Operations on polytopes: application to tolerance analysis

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 ≤ n ≤ 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. 1. INTRODUCTION The variational classes by Requicha were introduced at the beginning of the 80's and propose a model of tolerances of form, orientation, position and dimension [Requicha, 1983]. A generalization of specifications by volume envelopes is based on the works of Requicha [Srinivasan and al., 1989]. Fleming presents a model for geometric tolerances and constraints from contacts [Fleming, 1988]. Among the dimension-chains models based on Small Displacements Torsor concept [Clément and al., 1988], we note the Clearance Deviation Space by Giordano [Giordano and al., 1993]. The Clearance Deviation Space purposes an assembly method according to maximum material condition limit.