Design and implementation of a real-time compensation algorithm for nonlinear error based on ellipse fitting

Grating interferometers with high accuracy, strong robustness and multi-dimensional measurement ability have become a main approach in ultra-precision positioning. However, the installation errors, especially the nonideal rotation and translation of the grating, will affect the phase of diffraction beam, resulting in displacement calculation errors. In this paper, a common phase variation model of grating interferometer regarding to the mechanical errors is proposed. A simplified geometric model of general grating interferometers base on the principle of optical-path-unfolding is first established for convenience and universality. Optical path tracing algorithm and the ray vector analysis method are adopted to analyze the phase variations due to optical-path length change and Doppler effect. The experiment results indicate that when the grating moves 100 mm in X-direction with a rotation of 100 arcseconds around Y-axis, a non-linear error of 11.4 nm is generated. For deflection angles of 300 arcseconds around three axes, cross-talk errors between translation axes will reach 45.6 µm in a range of 30 mm for the measurement system illustrated. The Unfolding-Based model proposed in this paper can be applied to most of the grating interferometers. Calculation errors can be effectively quantified through the model and algorithm, which provide a reference for the optimization of optical path as well as the design of calibration algorithm. The results further improve the necessity of six degree-of-freedom measurement ability in ultra-precision positioning.

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

Liu,

Cui,

Yang

et al. 2024

Advanced Sensor Systems and Applications XIV

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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.

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Design and implementation of a real-time compensation algorithm for nonlinear error based on ellipse fitting

Cited by 4 publications

References 26 publications

A Real-Time Determination Method for Homodyne Grating Interferometer Signal Nonlinear Error Based on Duty Cycle Measurement of States

A Real-Time Determination Method for Homodyne Grating Interferometer Signal Nonlinear Error Based on Duty Cycle Measurement of States

An unfolding-based model for analysis of rotation and translation coupling in grating interferometers

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

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