Objective  The grating interferometer displacement measurement system, as one of the most precise measuring instruments in the measurement field, is not able to ensure the perfect assembly of the grating's attitude position during the measurement process, which makes the deviation between the grating's grating vector direction and the motion vector direction, leading to the periodic nonlinear error in the displacement measurement results. In previous studies, the assembly error of grating displacement measurement system is usually unfolded in the scalar state, ignoring the effect of incident azimuth angle on the system. Based on the grating vector diffraction theory, this paper analyses the attitude position error between the grating and the displacement stage as well as the readhead that occurs during the displacement measurement of the grating interferometer, and illustrates the possible displacement measurement error by analysing the amount of angular deviation of the three dimensions, so as to provide the theoretical basis for the improvement of the subsequent device.

  Methods  Ideally, the displacement measurement of a grating interferometer is based on the period of the core component, the grating. But due to the non-ideal assembly of the grating, the displacement stage, the readhead, the optics, and other system modules, there will be geometric errors in the system. The non-ideal assembly of the grating and the displacement stage, as well as the non-ideal assembly of the grating and the readhead, are the main factors leading to the geometrical error of the system. In this paper, we analyse the attitude position error between grating, displacement stage and readhead which occurs during the displacement measurement of grating interferometer, by establishing displacement coordinate system OXYZ and grating coordinate system OX'Y'Z', and referring to the attitude representation method of aircraft in the field of inertial navigation. We set the roll, pitch and yaw angles of one-dimensional grating to be α, β and γ respectively, which are common in describing the assembly state of the 1D grating relative to the translation stage. By analysing the amount of angular deviation in the three dimensions based on the grating vector diffraction theory, the possible displacement measurement errors are analysed and illustrated.

  Results and Discussions  The results of the analyses of the grating assembly errors show that the geometrical errors caused by the non-ideal assembly of the metrology grating are mainly due to the rotational error angles β and γ around the Y' and Z' axes, while the rotation of the grating around the X' axes does not cause any additional measurement errors. It can be seen from the error expressions that error angles β and γ have the same effect on the measurement error. When analysing the readhead assembly error, it was found that the biggest difference between this and the encoder assembly error is that the readhead assembly error causes the system not to satisfy the Littrow structure, further complicating the problem. However, because of this, it explores a more general conclusion based on the generalised one-dimensional grating equations in this paper, from which the relationship between the systematic measurement error and the three error angles α, β and γ and the angles θ₁, θ₂, Ψ₁, Ψ₂ affecting the relative states of the incident P-light and Q-light is discussed respectively.

  Conclusions  This paper analyses the measurement errors caused by clamping problems when using the grating displacement measurement system from two aspects of grating assembly errors and readhead assembly errors, and provides an analytical description of the possible displacement measurement errors. In the grating interferometer displacement measurement system, the analysis based on the grating vector diffraction theory shows that, when the Littrow incidence structure is satisfied, among the roll angle, pitch angle and yaw angle, the roll angle has no effect on the measurement results, and the expressions for the effects of pitch angle and yaw angle on the displacement measurement results are derived; When the Littrow incidence structure is not satisfied, the oblique incidence of laser light increases the azimuth angle. When the Littrow incidence structure is not satisfied, the laser will increase the azimuth angle after oblique incidence. According to the generalised one-dimensional grating equation, a more general conclusion in the presence of azimuth angle is deduced, which provides a theoretical basis for the subsequent improvement of the device.