Worst-case and statistical tolerance analysis based on quantified constraint satisfaction problems and Monte Carlo simulation

Abstract

This paper deals with the mathematical formulation of tolerance analysis. The mathematical formulation presented in this paper simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion (existential quantifier: “there exists”; universal quantifier: “for all”). It takes into account not only the influence of geometrical deviations but also the influence of the types of contacts on the geometrical behavior; these physical phenomena are modeled by convex hulls (compatibility hull, interface hull and functional hull) which are defined in parametric space. With this description by convex hulls, a mathematical expression of the admissible deviations of parts integrates the quantifier notion. This notion translates the concept that a functional requirement must be respected in at least one acceptable configuration of gaps (existential quantifier: “there exists”), or that a functional requirement must be respected in all acceptable configurations of gaps (universal quantifier: “for all”). To compute this mathematical formulation, two approaches based on Quantified Constraint Satisfaction Problem solvers and Monte Carlo simulation are proposed and tested.

Introduction

As technology improves and performance requirements continually tighten, the cost and the required precision of assemblies increase as well. There is a strong need for increased attention to tolerance design in order to enable high-precision assemblies to be manufactured at lower costs. Therefore, tolerance analysis is a key element in industry for improving product quality. To do so, a substantial amount of research has been devoted to the development of tolerance analysis. It can be either worst-case or statistical [1], [2], [3], [4].

Worst-case analysis (also called deterministic or high–low tolerance analysis) involves establishing the dimensions and tolerances such that any possible combination produces a functional assembly, i.e. the probability of non-assembly is identically equal to zero. It considers the worst possible combinations of individual tolerances and examines the functional characteristic. Consequently, worst-case tolerancing can lead to excessively tight part tolerances and hence high production costs [2], [4].

Statistical tolerancing is a more practical and economical way of looking at tolerances and works on setting the tolerances so as to ensure a desired yield. By permitting a small fraction of assemblies to not assemble or function as required, an increase in tolerances for individual dimensions may be obtained, and in turn, manufacturing costs may be reduced significantly [3]. Statistical tolerance analysis computes the probability that the product can be assembled and will function under a given individual tolerance.

This state of the art based on academic papers and also on four commercial systems (CATIA 3D FTA from Dassault Systèmes, CE/TOL 6 Sigma from Sigmetrix, e-TolMate from Tecnomatix and CAT_3DCS from DCS) points out a main difference between these commercial systems, the first two analyze one “sample” of an assembly and are based on a linear algebraic problem, whereas the later ones require a large number of “samples” to achieve reasonable accuracy and are based on statistics [2]. The models used within the systems are not clearly presented because it is very difficult to obtain information from CAT system vendors.

The analysis methods are divided into two distinct categories based on the type of accumulation input: displacement accumulation and tolerance accumulation.

• The aim of displacement accumulation is to simulate the influences of deviations on the geometrical behavior of the mechanism. Usually, tolerance analysis uses a relationship of the form [3]:

  $$Y=f(X_1,X_2,\dots ,X_n)$$

  where $Y$ is the response (characteristics such as gap or functional characteristics) of the assembly and $X=\{X_1,X_2,\dots ,X_n\}$ are the values of some characteristics (such as situation deviations or/and intrinsic deviations) of the individual parts or subassemblies making up the assembly. The part deviations could be represented by kinematic formulation [5], small displacement torsor (SDT) [6], matrix representation [7], vectorial tolerancing [8] etc.

  The function $f$ is the assembly response function which represents the deviation accumulation. The relationship can exist in any form for which it is possible to compute a value for $Y$ given values of $X=\{X_1,X_2,\dots ,X_n\}$. It could be an explicit analytic expression or an implicit analytic expression. In a particular relative configuration of parts of an assembly consisting of gaps without interference between parts, the composition relations of displacements in some topological loops of the assembly permit determining the function $f$. For hyperstatic assembly, determination of function $f$ is very complex, whereas this determination is easy for an open kinematic chain.

  For statistical tolerance analysis, the input variables $X=\{X_1,X_2,\dots ,X_n\}$ are continuous random variables which enable representing tolerances. In general, they could be mutually dependent. A variety of methods and techniques (Linear Propagation (Root Sum of Squares), Nonlinear propagation (Extended Taylor series), Numerical integration (Quadrature technique), Monte Carlo Simulation etc.) are available for estimation of the probability distribution of $Y$ and the probability $P\left(T\right)$ with respect to the geometrical requirement [3].

• The aim of tolerance accumulation is to simulate the composition of tolerances i.e. linear tolerance accumulation and 3D tolerance accumulation. Based on the displacement models, several vector space models map all possible manufacturing variations (geometrical displacements between manufacturing surfaces or between manufacturing surface and nominal surface) into a region of hypothetical parametric space. The geometrical tolerances or the dimensioning tolerances are represented by deviation domain [9], [10], [11], T-Map^® [12], [13] or specification hull [14], [15]. These three concepts are a hypothetical Euclidean volume which represents all possible deviations in size, orientation and position of features.

  For tolerance analysis, this mathematical representation of tolerances allows calculation of accumulation of the tolerances by Minkowsky sum of deviation and clearance domains [10], [11]: to calculate the intersection of domains for parallel kinematic chain and to verify the inclusion of a domain inside other one. The methods based on this mathematical representation of tolerances are very efficient for the tolerance analysis.

However, these two approaches do not take into account the quantifier notion. This notion translates the concept that a functional requirement must be respected in at least one acceptable configuration of gaps (existential quantifier: “there exists”), or that a functional requirement must be respected in all acceptable configurations of gaps (universal quantifier: “for all”) [16], [15]. A configuration is a particular relative position of parts of an assembly consisting of gaps without interference between parts.

The quantifier notion impacts the result of the tolerance analysis [16], [15]. Therefore, we propose a mathematical formulation of tolerance analysis which simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion. To compute this mathematical formulation, two approaches based on Quantified Constraint Satisfaction Problem solvers and Monte Carlo simulation are proposed and tested.