Issue 
Int. J. Metrol. Qual. Eng.
Volume 14, 2023



Article Number  12  
Number of page(s)  13  
DOI  https://doi.org/10.1051/ijmqe/2023010  
Published online  06 September 2023 
Research article
Perpendicularity assessment and uncertainty estimation using coordinate measuring machine
Mohammed V University of Rabat, ENSAM Rabat, PCMT Laboratory, Avenue de l'Armée Royale, 10100 Rabat Maroc, Morocco
^{*} Corresponding author: nabil_habibi@um5.ac.ma
Received:
7
April
2023
Accepted:
10
July
2023
The validation of the conformity of parts according to the ISO 984 standard, cannot be achieved without an accurate estimation of the measurement uncertainty, which can become difficult when it comes to a complex measurement strategy to control a geometrical specification of a mechanical part using a Coordinate Measuring Machine (CMM). The purpose of the study in this paper is to analyze the measurement strategy following the Geometric Product Specification (GPS) Standard, to estimate the associated uncertainty of the different parameters of each step, to be able to achieve the uncertainty of the measurement of a given specification (perpendicularity error in our study) using the Guide to the expression of uncertainty in measurement (GUM). This uncertainty will be thereafter validated by a Monte Carlo simulation, and an interlaboratory comparison will be conducted to compare the obtained results according to the ISO 13528 standard. Our contribution is based on a more accurate estimation of the measurement strategy's parameters uncertainties. This approach can also be used by accredited calibration laboratories (ISO 17025) or in the general case in the control of perpendicularity specification of mechanical parts using a coordinate measuring machine. A case study has been conducted, controlling a perpendicularity specification with a tolerance limit of 15 µm, after the calibration of the CMM to obtain the variancecovariance matrices. The mechanical part perpendicularity error (12.55 µm) was below the limit, however, was judged “not conform” when considering the estimated uncertainty (4.06 µm) and the interlaboratory comparison was satisfactory despite the difference of the acceptance criterion.
Key words: Measurement strategy / coordinate measuring machine (CMM) / ISO/IEC Guide 984 / perpendicularity error uncertainty / Geometric Product Specification (GPS / ISO 1101) / guide to the expression of uncertainty in measurement (GUM) / Interlaboratory Comparison (ISO 13528) / Monte Carlo simulation (MCS) / ISO 17025
© N. Habibi et al., Published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Coordinate measuring machines are very popular in the industrial field; it allows controlling dimensional and geometrical specifications of complex mechanical parts with great accuracy and precision of less than 1µm. Both hardware and software work simultaneously to collect and process data to generate measurement reports, hence the importance of estimating the uncertainty associated with the measurement. Equipped with a probing system, and following a specific measurement strategy, it collects the coordinates of the tolerance features, then proceeds to the surface fitting according to a given criterion, leastsquares method in our case, then proceeds to the verification of a dimensional or geometrical specification. This succession of steps is subject to a propagation of uncertainties, and if not estimated correctly, can lead to aberrant decisions. Evaluating the CMM's measurementassociated uncertainty is a challenging task, especially when examining geometrical error specifications, mainly due to the large number of factors that influence the measurement (Fig. 1).
Several studies have been made to estimate the influence of these parameters on the coordinate measuring machine's measurement uncertainty, such as geometric errors that goes up to 5.63 arcsec following the Y axis for Zeiss Opton CMM with a maximum permissible error of 1.3 [1], measuring probe errors estimated around almost ±0.9 µm for TP2 Renishaw probe head [2,3], thermal influence errors [4,5] that should be reduced by regulating the temperature homogeneously to 20 ± 1 °C, with a variation less than 0.5 °C per hour and less than 0.5 °C/m height, measurement strategy and fitting criterion [6] that are proven to show minimal influence by probing every 1/10 of the dimension of the surface feature, position, size and shape for point cloud data [7,8]. Rosenda et al. [9] proposed a simplified model, considering these parameters, to estimate the circularity and cylindricity measurement uncertainty using a coordinate measuring machine. Other studies have been oriented toward estimating the uncertainty of geometrical specifications. Wojciech et al. [10,11] developed different models for size, distance, angle, and geometrical deviation measurement uncertainty, including perpendicularity, fully consistent with the GPS [12] norm. Our contribution is positioned in this context, we seek to estimate the uncertainty of the orientation error following a measurement strategy that respects the normative guidelines.
The GUM and the Monte Carlo method are generally used to estimate the measurement uncertainty. Balasubramanian et al. [13] estimated uncertainty in angle measurement using the GUM considering the geometrical errors, temperature, vibrations, and measuring strategy. Moona et al. [14] developed a model using the Monte Carlo method to estimate the uncertainty for length measurement errors using an articulated arm coordinate measuring machine. Using a comparison between the GUM [15] approach and a Monte Carlo simulation [16] as a validation method has proven to give consistent results, it's within this framework that Jalid [17,18] proposed a comparison of these two methods estimating flatness uncertainty which showed satisfactory results with a gap less than 10^{−4}mm, then studied the influence that sample size has on it.
In this paper, we aim to review the process of validating the conformity of the mechanical parts inspected using CMM, by introducing and considering the measurement uncertainty as stated in the ISO/IEC Guide 984 [19]. Our model combines the experimental and the analytical methods to estimate the measurement associated uncertainty. The advantage of this approach is that the perpendicularity uncertainty can be estimated directly from the set of measured points and the calibration of the CMM. It is important to mention too that the uncertainty varies according to the number and position of the measured points and the chosen fitting criterion. To estimate the measurementassociated uncertainty, a deconstruction of the process has been realized, by identifying the different steps of the measurement strategy following the ISO 1101 [12] standard (GPS), and by estimating the variancecovariance matrices at the level of each step by considering the parameters which influence the results, to be able to estimate the final uncertainty of the measurement. This uncertainty will be thereafter validated by a Monte Carlo simulation, before finally proceeding to an interlaboratory comparison to compare the obtained results. Our contribution is based on a more accurate estimation of the measurement strategy's parameters uncertainties. This approach can be used by ISO 17025 [20] laboratories in the control of perpendicularity specification of mechanical parts using a CMM.
Fig. 1 CMM uncertainty sources. 
2 Materials and methods
To validate the conformity of a mechanical part using a coordinate measuring machine according to the ISO 984 standard, an estimation of the measurementassociated uncertainty is necessary, which can be particularly problematic considering the measurement strategy, mainly due to the number of unknown parameters that can influence the measurement. To do so, we studied a perpendicularity case following this approach:
Setting a perpendicularity error equation according to the ISO 1101 standard (GPS).
Estimation of the perpendicularity error associated uncertainty.
Validation of the proposed method.
Declaration of conformity according to the ISO 984 standard.
A verification of our results through an interlaboratory comparison according to the ISO 13528 standard will be accomplished in the results and discussion section.
2.1 Perpendicularity error modeling
Based on the Geometric Product Specification (ISO 1101 standard) [12], perpendicularity is an orientation tolerance; and can be defined as the minimum distance between two theoretical parallel elements, both perpendicular to the datum, within which all measured points lie inside, whether it is a plane or axis (Fig. 2).
We have studied a planetoplane perpendicularity case, the geometrical specification was summarized as follows:
Tolerance feature: probed points that belong to the tolerance surface.
Datum: theorical fitted plane P0.
Tolerance zone: Volume between two theorical parallel planes P1 and P2, both perpendicular to the datum.
Condition: all probed points must lie inside the tolerance zone.
Fig. 2 Perpendicularity error. 
2.1.1 Measurement strategy
The measurement strategy using CMM should be carefully planned and executed to achieve the desired level of accuracy and comply with the GPS norm. In order to measure the perpendicularity error, we applied the following strategy (Fig. 3).
Probing the datum surface then fitting a theorical plane using the leastsquares method, the choice of the number of measured points and fitting criterion has been chosen to represent the best the surface [6], then extracting the datum's associated plane normal vector . Probing the tolerance element using the same method and extracting the measured points coordinates as well as the tolerance surface associated plane normal vector . It is important to mention that and are not necessarily perfectly perpendicular, hence the need to calculate the vector . Calculating the datum and tolerance planes intersection vector to be able to calculate the vector . Calculating the two most distant measured points p_{max} and p_{min} along the vector , allowing us to set the planes P1 and P2, and deducting the perpendicularity error (Fig. 4).
This succession of steps is subject to a propagation of uncertainties, and if not estimated correctly, can lead to a false conformity declaration. According to equation (1), and following this measurement strategy, the main sources of the perpendicularity measurement uncertainty are the probed points, the associated datum normal vector and the intersection vector. In the following Section, we will quantify the uncertainties associated with these parameters for each step of the measurement strategy. Forbes [21,22] conducted other studies on the estimation of the variancecovariance matrix of the features with a finite set of points dispersed evenly over the surface being sampled, allowing to estimate uncertainties using the GUM method without knowing the measurement strategy, and reducing the effect of form effort, considering only the number of data points and geometry of the area being sampled.
Fig. 3 Measurement strategy steps. 
Fig. 4 Construction of the tolerance zone. 
2.1.2 Perpendicularity error
Let p_{i} be the coordinates of the i^{th} probed point, such as p_{max} (x_{max}, y_{max}, z_{max}) ∈ P1 and p_{min} (x_{min}, y_{min}, z_{min}) ∈ P2 the two most distant measured points, the perpendicularity error can be expressed as follows:
where represents the theorical parallel plane's P1 and P2 normal vector, (n_{dx}, n_{dy}, n_{dz}) the datum plane's normal vector, and (n_{ex}, n_{ey}, n_{ez}) the vector of the intersection between the datum and tolerance surface (unit vectors). Hence the final expression of the parallelism error:
2.2 Estimation of the perpendicularity error associated uncertainty
In order to estimate the perpendicularity error associated uncertainty using the GUM uncertainty propagation model, we applied the following procedure:
Applying the GUM method to the perpendicularity equation (1).
Estimating the parameters and their associated variancecovariance matrix.
Validation of the GUM results through a Monte Carlo simulation.
2.2.1 GUM uncertainty propagation model
The GUM (Guide to the Expression of Uncertainty in Measurement [15]) variance propagation method is widely used in different fields, especially in metrology, it provides an analytic approach for quantifying and expressing the uncertainty of measurement based on a firstorder Taylor expansion of a function through a linear approximation. To estimate the perpendicularity error default uncertainty, the GUM method is applied to the perpendicularity model in equation (1):
The modeling in matrix form will allow us thereafter to implement the calculations on Matlab, where J represents the Jacobian matrix:
And M represents the uncertainty variancecovariance associated matrix:
where the terms are respectively the associated variancecovariance matrix of the variables .
2.2.2 Estimation of the parameters associated variancecovariance matrix
Coordinate measuring machines are precise and accurate. However, various factors influence the measurement uncertainty (Fig. 1), and it is very difficult to quantify the influence of each of these parameters independently of the others. Several approaches have been made, Bahassou et al. [23,24] proposed an estimation of the variances according to the ISO 10360 standard [25]. We will assume that the errors along the axes are independent and linear:
Thereby measuring 5 gauge blocks, each block for 3 repetitions, along 3 of the 7 directions (Fig. 5), then calculating the error equations along each direction E_{x}=A_{x}x+B_{x} (error following the direction X for example), Then applying the law of propagation of uncertainties to estimate the associated uncertainty u_{x}.
Surface fitting is a critical step. To estimate the variancecovariance matrix associated to the datum normal vector, we must first select a mathematical model to associate the set of probed points to an ideal plane, representing the measured surface without overfitting or underfitting. It may be done using a variety of techniques, such as polynomial fitting, radial basis functions, and splines, each method has advantages and disadvantages. Several criteria [6] of surface fitting are commonly used and comply with the norms, of which we can mention: The leastsquares method, it consists of minimizing the sum of squared residuals: where such as (A,) are the substitute plane parameters and P_{i} are the measured points.. And the Chebyshev criterion, minimizing the maximum absolute difference from the data points to the fitted surface: (Fig. 6).
The leastsquares method tends to be more sensitive to outliers in the data because it squares the errors. Large errors have a more significant impact which provides a good overall fit to the data but may not guarantee the smallest maximum error across the entire data range. While the Chebyshev approximation method is less sensitive to outliers because it focuses on the maximum absolute error providing a more accurate fit in terms of the worstcase scenario but potentially sacrificing the overall fit. The choice between these methods depends on the specific requirements of the problem, the characteristics of the data, and the desired tradeoff between overall fit and worstcase accuracy. For the rest of this study, we will refer to the leastsquares consisting of minimizing equation (7):
To solve this equation, we used the “nlinfit” function in Matlab, which requires starting parameters . To achieve a stable result and avoid local solutions, we have chosen the center of mass of the measured points and an initial normal vector based on the most distant probed point apart a, b and c.
Once the associated plane (A, ) is estimated, we proceed to the estimation of the variancecovariance matrix associated with a . The objective of the introduction of this matrix, is to highlight the influence of the measurement strategy parameters: the chosen fitting criterion (LSM) as well as the number and distribution of the probed points, based on the principle that the greater the number and of points probed and the larger their coverage, the more we converge to the normal that better represents the real surface, assuming that it follows a normal distribution of the form:
We will not consider the influence of the uncertainty associated with the probed points, being already taken into account in the matrix [P_{i}] above (Eq. (6)). The feature is measured for N repetitions, and we then estimate the normal vector for each sample using the leastsquares algorithm coded above, before calculating their variancecovariance referring to the following equations:
Following this procedure above, we developed an algorithm on Matlab, starting from a set of data points, proceeding to the fitting process using the leastsquares method then giving us the fitted datum plane parameters (A, ) and its variancecovariance matrix associated with the associated measurement strategy:
Regarding the intersection vector , representing the direction of the intersection between the datum and tolerance planes, expressed as follows:
We will assume that , and we apply the law of propagation of uncertainties on each term of the vector:
Similarly, for and , to reach the final form of the variancecovariance matrix:
Once the estimation of the variancecovariance matrix of each parameter is done, respectively , we will obtain the final form of the matrix [M] (Eq. (5)).
Fig. 5 The ISO 10360 directions. 
Fig. 6 Surface fitting. 
2.2.3 Monte Carlo simulation
The estimation of measurement uncertainty using a Monte Carlo simulation [16] is a great alternative especially when other methods present some difficulties such as an inadequate linearization of the model resulting in unrealistic confidence intervals. It's a statistical propagation of distributions that uses random sampling through a mathematical model to determine the range of possible outcomes allowing us therefore to estimate the model's uncertainty. A Monte Carlo simulation could also be used to compare and validate the results using the GUM method following this procedure.
Calculating the limits of the confidence interval and resulting from the application of the GUM method, where “dp” represents the nominal value of the perpendicularity error and U(dp) it's associated uncertainty:
Running a Monte Carlo Simulation and extracting from the generated distribution both perpendicularity error mean value and its deviation, to be able to calculate and as they represent the limits for a 95.45% confidence interval (), then comparing the GUM and Monte Carlo confidence interval limits:
Setting the numerical tolerance ζ = 0.5x10^{r}where “r” is expressing the necessary number of accurate decimal digits. Then if the condition ζ ≥ max (d_{low}, d_{high}) is verified, the comparison is favorable, meaning that GUM framework has been validated in this instance.
2.3 Declaration of the conformity
The conformity assessment is a critical step, it can decide whether the mechanical part conforms to the given specification. If the measurement uncertainty is not considered, it can lead to aberrant decisions, especially if the measurement result is close to the specification limit. CMMs can automatically generate a conformity report based on the specification tolerance interval. If we take the perpendicularity as an example, the CMM validity assessment follows this procedure: where T_{L} represents the specification/tolerance limit (Fig. 7).
If we consider the uncertainty, two forms of incorrect decisions would appear inside the uncertainty zone. False acceptance, which is validating the nonconform specification part, known as consumer's risk. And false rejection, which is rejecting a conform specification part, also known as producer's risk (respectively Type I (α) and Type II (β) errors). The decisionmaking process was significantly impacted by the development of a probabilistic approach, introducing measurement uncertainty as a conformity parameter (Fig. 8).
To establish a conformity validation procedure associated with the measured dimensional or geometrical specifications, it will be necessary to calculate the risk zone, assuming that the uncertainty follows a normal distribution:
According to the ISO/IEC Guide 984 [17], if the tolerated risk limit is not specified by the customer, the risk p_{α} = {dp > z_{i}} = 1 − Ø (z_{i}) should not exceed 2.3%. Where z_{i} is the Gaussian coefficient using the standard normal distribution expressed as follows:
Fig. 7 Conformity assessment. 
Fig. 8 Perpendicularity error uncertainty distribution. 
3 Results and discussion
This experimental study aims to bring the previously developed theoretical model into practice. The tests were carried out in the PCMT metrology laboratory where the temperature is regulated at around 20 ± 2 °C, the coordinate measuring machine used is a Mitutoyo EuroC 544 coupled to a TP2 type probing head on which is mounted a Tungsten Carbide stylus of effective working length EWL = 14 mm and D=2 mm ruby ball diameter, altogether driven by Geopak software. The maximum permissible error is E_{L,MPE} = ± (4μm + L/200) with L in mm. The geometrical specification being studied is a perpendicularity error with a tolerance limit of 15 μm:
We started by estimating the variancecovariance matrix associated to this CMM's measured points by applying the GUM method to the error equations following the ISO 10360 directions [23,24]:
Most of researchers use uncertainties based on the MPE. The main purpose of the variance matrix proposed, is to make good use of the ISO 10360 calibration results of the CMM, generating a correction matrix and a plausible variance matrix consistent with the MPE statement (Fig. 9).
To control the mechanical part, we referred to the steps described in Section 2.2, we probed the reference plane, then the specified plane, it is important to note that in order to minimize probing error, the probe must be oriented in the same orientation as the normal vector while measuring all the data points (Tab. 1).
After extracting the cloud of points, we then proceeded to the construction of the required vectors as shown in Table 2:
Then constructing the tolerance zone between and to be able to evaluate the perpendicularity error:
It is important to mention that he problem with the ISO 1101 definition of perpendicularity is that the uncertainty associated with the measurement is directly related to the measurement strategy and form error, which influence on the parallelism error considerably.
Fig. 9 Used CMM. 
Tolerance plane measured points (in mm).
Construction of the vectors (in mm).
3.1 GUM application
The perpendicularity error associated uncertainty is obtained by propagating the parameters uncertainty across the measurement strategy process through a linear approximation. By that means, we started by estimating the variance covariance matrix associated to the datum normal vector, which evaluates the influence of the number of probed points and their distribution as a result of the randomization of the measured points to generate the different possible combinations (Tab. 3).
Therefore, allowing us to set the datum normal vector associated variancecovariance matrix:
However, it is important to mention that the probing error influence the matrix being based on a repeatability model. An interesting alternative approach to estimate the normal vector associated variancecovariance matrix would be for each “N” probed points , we proceed to a Monte Carlo randomization of the measured points to generate the different possible combinations, representing the same feature plane, with varied distributions and number of points N with 3 ≤ n < N. We then estimate the normal vector for each sample using a specific fitting criterion, before calculating their variance covariance.
Similarly, we estimated associated with the tolerance plane's normal vector, to be able to assess the intermediate vector's variancecovariance matrix, representing the direction of the intersection of the two measured planes:
The Jacobian matrix (Eq. (4)) is calculated with the following simplifications:
Consequently, we can estimate the perpendicularity error associated uncertainty using the GUM method developed in Section 2.2.
The uncertainty may seem relatively big compared to the error 32%, but it is mainly due to the low perpendicularity default compared to the capability of CMMs used.
Sample of datum plane normal vectors for 10 repetitions (in mm).
3.2 Monte Carlo simulation
We referred to a comparison between the GUM results and a Monte Carlo simulation to validate the perpendicularity uncertainty estimation. The Monte Carlo method can cope with nonsmooth inputoutput models and can be used to evaluate the uncertainty associated with the perpendicularity error. Supposing that the parameters follow a normal law distribution, the simulation was carried out in two stages, the first being to randomize the cloud of probed points, with known mean values and standard deviations σ = U/k extracted from their respective variance matrices with k = 2 as coverage factor, to determine the maximum and minimum points for each sample, followed by a second randomization of the vectors, in order to obtain the perpendicularity error output estimated referring to the parameters.
Figure 10 shows the distribution function obtained when generating a 10^{5} sample and 10^{3} classes of 0.02 µm:
We extract the following results (Tab. 4).
The numerical tolerance is ζ = 0, 5.10^{−3}mm, and (d_{low}, d_{high}) represent the difference of the limits for a 95.45% confidence interval (y_{mean} ± 2σ_{MCM}) of the generated distribution and the GUM method results calculated as follows:
The validation criterion max (d_{low}, d_{high}) ≤ ζ is verified, meaning that the comparison is favorable, and that the GUM framework estimating the perpendicularity uncertainty has been validated in this instance.
Fig. 10 Distribution Law of MCM. 
Results comparison.
3.3 Conformity assessment
In order to control the perpendicularity specification (15µm tolerance limit), we calculated the consumer risk: p_{α} = {dp > zi} = 1 − Ø (z_{i}) where (Tab. 5).
The risk alpha p_{α} = 11.3% is significantly higher than the 2.3% limit specified by the standard ISO/IEC Guide 984. We can then conclude that the part is “not conform” to the perpendicularity specification. However, it is important to note that the conformity assessment could show different results measuring the same part and estimating the uncertainty referring to the same model, using a more performant and precise CMM, hence the necessity of an interlaboratory comparison.
Conformity assessment.
3.4 Interlaboratory comparison
Interlaboratory comparison (ILC) is a procedure usually used to evaluate the accuracy and the consistency of results obtained by different laboratories realizing the same measurement or test on the same sample, it can also be used in our case to validate our perpendicularity assessment model. Although there are several evaluation techniques, the calculation of the normalized error is the most often used [26,27]:
where dp_{L} and U_{L} are respectively the perpendicularity error and its associated uncertainty measured by the participant laboratory. The comparison would show satisfactory results if E_{n} ≤ 1.
The ILC was realized with the Measurement Control Center (MCC) laboratory where the temperature is regulated around 20 °±2 °C, the coordinate measuring machine used is a Zeiss Duramax coupled to a Vastxxttl3 type probing head on which is mounted a Tungsten Carbide stylus of effective working length EWL = 14 mm and D = 2 mm ruby ball diameter, altogether driven by Calypso software. The same industrial part was controlled under the same conditions and following the same measurement strategy, resulting in a perpendicularity error dp_{MCC} = 11.9 μm, and the mechanical part was judged to be compliant with the given specification. The CMMassociated measurement uncertainty is U_{MCC} = 3.3 μm estimated by manufacturer calibration. The normalized error is significantly inferior to 1:
The ILC showed very satisfactory results (E_{n} ≪ 1), we can then conclude that our CMM is accurate and that our uncertainty estimation is suitable for perpendicularity measurement. Both laboratories evaluated the that the part is “not conform” to the given specification, however, the MCC laboratory judgment was based on the application of the acceptance criterion: dp_{L} + U_{L} < T_{s}, which in this case, did not alter the decision.
4 Conclusion
The proposed article presents a different approach for the perpendicularity conformity validation of mechanical parts using the coordinate measuring machine, by estimating the measurement uncertainty and including it in the assessment as stated in the ISO/IEC 984 standard. The main purpose is to provide the perpendicularity error, its associated uncertainty, and the conformity risk, directly from the set of data points.
In order to evaluate the perpendicularity error, a measuring strategy was set according to the ISO 1101 specifications, then the error mathematical model was developed (Eq. 2). To estimate it's associated uncertainty, a deconstruction of the process has been realized and the GUM propagation of uncertainties was applied, then put together in matrix form (Eq. 3). The uncertainty variancecovariance matrices were then estimated in Section 2.3, highlighting the influence of the measurement strategy parameters: the chosen fitting criterion as well as the distribution and number of the measured points. Then a Monte Carlo simulation was used to compare and validate the uncertainty estimation and showed complying results (gap less than 10^{−4}mm) which validates our developed model. The uncertainty may seem relatively big compared to the error , but it is mainly due to the low perpendicularity default compared to the capability of CMMs used.
The interlaboratory comparison was satisfactory, the normalized error confirms the concordance between the perpendicularity error and its associated uncertainty of the measured mechanical part for both laboratories. However, despite the difference of the acceptance criterion, the conformity assessment was the same.
References
 P. Fangyu, L. Nie, Y. Bai, X. Wang, Geometric errors measurement for coordinate measuring machines, IOP Conf. Ser.: Earth Environ. Sci. 81, 28–30 (2017) [Google Scholar]
 J. Stone, B. Muralikrishnan, C. Sahay,Geometric effects when measuring small holes with micro contact probes, J. Res. Natl. Inst. Stand. Technol. 116, 573–587 (2011) [CrossRef] [Google Scholar]
 L. Laaouina, A. Nafi, A. Mouchtachi, Application of CMM separation method for identifying absolute values of probe errors and machine errors, Int. Conf. Eng. & MIS 2016, Agadir, Morocco [Google Scholar]
 S. Branko, B. Acko, S. Havrlisan, I. Matin, B. Savkovic, Investigation of the effect of temperature and other significant factors on systematic error and measurement uncertainty in CMM measurements by applying design of experiments, Measurement 158, 107692 (2020) [CrossRef] [Google Scholar]
 M. Mussatayev, M. Huang, S. Beshleyev, Thermal influences as an uncertainty contributor of the coordinate measuring machine (CMM), Int. J. Adv. Manuf. Technol. 111, 537–547 (2020) [CrossRef] [Google Scholar]
 M. Djezoul, E. Pairel, H. Favreliere, Influence of the probing definition on the flatness measurement, Int. J. Metrol. Qual. Eng. 9, 15 (2018) [CrossRef] [EDP Sciences] [Google Scholar]
 A.B. Forbes, Uncertainties associated with position, size and shape for point cloud data, J. Phys.: Conf. Ser. 1065, 142023 (2018) [CrossRef] [Google Scholar]
 A.B. Forbes, Approximate models of CMM behaviour and point cloud uncertainties, Meas. Sens. 18, 100304 (2021) [CrossRef] [Google Scholar]
 V.A. Rosenda, C. Costa, Souza, H.L. Costa, A. P. PiratelliFilho, Simplified model to estimate uncertainty in CMM, J. Braz. Soc. Mech. Sci. Eng. 37, 411–521 (2015) [CrossRef] [Google Scholar]
 P. Wojciech, Uncertainty of coordinate measurement of geometrical deviations, Procedia CIRP 75, 361–366 (2018) [CrossRef] [Google Scholar]
 J. Wladyslaw, P. Wojcich, First coordinate measurement uncertainty evaluation software fully consistent with the GPS Philosophy, Procedia CIRP 10, 317–322 (2013) [CrossRef] [Google Scholar]
 International Organization for Standardization ISO 1101: 2017–02, Geometrical product specifications (GPS) − geometrical tolerancing − tolerances of form, orientation, location and runout [Google Scholar]
 R. Rajamani, R. Vignesh, B. Mouliprasanth, Evaluation of uncertainty in angle measurement performed on a coordinate measuring machine, Proceedings of the First International Conference on Combinatorial and Optimization, ICCAP 2021, December 7–8 2021, Chennai, India [Google Scholar]
 G. Moona, V. Kumar, M. Jewariya, H. Kumar, R. Sharma, Measurement uncertainty assessment of articulated arm coordinate measuring machine for length measurement errors using Monte Carlo simulation, Int. J. Adv. Manuf. Technol. 119, 5903–5916 (2022) [CrossRef] [Google Scholar]
 JCGM 100:2008, Evaluation of measurement data − guide to the expression of uncertainty in measurement [Google Scholar]
 JCGM 101:2008, Evaluation of measurement data − supplement 1 to the guide to the expression of uncertainty in measurement − propagation of distributions using a Monte Carlo method [Google Scholar]
 A. Jalid, S. Hariri, A. El Gharad, J.P. Senelaer, Comparison of the GUM and Monte Carlo methods on the flatness uncertainty estimation in coordinate measuring machine, Int. J. Metrol. Qual. Eng. 7, 302 (2016) [CrossRef] [EDP Sciences] [Google Scholar]
 A. Jalid, S. Hariri, N.E. Laghzale, Influence of sample size on flatness estimation and uncertainty in threedimensional measurement, Int. J. Metrol. Qual. Eng. 6, 102 (2015) [CrossRef] [EDP Sciences] [Google Scholar]
 ISO/IEC GUIDE 98–4:2012, Uncertainty of measurement − Part 4: role of measurement uncertainty in conformity assessment [Google Scholar]
 ISO 17025:2017 General Requirements for Competence of Testing and Calibration Laboratories, International Organization for Standardization [Google Scholar]
 A.B. Forbes, Sensitivity analysis for Gaussian associated features, Appl. Sci. 12, 2808 (2022) [CrossRef] [Google Scholar]
 A.B. Forbes, Verification of sensitivity analysis method of measurement uncertainty evaluation, Meas. Sens. 18, 100274 (2021) [CrossRef] [Google Scholar]
 K. Bahassou, A. Salih, A. Jalid, M. Oubrek, Modeling of the correction matrix for the calibration of measuring machines, Int. J. Mech. Eng. Tech. 8, 862–870 (2017) [Google Scholar]
 K. Bahassou Salih, M. Oubrek, A. Jalid, Measurement uncertainty on the correction matrix of the coordinate measuring machine, Int. J. Adv. Res. Eng. Tech. 10, 669–676 (2019) [Google Scholar]
 International Organization for Standardization ISO 103602:2009, CMMs Used for Measuring Linear Dimensions [Google Scholar]
 S. Almira, H. Bašić, Proficiency testing and interlaboratory comparisons in laboratory for dimensional measurement, J. Trends Dev. Mach. Assoc. Technol. 16, 115–118 (2012) [Google Scholar]
 International Organization for Standardization ISO 13528:2015, Statistical Methods for Use in Proficiency Testing by Interlaboratory Comparison [Google Scholar]
Cite this article as: Nabil Habibi, Abdelilah Jalid, Abdelouahab Salih, Mohamed Zeriab Essadek, Perpendicularity assessment and uncertainty estimation using coordinate measuring machine, Int. J. Metrol. Qual. Eng. 14, 12 (2023)
All Tables
All Figures
Fig. 1 CMM uncertainty sources. 

In the text 
Fig. 2 Perpendicularity error. 

In the text 
Fig. 3 Measurement strategy steps. 

In the text 
Fig. 4 Construction of the tolerance zone. 

In the text 
Fig. 5 The ISO 10360 directions. 

In the text 
Fig. 6 Surface fitting. 

In the text 
Fig. 7 Conformity assessment. 

In the text 
Fig. 8 Perpendicularity error uncertainty distribution. 

In the text 
Fig. 9 Used CMM. 

In the text 
Fig. 10 Distribution Law of MCM. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.