# An Improved Measurement Uncertainty Calculation Method of Profile Error for Sculptured Surfaces

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## 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 non-linear. An improved second-order 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 k-factor 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 second-order GUMM is better. Two practical examples are listed and the conclusion is drawn by comparing and discussing the second-order GUMM and AMCM. The results show that the difference between the improved second-order GUM and the AMCM is smaller than the difference between the standard GUM and the AMCM. The improved second-order GUMM is more precise in consideration of the nonlinear mathematical model of profile error.

## Keywords

Second-order GUMM Adaptive Monte Carlo method Uncertainty Converge factor## 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 non-uniform rational B-splines (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 decomposition-iterative 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 first-order 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 random-fuzzy and fuzzy-random 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 non-linear 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 second-order GUMM was proposed to estimate the uncertainty considering the difficulty of the prior information. In 2011, the second-order and third-order Taylor series expansion in GUMM were adopted to calculate the uncertainty in a simple non-linear 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 non-linear, so the first-order 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 second-order 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

*t*is the diameter of the enveloping spheres.

## 3 Uncertainty Calculation Methods

### 3.1 Gumm

### 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).

Symbolic representation

Symbol | Variable | Symbol | Variable | Symbol | Variable |
---|---|---|---|---|---|

| \(x_{i}\) |
| \(y_{i}\) |
| \(z_{i}\) |

| \(\alpha\) |
| \(\beta\) |
| \(\gamma\) |

| \(\Delta x\) |
| \(\Delta y\) |
| \(\Delta z\) |

### 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 t-distribution, the k-factor conversion method is proposed to calculate the converge factor. The conversion method is to approximate the coverage factor of the non-normal distribution.

*f*is the profile error, \(\bar{f}\) is the mean of profile error, \(Var(f)\) is the variance of profile error.

## 4 Experiment

### 4.1 Profile Error and Uncertainty Calculation in Experiment 1

- (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 S-shaped 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.Table 2Correlation coefficient matrix of \(\alpha ,\;\beta ,\;\gamma ,\;\Delta x,\;\Delta y,\;\Delta z\)

Correlation

\(\alpha\)

\(\beta\)

\(\gamma\)

\(\Delta x\)

\(\Delta y\)

\(\Delta z\)

\(\alpha\)

1

0.086

− 0.002

0.049

− 0.060

0.131

\(\beta\)

0.086

1

− 0.018

− 0.094

− 0.006

0.203

\(\gamma\)

− 0.002

− 0.018

1

− 0.029

0.040

0.027

\(\Delta x\)

0.049

− 0.094

− 0.029

1

− 0.728

0.092

\(\Delta y\)

− 0.060

− 0.006

0.040

− 0.728

1

− 0.362

\(\Delta z\)

0.131

0.203

0.027

0.092

− 0.362

1

Table 3Covariance matrix of \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) (mm

^{2})Covariance

\(\alpha\)

\(\beta\)

\(\gamma\)

\(\Delta x\)

\(\Delta y\)

\(\Delta z\)

\(\alpha\)

9.44 × 10

^{−8}3.90 × 10

^{−9}− 4.01 × 10

^{−9}− 3.12 × 10

^{−7}6.30 × 10

^{−7}− 8.2 × 10

^{−6}\(\beta\)

3.90 × 10

^{−9}5.78 × 10

^{−8}− 7.06 × 10

^{−10}− 2.89 × 10

^{−7}− 8.23 × 10

^{−7}7.10 × 10

^{−6}\(\gamma\)

− 4.01 × 10

^{−9}−7.06 × 10

^{−10}3.63 × 10

^{−9}3.54 × 10

^{−7}− 6.38 × 10

^{−7}4.83 × 10

^{−7}\(\Delta x\)

− 3.12 × 10

^{−7}− 2.89 × 10

^{−7}3.54 × 10

^{−7}4.23 × 10

^{−5}− 5.62 × 10

^{−5}2.59 × 10

^{−5}\(\Delta y\)

6.30 × 10

^{−7}− 8.23 × 10

^{−7}− 6.38 × 10

^{−7}− 5.62 × 10

^{−5}1.41 × 10

^{−4}− 1.90 × 10

^{−4}\(\Delta z\)

− 8.24 × 10

^{−6}7.10 × 10

^{−6}4.83 × 10

^{−7}2.59 × 10

^{−5}− 1.90 × 10

^{−4}1.877 × 10

^{−3}According to the GUMM, the uncertainty of the first-order Taylor expansion is calculated with Eq. (6) and the value is \(u1 = 0.0028968\) mm. The uncertainty of the second-order 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*p*-values (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.Table 4One-Sample Kolmogorov–Smirnov test

Parameter or measuring points

*x**y**z*Parameters-Mean (mm)

26.631907

97.468943

35.192155

Parameters-Std. deviation (mm)

6.50275 × 10

^{−3}6.1498 × 10

^{−4}1.48275 × 10

^{−3}Kolmogorov–Smirnov Z

0.669

0.699

0.529

Asymp. Sig. (2-tailed)

0.762

0.712

0.943

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

- (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.Table 5Covariance matrix of \(\alpha ,\beta ,\gamma ,\Delta x,\Delta y,\Delta z\) (mm

^{2})Covariance

\(\alpha\)

\(\beta\)

\(\gamma\)

\(\Delta x\)

\(\Delta y\)

\(\Delta z\)

\(\alpha\)

1.46 × 10

^{−7}6.95 × 10

^{−8}6.74 × 10

^{−9}2.12 × 10

^{−7}− 1.52 × 10

^{−6}− 4.23 × 10

^{−6}\(\beta\)

6.95 × 10

^{−8}5.74 × 10

^{−8}7.05 × 10

^{−9}4.28 × 10

^{−7}− 1.75 × 10

^{−6}1.66 × 10

^{−6}\(\gamma\)

6.74 × 10

^{−9}7.05 × 10

^{−9}3.97 × 10

^{−9}4.27 × 10

^{−7}− 7.35 × 10

^{−7}1.08 × 10

^{−6}\(\Delta x\)

2.12 × 10

^{−7}4.28 × 10

^{−7}4.27 × 10

^{−7}5.20 × 10

^{−6}− 7.39 × 10

^{−6}1.28 × 10

^{−4}\(\Delta y\)

− 1.52 × 10

^{−6}− 1.75 × 10

^{−6}− 7.35 × 10

^{−7}− 7.39 × 10

^{−5}1.47 × 10

^{−4}− 2.20 × 10

^{−4}\(\Delta z\)

− 4.23 × 10

^{−6}1.66 × 10

^{−6}1.08 × 10

^{−6}1.28 × 10

^{−4}− 2.20 × 10

^{−4}9.21 × 10

^{−4}Based on the values, the uncertainty of the first-order 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 first-order and the second-order GUMM. They are written as GUMM-1 and GUMM-2 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 first-order Taylor series expansion is considered in GUMM, the values of*d*_{low}and*d*_{high}are far larger than those in GUMM-2 (0.0419 > 0.0083 & 0.09914 > 0.0537). In a word, the second-order GUMM is more precise to calculate the uncertainty of profile error.Table 6Uncertainty evaluation results in experiment 1 (mm)

Methods

\(\bar{f}\)

*u*95% confidence interval

*d*_{low}*d*_{high}GUMM-1

0.0656

0.002897

[0.060675, 0.07226]

0.0419

0.09914

GUMM-2

0.0656

0.02265

[0.027095, 0.11770]

0.0083

0.0537

AMCM

0.0875

0.0433

[0.0188, 0.1714]

Table 7Uncertainty evaluation results in experiment 2 (mm)

Methods

\(\bar{f}\)

*u*95% confidence interval

*d*_{low}*d*_{high}GUMM-1

0.0993

0.00199

[0.094524, 0.102484]

0.0348

0.1037

GUMM-2

0.0993

0.01945

[0.05262, 0.13042]

0.00708

0.07578

AMCM

0.1233

0.0416

[0.0597, 0.2062]

- (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 GUMM-2 are less than those in GUMM-1 (0.00708 < 0.0348 & 0.07578 < 0.1037). It indicates that the second-order 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 non-linear, the second-order Taylor series expansion needs to be considered. The third order or a higher-order 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].

## Notes

### 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 vice-director 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. 2014ZX04014-031).

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