Operations on polytopes: application to tolerance analysis

TL;DR

This paper introduces numerical methods using polytope operations, specifically Minkowski sums and intersections, to analyze tolerances and verify functional requirements in mechanical systems.

Contribution

It develops a geometric approach applying polytope operations to solve tolerance analysis problems, linking geometric constraints to polytope manipulations.

Findings

Polytope operations effectively model tolerance constraints.

The method verifies functional requirements through polytope inclusion.

Numerical examples demonstrate practical application.

Abstract

This article presents numerical methods in order to solve problems of tolerance analysis. A geometric specification, a contact specification and a functional requirement can be respectively characterized by a finite set of geometric constraints, a finite set of contact constraints and a finite set of functional constraints. Mathematically each constraint formalises a n-face (hyperplan of dimension n) of a n-polytope (1 {\leq} n {\leq} 6). Thus the relative position between two any surfaces of a mechanism can be calculated with two operations on polytopes : the Minkowski sum and the Intersection. The result is a new polytope: the calculated polytope. The inclusion of the calculated polytope inside the functional polytope indicates if the functional requirement is satisfied or not satisfied. Examples illustrate these numerical methods.