Statistical geometric computation on tolerances for dimensioning

• A generalized RSS method is proposed for modeling geometric representations of tolerances in the statistical way.
• A set of basic operations over the new tolerance model are proposed to enable tolerance compositing and cascading.
• A set of examples demonstrate applications of the new model in tolerance estimation.
• A tolerance allocation framework based on optimization is also proposed by utilizing analytical forms of the new model.

Dimensions are used to specify the distances between different features in geometric models. These dimensions will often be expressed as a range of allowable dimensions. When considering a group of toleranced dimensions, these ranges can be analyzed as either a worst-case bound on allowable ranges, or as a statistical measure of expected distribution. This paper presents a new geometric model for representing statistically-based tolerance regions. Methods for tolerance estimation and allocation on a geometric model are provided by generalizing root sum square (RSS) methods for compositing and cascading over tolerance zones. This gives us a geometric interpretation of a statistical analysis. Tolerance regions are determined by probabilities of variations of dimensions falling into the region. A dependency graph over dimensions can be represented by a topological graph on which the tolerance cascading and tolerance allocation can be processed. To illustrate applications of this geometric method, we provide examples of tolerance estimation and tolerance allocation on our model. The estimation examples utilize the compositing and cascading operations provided in the analysis method. The allocation examples present an automatic tolerance allocation procedure on the tolerance model. As opposed to existing methods, our allocation method allows us to specify not only a numerical objective of the optimization, but also a statistically-based objective for the geometric shape of the tolerance.