Uncertainty evaluation and reduction in three-probe roundness profile measurement based on the system transfer function

Abstract

The three-probe method for separating the spindle rotation error and the specimen form error is extensively described in the literature. An attractive feature is its application in in-process measurement. However, the resulting uncertainty is studied far less extensively. In this paper, an evaluation and propagation method for the uncertainty, as well as for an uncertainty reduction, is given based on the system transfer function (S-function). First, utilizing the Laplace transform, the measurement system model is developed and expressed by an S-function. Second, the propagation laws of input uncertainties are analytically deduced by computing the partial derivatives of the S-function of roundness. Then, the laws are numerically validated by Monte Carlo simulations. The uncertainty propagation laws show that the uncertainties propagate with varying amplification over the harmonic domain, and moreover, they enable the quantification of both the harmonic and the total uncertainty of roundness. Taking the roundness uncertainty as a decisive parameter, three approaches are proposed for uncertainty reduction: (1) the hybrid 3-PM, where two roundness estimates are combined by taking individual harmonic estimate with the lowest uncertainty, (2) the fusion 3-PM, where the weighted average is taken over the harmonic domain, and (3) the angle optimization, which minimizes the total roundness uncertainty by properly arranging the sensor angles. The angle optimization is applied to the conventional 3-PM, as well as to the hybrid and the fusion 3-PMs. The genetic algorithm is adopted to speed up the optimization process. Finally, practical roundness measurements are performed.