Flatness, Cylindricity, and Sphericity Assessment Based on theSeven Classes of Symmetry of the Surfaces

Dimensional inspection of a manufactured surface by means of a coordinate measuring machine (CMM) produces a set of Cartesian coordinates. The coordinates are processed to yield the geometric tolerance of the surface. This paper presents a new approach to the evaluation of flatness, cylindricity and sphericity tolerance based on surface invariance with regard to the rigid motions. The proposed algorithm transforms, through homogeneous transformation matrices, the coordinates measured to best fit the reference element of the surface class from which the actual measurements were sampled. The transformation matrix is simplified taking into account the invariance of the sum of the squared normal distances of the measured points from the nominal surface as regards some rigid motions. This invariance is a consequence of the invariance as regards some displacements of the nominal surface from which the data points were sampled. In this way, the number of parameters to be optimised is reduced in comparison with the six parameters characterizing the general homogeneous transform matrix. The methodology was computer implemented and numerical simulations were performed for planes, cylinders, and spheres in order to validate the effectiveness of the approach. The results indicate that the proposed algorithm provides accurate and quick assessments.