Workpiece Positioning Analyses: The Exact Solutions and a Quadratic Variation Approximation Using the Method of Moments

Abstract

This paper presents algorithms for workpiece positioning analysis under locating errors. Workpiece constraint equations are first constructed using the method of homogenous coordinate transformation. These constraint equations are solved numerically for exact workpiece positional deviations by means of deterministic analysis (using the Newton–Raphson method) and variation analysis (i.e., random analysis using a Monte Carlo simulation). To enhance numerical efficiency in variation analysis, we further propose a quadratic approximation solution using the method of moments instead of the Monte Carlo method. Several case studies are presented to exemplify the proposed algorithms, with comparisons to prior literature results on linear and quadratic analyses. The criterion for using the proposed quadratic variation analysis versus the linear method and Monte Carlo simulation is also presented. By using the proposed algorithms, the exact workpiece positioning errors or quadratic variation approximations can be calculated, with consideration of workpiece surface nonlinearity, interactions between locating errors, and the impact of noninfinitesimal locating errors. This paper represents algorithmic advancement in the field where exact solutions and approximations can all be obtained at users’ choice.