 Original Article
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An Improved Measurement Uncertainty Calculation Method of Profile Error for Sculptured Surfaces
Chinese Journal of Mechanical Engineering volume 32, Article number: 90 (2019)
Abstract
The current researches mainly adopt “Guide to the expression of uncertainty in measurement (GUM)” to calculate the profile error. However, GUM can only be applied in the linear models. The standard GUM is not appropriate to calculate the uncertainty of profile error because the mathematical model of profile error is strongly nonlinear. An improved secondorder GUM method (GUMM) is proposed to calculate the uncertainty. At the same time, the uncertainties in different coordinate axes directions are calculated as the measuring points uncertainties. In addition, the correlations between variables could not be ignored while calculating the uncertainty. A kfactor conversion method is proposed to calculate the converge factor due to the unknown and asymmetrical distribution of the output quantity. Subsequently, the adaptive Monte Carlo method (AMCM) is used to evaluate whether the secondorder GUMM is better. Two practical examples are listed and the conclusion is drawn by comparing and discussing the secondorder GUMM and AMCM. The results show that the difference between the improved secondorder GUM and the AMCM is smaller than the difference between the standard GUM and the AMCM. The improved secondorder GUMM is more precise in consideration of the nonlinear mathematical model of profile error.
1 Introduction
Profile error is an important feature to evaluate the machining quality of sculptured surfaces. More and more studies have been done to improve the algorithms of calculating the profile error [1]. The least squares fitting was adopted to evaluate the profile error of the ellipse. The data was obtained by the coordinate measuring machine (CMM) [2]. Geometry optimization approximation algorithm was proposed to calculate the elliptical profile error considering the geometric characteristics [3]. Profile error could be obtained by nonuniform rational Bsplines (NURBS) surface fitting according to the data points. NURBS surface fitting was accepted when the center axis was difficult to obtain [4]. The singular value decompositioniterative closest point method was proposed to match the measurement points and the ideal section curve. Based on this, the profile error of the blade surface was calculated and balanced [5]. Surface reconstruction was implemented by the genetic algorithm. The profile error was evaluated by calculating the shortest distance between measurement points and sculptured surface in the split spherical approximation method [6]. In our lab, Lang et al. [7] proposed the sequential quadratic programming (SQP) algorithm to calculate the profile error. The computing speed is faster, and the result is more approach to the optimal solution.
However, the method of calculating the profile error could not get the corresponding uncertainty. The “Guide to the expression of uncertainty in measurement” is a standard of evaluating the uncertainty. The international organization revised the GUM in 2008, which has been accepted all over the world [8]. GUMM is a method to estimate the uncertainty by calculating the firstorder Taylor series expansion of profile error. The GUMM was used to analyze the uncertainty about the location of a hole [9]. Though the GUMM is a standard method to calculate the uncertainty, it also has some limitations. GUMM can only be applied in the linear models [10]. When the actual situation does not meet the requirement, some alternative methods are put forward. The idea of the randomfuzzy and fuzzyrandom uncertainties was proposed to estimate the uncertainty [11]. GUM S1 [12] used the Monte Carlo method (MCM) to analyze the uncertainty without considering the distribution of the output variable and the format of the mathematical model. The MCM is mainly based on the statistical analysis to calculate the uncertainty [13]. When the partial derivatives of the model could not be calculated, Cox and Siebert [14] adopted the MCM to calculate the expanded uncertainty. The MCM was used to evaluate the uncertainty in an experimental model, which reduced the work load of calculating the partial derivatives in a nonlinear model [15]. The proper number of the experiment is difficult to determine in the Monte Carlo simulation. The AMCM was proposed because it could adaptively select the proper number of the experiment [16]. However, the prior information of the distributions is often unknown or inaccurate in AMCM. In this situation, AMCM is not suitable to calculate the uncertainty. It can still be used as a suitable method to evaluate the GUMM. The secondorder GUMM was proposed to estimate the uncertainty considering the difficulty of the prior information. In 2011, the secondorder and thirdorder Taylor series expansion in GUMM were adopted to calculate the uncertainty in a simple nonlinear model [17]. However, the relativity of variables was neglected, which led to the imprecise result. In 2013, Wen et al. [18] calculated the uncertainty of cylindricity errors in AMCM and GUMM. The model is nonlinear, so the firstorder Taylor series expansion is inadequate. The uncertainty of measuring points in different coordinate axes directions was not calculated, and on the contrary the measuring points uncertainty was estimated by considering the limited factors, which was imprecise and complex. Although virtual coordinate measuring machine has been developed to estimate the uncertainty [19], virtual CMM is usually adopted when the actual data is difficult to be obtained.
The paper structure is as follows: Section 2 establishes a mathematical model of profile error. In Section 3, the secondorder Taylor series expansion of GUMM is proposed to calculate the uncertainty. Some improvements of GUMM are also listed. Subsequently, two practical examples are presented in Section 4. Section 5 compares and discusses the AMCM and GUMM. AMCM is used for evaluating the GUMM. The conclusion is drawn in Section 6.
2 Mathematical Model of Profile Error
Profile error is the greatest deviation between the actual sculptured surfaces and the theoretical sculptured surfaces [20], as shown in Figure 1, where t is the diameter of the enveloping spheres.
The model of sculptured surfaces profile error is shown as follows [7]:
where \(\alpha ,\beta ,\gamma\) are the rotation angles of measuring points along the directions of \(X\) axis, \(Y\) axis and \(Z\) axis respectively, \(\Delta x,\Delta y,\Delta z\) are the offsets along the directions of \(X\) axis, \(Y\) axis and \(Z\) axis respectively, \(s_{i} (x_{i} ,y_{i} ,z_{i} )\) is the initial measuring point, \(s_{i}^{\prime } (x_{i}^{\prime } ,y_{i}^{\prime } ,z_{i}^{\prime } )\) is the closest point. The meaning of the symbols \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) is shown in Figure 2.
3 Uncertainty Calculation Methods
3.1 Gumm
Taylor expansion of the nonlinear model \(f = f(m_{ 1} ,m_{ 2} , \ldots ,m_{n} )\) in the neighborhood of the point \((m_{ 1}^{ 0} ,m_{ 2}^{ 0} , \cdot \cdot \cdot ,m_{n}^{ 0} )\) is given as follows:
where partial derivative \(\partial f/\partial ( \cdot )\) is the sensitivity coefficient of each variable, \(\delta^{p} m_{i}\) is defined as \(\delta^{p} m_{i} { = }(m_{i}  m_{i}^{0} )^{p} ,i = 1, \ldots ,n,p = 1,2,3\).
Then strive for the variance or covariance of Eq. (2) at the left and right sides. The square of uncertainty calculation formula [21] is given as follows:
where \(u( \cdot )\) is the standard uncertainty of each variable and \(r( \cdot {\kern 1pt} \,,\, \cdot )\) is the correlation coefficient between two different variables.
3.2 Uncertainty Evaluation Formulas
In order to calculate the uncertainty of profile error, we need to consider the calculation of the sensitivity coefficient and the standard uncertainty of variables [22].
The closest point \(s_{i}^{\prime } \left( {x_{i}^{\prime } ,y_{i}^{\prime } ,z_{i}^{\prime } } \right)\) can be determined by measuring points \(s_{i} (x_{i} ,y_{i} ,z_{i} )\) and the transformation matrix \(\varvec{T}(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z)\), so it is no need to consider the uncertainty of the three variables \(x_{i}^{\prime } ,y_{i}^{\prime } ,z_{i}^{\prime }\). \(x_{i} ,y_{i} ,z_{i}\) are the coordinates of measuring points, so they are independent in theory. \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) are the rotation angles and offsets of the measuring points, and the change of one variable can directly cause the changes of other variables. \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) interact on each other. Assuming \(x_{i} ,y_{i} ,z_{i}\)are independent and \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\)are related, Eq. (3) will be turned into Eq. (4) and Eq. (5).
According to the uncertainty propagation law, we can get the square of uncertainty calculation formula [23, 24]. The firstorder Taylor expansion:
The secondorder Taylor expansion:
The symbols of the nine variables \((x_{i} ,y_{i} ,z_{i} ,\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z)\) are shown in Table 1.
According to Table 1, Eq. (4) and Eq. (5) can be turned into Eq. (6) and Eqs. (6)–(12) respectively. The firstorder Taylor expansion to calculate the uncertainty is shown as follows:
For the secondorder Taylor expansion except for the part of the firstorder, the formulas are shown as follows:
Finally, we get the final uncertainty Eq. (13) of the secondorder Taylor expansion:
3.3 Improvements of GUMM

(1)
For measuring points uncertainty:
The uncertainties in the directions of \(X\) axis, \(Y\) axis and \(Z\) axis are written as \(ux,\;uy,\;uz\) respectively. They are different and have their own trends, so we cannot use \(u0\) to replace \(ux,uy,uz\). The \(u0\) is obtained by estimating some influence factors. The \(u0\) is not accurate, because the factors that are considered are limited. \(u0\) is difficult to estimate in this situation of this paper. Whereas \(ux,uy,uz\) can be easily calculated according to the measuring points \(s_{i} (x_{i} ,y_{i} ,z_{i} )\). We adopt \(ux,uy,uz\) to calculate the uncertainty.

(2)
For the uncertainty of \(\varvec{T}(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z)\):
In order to calculate the uncertainty, the uncertainty of \(\varvec{T}(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z)\) is also a necessary value. In most cases, the relativity of variables is too high to neglect. The relativity of variables \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) must be considered.

(3)
For the converge factor about the unknown distribution:
When the probability density function (PDF) of profile errors is not the Gaussian distribution or tdistribution, the kfactor conversion method is proposed to calculate the converge factor. The conversion method is to approximate the coverage factor of the nonnormal distribution.
The skewness coefficient is a parameter to describe the degree of deviation from the center. The formula of the skewness coefficient is shown in Ref. [25]. The skewness coefficient of the profile error is given as follows:
where f is the profile error, \(\bar{f}\) is the mean of profile error, \(Var(f)\) is the variance of profile error.
When the skewness coefficient is unequal to zero, the converge factor \(k_{down}\) is unequal to \(k_{up}\). When the distribution of the output variable is the normal distribution, \(k_{down} ,\,\,k_{up}\) are equal to 2 for the 95% confidence interval [26]. When the distribution of the output variable is unknown, assuming \(k_{down} + k_{up}\) is equal to 4. It is the same as the situation of normal distribution. According to the sample values, we can get the mean of profile error \(\bar{f}\), 2.5%quantile \(Q_{0.025}\) and 97.5%quantile \(Q_{0.975}\). Then calculate \(k_{down}\) and \(k_{up}\) in the kfactor conversion method.
We can get the 95% confidence interval of profile error in GUMM as follows:
where \(u\) is the uncertainty of the profile error.
4 Experiment
Sshaped test specimen has been put off in recent years and it has been proved to be a good specimen to detect the machining quality [27, 28]. Sshaped test specimen is adopted as examples to analyze the secondorder GUMM. Two practical Sshaped test specimens were measured many times by Edward DAISY564 CMM in the same environment condition. The temperature is controlled at 20 °C and the fluctuation is less than 1 °C. The edition of measuring software is ACDIMSv5.2, and the maximal measuring distances are 500 mm, 600 mm, 400 mm in the directions of \(X\) axis, \(Y\) axis and \(Z\) axis respectively. The maximum permissible error is \({\text{MPE}}_{\text{E}} \le (2 + L/300)\;\upmu{\text{m,}}\) the maximum allowable space detection error is \({\text{MPE}}_{\text{P}} \le 2.2\;\upmu{\text{m}} .\) Before the experiment, the Edward DAISY564 CMM should be initialized. The probe should be installed and inspected according to the standard. Two Sshaped test specimens have different distributions of deviations shown in Figure 3. Different colors represent different deviations. The measuring experiment in CMM is shown in Figure 4.
4.1 Profile Error and Uncertainty Calculation in Experiment 1
In Unigraphics NX, we set 5 rows along the Z directions and each row has 32 points. We select 160 points in the whole surface as the theoretical points. 100 groups of measuring points can be obtained in CMM according to the theoretical points. In the procedure, the points in the edge of the surface must avoid to be selected. The theoretical points are uniformly sampled on the sculptured surfaces as the green ones in Figure 5. The average measuring points of 100 groups are presented as the red ones in Figure 5. SQP is adopted to search the nearest points [7].
In SQP, we can get the profile error, the best transformation matrix and the nearest points. The mean of the profile errors \(\bar{f}\) is calculated as the final profile error. The value is 0.0656 mm. Then GUMM and AMCM are adopted to calculate the uncertainty.
 (1)
GUMM
According to the formula of uncertainty, we must get the uncertainties of measuring points and the uncertainties of transformation matrix in order to get the uncertainty of profile error. By measuring 160 points of the Sshaped test specimen 100 times, we get the uncertainties of the measuring points. The uncertainties in the directions of \(X\) axis, \(Y\) axis and \(Z\) axis have different values. The average values of the \(ux,\;uy,\;uz\)are \(\overline{ux} = 6.5\;\upmu {\text{m}},\) \(\overline{uy} = 0.615\;\upmu {\text{m}}, \,\)\(\overline{uz} = 1. 5\;\upmu {\text{m,}}\) respectively. The correlation coefficients \(\alpha \beta ,\;\alpha \gamma ,\;\alpha \Delta x,\;\alpha \Delta y,\;\alpha \Delta z,\;\beta \gamma ,\;\beta \Delta x\) \(\beta \Delta y,\;\beta \Delta z,\;\gamma \Delta x,\;\gamma \Delta y,\;\gamma \Delta z,\;\Delta x\Delta y,\;\Delta x\Delta z,\;\Delta y\Delta z\) can be calculated in SPSS and the values are shown in Table 2. It’s obvious that the correlation between these variables \(\alpha ,\;\beta ,\;\gamma ,\;\Delta x,\;\Delta y,\;\Delta z\) cannot be ignored. The covariance matrix is shown in Table 3.
According to the GUMM, the uncertainty of the firstorder Taylor expansion is calculated with Eq. (6) and the value is \(u1 = 0.0028968\) mm. The uncertainty of the secondorder Taylor expansion is \(u2 = 0.02265\) mm calculated by Eqs. (6)–(13).
 (2)
AMCM
Estimating the distributions of variables is an important step to use AMCM [29]. The distributions of the measuring points coordinates follow the Gaussian distributions, which is tested by Kolmogorov–Smirnov Test in SPSS shown in Table 4. Because all the pvalues (0.762, 0.712, 0.943) are greater than 0.05 (p > 0.05), the distributions of x, y, z all follow the Gaussian distributions. x, y, z are independent. They all follow the Gaussian distributions \(N(\mu ,\sigma^{2} )\), where \(\mu\) is the mean of each variable and \(\sigma^{2}\) is the variance of each variable shown in Table 4. The distribution of \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) follows the multivariate Gaussian distribution \(N(\mu ,\sigma^{2} )\), where \(\mu\) is the expectation vector, \(\mu = \left( {0.0005,  0.0004,0.0001,  0.0315,0.0320,  0.0395} \right)\); \(\sigma^{2}\) is the covariance matrix shown in Table 3. The number of experiments is \(M = 10^{6}\), the frequency distribution of profile error in AMCM is shown in Figure 6, where the dashed lines are drawn in correspondence of the 95% confidence interval.
Using AMCM, the profile error is 0.0875 mm, the uncertainty is 0.0433 mm and the confidence interval of 95% is [0.0188, 0.1714] mm.
4.2 Profile Error and Uncertainty Calculation in Experiment 2
In Unigraphics NX, adaptive placement strategy is adopted to select 50 points. It is shown in Figure 7. The CMM measures these points 100 times repeatedly.
Using SQP, the profile error is 0.0993 mm. The best transformation matrix \(\varvec{T}(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z)\) and the nearest points \(s_{i}^{\prime } (x_{i}^{\prime } ,y_{i}^{\prime } ,z_{i}^{\prime } )\) are also obtained. The nine variables \(x_{i} ,y_{i} ,z_{i} ,\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) have been known, the covariance and variance between the variables can also be obtained. Then GUMM and AMCM are adopted to calculate the uncertainty.
 (1)
GUMM
The uncertainties of measuring points \(ux ,\;uy ,\;uz\) are \(\overline{ux} =5.97\;\upmu {\text{m,}}\) \(\overline{uy}=0.40697\;\upmu {\text{m,}}\) \(\overline{uz}=1.3\;\upmu {\text{m}} .\) Covariance matrix of the best transformation matrix is shown in Table 5.
Based on the values, the uncertainty of the firstorder GUMM is \(u1\, =\, 0.00199\;{\text{mm,}}\) the second order is \(u2 \, =\, 0.01945\;{\text{mm}} .\)
 (2)
AMCM
Refer to the principle of AMCM in experiment 1, the profile error is 0.1233 mm, the uncertainty is 0.0416 mm, the confidence interval of 95% is [0.0597, 0.2062] mm. The distribution of profile error in AMCM is shown in Figure 8.
5 Results and Discussion

(1)
According to experiment 1:
In GUMM, the skewness coefficient of profile error \(SK\) is 0.6287. The distribution of the profile error is asymmetrical and unknown. According to the 100 profile errors, the \(\bar{f} = 0.065574\;{\text{mm,}}\) \(Q_{0.025} = 0.05849\;{\text{mm}}\) and \(Q_{0.975} = 0.075034\;{\text{mm}}\) are obtained. According to Eqs. (15), (16), \(k_{down}\) and \(k_{up}\) are shown as follows:
$$\begin{aligned} k_{{down}} = 4 \times \frac{0.065574  0.05849}{0.075034  0.05849} \approx 1.7, \hfill \\ k_{{up}} = 4 \times \frac{0.075034  0.065574}{0.075034  0.05849} \approx 2.3. \hfill \\ \end{aligned}$$The uncertainties are calculated in the firstorder and the secondorder GUMM. They are written as GUMM1 and GUMM2 respectively in Table 6 and Table 7. The coverage probability p is 0.95. d_{low} and d_{high} are the difference of the GUM and AMCM in 95% confidence interval. When the firstorder Taylor series expansion is considered in GUMM, the values of d_{low} and d_{high} are far larger than those in GUMM2 (0.0419 > 0.0083 & 0.09914 > 0.0537). In a word, the secondorder GUMM is more precise to calculate the uncertainty of profile error.

(2)
According to experiment 2:
In GUMM, the skewness coefficient is \(SK =  0.4356\). The distribution of profile error is asymmetrical. Then the converge factors are shown as follows.
$$\begin{aligned} k_{{down}} = 4 \times \frac{0.099314192  0.089258703}{0.106076038  0.089258703} \approx 2.4, \hfill \\ k_{{up}} = 4 \times \frac{0.106076038  0.099314192}{0.106076038  0.089258703} \approx 1.6. \hfill \\ \end{aligned}$$In Table 7, d_{low} and d_{high} in GUMM2 are less than those in GUMM1 (0.00708 < 0.0348 & 0.07578 < 0.1037). It indicates that the secondorder GUMM is more precise to calculate the uncertainty of profile error.
6 Conclusions

(1)
When the amount of data is limited, SQP can obtain the profile error accurately and quickly. However, when the amount of data is very large, SQP is time consuming. SQP is an appropriate method according to the proper CMM data in this paper.

(2)
When the mathematical model of profile error is nonlinear, the secondorder Taylor series expansion needs to be considered. The third order or a higherorder requires the calculation and analysis of tensors, which is difficult to achieve at present. The precise prior distributions of variables in AMCM are often difficult to obtain in most cases, so the AMCM is not widely used.

(3)
If you want to get different confidence intervals, you need to find the corresponding endpoints of the confidence intervals. If you want to obtain the shortest 99% confidence interval, you need to sort the profile error from the smallest to the largest. Then find the location of the left endpoint n in MATLAB. Next the right endpoint f(M * 99/100 + n) is obtained, where M is the number of experiments. The 99% confidence interval in AMCM is [f(n), f(M * 99/100 + n)]. In GUMM, you need to select the corresponding converge factors (k = 3 in 99% confidence interval). The details are shown in Ref. [30].
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Authors’ Contributions
CL wrote the initial manuscript; YS assisted with the experimental process; GH and ZS revised the manuscript. All authors read and approved the final manuscript.
Authors’ Information
Chenhui Liu, born in 1995, is currently a graduate student, majoring in probability theory and mathematical statistics at School of Mathematics, Tianjin University, China. She is now participating in projects at Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, China.
Zhanjie Song, was born in Hebei Province, China, in 1965. He received his PhD. degree from Nankai University, in 2006. He was a Postdoctoral Fellow in signal and information processing, with School of Electronic and Information Engineering, Tianjin University, China. He is currently a Professor with the School of Mathematics, a Fellow of Visual Pattern Analysis Research Lab, and a vicedirector with the Institute of TV and Image Information, all in Tianjin University. His current research interests are in sampling, approximation and reconstruction of random signals and random fields.
Yicun Sang, born in 1991, is currently a PhD candidate. He is now participating in projects at Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, China.
Gaiyun He, born in 1965, is currently a professor at Tianjin University, China. She received her PhD degree from Tianjin University, China, in 2006. Her research interests include modern manufacturing quality control, evaluation methods of geometric errors and CAD/CAM/CAI integration technology.
Acknowledgements
The authors gratefully acknowledge the supports of the open fund of Tianjin Key Laboratory of Equipment Design and Manufacturing Technology (Tianjin University).
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Natural Science Foundation of China (Grant No. 51675378) and National Science and Technology Major Project of China (Grant No. 2014ZX04014031).
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Liu, C., Song, Z., Sang, Y. et al. An Improved Measurement Uncertainty Calculation Method of Profile Error for Sculptured Surfaces. Chin. J. Mech. Eng. 32, 90 (2019). https://doi.org/10.1186/s1003301904040
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DOI: https://doi.org/10.1186/s1003301904040
Keywords
 Secondorder GUMM
 Adaptive Monte Carlo method
 Uncertainty
 Converge factor