A geometrical method to assess the quality of a part is to sample points on the surface of the part and to bring an ideal computer representation of the part as close as possible to the sampled locations. The points are sampled with a coordinate measuring machine (CMM) that gives the coordinates, with respect to the machine's own three-dimensional system of axis, of the sampled locations. Computer-assisted design (CAD) gives the ideal representation of the part. The matching involves estimating a rotation and a translation vector characterizing the rigid body transformation needed to go from the computer model to the sampled locations. The sum of the squared residual distances between the sampled locations and the fitted computer part is a measure of the quality of the part. This article investigates the statistical properties of this method when the surface of the part is made of planar regions. Estimating the translation vector and the rotation matrix is shown to be “locally” equivalent to fitt...

Some Linear Models for Estimating the Motion of Rigid Bodies with Applications to Geometric Quality Assurance

To guarantee the reliability of the measurements made with CMM, the machines must be periodically checked and calibrated, to evaluate the performance and obtain optimized measurement results. However, the main obstacles encountered to ensure these accuracies are the development and implementation of periodic calibration practices and interim checking methods that guarantee the traceability of measurements. Therefore, this systematic review aims to find scientific evidence that contributes to answering the research question: “What type of device is more suitable for performing interim checks on CMM?”. For this, the study will use the methodology of the Systematic Review. In this respect, it was found eleven works (24%) of the final textual corpus, which show the development of checking CMMs, related standards, and measuring instruments. Finally, of the articles with adherence to the research question, only three are highlighted for developing new interim check devices for CMM.

Devices for Interim Check of Coordinate Measuring Machines: A Systematic Review

Coordinate measuring machines (CMMs) are used to check the geometric integrity of component parts. The geometric constraints to which a part must conform, as defined e.g., by The American National Standards Institute, assume the use of some type of gauging system when inspecting the part. Statistical issues arise in interpretting CMM data in the inspection of part tolerances. We consider a set of n planar regions on the surface of a part. The unit vector normal to each plane is estimated by orthogonal least squares. The small-sample density of this estimator (on the unit sphere S2) is determined asymptotically as the variance of the CMM error approaches 0. To a first-degree approximation, this density is Fisher-von Mises. Diagnostics are reviewed to test the geometric constraint that the n planar regions are oriented correctly with respect to one another, and to test the flatness of planar regions.



Des Machines de Mesure de Coordonnees (MMC) sont utilisees pour verifier la conformite geometrique de pieces composantes. Les contraintes geometriques qu'une piece doit satisfaire, telles que definies par The Amarican National Standards Institute par exemple, supposent l'utilisation de certains types de systes de mesure lors de l'inspection des pies. Des questions statistiques se posent alors pour l'interpretation des donees de MMC dans l'inspection de la tolerance des pieces. Dans cet article nous considerons un ensemble de n regions planaires sur la surface d'une piece. Le vecteur unitaire normal a chaque plan est estime par les Moindres Carres Orthogonaux. La densite de petits echantillons de cet estimateur (sur la sphere unite S2) est determinee asymptotiquement lorsque la variance de l'erreur de la MMC tend vers zero. A un premier degre d'approximation, cette densite est de Fisher-von Mises. Les diagnostiques sont revus pour tester la contrainte geometrique que les n regions planaires sont correctement orientes les unes par rapports aux autres, et pour tester la rectitude lineaire de regions planaires.

G. R. Chapman

Papers

Perturbation analysis for an orthogonal least‐squares estimator