Application of a Unified Jacobian—Torsor Model for Tolerance Analysis

Abstract

Because of uncertainties in manufacturing processes, a mechanical part always shows variations in its geometrical characteristics (ex. form, dimension, orientation and position). Quality then often reflect how well tolerances and hence, functional requirements, are being achieved by the manufacturing processes in the final product. From a design perspective, efficient methods must be made available to compute, from the tolerances on individual parts, the value of the functional requirement on the final assembly. This is known as tolerance analysis. To that end, existing methods, often based on modeling of the open kinematic chains in robotics, are classified as deterministic or statistical. These methods suppose that the assembled parts are not perfect with regard to the nominal geometry and are rigid. The rigidity of the parts implies that the places of contacts are regarded as points. The validation or the determination of a tolerance zone is therefore accomplished by a series of simulation in specific points subjected to assembly constraints. To overcome the limitations and difficulties of point based approaches, the paper proposes the unification of two existing models: the Jacobian’s matrix model, based on the infinitesimal modeling of open kinematic chains in robotics, and the tolerance zone representation model, using small displacement screws and constraints to establish the extreme limits between which points and surfaces can vary. The approach also uses interval algebra as a novel method to take tolerance boundaries into account in tolerance analysis. The approach has been illustrated on a simple two parts assembly, nevertheless demonstrating the capability of the method to handle three-dimensional geometry. The results are then validated geometrically, showing the overall soundness of the approach.

Three-Dimensional Assembly Tolerance Analysis Based on the Jacobian-Torsor Statistical Model

“…The part 1 (see Fig. 1) is divided into two surfaces: plane (1, 1) and cylinder (1,2). The vertex (1, 0) corresponds to nominal surfaces of part 1.…”

“…Based on the combination of the Jacobian model and the torsor based representation of tolerance zones, the unified Jacobian-Torsor model proposed by Desrochers et al [2] is represented as:…”

“…, n) functional element pair; n is the number of torsors in the tolerancing chain; (1) uses the interval arithmetic [2].…”

“…Geometric tolerances and dimensional tolerances, as well as the interaction between them in the tolerance zone can be taken into consideration by 3D tolerance analysis methods. Desrochers et al [2] proposed the unified Jacobian-Torsor model for deterministic tolerance analysis by the unifying of two existing models: the torsor model and the Jacobian matrix model in which the torsor model is used for tolerance representation and the Jacobian matrix for tolerance propagation. Currently, the research on combinations of 3D tolerance analysis technology and statistical analysis method is attracting the attention of many researchers [3,4].…”

Statistical Variation Analysis Using Pearson Distribution Family Based on Jacobian-Torsor Model

“…In addition, Jacobian-Torsor theory [13] will be used to model the variation propagation, which is well suited to a complex assembly that contains large numbers of joints and geometric tolerances. It combines the advantages of the torsor model which is suitable for tolerance representation and the Jacobian matrix which is suitable for tolerance propagation.…”