## 2 Principle  
The large-aperture coordinate precision measurement instrument used in this study was designed based on the Abbe principle, as shown in Figure 1. The base assembly consisted of a granite assembly that supported the spindle and measuring frame. It provided guiding surfaces for the Y-axis and Z-axis linear bearings. The kinematic system included Y and Z linear axes, an A-axis, and a C-axis rotary axis. The non-contact probe was installed on the A-axis, which was perpendicular to the workpiece to be measured through the rotation axis and moved in the radial and vertical directions along the Y-axis and Z-axis, respectively. The strokes of the A-axis, Y-axis, and Z-axis were 130\({}^{\circ}\), 300 mm, and 80 mm, respectively, allowing measurement of various workpieces. Y- and Z-direction laser interferometers were mounted on either side of the Y-axis table to measure the displacement of the A-axis relative to the measurement frame. The C-axis was bolted directly to the base. During the measurement, the workpiece was placed at the center of the C-axis, and the face shape was measured by moving the table with the motors of each axis. The Y- and Z-axis laser interferometers were mounted on the measuring frame through the standard ruler reflecting the measurement signal to sense the movement distance of the probe. The laser interferometer display value, the probe coordinate value, and the rotary value of the rotary table were concurrently recorded through algorithm processing to obtain the three-dimensional spatial coordinate value and the picking point measurement of the measured point. In a traditional CMM, the main reason affecting the measurement uncertainty is the difficulty of simultaneously conforming to the Abbe principle in the measurement direction, which seriously limits the improvement of CMM measurement accuracy. In order to solve this difficulty and avoid the influence of Abbe error on the measurement, the above measurement device is designed, which consists of Y- and Z-laser interferometers with the measurement surface parallel to the Y-axis and Z-axis guide surfaces. By installing mutually perpendicular high-precision mirrors on the measuring frame and adjusting the angle of the two-axis laser interferometer so that the laser beams of the laser interferometer and the axes of the two rotary axes intersect at the center of rotation of the probe, the measurement system of the measuring device is constituted, so that the designed measuring device has an Abbe arm of zero in space. To analyze the rigid geometry error of the measuring instrument, we assumed that the mirrors were perpendicular and devoid of topographical errors, that the laser scale lines were perpendicular, and that the influence of the guideway motion error was considered separately. The main reason for the positioning error in the measuring machine was the indication error of the laser interferometer. The indication error includes the system error of the laser interferometer and the Abbe error. However, the system error of the laser interferometer is not discussed in this study. For the linear axis, the straightness motion and angular motion errors are caused by imperfections in the linear guide system [28]. The movement of the Y-axis guide rail produces positioning error (\(\delta_{yY}\)) and Y- and Z-direction straightness errors (\(\delta_{xY}\) and \(\delta_{zY}\)), as well as three angular motion errors around the X-, Y-, and Z-axes: \(\varepsilon_{xY}\), \(\varepsilon_{yY}\), and \(\varepsilon_{zY}\). These angular motion errors cause micro-displacement along the other axes. Therefore, the Y-direction positioning error and Z-direction straightness error generated by the movement of the Y-axis guideway cause the Y- and Z-direction laser interferometer signals to have an error of \(\delta_{yY}\) and \(\delta_{zY}\). For the change in laser interferometer signals caused by angular motion errors, see Figure 2a (taking the effect of angular error \(\varepsilon_{xY}\) on the Z-direction laser interferometer signals as an example).