Form Deviation Uncertainty and Conformity Assessment on a Coordinate Measuring Machine

Abstract

Coordinate measuring machines are widely used in the industrial field due to their ease of automation. However, estimating the measurement uncertainty is a delicate task, especially when controlling for deviation, given the large number of factors that influence the measurement. A precise estimate of the uncertainty is crucial to avoid incorrect conformity assessments. The purpose of this study is to control geometrical-form tolerance specifications, taking into consideration their associated uncertainty. A surface fitting model based on the least squares criterion is proposed, allowing one to obtain the variance–covariance matrix by iterative calculation according to the Levenberg–Marquard optimization method. The form deviation is then evaluated following the Geometrical Product Specifications (GPS) Standard, and its associated uncertainty is estimated using the guide to the expression of uncertainty in measurement (GUM) propagation of the uncertainty law. Finally, the conformity assessment is performed based on the measured deviation and its associated uncertainty. Different results for the measurement of straightness, flatness, circularity, roundness, and cylindricity are presented and detailed. This model is thereafter validated by a Monte Carlo simulation, and interlaboratory comparisons of the obtained results were performed, which showed satisfactory outcome. This contribution is of great use to manufacturing companies and metrology laboratories, allowing them to meet the normative guidelines, which stipulates that each measurement result must be accompanied by its associated uncertainty.

1. Introduction

The precision and accuracy of coordinate measuring machines (CMMs) are crucial for quality control in manufacturing processes, as they ensure that components meet stringent design specifications. Understanding and quantifying form deviation uncertainty are essential for ensuring correct conformity assessments following the normative guidelines [1], which stipulate that each measurement result must be accompanied by its associated uncertainty.

Form deviation refers to the difference between a measured surface and its nominal geometry, as stated in the Geometric Product Specifications norm [2,3,4]. Numerous studies have explored methods to assess and estimate form deviation in CMM measurements. For instance, Forbes [5,6] described semi-analytical approaches based on rectangular distributions and Gaussian process models, which take into consideration the likely spatial correlation of form deviations. Shichang Du et al. [7] presented a method based on multivariate spatial statistics for estimating the form deviation considering the influence of machining conditions and spatial correlation.

In a related study, Habibi et al. [8,9] examined parallelism and perpendicularity assessment and uncertainty estimation based on the propagation of uncertainties [10,11]. However, the estimation of the surface parameters’ variances was based on repeatability. Jalid et al. [12,13,14] used the same method, based on orthogonal distance regression, to estimate the fitting parameters uncertainties.

Recent advancements in measurement technology have also led to the development of sophisticated algorithms for form deviation analysis. Jakubiec et al. [15,16] introduced a software fully compliant with the GPS [2,3,4] philosophy for the evaluation of dimensional and geometrical specification tolerances, which demonstrated significant improvements compared to conventional CMM methods. Furthermore, Cui et al. [17] focused on the comparison between the results obtained using the least squares method and the genetic optimization algorithm, applied to flatness and roundness, which showed almost identical results.

Despite major advances and the various methods in the form deviation uncertainty estimation [18,19,20], there remains a need for more comprehensive studies that integrate form deviation uncertainty analysis with practical conformity assessment procedures [21]. This research aims to bridge this gap by proposing a model, as shown in Figure 1, for estimating the uncertainty using the GUM [10] law or propagation, where surface fitting [22,23] and CMM probing uncertainty [24,25,26,27] are two key components to estimate the form deviation’s measurement uncertainty.

Accordingly, the conformity assessment is performed based on the measured deviation and its associated uncertainty according to the ISO 98-4 [28]. Different results for the measurement of straightness, flatness, circularity, roundness, and cylindricity are presented and detailed. The surface fitting algorithm based on the least squares method is validated by comparison [29] with the National Institute of Standards and Technology [30] database. The form deviation calculation and its associated uncertainty estimation will thereafter be validated by a Monte Carlo simulation [11] and interlaboratory comparisons of the obtained assessment results will be conducted with Measurement Control Center [31] according to the ISO13528 [32].

2. Materials and Methods

The objective is to propose a model that enables the estimation of form deviation, its associated uncertainty, and the conformity assessment with the specified standards, as shown in Figure 2. We start by probing a cloud of points P_i (x_i, y_i, z_i) using the coordinate measuring machine. Initially, we performed a surface fitting, which generates the parameters $\beta$ and its covariance matrix ${\widehat{V}}_{\beta }$. This covariance matrix is incorporated into the estimation of the form deviation uncertainty. Following this, we calculate the form deviation and its associated uncertainty, accounting for the probing error of the coordinate measuring machine. Finally, we declare conformity, taking into consideration the calculated uncertainty.

A specific measurement process must be followed in order to control a geometrical specification:

• Surface probing.
• Surface fitting.
• Form deviation evaluation.
• Estimation of the form deviation associated uncertainty.
• Declaration of conformity.

The details of each of these steps will be provided in the following paragraphs.

2.1. Machine Probing Uncertainty

Coordinate measuring machines are precise and accurate. However, various factors influence the measurement uncertainty (Figure 3), and it is very difficult to quantify the influence of each of these parameters independently of the others.

Several approaches have been used for the estimation of the uncertainty associated with probing [33]. We will assume that the errors along the axes are independent and linear, and the uncertainty matrix associated with each probed point $P_i$ is given as follows:

$$\left[P_i\right]=\left(\begin{array}{ccc}var\left(x_P\right)& 0& 0\\ 0& var\left(y_P\right)& 0\\ 0& 0& var\left(z_P\right)\end{array}\right).$$

(1)

where $var\left(x\right)={\sigma }_x^2$, $var\left(y\right)={\sigma }_y^2$, $var\left(z\right)={\sigma }_z^2$, with ${\sigma }_L={\displaystyle \frac{\sqrt{a^2+{\left(bL\right)}^2}}{k}}$, where a and b are constants, and k the coverage factor defined by the law of the selected distribution. The variables a and b are determined by measuring 5 gauge blocks, each block for 3 repetitions, along 3 of the 7 directions specified in ISO 10360-2 [34], then calculating the error equations along each direction $E_x$ = $A_x$x + $B_x$ (error following the direction X for example), then estimating the associated uncertainty $u_x$.

2.2. Surface Fitting Parameters and Associated Variance–Covariance Matrix

Surface fitting is a mathematical modeling technique that enables the transition from a cloud of points extracted from the actual surface to an associated theoretical element that best represents it. Basic geometric elements can be represented in terms of their parameters. Let P be a point on the surface in question (as shown in Figure 4); it satisfies the following equations in

Table 1. Geometrical elements’ equations.

Geometrical Elements		Point $\mathit{A}({\mathit{x}}_{\mathit{A}},{\mathit{y}}_{\mathit{A}},{\mathit{z}}_{\mathit{A}})$	Normal Vector $\overrightarrow{\mathit{n}}({\mathit{n}}_{\mathit{x}},{\mathit{n}}_{\mathit{y}},{\mathit{n}}_{\mathit{z}})$	$\mathbf{Direction}\mathbf{Vector}\overrightarrow{\mathit{d}}({\mathit{d}}_{\mathit{x}},{\mathit{d}}_{\mathit{y}},{\mathit{d}}_{\mathit{z}})$	Radius R	Equation
Line		∗		∗		$‖\overrightarrow{AP}\wedge \overrightarrow{d}‖=0$
Plane		∗	∗			$\overrightarrow{AP}.\overrightarrow{n}=0$
Circle		∗	∗		∗	$\left\{\begin{array}{c}\overrightarrow{AP}.\overrightarrow{n}=0\\ ‖\overrightarrow{AP}\wedge \overrightarrow{n}‖=R\end{array}\right.$
Cylinder		∗		∗	∗	$‖\overrightarrow{AP}\wedge \overrightarrow{d}‖=R$
Sphere		∗			∗	$‖\overrightarrow{AP}‖=R$

(*) indicates the necessary properties for parameterizing the geometrical element.

.

We will base our approach on the least squares criterion, which involves minimizing the sum of squared residuals $min{\sum }_{i=1}^N{e}_i^2$, where $e_i$ represents the orthogonal distance between the measured points and the substitute feature, as shown in Figure 5.

Consider a point cloud $M_i$ probed using a coordinate measuring machine, on elements that are nominally linear, planar, circular, cylindrical, and spherical. We propose the following objective functions to obtain the substitute features parameters:

• Line:

$$\begin{gathered} \mathrm{The}\mathrm{deviation}\mathrm{is}\mathrm{represented}\mathrm{by}{e}_i=∆r_i=‖\overrightarrow{A{P}_i}\wedge \overrightarrow{d}‖. \\ \mathrm{The}\mathrm{optimization}\mathrm{problem}\mathrm{reduces}\mathrm{to}\left\{\begin{array}{c}min{\sum }_{i=1}^N{e}_i^2\\ {d_x}^2+{d_y}^2+{d_z}^2=1\end{array}\right.. \end{gathered}$$

where $∆r_i=$ $\sqrt{{\left(\left(y_0-y_i\right)d_z-\left(z_0-z_i\right)d_y\right)}^2+{\left(\left(z_0-z_i\right)d_x-\left(x_0-x_i\right)d_z\right)}^2+{\left(\left(x_0-x_i\right)d_y-\left(y_0-y_i\right)d_x\right)}^2}$.

• Plane:

$$\begin{gathered} \mathrm{The}\mathrm{deviation}\mathrm{is}\mathrm{represented}\mathrm{by}{e}_i=∆h_i=\left|\overrightarrow{A{P}_i}.\overrightarrow{n}\right|. \\ \mathrm{The}\mathrm{optimization}\mathrm{problem}\mathrm{reduces}\mathrm{to}\left\{\begin{array}{c}min{\sum }_{i=1}^N{e}_i^2\\ {n_x}^2+{n_y}^2+{n_z}^2=1\end{array}\right.. \end{gathered}$$

where $∆h_i=\left(x_0-x_i\right)n_x+\left(y_0-y_i\right)n_y+\left(z_0-z_i\right)n_z$.

• Circle:

$$\begin{gathered} \mathrm{The}\mathrm{deviation}\mathrm{is}\mathrm{represented}\mathrm{by}{e}_i=\sqrt{{(∆r_i)}^2+{(∆h_i)}^2}. \\ \mathrm{Such}\mathrm{as}∆r_i=r_i-R=‖\overrightarrow{A{P}_i}\wedge \overrightarrow{n}‖-R\mathrm{et}∆h_i=\left|\overrightarrow{A{P}_i}.\overrightarrow{n}\right|. \\ \mathrm{The}\mathrm{optimization}\mathrm{problem}\mathrm{reduces}\mathrm{to}\left\{\begin{array}{c}min{\sum }_{i=1}^N{e}_i^2\\ {n_x}^2+{n_y}^2+{n_z}^2=1\end{array}\right.. \end{gathered}$$

where $∆h_i=\left(x_0-x_i\right)n_x+\left(y_0-y_i\right)n_y+\left(z_0-z_i\right)n_z$and $∆r_i=$ $\sqrt{{\left(\left(y_0-y_i\right)n_z-\left(z_0-z_i\right)n_y\right)}^2+{\left(\left(z_0-z_i\right)n_x-\left(x_0-x_i\right)n_z\right)}^2+{\left(\left(x_0-x_i\right)n_y-\left(y_0-y_i\right)n_x\right)}^2}-R$.

• Sphere:

$$\begin{gathered} \mathrm{The}\mathrm{deviation}\mathrm{is}\mathrm{represented}\mathrm{by}{e}_i=∆r_i=r_i-R=‖\overrightarrow{A{P}_i}‖-R. \\ \mathrm{The}\mathrm{optimization}\mathrm{problem}\mathrm{reduces}\mathrm{to}min{\sum }_{i=1}^N{e}_i^2. \end{gathered}$$

where $∆r_i=$ $\sqrt{{\left(x_0-x_i\right)}^2+{\left(y_0-y_i\right)}^2{+\left(z_0-z_i\right)}^2}-R$.

• Cylinder:

$$\begin{gathered} \mathrm{The}\mathrm{deviation}\mathrm{is}\mathrm{represented}\mathrm{by}{e}_i=∆r_i=‖\overrightarrow{A{P}_i}\wedge \overrightarrow{d}‖-R. \\ \mathrm{The}\mathrm{optimization}\mathrm{problem}\mathrm{reduces}\mathrm{to}\left\{\begin{array}{c}min{\sum }_{i=1}^N{e}_i^2\\ {d_x}^2+{d_y}^2+{d_z}^2=1\end{array}\right.. \end{gathered}$$

where $∆r_i=\sqrt{{\left(\left(y_0-y_i\right)d_z-\left(z_0-z_i\right)d_y\right)}^2+{\left(\left(z_0-z_i\right)d_x-\left(x_0-x_i\right)d_z\right)}^2+{\left(\left(x_0-x_i\right)d_y-\left(y_0-y_i\right)d_x\right)}^2}-R$.

The least squares method tends to be more sensitive to outliers because it squares the errors. Significant errors have a greater impact, leading to a good overall fit to the data, but it does not necessarily ensure the smallest maximum error across the entire data range. Our objective is to minimize $min{\sum }_{i=1}^N{e}_i^2$ for the different values of the parameters $\beta$:

$${\beta }^T=\left({\beta }_1,{\beta }_2,\dots ,{\beta }_p\right).$$

(2)

A weighting ${\omega }_{{\mathrm{e}}_i}=k$ $\left(\begin{array}{ccc}{\displaystyle \frac{1}{{{\sigma }_x}^2}}& 0& 0\\ 0& {\displaystyle \frac{1}{{{\sigma }_y}^2}}& 0\\ 0& 0& {\displaystyle \frac{1}{{{\sigma }_z}^2}}\end{array}\right)$ can be introduced into the model such as:

• ${\partial }^2$ is the measurement uncertainty of the probed point.
• $k>1$ for points located in a critical area.
• $k<1$ for outliers.

Our optimization problem translates to the following nonlinear regression equation:

$$\left\{\begin{array}{c}\underset{\beta }{\min }{\sum }_{i=1}^n\left(\left[{{\mathrm{e}}_i}^T\right]\left[{\omega }_{{\mathrm{e}}_i}\right]\left[{\mathrm{e}}_i\right]\right)\\ f_i\left(x_i;\beta \right)=0\end{array}\right..$$

(3)

Solving the equation requires minimizing $S\left(\eta \right)$ expressed by the equation:

$$S\left(\beta \right)={G\left(\widehat{\beta }\right)}^T{\omega }_{{\mathrm{e}}_i}G\left(\widehat{\beta }\right),$$

(4)

with $G\left(\widehat{\beta }\right)=\left({\mathrm{e}}_x\left(\beta \right),{\mathrm{e}}_y\left(\beta \right),{\mathrm{e}}_z\left(\beta \right)\right)$.

The variance–covariance matrix associated with the estimated parameters $\beta$ is expressed as

$${\widehat{V}}_{\beta }={\sigma }^2{\left[{\dot{G}\left(\widehat{\beta }\right)}^T{\omega }_{e_i}\dot{G}\left(\widehat{\beta }\right)\right]}^{-1},$$

(5)

as $\dot{G}\left(\widehat{\beta }\right)$ represents the Jacobian matrix of $G\left(\widehat{\beta }\right)$ evaluated at the estimated parameters ${\beta }^T$, and ${\sigma }^2$ the residual variance estimated by the following equation:

$${\sigma }^2=\left[{G\left(\widehat{\beta }\right)}^T{\omega }_{e_i}G\left(\widehat{\beta }\right)\right]/\left(n-p\right),$$

(6)

where $p$ is the number of parameters and $n$ is the number of measured points.

The estimated parameter ${\beta }_i$ associated uncertainty is given by $u^2\left({\beta }_i\right)={\widehat{V}}_{\beta }(i,i)$.

2.3. Form Deviation Evaluation

After surface fitting, it is possible to evaluate the form deviation $fe$ based on the fitting parameters (Section 2.2) of the geometric shapes and the probed points $P_i$. We propose the following models for each specification:

$$\mathit{Straightness}\mathit{:}fe=‖\overrightarrow{Mm}\wedge \overrightarrow{d}‖$$

where $M$ and $m$ are the extreme points that are the furthest apart perpendicularly to the vector $\overrightarrow{d}$:

$$M,m=\left\{P_i,P_j|\mathit{max}\left(‖\overrightarrow{P_i{P}_j}\wedge \overrightarrow{d}‖\right)fori=\mathrm{1,2},\dots ,n-1andj=i+1,\dots ,n\right\}.$$

(7)

$$\mathit{Flatness}\mathit{:}fe=\left|\overrightarrow{Mm}.\overrightarrow{n}\right|$$

where $M$ et $m$ are the extreme points that are the furthest apart along the vector $\overrightarrow{n}$:

$$M,m=\left\{P_i,P_j|\max \left(\overrightarrow{P_i{P}_j}.\overrightarrow{n}\right)fori=\mathrm{1,2},\dots ,n-1andj=i+1,\dots ,n\right\}.$$

(8)

$$\mathit{Circularity}\mathit{:}fe=‖\overrightarrow{AM}\wedge \overrightarrow{n}‖-‖\overrightarrow{Am}\wedge \overrightarrow{n}‖$$

where $M$ and $m$ are the extreme points that are, respectively, the farthest and the closest relative to the axis defined by the vector $\overrightarrow{n}$ passing through the center $A$:

$$\left\{\begin{array}{c}M=\left\{P_i|\mathit{max}\left(‖\overrightarrow{A{P}_i}\wedge \overrightarrow{n}‖\right)fori=\mathrm{1,2}\dots n\right\}\\ m=\left\{P_i|\mathit{min}\left(‖\overrightarrow{A{P}_i}\wedge \overrightarrow{n}‖\right)fori=\mathrm{1,2}\dots n\right\}\end{array}\right..$$

(9)

$$\mathit{Sphericity}\mathit{:}fe=‖\overrightarrow{CM}‖-‖\overrightarrow{Cm}‖$$

where $M$ and $m$ are the extreme points that are, respectively, the farthest and the closest relative to the center $A$:

$$\left\{\begin{array}{c}M=\left\{P_i|\mathit{max}\left(‖\overrightarrow{A{P}_i}‖\right)fori=\mathrm{1,2}\dots n\right\}\\ m=\left\{P_i|\mathit{min}\left(‖\overrightarrow{A{P}_i}‖\right)fori=\mathrm{1,2}\dots n\right\}\end{array}\right..$$

(10)

$$\mathit{Cylindricity}\mathit{:}fe=‖\overrightarrow{AM}\wedge \overrightarrow{d}‖-‖\overrightarrow{Am}\wedge \overrightarrow{d}‖$$

where $M$ and $m$ are the extreme points that are, respectively, the farthest and the closest relative to the axis defined by the vector $\overrightarrow{d}$ passing through the center $A$:

$$\left\{\begin{array}{c}M=\left\{P_i|\mathit{max}\left(‖\overrightarrow{A{P}_i}\wedge \overrightarrow{d}‖\right)fori=\mathrm{1,2}\dots n\right\}\\ m=\left\{P_i|\mathit{min}\left(‖\overrightarrow{A{P}_i}\wedge \overrightarrow{d}‖\right)fori=\mathrm{1,2}\dots n\right\}\end{array}\right..$$

(11)

2.4. Estimation of Form Deviation Associated Uncertainty

The determination of the combined uncertainty is the most critical step in the process and involves propagating the uncertainties of the error sources $\left(X_1,X_2,\dots ,X_N\right)$ through the mathematical equation of the measurand $Y=fe\left(X_1,X_2,\dots ,X_N\right)$, with $X_i={\mu }_i+e_i$ based on a first-order Taylor series expansion. We assume that $e_i$ is statistically zero, such that $E\left(e_i\right)=0$ and $E\left(X_i\right)={\mu }_i$, and that the partial derivatives at the central point are constant. The variance can then be deduced as $V\left(Y\right)=E\left({\left[Y-E\left(Y\right)\right]}^2\right)=E\left({\left({\sum }_{i=1}^k{\left({\displaystyle \frac{\partial fe}{\partial X_i}}\right)}_{X=\mu }{e}_i\right)}^2\right)$ leading to the final form of the GUM uncertainty propagation law:

$$\begin{array}{c}{u}_c^2={\sum }_{i=1}^N{\left[{\left({\displaystyle \frac{\partial fe}{\partial X_i}}\right)}_{X=\mu }\right]}^{2}var\left(X_i\right)+2{\sum }_{i=1}^{N-1}{\sum }_{j=i+1}^N{\left({\displaystyle \frac{\partial fe}{\partial X_i}}\right)}_{X=\mu }{\left({\displaystyle \frac{\partial fe}{\partial X_j}}\right)}_{X=\mu }cov\left(X_i,X_j\right)\hfill \\ ={\sum }_{i=1}^N{\sum }_{j=1}^N{\left({\displaystyle \frac{\partial fe}{\partial X_i}}\right)}_{X=\mu }{\left({\displaystyle \frac{\partial fe}{\partial X_j}}\right)}_{X=\mu }cov\left(X_i,X_j\right)\hfill \\ =\left[J_{fe}\right]\left[\Omega \right][{J_{fe}]}^T,\hfill \end{array}$$

(12)

where $\left[\Omega \right]$ represents the variance–covariance matrix, and $\left[J_{fe}\right]$ is the Jacobian expressed as:

$$\left[J_{fe}\right]=\left[{\displaystyle \frac{\partial fe}{\partial X_1}}{\displaystyle \frac{\partial fe}{\partial X_2}}\dots {\displaystyle \frac{\partial fe}{\partial X_N}}\right].$$

(13)

We then obtain the expanded uncertainty $U_{GUM}=k.\sqrt{u_c^2}$, where $k$ is the coverage factor and $u_c$ is the standard uncertainty. For a 95.45% confidence interval, the coverage factor is equal to 2.

2.5. Conformity Assessment

To establish a conformity assessment procedure for dimensional and geometric specifications taking into account the measurement uncertainty, it will be necessary to calculate the risk zone, assuming that uncertainty follows a normal distribution:

$$Ø\left(x\right)={\int }_x^{+\infty }F\left(Z\right)dZ.$$

(14)

According to ISO/IEC Guide 98-4 [30], if the acceptable risk limit is not specified by the client, the risk $p_{\mathsf{\alpha }}=\left\{df>z_i\right\}=1-Ø\left(z_i\right)$ n must not exceed 2.3%, where, $z_i$ is the Gaussian coefficient using the standard normal distribution, expressed as $z_i={\displaystyle \frac{Ts-df}{U/2}}$.

By incorporating measurement uncertainty into the conformity assessment process, the decisions made become more aligned with real-world situations and imposed standards, thereby improving quality control.

3. Proposed Algorithm

After modeling the steps outlined in the previous section, a data processing program was developed in MATLAB R2024b. Figure 2 presents the algorithm that illustrates the process. This program involves reading the data file containing the coordinates of the measured points. It will also require the type of the controlled specification, and its tolerance limit. As output, we obtain the measured form deviation, its associated uncertainty, and the conformity statement.

Depending on the geometry being controlled, the algorithm can require up to 7 starting parameters ${\beta }_0=(x_0,y_0,z_0,v_{x0},v_{y0},v_{z0},R_0)$ that can approximate the solution to avoid local optima and converge more effectively. $(x_0,y_0,z_0)$ represent the coordinates of a point, $(v_{x0},v_{y0},v_{z0})$ a vector, and $R_0$ a radius.

To determine the starting parameters [35,36], we will assume that the studied geometries are complete; therefore, the starting point will be the centroid, such as $x_0={\displaystyle \frac{1}{n}}\sum x_i$, $y_0={\displaystyle \frac{1}{n}}\sum y_i$, $z_0={\displaystyle \frac{1}{n}}\sum z_i$, where $n$ represents the total number of probed points.

In order to determine the starting normal vector for the plane and circle, and the starting direction vector for the line and the cylinder, we are going to use singular value decomposition. For example, $(v_{x0},v_{y0},v_{z0})$ is the eigenvector associated with the largest eigenvalue of $B=A^{T}A$, where A is the $n\times 3$ matrix whose $i^{th}$ row is $(x_i-x_0,y_i-y_0,z_i-z_0)$. Alternatively, $(v_{x0},v_{y0},v_{z0})$ is the singular vector corresponding to the largest singular value of A for the direction vector and the smallest singular value of A for the normal vector.

Regarding the starting radius, it will be based on the following mean value of the radius; for example, the sphere and circle starting radius is given by the following equation:

$$R_0={\displaystyle \frac{1}{n}}\sum \sqrt{{\left(x_0-x_i\right)}^2+{\left(y_0-y_i\right)}^2{+\left(z_0-z_i\right)}^2}.$$

(15)

Based on the probed points $P_i$ and the objective function (error function, detailed in Section 2.2), once the initial vector ${\beta }_0$ is determined, we can proceed to surface fitting. A nonlinear regression process is then applied, allowing us to obtain both the fitting parameters $\beta$ and the associated covariance matrix ${\widehat{V}}_{\beta }$ as shown in Figure 6.

The optimization method followed is the Levenberg–Marquardt algorithm [37], assuming that the objective function is smooth and differentiable allowing gradient information to be used effectively, and that the starting vector ${\beta }_0$ is a reasonable initial guess avoiding local optima. The algorithm combines the stability of gradient descent with the efficiency of the Gauss–Newton approach, making it particularly useful for curve fitting. This hybrid approach provides both stability and speed, addressing the weaknesses of each individual technique.

The form deviation $fe$ is subsequently calculated from the data sets $P_i$ and the fitting parameters $\beta$, by determining the extreme points M and m as detailed in Section 2.3. Therefore, based on the form deviation equation $fe\left(X_1,X_2,\dots ,X_N\right)$, the GUM law of uncertainty propagation is applied, as shown in Figure 7, and the result is expressed in matrix form as $U_{fe}=2\sqrt{J_{fe}\Omega {J_{fe}}^T}$.

The calculation of the Jacobian $J_{fe}$ is detailed in Annex B, and the matrix $\Omega$ is constructed from the uncertainty associated with the extreme points M and m, resulting from the CMM probing error, as well as the uncertainty from the surface fitting, depending on the parameters influencing the form deviation.

$$\Omega =\left[\begin{array}{ccc}\left[{\widehat{V}}_{\beta }\right(\left[X_1,X_2,\dots ,X_N\right],\left[X_1,X_2,\dots ,X_N\right]\left)\right]& 0& 0\\ 0& \left[M\right]& 0\\ 0& 0& \left[m\right]\end{array}\right]$$

(16)

It is important not to confuse the parameters $X_1,X_2,\dots ,X_N$ with ${\beta }_1,{\beta }_2,\dots ,{\beta }_p$, as $N$ represents the number of parameters in the deviation equation and $p$ represents the number of fitting parameters related to the fitted feature.

Finally, the declaration of conformity will be made based on the calculated deviation and its associated uncertainty, in accordance with ISO standard 98-4 [28], by applying Equation (14), with a risk not exceeding 2.3% unless specified by the customer.

4. Results

In this paragraph, we will address examples of geometric form specifications on different mechanical parts, namely, 10 μm straightness, 10 μm flatness, 15 μm circularity, 15 μm sphericity, and 15 μm cylindricity deviation tolerance limits from features that are, respectively, nominally linear, plane, circular, spherical, and cylindrical. The probed point datasets for each experiment are provided in

Table A1. Coordinates of the points probed from the line and the circle.

	Line			Circle
x	y	z	x	y	z
−9.9856	30.0019	0.0061	24.9996	15.0007	10.0001
−4.9842	30.0	0.0077	23.5398	18.5305	10.0002
−0.0347	30.0031	0.0072	20.0006	19.9997	9.9999
5.0178	29.9974	0.0063	16.4797	18.5391	10.0001
9.9783	30.0016	0.0058	14.9994	15.0001	9.9998
14.9966	30.0	0.0072	16.4694	11.4602	9.9998
19.9726	30.0024	0.0047	20.0004	10.0001	9.9999
25.0317	29.9998	0.0036	23.5396	11.4699	9.9999

,

Table A2. Coordinates of the points probed from the plane and the cylinder.

	Plane			Cylinder
x	y	z	x	y	z
4.9219	9.972	0.0026	4.9994	0.0002	−0.0003
5.2049	20.0021	0.0047	3.5351	3.545	0.0003
5.0133	30.011	0.0044	−0.0001	5.0005	−0.0004
4.9998	40.018	0.0049	−3.5348	3.5349	0.0006
4.9629	50.0275	0.0065	−5.0003	−0.0003	−0.0002
10.0416	10.0348	0.005	−3.5351	−3.5352	−0.0006
10.2183	20.0059	0.0069	0.0	−5.0001	−0.0
10.006	30.02	0.005	3.535	−3.5347	0.0
9.7648	39.9959	0.0086	5.0	0.0001	10.0
9.9785	50.0021	0.0076	3.5348	3.5348	10.0002
15.0064	9.9988	0.0083	0.0001	5.0002	10.001
14.7649	20.0022	0.0099	−3.535	3.5348	9.9999
15.111	30.0036	0.0085	−4.9999	−0.0002	10.0004
15.02	40.0082	0.0097	−3.5349	−3.5352	9.9997
14.9722	50.0027	0.0065	−0.0001	−4.9999	10.0001
19.9854	10.0043	0.0054	3.5351	−3.5351	9.9996
19.9845	20.0014	0.0091	5.0001	−0.0002	20.0008
20.0342	30.06	0.0066	3.5352	3.5349	20.0005
20.0171	40.0035	0.0074	−0.0	5.0004	19.9997
19.9783	50.9976	0.0074	−3.5348	3.5347	19.9991
24.9961	10.0013	0.0061	−5.0001	−0.0004	20.0
25.0317	20.06	0.0057	−3.5351	−3.5354	20.0002
25.2187	30.0026	0.0077	−0.0001	−4.9996	20.0001
25.0159	39.9991	0.0042	3.5349	−3.5352	19.9999
24.9788	50.0051	0.0037

and

Table A3. Coordinates of the points probed from the sphere.

	Sphere
x	y	z
5.0022	−0.0091	−0.0041
3.5407	3.5373	−0.0041
−0.0041	5.0025	0.0003
−3.5352	3.5328	−0.0002
−5.005	−0.0056	0.002
−3.5363	−3.5327	0.0097
−0.0066	−5.0023	0.0042
3.5361	−3.5401	0.0028
0.0032	5.0029	0.0017
0.0007	3.5275	3.5362
−0.0009	0.0077	4.9918
−0.0003	−3.5393	3.5294
−0.0061	−4.995	−0.004
−0.0012	−3.5358	−3.528
−0.0031	−0.0038	−5.0032
−0.0033	3.5244	−3.5393
5.004	0.0101	0.0015
3.536	−0.0097	3.5311
−0.0008	−0.0066	5.0035
−3.5337	0.0055	3.5396
−5.0068	−0.0011	0.0043

.

As shown in Figure 8, the CMM used is a Mitutoyo Euro-C 544 (Kanagawa, Japan), with a measuring volume of 500 × 400 × 400 mm³. It offers an MPE (Maximum Permissible Error) for length measurement of 2.9 + 4 L/1000 µm within the temperature range of 18 °C to 22 °C coupled with a TP2 type probe head mounted with a tungsten carbide stylus of effective working length EWL = 14 mm and a ruby tip of diameter D = 2 mm, all controlled by Mcosmos Geopack software. The temperature is regulated at 20° ± 2 °C and the probing speed is set to 10 mm/s.

We begin by probing the point dataset, accounting for the probing uncertainty associated with each measured point $P$, where ${\sigma }_L=\sqrt{3^2+{(L/200)}^2}$.

Subsequently, we perform surface fitting to obtain the fitting parameters β. The resulting fitting covariance matrix ${\widehat{V}}_{\beta }$ is then utilized for uncertainty estimation. Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 illustrate the probed points and the fitted features for the line, plane, circle, sphere, and cylinder. The form deviation $fe$ is evaluated by identifying the two extreme points $M$ and $m$. The Jacobian matrix $J_{fe}$ is computed, and the variance–covariance matrix $\Omega$ is constructed, allowing us to accurately estimate the form deviation and its associated uncertainty calculated as $U_{fe}=2\sqrt{J_{fe}\Omega {J_{fe}}^T}$.

More detailed calculations of the different form deviations and the Jacobian are provided in Appendix B. In the following examples, we will briefly outline the key results relevant for each step to control the different specifications.

4.1. Straightness

-

Probing of the point dataset and surface fitting:

Figure 9. Line fitting.

Figure 9. Line fitting.

$$\beta =\left[7.498930.00040.00560.9999-0.0000-0.0001\right]$$

$${\widehat{V}}_{\beta }=1\mathrm{E}-05\left(\begin{array}{cccccc}0.0000& 0.0000& 0.0000& -0.0000& 0.0000& -0.0000\\ 0.0000& 0.0798& -0.0012& -0.0000& 0.0010& 0.0001\\ 0.0000& -0.0012& 0.2625& -0.0000& 0.0010& -0.0066\\ -0.0000& -0.0000& -0.0000& 0.0000& -0.0000& 0.0000\\ 0.0000& 0.0010& 0.0010& -0.0000& 0.0014& -0.0002\\ -0.0000& 0.0001& -0.0066& 0.0000& -0.0002& 0.0019\end{array}\right)$$

-

Determination of the two extreme points:

$$M=\left[5.017029.99700.0060\right]\mathrm{and}m=\left[-4.9840300.0070\right].$$

-

Calculation of the Jacobian, and construction of the variance–covariance matrix:

$$J=\left[0.00000.9990-0.04550.0034-2.47930.1129-0.0000-0.99900.0455\right]$$

$$\Omega =\left[\begin{array}{ccc}\left[{\widehat{V}}_{\beta }\right(\left[4,5,6\right],\left[4,5,6\right]\left)\right]& 0& 0\\ 0& \left[M\right]& 0\\ 0& 0& \left[m\right]\end{array}\right]$$

-

Estimation of straightness deviation and its associated uncertainty:

$$fe=0.0069\mathrm{mm}\mathrm{and}{U}_{fe}=0.0035\mathrm{mm}.$$

4.2. Flatness

-

Probing of the point dataset and surface fitting:

Figure 10. Plane fitting.

Figure 10. Plane fitting.

$$\beta =\left[15.007830.04360.0060-0.0001-0.00001.0000\right]$$

$${\widehat{V}}_{\beta }=1\mathrm{E}-07\left(\begin{array}{cccccc}0.0000& 0.0000& -0.0000& -0.0000& 0.0000& -0.0000\\ 0.0000& 0.0000& -0.0000& -0.0000& 0.0000& -0.0000\\ -0.0000& -0.0000& 0.9613& 0.0213& -0.0071& 0.0000\\ -0.0000& -0.0000& 0.0213& 0.0245& -0.0064& 0.0000\\ 0.0000& 0.0000& -0.0071& -0.0064& 0.0062& -0.0000\\ -0.0000& -0.0000& -0.0000& 0.0000& -0.0000& 0.0001\end{array}\right)$$

-

Determination of the two extreme points:

$$M=\left[14.764020.00200.0090\right]\mathrm{and}m=\left[24.978050.00500.0030\right].$$

-

Calculation of the Jacobian and construction of the variance–covariance matrix:

$$J=\left[10.214030.0030-0.0060-0.0001-0.00001.00000.00010.0000-1.0000\right]$$

$$\Omega =\left[\begin{array}{ccc}\left[{\widehat{V}}_{\beta }\right(\left[4,5,6\right],\left[4,5,6\right]\left)\right]& 0& 0\\ 0& \left[M\right]& 0\\ 0& 0& \left[m\right]\end{array}\right]$$

-

Estimation of flatness deviation and its associated uncertainty:

$$fe=0.0068\mathrm{mm}\mathrm{and}{U}_{fe}=0.0044\mathrm{mm}.$$

4.3. Circularity

-

Probing of the point dataset and surface fitting:

Figure 11. Circle fitting.

Figure 11. Circle fitting.

$$\beta =\left[19.998514.99649.99140.0001-0.00011.00004.9984\right]$$

$${\widehat{V}}_{\beta }=1\mathrm{E}-06\left(\begin{array}{ccccccc}0.0440& -0.0044& 0.0091& 0.0057& -0.0019& -0.0000& 0.0000\\ -0.0044& 0.0482& -0.0738& -0.0012& -0.0017& -0.0006& -0.0000\\ 0.0091& -0.0738& 0.3820& -0.0090& 0.0043& 0.0019& -0.0097\\ 0.0057& -0.0012& -0.0090& 0.0226& 0.0014& 0.0001& -0.0002\\ -0.0019& -0.0017& 0.0043& 0.0014& 0.0175& 0.0047& -0.0014\\ -0.0000& -0.0006& 0.0019& 0.0001& 0.0047& 0.0013& -0.0004\\ 0.0000& -0.0000& -0.0097& -0.0002& -0.0014& -0.0004& 0.0201\end{array}\right)$$

-

Determination of the two extreme points:

$$M=\left[20.000119.99999.9999\right]\mathrm{and}m=\left[16.475018.534910.0001\right].$$

-

Calculation of the Jacobian and construction of the variance–covariance matrix:

$$J=\left[-0.7059-0.29140.0000-0.0056-0.00230.00990.7056-0.7086-0.00010.00031.00000.0001\right]$$

$$\Omega =\left[\begin{array}{ccc}\left[{\widehat{V}}_{\beta }\right(\left[1,2,3,4,5,6\right],\left[1,2,3,4,5,6\right]\left)\right]& 0& 0\\ 0& \left[M\right]& 0\\ 0& 0& \left[m\right]\end{array}\right]$$

-

Estimation of circularity deviation and its associated uncertainty:

$$fe=0.0099\mathrm{mm}\mathrm{and}{U}_{fe}=0.0044\mathrm{mm}.$$

4.4. Sphericity

-

Probing of the point dataset and surface fitting:

Figure 12. Sphere fitting.

Figure 12. Sphere fitting.

$$\beta =\left[-0.00030.0004-0.00064.9996\right]$$

$${\widehat{V}}_{\beta }=1\mathrm{E}-05\left(\begin{array}{cccc}0.0961& -0.0018& -0.0113& 0.0089\\ -0.0018& 0.1367& -0.0062& -0.0049\\ -0.0113& -0.0062& 0.1047& -0.0049\\ 0.0089& -0.0049& -0.0049& 0.0370\end{array}\right)$$

-

Determination of the two extreme points:

$$M=\left[-5.0069-0.00120.0044\right]\mathrm{and}m=\left[-0.0009440.00784.9919\right].$$

-

Calculation of the Jacobian and construction of the variance–covariance matrix:

$$J=\left[0.99990.00180.99900.0001-0.0015-1.0000-1.0000-0.00030.0010\right]$$

$$\Omega =\left[\begin{array}{ccc}\left[{\widehat{V}}_{\beta }\right(\left[1,2,3\right],\left[1,2,3\right]\left)\right]& 0& 0\\ 0& \left[M\right]& 0\\ 0& 0& \left[m\right]\end{array}\right]$$

-

Estimation of sphericity deviation and its associated uncertainty:

$$fe=0.01399\mathrm{mm}\mathrm{and}{U}_{fe}=0.0050\mathrm{mm}.$$

4.5. Cylindricity

-

Probing of the point dataset and surface fitting:

Figure 13. Cylinder fitting.

Figure 13. Cylinder fitting.

$$\beta =\left[-0.0000-0.00019.99990.0001-0.00001.00005.0001\right]$$

$${\widehat{V}}_{\beta }=1\mathrm{E}-06\left(\begin{array}{ccccccc}0.1199& 0.0057& -0.0000& 0.0004& -0.0006& -0.0000& 0.0083\\ 0.0057& 0.1044& 0.0000& 0.0001& 0.0005& 0.0001& -0.0513\\ -0.0000& 0.0000& 0.0010& -0.0000& 0.0000& 0.0000& -0.0000\\ 0.0004& 0.0001& -0.0000& 0.0017& 0.0003& 0.0000& -0.0007\\ -0.0006& 0.0005& 0.0000& 0.0003& 0.0009& 0.0000& -0.0004\\ -0.0000& 0.0001& 0.0000& 0.0000& 0.0000& 0.0000& -0.0001\\ 0.0083& -0.0513& -0.0000& -0.0007& -0.0004& -0.0001& 0.0656\end{array}\right)$$

-

Determination of the two extreme points:

$$M=\left[3.53553.53550.000038\right]\mathrm{and}m=\left[4.99590.000021-0.000033\right].$$

-

Calculation of the Jacobian and construction of the variance–covariance matrix:

$$J=\left[0.2927-0.7069-0.0000-5.149912.43710.0039-1.0000-0.00000.00010.70730.7069-0.0001\right]$$

$$\Omega =\left[\begin{array}{ccc}\left[{\widehat{V}}_{\beta }\right(\left[1,2,3,4,5,6\right],\left[1,2,3,4,5,6\right]\left)\right]& 0& 0\\ 0& \left[M\right]& 0\\ 0& 0& \left[m\right]\end{array}\right]$$

-

Estimation of cylindricity deviation and its associated uncertainty:

$$fe=0.0070\mathrm{mm}\mathrm{and}{U}_{fe}=0.0045\mathrm{mm}.$$

5. Validation of the Proposed Model

In this section, we will initially validate our algorithm, proposed by the Mechanical and Thermal Processes and Controls (PCMT) laboratory for estimating the parameters of the fitted geometrical features, by comparing our results with those provided by National Institute of Standards and Technology (NIST). Subsequently, we will conduct Monte Carlo simulations to validate the measurement uncertainty estimation method, based on the results obtained in Section 4. We will then conclude with an inter-laboratory comparison with the Measurement Control Center (MCC) laboratory. This combined approach, as shown in

Table 2. The validation methods employed.

Validation Method	Fitting Parameters	Form Deviation and Associated Uncertainty	Conformity Assessment
Inter-laboratory comparison	NIST	MCC	MCC
Numerical simulation	-	Monte Carlo	-

, ensures a comprehensive and rigorous validation of uncertainty estimates, thereby ensuring the consistency of our model.

5.1. Validation of the Surface Fitting Parameters Algorithm

NIST (National Institute of Standards and Technology) provides point datasets sampled from various surfaces along with their corresponding parameters. We used their results as references to validate our method, highlighting common optimization problems such as convergence, local optima, and computation time. We studied different cases for each geometry (Figure 14). The results in

Table 3. Results of the surface fitting parameters comparison (all dimensions in mm).

Element	Parameters	NIST	PCMT	Deviations
Line	$\left(x_a,y_a,z_a\right)$	(−387.257010, −301.481241, 712.848232)	(−387.257010, 3.01481241, 712.848232)	(0, 0, 0)
	$\left(d_x,d_y,d_z\right)$	(−0.356058, 0.751045, 0.556015)	(−0.356058, 0.751045, 0.556015)	(0, 0, 0)
Circle	$\left(x_c,y_c,z_c\right)$	(−71.807490, 649.102784, −492.439240)	(−71.807458, 649.102763, −492.439283)	(3.2 × 10−5, 2.1× 10−5, 4.3 × 10−5)
	$\left(n_x,n_y,n_z\right)$	(0.012021, −0.261410, 0.965152)	(0.012021, −0.261410, 0.965152)	(0, 0, 0)
	R	6.441995	6.441995	3 × 10−6
Plane	$\left(x_a,y_a,z_a\right)$	(23.199058, 20.545721, 4.654758)	(23.199058, 20.545721, 4.654758)	(0, 0, 0)
	$\left(n_x,n_y,n_z\right)$	(−0.737659, −0.015306, 0.674999)	(0.737659, 0.015306, −0.674999)	(0, 0, 0)
Sphere	$\left(x_c,y_c,z_c\right)$	(−24.376673, 15.038779, −5.784704)	(−24.376673, 15.038779, −5.784704)	(0, 0, 0)
	R	47.462416	47.462416	0
Cylinder	$\left(x_a,y_a,z_a\right)$	(−324.756973, 752.733750, −563.110457)	(−324.756974, 752.733749, −563.110466)	(1 × 10−6, 1 × 10−6, 9 × 10−6)
	$\left(d_x,d_y,d_z\right)$	(−0.360737, −0.893967, 0.265877)	(0.360740, 0.8939637, −0.265885)	(3 × 10−6, 3 × 10−6, 8 × 10−6)
	R	1.323236	1.323231	5 × 10−6

below represent the cases with the largest deviations found for each geometry (sphere, plane, line, cylinder, circle) for a resolution of ${10}^{-6}$ mm.

We observe that the direction vector of the fitted cylinder is not equal to its reference vector (same norm, opposite direction). However, this will not impact the form deviation calculation and its associated uncertainty estimation.

Comparisons with the various reference geometrical elements yielded positive results. According to the classification proposed by NIST, our model falls under “Category 1”, characterized by its precision (deviations of less than 0.1 µm for linear distances, and less than 0.1 arc seconds for angular measurements), thus ensuring the reliability and accuracy of the results obtained. Compliance with NIST standards reinforces the credibility of our model, demonstrating that our approach aligns with best practices established by recognized experts in the field.

5.2. Validation of the Form-Deviation-Associated Uncertainty Estimation Method (Monte Carlo Simulation)

A Monte Carlo simulation is often used to validate the results of the uncertainty propagation method using the GUM method by determining if the confidence intervals obtained by the two methods are in agreement with a numerical tolerance. The validation procedure involved determining the confidence interval limits, $y_{{low}_{GUM}}$ and $y_{{high}_{GUM}}$, using the GUM method ($y_{mesured}\pm U_{GUM}$), where $U_{GUM}$ represents the expanded uncertainty obtained by the GUM method, and $y_{mesured}$ is the measured value of the form deviation. A Monte Carlo simulation is then conducted to obtain the confidence interval limits $y_{{low}_{MCS}}$ and $y_{{high}_{MCS}}$, corresponding to the 95.45% confidence interval limits (${\overline{y}}_{MCS}\pm {2\mathsf{\sigma }}_{MCS}$), where ${\overline{y}}_{MCS}$ is the mean and ${\mathsf{\sigma }}_{MCS}$ is the standard deviation from the obtained probability density. We then calculate $d_{low}$ and $d_{high}$, the absolute differences between the respective limits of the GUM and Monte Carlo confidence intervals:

$$\left\{\begin{array}{c}{d}_{low}=\left|y_{{low}_{GUM}}-y_{{low}_{MCS}}\right|\\ d_{high}=\left|y_{{high}_{GUM}}-y_{{high}_{MCS}}\right|\end{array}\right.$$

(17)

The validation was based on the comparison of these confidence interval limits. To ensure the adequacy of the significant digits provided by the GUM uncertainty, we calculated the numerical tolerance associated with the measurement process, $\zeta =\left({\displaystyle \frac{1}{2}}\right).{10}^{-r}$, where $r$ is the number of significant digits after the decimal point. The validation criterion was that both $d_{low}$ and $d_{high}$ should be strictly less than $\zeta$:

$$max\left(d_{low},d_{high}\right)<\zeta$$

(18)

If this condition is met, the comparison is favorable, confirming the validity of the GUM framework in this case.

We generated a distribution for a sample size of ${10}^{-6}$. The Box–Muller [38] algorithm was employed for the generation of the random values. For instance, in the case of the studied flatness deviation, we obtained the following distribution in Figure 15.

From this generated distribution, we extracted the confidence interval limits $y_{low}=0.002165\mathrm{m}\mathrm{m}$ and $y_{high}=0.011440\mathrm{m}\mathrm{m}$, as well as the mean $y_{mean}=0.006839\mathrm{m}\mathrm{m}$, which allowed us to deduce the standard deviation $\mathsf{\sigma }=0.0023188\mathrm{m}\mathrm{m}$. Similarly, for each of the other studied specifications, distributions were generated, enabling us to compare the results obtained by our GUM-based model with the Monte Carlo numerical method, as presented in

Table 4. Monte Carlo simulation results.

Geometrical Specification	Method	Form Deviation (in mm)	Associated Uncertainty (in mm)	Confidence Interval Limits (in mm)		Deviations (in mm)
				Upper Bound	Lower Bound	${\mathit{d}}_{\mathit{h}\mathit{i}\mathit{g}\mathit{h}}$	${\mathit{d}}_{\mathit{l}\mathit{o}\mathit{w}}$
Straightness	Gum	$0.0069$	$0.0035$	0.010456	0.00334	3.429 × 10−4	3.562 × 10−4
	Monte Carlo	$0.0069$	0.0039	0.010798	0.002983
Flatness	Gum	$0.0068$	$0.0044$	0.011237	0.002363	2.035 × 10−4	1.978 × 10−4
	Monte Carlo	$0.0068$	0.0046	0.011440	0.002165
Circularity	Gum	$0.0099$	$0.0044$	0.014375	0.005424	2.245 × 10−4	2.555 × 10−4
	Monte Carlo	$0.0099$	0.0047	0.014599	0.005168
Sphericity	Gum	$0.0139$	$0.0050$	0.018911	0.008889	1.826 × 10−4	8.256 × 10−5
	Monte Carlo	$0.0139$	0.0051	0.019093	0.008806
Cylindricity	Gum	$0.0070$	$0.0045$	0.011581	0.002419	3.964 × 10−4	3.813 × 10−4
	Monte Carlo	$0.0070$	0.0049	0.011977	0.002037

.

We note that in our case, the uncertainty estimated by the Monte Carlo simulation is greater than or equal to that of our proposed model, due to negative covariance components in our model, unlike the Monte Carlo numerical model based solely on variances (positive).

The machine’s numerical tolerance ζ = 0.5 × 10⁻³ mm is greater than all the calculated $d_{high}$ and $d_{low}$ values. The criterion $\mathsf{\zeta }>\max (d_{high},d_{low})$ is therefore satisfied, thus validating our uncertainty estimation model.

5.3. Inter-Laboratory Comparison

The inter-laboratory comparison was conducted with Measurement Control Center. The CMM used is a Zeiss Duramax (Oberkochen, Germany), with a measuring volume of 500 × 500 × 500 mm³. It complies with the ISO 10360-2:2009 standard [34], offering an MPE for length measurement (E0/E40) of 2.4 + L/300 µm within the temperature range of 18 °C to 22 °C coupled with a Vast-xxt-tl3 probe head mounted with a tungsten carbide stylus of effective working length EWL = 30 mm and a ruby tip of diameter D = 2 mm, all controlled by Calypso software. The temperature was regulated at 20° ± 2 °C and the probing speed was set at 10 mm/s. Although various evaluation techniques exist, the calculation of normalized error [32] is the most commonly employed method:

$${\mathrm{E}}_n={\displaystyle \frac{\left|y_{PCMT}-y_{MCC}\right|}{\sqrt{{U_{PCMT}}^2+{U_{MCC}}^2}}}.$$

(19)

where $y_{PCMT}$ and $y_{MCC}$ are the form deviations measured by the two laboratories, and $U_{PCMT}$ and $U_{MCC}$ are, respectively, the uncertainties associated with these measurements. For the results to be considered valid, the normalized error must be less than 1. This ensures that the differences between the measurements from the two laboratories are within acceptable limits, accounting for the associated uncertainties.

The same industrial parts (Section 4) were measured under the same conditions and measurement range, we obtained the results summarized in

Table 5. Inter-laboratory comparison results.

Geometrical Specification	Tolerance (in mm)	Laboratory	Form Deviation (in mm)	Associated Uncertainty (in mm)	Normalized Error	Conformity Assessment
Straightness	$0.01$	PCMT	$0.0069$	$0.0035$	0.247352	NC
		MCC	$0.0081$	0.0033		NC
Flatness	$0.01$	PCMT	$0.0068$	$0.0044$	0.271245	NC
		MCC	0.0053	0.0033		C
Circularity	$0.015$	PCMT	$0.0099$	$0.0044$	0.143876	C
		MCC	0.0107	0.0033		C
Sphericity	$0.015$	PCMT	$0.0139$	$0.0050$	0.349955	NC
		MCC	0.0118	0.0033		C
Cylindricity	$0.015$	PCMT	$0.0070$	$0.0045$	0.141693	C
		MCC	0.0078	0.0033		C

NC: Not conform. C: Conform.

.

It is noteworthy that the values measured by MCC fall within the uncertainty interval we estimated, thus supporting the consistency of our model.

The normalized deviations calculated between the results of PCMT and MCC laboratories are strictly less than 1. The criterion $\left|{\mathrm{E}}_{\mathit{n}}\right|\le 1$ is therefore satisfied, validating our uncertainty estimation model, as well as the consistency of the results.

Despite the conclusive results of the normalized deviations, we notice contradictions in the conformity assessment. This is mainly due to an underestimation of measurement uncertainty by the MCC laboratory, which refers to the machine calibration to propose an uncertainty considering only the measurement ranges, as well as a difference in the conformity declaration criterion $\left|d\right|+U\le Ts$, where $d$ and $U$ are, respectively, the measured deviation and its associated uncertainty and the $Ts$ is the specified tolerance.

6. Conclusions

In conclusion, this study offers a framework for controlling geometrical form tolerance specifications by focusing on the accurate estimation of measurement uncertainty in coordinate measuring machines, thus aligning with ISO 17025 standards [1]. Based on the GUM law of propagation, it integrates surface fitting and CMM probing uncertainty as key components in estimating the measurement uncertainty of form deviations. Thereafter, the conformity assessment following ISO 98-4 is conducted considering the estimated uncertainty.

Coordinate measuring machines generally exhibit strong performance when measuring large to medium-sized dimensional features. However, when it comes to relatively small geometrical features, we observe a high repeatability relative to the size of the measured deviation, which aligns with the results obtained. Consequently, we focused on parts with minor form deviations, ranging from a $0.0068$ mm flatness to a $0.0139$ mm sphericity deviation; the estimated uncertainty reached nearly $65\%$ of the measured deviation value $\left(max{\displaystyle \frac{U_{fe}}{fe}}\right)$, mainly due to the machine’s capability. To reduce this uncertainty, several actions can be taken:

-

Increasing the number of probed points $n$, leading to a decrease in the residual variance ${\sigma }^2=\left[{G\left(\widehat{\beta }\right)}^T{\omega }_{e_i}G\left(\widehat{\beta }\right)\right]/\left(n-p\right)$, which has a direct impact on the covariance matrix ${\widehat{V}}_{\beta }={\sigma }^2{\left[{\dot{G}\left(\widehat{\beta }\right)}^T{\omega }_{e_i}\dot{G}\left(\widehat{\beta }\right)\right]}^{-1}$, ultimately resulting in lower measurement uncertainty.

-

Ensuring better distribution of probed points on the controlled surface, which enhances the accuracy of the surface representation, reducing the likelihood of biases and gaps in the data that could lead to poor initial estimates for the fitting parameters.

-

Applying a low weighting factor (k < 1) to outliers. Significant errors can disproportionately affect the results when using the least squares method, as it tends to be particularly sensitive to outliers because it squares the errors, magnifying their impact on the overall fitting process.

-

Calibrating the machine, reducing probing speed, maintaining the temperature regulated at 20° ± 2 °C, choosing a stylus with short effective working length, etc., thus allowing for the minimization of the uncertainty associated with probing $\left[P_i\right]$ and consequently lowering the measurement uncertainty.

The validation of this model through Monte Carlo numerical simulations confirmed its accuracy and reliability, as the maximum error between the confidence intervals compared to the GUM analytic method did not exceed $3.964\times {10}^{-4}$ mm, which remained significantly lower than the machine’s numerical tolerance ζ = 0.5 × 10⁻³ mm. The interlaboratory comparison conducted in accordance with ISO 13528 [32] yielded satisfactory results, as the largest normalized error ${\mathrm{E}}_{\mathit{n}}=0.349955\le 1$ was within the standards, further substantiating the model’s effectiveness. These findings underscore the model’s potential to enhance measurement practices in industrial settings, particularly in manufacturing companies and metrology laboratories, improving measurement accuracy and conformity assessments in various applications.