Simulated variogram-based error inspection of manufactured parts

Industrial parts are routinely affected by dimensional and geometric errors originated in the course of manufacturing processes. These errors, whose pattern is typically related to a specific machining or forming process, are controlled in terms of dimensional and geometrical tolerances (such as e.g. straightness, roundness, flatness, profile) that require verification. In the last two decades, the Kriging model has been put forward for predicting errors of surfaces, whose tolerances are verified using coordinate measuring machines (CMMs), commonly used for 3D measurement on account of both accuracy and flexibility. Kriging is stochastic linear interpolation technique based on an evaluation of how the variance between responses at different points depends on the distance between the two locations. This can be expressed as a variogram which shows how the average difference between values at points changes; it is a function of the distance and of the corresponding direction of any pair of points depicting their correlation extent. The use of the variogram for identifying the correlation structure is recommended by the geostatisticians even if the stochastic process is not stationary. In this paper we resort to empirical variograms to detect possible manufacturing signatures, i.e. systematic pattern that characterises all the features manufactured with a particular production process, and systematic errors of the CMM measurement process. We simulate planar surfaces affected by different and typical manufacturing signatures and possible errors of a measurement process. The behaviour of the variogram gives evidence of spatial correlations, enlightening possible non isotropy. More specifically, we recommend the variograms in different directions of the axis because they draw attention to trend components with a specified direction.