Evaluation and uncertainty analysis of vectorial tolerances

Recently, the integration of coordinate measuring machines (CMMs) and computer-aided design (CAD) systems has promoted a new approach to the evaluation of workpiece geometry; namely, vectorial dimensioning and tolerancing. This new approach is promising, because it defines process-related dimensions and tolerances clearly and distinctly. Therefore, it enables proper manufacturing control and process diagnosis. However, current proposal of vectorial tolerancing has several limitations. First, the current orientation vector is inadequate for representing true three-dimensional (3D) orientation. As a result, the orientation of a free-form surface cannot be properly established. Second, there has been little discussion of vectorial tolerance evaluation. This paper proposes a new rotation vector that provides a more general mathematical basis for representing vectorial tolerances. A nonlinear, best-fit algorithm has been developed to evaluate vectorial tolerances for both analytical geometric elements and free-form surfaces. To study the uncertainty of the best-fit result caused by sampling strategy and dimensional errors, sensitivity analysis of the evaluated parameters was investigated. Simulation and experiment showed that the developed model can predict the uncertainty of the evaluated parameters accurately.