A unified least-squares approach to the evaluation of geometric errors using discrete measurement data

This paper presents a unified least-squares approach to the best fit of geometric features and evaluation of dimensional errors. The study originated from the need for advanced algorithms for the dimensional measurement of high precision manufactured parts. The proposed algorithm differs from traditional least-squares in that no linearization or approximation is employed and that it is general to all kinds of geometric features. Instead of computing a substitute best fit feature, the algorithm inversely transforms the measurement coordinates to best fit the nominal geometry. The sum of the squared errors in the surface normal direction is thus minimized with respect to the parameters of a rigid body transformation. Form tolerances are then evaluated using the peak-to-valley deviations after the best fit. To examine the uncertainty of the transformation obtained, sensitivity analysis was investigated to relate transformation errors to dimensional errors. A sensitivity measure is used to estimate the joint effect of the measurement locations and the number of measurement data on the accuracy of the coordinate transformation. Computer simulations have been performed on different geometric features to study the robustness and efficiency of the algorithm. Application to the measurement of high precision fuel injectors is also presented.