Angle measurement is an essential component of precision measurement and serves as a crucial prerequisite for high-end manufacturing. It guides the implementation of precision manufacturing and assembly. The current angle measurement methods mainly focus on multiple axes, high precision, and large measurement ranges. This article introduces the technology of angle measurement from the perspectives of single-axis and multi-axis measurement schemes. Firstly, the single-axis measurement scheme is primarily achieved through optical methods, such as encoder discs that measure energy changes and interferometric phase changes, as well as mechanical, electromagnetic, and inertial angle measurement methods, among which interferometric methods offer the highest accuracy, with high cost, and encoder discs provide the largest measurement range with an ordinary price. Secondly, in the multi-axis measurement scheme, autocollimation instruments, including plane mirrors, gratings, and self-designed targets, are the main options. Although grating encoders can achieve three degrees of freedom in angle measurement with an ordinary price, they are limited in terms of measurement range and sensitivity compared to self-designed targets. Lastly, artificial intelligence assistance precision measurement is increasingly being embraced due to significant advancements in computer performance, making it more convenient to identify the relationship between measured values and detection values. In conclusion, angle measurement plays a crucial role in precision manufacturing, and the evolving and improving technologies provide the manufacturing industry with greater choices. The purpose of this review is to help readers quickly find more suitable technical solutions according to current application requirements, such as single/multiple axes, accuracy level, measuring range, budget, etc.

Grating interferometers highly valued for high resolution, straightforward benchmark and insensitivity to environmental disturbances, have become one of the most mainstream methods for ultra-precision positioning. Traditional heterodyne grating interferometers which use a dual-frequency orthogonally polarized laser source suffer nanometer-level periodic nonlinear error due to frequency and polarization mixing, thus becoming incompetent to undertake sub-nanometer measurement. Based on the design of two separated laser beams with a small frequency difference for heterodyne beating, a wavelength-stabilized and quasi-common-path heterodyne grating interferometer is proposed. It incorporates a symmetrical oblique incidence structure where the residual dead path is calibrated and the laser source wavelength is stabilized to the rubidium atomic transition line, diminishing heterodyne phase errors resulted from the optical path difference. This approach not only significantly mitigates periodic nonlinear error to below 0.3 nm, but also enhances measurement resolution, repeatability and interference robustness. The proposed heterodyne grating interferometer is determined for ultra-precision positioning serving for precision machinery manufacturing and scanning beam interference lithography and so forth.

A heterodyne grating interferometer (HTGI) able to simultaneously measure the three-axis translational displacements of ultraprecision motion stages with a subnanometer resolution is demonstrated. The HTGI is composed of two main parts: a reflective planar grating with a uniform micron-scale period in two orthogonal directions and a reading head. Thanks to the reflective grating, the reading head design manages to make main optical paths common and located on the same side of this compact structure, potentially resulting in low Abbe errors and environmental variations. Relying on a laboratory-made dual-frequency laser source at 780 nm, two heterodyne interferometric measurement modules for X- / Y- direction motions and Z- direction motion were realized based on first-order and zeroth-order diffractions, respectively. Experiments show that the proposed HTGI, in all three axes, can distinguish a 0.5 nm step or even better, perform an up to $2.5\times10$ -5 linearity in a range of $80 \mu \text{m}$ , and hold a 5 nm stability within 300 s. These performances confirm that the proposed HTGI can greatly benefit nanoscience and technology, where multiaxis ultraprecision positioning is required, such as wafer stage in next-generation lithography machine, and so forth.

An absolute-type four-degree-of-freedom (four-DOF) grating encoder that can simultaneously measure the three-axis pose (θx, θy, θz) and one-axis out-of-plane position (Z) of an object with high accuracy is demonstrated for the first time in this research. This grating encoder is composed of a stationary reading head and a movable grating reflector. A light beam from the reading head is projected onto the grating, and three diffracted beams (0th-, +1st-, and −1st-order) are generated, collimated, and received by three separate quadrant photodetectors (QPDs). The information of θx, θy, θz, and Z is coded into spot positions of these three diffracted beams on the QPDs. Thus, the modeling and decoupling algorithms were investigated, and an independent calculation of these four-DOF absolute positions was theoretically guaranteed. A prototype was then designed, constructed, and evaluated. Experimental results verified that the proposed grating encoder could achieve the absolute measurement of four-DOF θx, θy, θz, and Z with an accuracy of sub-arcseconds and sub-micrometers. To the best of our knowledge, the proposed encoder in this research is the first one to achieve absolute simultaneous measurements of four-DOF position and pose with a large measurement range. The success of this new grating encoder can benefit various multi-DOF positioning applications, especially for large-scale synthetic aperture optics (SAO), including stitching off-axis parabolic mirrors and pulse compression grating.

A compact and high-precision three-degrees-of-freedom (DOF; X, Y, and Z directions) grating encoder based on the quadrangular frustum pyramid (QFP) prisms is proposed in this paper to solve the insufficient installation space problem of the reading head of the multi-DOF in high-precision displacement measurement applications. The encoder is based on the grating diffraction and interference principle, and a three-DOF measurement platform is built through the self-collimation function of the miniaturized QFP prism. The overall size of the reading head is 12.3 × 7.7 × 3 cm³ and has the potential for further miniaturization. The test results show that three-DOF measurements can be realized simultaneously in the range of X-250, Y-200, and Z-100 μm due to the limitations of the measurement grating size. The measurement accuracy of the main displacement is below 500 nm on average; the minimum and maximum errors are 0.0708% and 2.8422%, respectively. This design will help further popularize the research and applications of multi-DOF grating encoders in high-precision measurements.