis. After the rotation, each datum was measured and the workpiece coordinate system was reconstructed. The measurement in the next step was executed in the new coordinate system. 4.2. Results of the calibration In the calibration, single pitch deviations on both left and right flanks were measured. Fig. 9 shows the results of the measurement at every step. In accordance with the proposed method, the uncertainties of the calibration were calculated from the measurement results. Fig. 10 shows the results of the calibration. The values of the calibration uncertainty were 0.15 m for the left flank and 0.16 m for the right flank (see Table 1). 4.3. Comparison with results using other measuring devices 4.3.1. Comparison with other CMMs First, authors compared the calibration results to those obtained using two other CMMs, a highly accurate CMM (CMM-I) and a midlevel CMM (CMM-II). Both CMMs were correctly calibrated. The
Fig. 8. Overview of the calibration of single pitch measurement using the CMM. (a) Artefact on the CMM. (b) Workpiece coordinate system in the calibration.
Fig. 9. Results of the multiple-measurement. (a) Left flank. (b) Right flank.
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Fig. 10. Results of the calibration. The measurement uncertainties of single pitch deviation are 0.15 m and 0.16 m. (a) Left flank. (b) Right flank. Table 1 Results of the calibration. Single pitch deviation (Pi ) and its uncertainty.
Max. Pi [m] Min. Pi [m] uP,rand [m] uP,sys [m] UP,95 [m]
Left flank
Right flank
0.57 −0.69
0.69 −0.68
0.02 0.08 0.15
0.02 0.08 0.16
same artefact was measured using both CMMs and the uncertainty was estimated using the same protocol. Fig. 11 and Table 2 show the comparison results; the measurement result obtained using each CMM is about the same. This means that the multiple-measurement
Fig. 11. Comparison with other CMMs. (a) Left flank. (b) Right flank.
technique works effectively for the measurement of single pitch deviation. Furthermore, taking into consideration that the measurement results obtained using CMMs include their uncertainties, the measurement results obtained using CMM-I and CMM-II lie inside the confidence interval of the calibration value which authors derived. This demonstrates the adequacy of our proposed uncertainty evaluation procedure. 4.3.2. Comparison with other GMIs Next, authors compared the calibration results with those obtained using two GMIs manufactured by different companies (GMI-I and GMI-II). The same artefact was measured using both GMIs, however, the measurement and uncertainty evaluation protocol were different from those used with the CMMs. Fig. 12 and Table 3 show the comparison results. Comparing the results shown in Fig. 12 with those shown in Fig. 11, there seems to be some differ-
Table 2 Comparison with other CMMs. Single pitch deviation (Pi ) and its uncertainty. Left flank
Max. Pi [m] Min. Pi [m] uP,rand [m] uP,sys [m] UP,95 [m]
Table 3 Comparison with other GMIs. Single pitch deviation (Pi ) and its uncertainty.
Right flank
NMIJ
CMM-I
CMM-II
NMIJ
CMM-I
CMM-II
0.57 −0.69
0.62 −0.79
0.62 −0.63
0.69 −0.68
0.44 −0.58
0.46 −0.63
0.02 0.08 0.15
0.03 0.07 0.15
0.08 0.53 1.06
0.02 0.08 0.16
0.03 0.07 0.15
0.07 0.50 1.01
Left flank
Max. Pi [m] Min. Pi [m] UP,95 [m]
Right flank
NMIJ
GMI-I
GMI-II
NMIJ
GMI-I
GMI-II
0.57 −0.69
0.84 −0.93
0.60 −0.60
0.69 −0.68
1.01 −1.46
0.60 −0.80
0.15
1.00
1.00
0.16
1.00
1.00
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• applying the multiple-measurement technique to eliminate the systematic errors attributed to CMMs, • using ANOVA to estimate the calibration uncertainty, and • optimizing the uncertainty evaluation by considering the symmetry in the calibration. First, the performance of the multiple-measurement technique was demonstrated through the experiments. Regardless of the geometrical errors of the measuring devices, the same calibration values of single pitch deviation were derived from the measurement results obtained using CMMs. Next, the uncertainty of the measurement was rigorously formulated on the basis of ANOVA. As factors contributing to the uncertainty, two effects, the random errors and the systematic errors of the CMM, were considered. Also, the strategy of the number of measurement trials reduction by optimizing the use of the symmetry was proposed and demonstrated. Finally, the reliability of the proposed method was verified by comparing the calibration results to the measurement results obtained using other instruments. Throughout the comparative experiment, it was confirmed that the calibration result obtained by the proposed method corresponded to the measurement results obtained using other instruments. Acknowledgment The authors would like to thank Mr. Takeshi Hagino and Dr. Makoto Abbe (Mitutoyo Corporation) and Mr. Kinya Fujiwara (Honda R&D Co., Ltd.) for the providing the measurement data. References
Fig. 12. Comparison with other GMIs. (a) Left flank. (b) Right flank.
ence between the measurement results obtained using GMIs and CMMs. Taking into account the fact that the measurement results obtained using GMIs include their uncertainties, which are estimated 1.00 m, it is considered within the allowance. 5. Conclusion In this article, authors have described a calibration procedure for single pitch measurement. The features of this method include
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