Coordinate transformation and its uncertainty under consideration of a non-orthogonal coordinate base

Nanopositioning and nanomeasuring machines are 3D coordinate measuring systems with nanometer precision at measurement volumes in the cubic centimeter range whose coordinate base is formed by an interferometer system with a common mirror corner. This mirror corner shows a typical deviation from the orthogonal coordinate base of up to 17 µrad with an uncertainty of 0.07 µrad. After a brief description of nanomeasuring and nanopositioning machines, a transformation model from the skewed coordinate system into the orthogonal coordinate system is determined. Starting from this complete transformation model, the result of model simplifications on the transformation behaviour is analysed and discussed. A GUM-based uncertainty calculation shows that the linearised coordinate transformation does not increase the error and the uncertainty significantly.