Adapting polytopes dimension for managing degrees of freedom in tolerancing analysis

TL;DR

This paper introduces a screw theory-based method to adapt the dimension of polytopes in tolerancing analysis, reducing computational complexity by projecting polytopes onto relevant subspaces, thereby improving efficiency in Minkowski sum calculations.

Contribution

It proposes a novel approach using screw theory to dynamically adapt polytope dimensions, avoiding unbounded displacements and decreasing computation time in tolerancing analysis.

Findings

Significant reduction in computation time demonstrated.

Method effectively limits the displacements to relevant subspaces.

Example validates the efficiency of the proposed approach.

Abstract

In tolerancing analysis, geometrical or contact specifications can be represented by polytopes. Due to the degrees of invariance of surfaces and that of freedom of joints, these operand polytopes are originally unbounded in most of the cases (i.e. polyhedra). Homri et al. proposed the introduction of virtual boundaries (called cap half-spaces) over the unbounded displacements of each polyhedron to turn them into 6-polytopes. This decision was motivated by the complexity that operating on polyhedra in R6 supposes. However, that strategy has to face the multiplication of the number of cap half-spaces during the computation of Minkowski sums. In general, the time for computing cap facets is greater than for computing facets representing real limits of bounded displacements. In order to deal with that, this paper proposes the use of the theory of screws to determine the set of displacements that defines the positioning of one surface in relation to another. This set of displacements defines the subspace of R6 in which the polytopes of the respective surfaces have to be projected and operated to avoid calculating facets and vertices along the directions of unbounded displacements. With this new strategy it is possible to decrease the complexity of the Minkowski sums by reducing the dimension of the operands and consequently reducing the computation time. An example illustrates the method and shows the time reduction during the computations.