On the assessment of the form error using Probabilistic Approach based on Symmetry Classes

Nowadays, a Coordinate Measuring Machine (CMM) is one of the essential tools used in the product verification process. Measurement points provided by a CMM are conveyed to the CMM data analysis software. As a matter of fact, the software can contribute significantly to the measurement uncertainty, which is very important from the metrological point of view. Mainly, it is related to the association algorithm used in the software, which is intended to find an optimum fitting solution necessary to ensure that the calculations performed satisfy functional requirements. There are various association methods, which can be used in these algorithms (such as Least squares, Minimum zone, etc.). However, the current standards do not specify any of the methods that have to be established. Moreover, there are different techniques for the evaluation of uncertainty (such as experimental resamplings, Monte Carlo simulations, theoretical approaches based on gradients, etc.), which can be used with association methods for the further processing. Uncertainty evaluated by a combination of an association method and uncertainty evaluation technique is a term of implementation uncertainty, which in its turn is a contributor to measurement uncertainty according to the Geometrical Product Specification and Verification project (GPS). This work is focused on the analysis of the impact of the association method on the implementation uncertainty, by assuming that all the other factors (such as the sampling strategy, the measurement equipment parameters, etc.) are fixed and chosen according to standards, within the GPS framework. The objective of the study is Probabilistic method (PM), which is based on the classification of continuous subgroups of a rigid motion (a mathematical principle of the GPS language) and non-parametric density estimation techniques. The method has essentially been developed to decompose complex surfaces and showed promising future in the shape partitioning. However, it comprises geometric fitting procedures, which are considered in this work in more detail. The methodology of the research is based on the comparison of PM with another statistical association method, namely the Least squares method (LS) by means of the parameter estimation and uncertainty evaluation. For the uncertainty evaluation two different techniques, the Gradient-based and Bootstrap methods are used in a combination with the both association methods, PM and LS. The comparison is performed through both the analysis of the results obtained by the parameter estimation and analysis of variance. Variances of the estimated parameters and estimated form error are considered as the response variables in the analysis of variance. The case study is restricted to the roundness geometric tolerance evaluation. Despite the measurement process was simulated, the methodology can be applied for real measurement data. The obtained results during the work can be interesting both in the theoretical and in the practical points of view.