An ultra-precision and real-time algorithm for 6-DOF displacements based on four grating interferometers

The grating interferometer possesses inherent advantages such as high robustness , compact structure, and the ability to measure multiple degrees of freedom in displacements. However, traditional models primarily focus on translational displacements while overlooking rotational displacements. This oversight leads to micrometer-range errors that diminish the precision of measurement algorithms in multiple interferometers, hindering accurate positioning in motion stages.To address this problem First, a mathematical model for the phase-shift of the interference signal is formulated based on the principles of spatial analytical geometry and the Doppler effect. The model is subsequently simplified through Taylor series expansion to eliminate non-linear terms. By utilizing four grating interferometers symmetrically arranged in a square configuration around a centrally positioned motion stage, the redundant measurement data facilitates the computation of the 6-DOF displacements of the motion stage. Simulation results confirm that the algorithm effectively reduces the translational error to below ±4.85 × 10⁻² nm and the rotational error to below ±2.28 × 10⁻¹⁰ rad, while requiring computational times of only 6.257 × 10⁻² ms and 2.906 × 10⁻² ms, respectively.