The harmonic fitting method for the assessment of the substitute geometry estimate error. Part I: 2D and 3D theory

The dimensional inspection of a part consists of the measurement of several points on the manufactured surface and the fitting of a substitute geometry. The estimated parameters of the substitute geometry, which are related to the position and dimensions of the measured surface, are then compared with the specifications, leading to the acceptance or rejection of the manufactured part. Consequently, the reliability of the dimensional inspection depends on the precision of the estimated substitute geometry. The error that affects the estimate (estimate error) is a function of several factors (number and distribution of sampled points, measurement error, machining deviations, etc.), but the analytic expression of this relationship is still unknown.The first part of this work presents the Harmonic Fitting Method (HFM) for the assessment of the estimate error. It will be demonstrated that with the HFM the closed analytic relationship between the estimate error and the most important factors can be determined. This relationship is extremely simple when using the matrix notation, since the estimate error can be expressed as the product of different matrices, each one accounting for a single factor. In the first part of the work the general theory of the HFM will be presented and discussed for 2D features (line and circle) and 3D features (plane, cylinder and cone). In the second part the statistic approach and the machining process analysis will be presented, which are the basis for the practical use of the HFM for the inspection plan optimisation. The HFM will be applied to real data to optimise the inspection of circular geometric elements, which are particularly relevant for the mechanical industry.