## 1 Introduction
### Measurement Uncertainties Associated with Dimensional and Geometric Measurement Using CMM  
During the last three decades, the coordinate measuring machine (CMM) saw progress in terms of accuracy and repeatability, which, in turn, resulted in productivity improvements. Currently, CMM plays a major role in GD&T standards, such as [1; 2; 3; 4; 5; 6; 7; 8], which call for crucial measuring equipment needed for manufacturing quality control [12]. Notwithstanding such improvements, however, uncertainty can be induced not only by the equipment used, but also by the algorithmic choices and the measurement methodology adopted [13; 14; 15; 16]. Measurement uncertainty evaluation (quantification) is a crucial step in characterizing and certifying the consistency of the inspection results [17; 18]. Measurement uncertainty evaluation must be carried out to ensure advances in measurement science. CMM measurement uncertainty evaluation has become a key focus area for research by many institutions around the world. The Physikalisch-Technische Bundesanstalt (PTB) in Germany, for instance, suggested an expert system scheme for CMM uncertainty evaluation and investigated the impact of the measurement strategy on the overall CMM uncertainty [19]. The National Physical Laboratory (NPL) in the UK standardized the measurement strategies for CMM in order to ensure that the measurement results are reliable [20]. A few authors have employed the design of experiment techniques to estimate the CMM measurement uncertainty. The factorial design of experiments was applied by [PERSON] et al. [21] to study the measurement uncertainty of the position of a hole measured by CMM. They analyzed the effect of variables and their interactions on the uncertainty, while complying with the fundamental rules of the Guide to the Expression of Uncertainty (GUM) [9]. [PERSON] et al. [22] designed and implemented a random factorial design of experiments in order to analyze and quantify the influence of different factors (stylus diameter, step width and speed) and their interactions on the CMM measurements' uncertainty of the variable's parallelism, angularity, roundness, diameter, and distance. Other authors, such as [PERSON] et al. [23] and [PERSON] et al. [24], proposed methods to determine uncertainties using the Monte Carlo method for feature measurements on CMM. [PERSON] et al. [25] proposed the Simplified Virtual Coordinate Measuring Machine (SVCMM) method, which makes full use of the CMM acceptance or reinspection report and the Monte Carlo simulation method. For dimensional metrology with CMM measurements, a task-specific uncertainty estimation was suggested by [PERSON] [26], and can be extended to other measurement types as well as linear dimensions, forms (flatness, cylindricity, etc.) and roughnesses. [PERSON] and [PERSON] [27] performed an experimental evaluation of the software estimation of the task-specific measurement uncertainty for CMMs. [PERSON] et al. [28] addressed this topic and proposed an evaluation of CMM uncertainty, not by studying each axis of the machine, but by proceeding based directly on key specifications expressed in the GD&T standard. [PERSON] et al. [29] suggested a benchmark including several geometrical and dimensional features for the algorithm efficiency comparison of different Computer-Aided Inspection (CAI) software applications. A comprehensive review of different methods, techniques, and various artifacts for monitoring CMM performance can be found in the research work conducted by [30; 31; 32; 33]. In coordinate metrology, [PERSON] et al. [34] mentioned the main contributors to uncertainty, which they subdivided into five groups: measuring devices, environment, workpieces, software, operators, and measurement strategy. A great deal of work has been carried out by the metrology community in terms of investigating the measuring devices, environment, and workpiece components.