Efficient uncertainty estimation of indirectly measured geometric errors of five-axis machine tools via Monte-Carlo validated GUM framework

Abstract Whether using a machine tool as a machining or a measuring system, its accuracy has a key role in ensuring product quality. As a result, conducting periodic geometric calibration of a machine tool to quantify and to compensate for the machine's geometric errors is highly desirable. The measured geometric errors must be accompanied by uncertainty estimates, reflecting the lack of exact knowledge of the value of measurands. Indirect calibration approaches, as opposed to direct ones, have gained considerable attention recently because they can be automated and take considerably less time to conduct. However, the uncertainty of the indirectly measured error parameters is more challenging to estimate because of the complexity of the used mathematical models and the large number of machine error parameters identified simultaneously. The use of uncalibrated artifacts also adds to the wariness towards such approaches. In this paper, two different approaches are used to evaluate the uncertainty of machine geometric errors identified by the scale and master ball artefact (SAMBA), an indirect approach relying on the on-machine probing of uncalibrated and calibrated artifacts. First, a simulator of an adaptive Monte Carlo method (MCM) determines the uncertainty of machine tool geometric errors. A multivariate sample generator draws at random the input vectors from a joint probability distribution function (PDF) obtained through experimental replications over 15 days. Supplement 2 to the Guide to the expression of uncertainty in measurement (GUM) specifies the conditions where an adaptive MCM validates the GUM uncertainty framework (GUF). Adhering to the instructions given by this standard, the validity of GUF, a much faster uncertainty evaluation method, is examined. Results show that the adaptive MCM procedure, which takes 24 h to run, validates the alternative GUF approach, which takes a computation time of only 10 min. A maximum disagreement of 1% between the geometric errors from experimental replications and their estimates identified by the adaptive MCM implies that the marginal means of the estimated output joint PDF in the adaptive MCM well represent the average of the output quantities. High dependencies of the linear terms (slopes) of the linear positioning errors of prismatic axes X, Y, and Z on the SAMBA scale bar length cause high correlations between these errors with correlation coefficients up to 0.965. The largest standard uncertainty estimated for the angular geometric errors is 1.26 μrad (associated with the out-of-squareness error of axis Z relative to axis X, EB0Z) and that for the translational geometric errors is 1.03 μm (associated with the distance between the spindle axis and axis B along axis X, EX0S). Axis X is estimated to have the largest standard uncertainty (6.1 μm/m) of the linear terms (slopes) of the linear positioning errors among the machine's prismatic joints.