Representation of geometric variations using matrix transforms for statistical tolerance analysis in assemblies

TLDR: In this article, the authors developed a tolerance representation for assemblies compatible with tolerance analysis based on a closed-form algorithm used in robotic applications, which is well suited to integration with CAD systems and feature-based design.

Abstract: The goal of this article is to develop a tolerance representation for assemblies compatible with tolerance analysis based on a closed-form algorithm used in robotic applications. A methodology is described that represents standard Y14.5M-1982 tolerances using homogeneous 4×4 matrix transforms. Transforms represent both the nominal relations between parts and the variations caused by geometric deviations allowed by the tolerances. The analysis calculates a statistical estimate of the location of theNth part in an assembly starting from the first part or a fixture. Except forform tolerances, most types of tolerance specifications are compatible with the proposed representation. This approach is well suited to integration with CAD systems and feature-based design. Since assembly apparatus errors can be calculated using the same methodology, one can predict the relative position and angle errors between two parts about to be mated. This permits useful evaluation of assembly equipment errors, comparison of different product tolerance assignments, and calculations of assembly process capability.