A mathematic error model of a linear moving mechanism based on the transmission medium of a diffraction grating
This study applied laser diffraction theory and forward kinematic analysis to propose a mathematical model and a verification system for the four-degree-of-freedom motion errors of a linear transmission mechanism. The mathematical model is derived using the three-dimensional diffraction grating equation and a homogeneous transformation matrix. The verification system consists of a laser diode, a piece of transmission grating, and two quadrant photodiode detectors. This system can measure the straightness errors in the horizontal and vertical directions of a one-dimensional linear moving stage, as well as the angular errors of pitch and yaw of this stage, based on the proposed mathematical model. The experiment correction results for the mathematical model showed that the accuracy of this system in measuring the straightness errors in horizontal and vertical directions within a  50?m measurement range is  1?m, and the accuracy in measuring the angular errors of pitch and yaw within a 50arc sec measurement range is 1 arc sec.
diffraction grating grating equation multi-degree of freedom geometrical error