Uncertainty analysis of two-dimensional self-calibration with hybrid position using the GUM and MCM methods

Abstract

Two-dimensional (2D) self-calibration is more suitable for ultra-precision engineering than conventional calibration, as it does not require higher-precision device calibration. A grid plate is translated or rotated in various positions on a stage, and the translation and rotation can be combined into a hybrid position. Using 2D self-calibration, the stage and plate errors are separated from the overall measurement errors obtained by measuring the grid plate on the stage. During this process, random noise propagates to the separated stage errors through the self-calibration model, causing propagation of the uncertainty of the errors. Accordingly, in this study, least squares-based 2D self-calibration was investigated. An overdetermined linear equation system was established based on the relationships among the variables for self-calibration. The Guide to the Expression of Uncertainty in Measurement was used to calculate the uncertainty propagation ratio after obtaining the least squares solutions for the self-calibration model, and the uncertainty was further analyzed using the Monte Carlo method. The uncertainty propagation ratios were less than 1, indicating that the self-calibration model had a favourable noise-suppression ability. The robustness of this self-calibration approach was mathematically determined. The effects of various position schemes (with and without a hybrid position) on the uncertainty propagation ratio were examined to support the design optimisation of the self-calibration program. The experiments revealed that the use of hybrid position schemes was advantageous over not using such schemes, although the self-calibration performance was comparable under both types of schemes.