二维光栅制作与装配误差综合建模与实验研究

针对二维位移测量系统中光栅制作与装配非理想所引起的几何误差，基于多普勒频移理论和坐标变换方法，建立了同时包含光栅非正交角与装配角在内的通用几何误差模型，定量研究了各误差角对系统性能的影响程度，仿真分析了X与Y方向余弦误差和耦合误差随误差参数的变化规律。结果表明，光栅制作和装配误差与系统所用衍射级次、衍射次数和光学细分倍数无关，只与各误差角和被测位移有关。与此同时，四个误差角都会导致余弦误差的产生，而耦合误差则主要受光栅非正交角和偏航角的影响。此外，相同误差角所引起的耦合误差要明显严重于余弦误差，是系统几何误差的主要构成成分。通过搭建基于二维交叉光栅的平面位移测量系统，利用10 mm方形运动轨迹实验验证了理论分析与数值仿真的正确性。

Synthetical modeling and experimental study of fabrication and assembly errors of two-dimensional gratings

The geometrical errors resulting from imperfection of the fabrication and assembly of two-dimensional(2D) grating were investigated. Based on the Doppler frequency shift theory and the coordinate transformation method, the general error model versus non-orthogonal and misalignment angles of 2D gratings was established. Then the effects of these error angles were quantitatively studied, and the cosine and the cross-talk errors of both directions were simulated and analyzed. The results illustrated that the geometrical errors were independent from the diffraction orders, diffraction times and the optical subdivision multiple, just determined by the error angles and the 2D displacement. Meanwhile, the cosine errors could be induced by all the error angles, but the cross-talk errors were mainly influenced by the non-orthogonal and raw angles. Besides, the cross-talk errors were more serious than the cosine errors, which is the dominant component of the geometrical errors. A planar displacement measurement system with 2D cross-grating was constructed, and the theoretical analyses and the numerical simulation were verified with the 10 mm square displacement measurement.