Uncertainty Calculation of Flatness in Three-Dimensional Measurements

The least-square method is commonly employed to verify the flatness in actual three-dimensional measurement processes, but the uncertainty of the verification results is usually not given by the coordinate measuring machines. According to basic principles of flatness least-square verification and the uncertainty propagation formula given by ISO/TS 14253-2, a calculation method for the uncertainty of flatness least-square verification was proposed. By this method, coefficients of the plane equation were regarded as a statistical vector, so that the plane equation, the results of flatness verification and the uncertainty of the results can be obtained after the expected values and covariance matrix of vector were determined. The method assured the integrity of the verification results, and accorded with requirements of new generation of GPS standards, which can improve the veracity of verification. Experimental results indicate that in conformity to the results of flatness least-square verification and the uncertainty, the plane can be accepted or rejected quantitatively in accordance with the decision rules given by ISO 14253-1.