A New Variational Association Process for the Verification of Geometrical Specifications

Abstract

When this new association process of a datum is performed to verify a geometrical specification, measured points are considered as perturbations which generate modifications of the nominal geometry by variation of its location, orientation, and intrinsic dimensional characteristics, without requiring rotation and translation variables as the traditional methods usually do (Bourdet, et al., 1996, Advanced Mathematical Tools in Metrology II, World Scientific) with torsors or matrices. This new association process (Choley, 2005, Ph.D. thesis, Ecole Central, Paris; Choley, et al., 2006, Advanced Mathematical and Computational Tools in Metrology, VII, World Scientific) is based on both a reduced modeling of the geometry, taken out of the computer aided design system database, and a variational distance function. The whole measured points set influence is taken into account as an optimization criterion is applied (Bourdet and Clement, 1988, Ann. CIRP 37(1), p. 503; Srinivasan, DIMACS Workshop on Computer Aided Design and Manufacturing, Rutgers University, NJ, October 7–9). Thus, the least squares optimization is achieved using the pseudo-inverse matrix, whereas the minimax optimization is treated with an algorithm developed by the Physikalisch-Technische Bundesanstalt and adapted for this purpose. In this paper, it is explained how this association process may be applied to planes and cylinders, used as single datum, datum systems, or common datum, with the least squares and minimax criteria.