A Unified Jacobian-Torsor Model for Analysis in Computer Aided Tolerancing

This paper presents a unified model for computer-aided tolerancing. It combines the benefits of the Jacobian matrix model and the torsor (or screw) model. The former is based on small displacements modeling of points using 6x6 transformation matrices of open kinematic chains in robotics. The latter models the boundaries of 3-D tolerance zones resulting from a feature’s small displacements using a torsor representation with constraints. The proposed unified model expands the functionalities of the Jacobian model under two important aspects. First, the punctual small displacement variables of the former Jacobian formulation are now considered as intervals formulated and solved using interval-based arithmetic. The equations describing the bounds within which the feature is permitted to move, which are the constraint equations of the torsor formulation, are applied on the unified model. Second, some of the small displacement variables used in the model are eliminated due to the invariant nature of the movements they generate with respect to the toleranced feature. This standard result of the torsor formulation is applied to the unified model. The effect of this is to significantly reduce the unified model size. A example application is also presented.