Uncertainty analysis in geometric best fit

In computing geometric tolerances using point data from a coordinate measuring machine (CMM), a best fit process is needed to match the measurement data to the substitute geometry. Since the measurement data does not precisely conform to the substitute geometry, the best fit result always contains uncertainties resulting from surface deviations, different point locations, and various sample size. Because the best fit uncertainties reduce accuracy of the evaluated tolerances, it is important to estimate and control the best fit uncertainties. In this paper, a theoretical model is presented to predict the uncertainty zones of the best fit parameters. Supported by simulations and experiments, the proposed model is proved to be effective in determining the best fit uncertainties. To explore factors that affect uncertainty variations, relationships between geometric variables and uncertainties are investigated. Then, to understand the effect of point location on the best fit uncertainty, an optimization scheme which minimizes the total uncertainties is used to find the best measurement locations. Simulations also show that the best fit uncertainty is inversely proportional to the squared root of the number of measurement points. This result can be used to estimate the CMM sample size that can control the best fit uncertainty under specified tolerances.