A Finite-Differences Derivative-Descent Approach for Estimating Form Error in Precision-Manufactured Parts

Background: Form-error measurement is mandatory for the quality assurance of manufactured parts and plays a critical role in precision engineering. There is now a significant literature on analytical methods of form-error measurement, which either use mathematical properties of the relevant objective function or develop a surrogate for the objective function that is more suitable in optimization. On the other hand, computational or numerical methods, which only require the numeric values of the objective function, are less studied in the literature on form-error metrology. Method of Approach: In this paper, we develop a methodology based on the theory of finite-differences derivative descent, which is of a computational nature, for measuring form error in a wide spectrum of features, including straightness, flatness, circularity, sphericity, and cylindricity. For measuring form-error in cylindricity, we also develop a mathematical model that can be used suitably in any computational technique. A goal of this research is to critically evaluate the performance of two computational methods, namely finite-differences and Nelder-Mead, in form-error metrology. Results: Empirically, we find encouraging evidence with the finite-differences approach. Many of the data sets used in experimentation are from the literature. We show that the finite-differences approach outperforms the Nelder-Mead technique in sphericity and cylindricity. Conclusions: Our encouraging empirical evidence with computational methods (like finite differences) indicates that these methods may require closer research attention in the future as the need for more accurate methods increases. A general conclusion from our work is that when analytical methods are unavailable, computational techniques form an efficient route for solving these problems.