Tolerance Analysis by Polytopes

TL;DR

This paper presents a method for tolerance analysis of mechanical systems using polytopes, involving Minkowski sums and intersections of constrained sets to ensure system compliance.

Contribution

It introduces a novel approach to bounded polytope operations by adding cap half-spaces, enabling effective tolerance analysis of unbounded operand sets.

Findings

Cap half-spaces successfully bound unbounded polyhedra

Polytope operations validate geometric tolerances

Method improves system compliance verification

Abstract

To determine the relative position of any two surfaces in a system, one approach is to useoperations (Minkowski sum and intersection) on sets of constraints. These constraints aremade compliant with half-spaces of R^n where each set of half-spaces defines an operandpolyhedron. These operands are generally unbounded due to the inclusion of degrees ofinvariance for surfaces and degrees of freedom for joints defining theoretically unlimiteddisplacements. To solve operations on operands, Minkowski sums in particular, "cap" halfspacesare added to each polyhedron to make it compliant with a polytope which is bydefinition a bounded polyhedron. The difficulty of this method lies in controlling the influenceof these additional half-spaces on the topology of polytopes calculated by sum or intersection.This is necessary to validate the geometric tolerances that ensure the compliance of amechanical system in terms of functional requirements.