# Multi-Probe Measurement Method for Error Motion of Precision Rotary Stage Based on Reference Plate
## 3 Experimental Setup  
The experimental setup for the reference plate-based multi-probe measurement method is illustrated in Figure 4. The configuration comprises several essential components: the test rotary stage (an externally pressurized air-static bearing rotary stage developed by Chang Guang Satellite Technology Co., Ltd., Changchun, China, R200), leveling fixture, precision reference plate, capacitive displacement sensors, and data acquisition apparatus. The reference plate is affixed to the rotary stage surface, with its alignment meticulously adjusted using the leveling fixture. This alignment process ensures that the end face of the reference plate remains as perpendicular as possible to the rotational axis of the rotary stage, thus effectively minimizing first-order harmonic error in the acquired measurement data.  
The reference plate is an aluminum alloy component, 100 mm in diameter, machined using single-point diamond turning. The surface profile measurement findings were acquired using an interferometer (ZYGO 4 inch, 6 nm PV precision, Middlefield, CT, USA) over an annular region (75 mm diameter, 10 mm width) of the plate. The processed surface exhibits a peak-to-valley (PV) form accuracy of 58.2 nm. The high precision of the reference plate renders it especially appropriate for error separation in rotary stage error motion measurements.  
Four high-precision capacitance sensors are affixed above the reference plate, utilizing a universal fixture. The fixture has accurately machined mounting holes arranged in a specific geometric configuration: three sensor probes are positioned along a circumference with a 75 mm diameter, while the fourth probe is located at the exact center of this circle. By precisely adjusting the fixture, the fourth probe is aligned to be coaxial with the rotational axis of the rotary stage. To avoid or reduce measurement inaccuracies induced by harmonic  
Figure 4: Experimental setup for multi-probe measurement system based on reference plate.  
suppression, the probe mounting holes are deliberately located at prime-numbered angular intervals around the circumference. This configuration ensures that when either angle \(\varphi\) or angle \(\psi\) is a prime number, the greatest common divisor (GCD) of \(\varphi\), \(\psi\), and \(2\pi\) equals 1 (\(\Delta=1\)). Consequently, the determinant \(W_{k}\) is zero exclusively when \(k\) fulfills the condition \(k=360n\pm 1\). In practical testing circumstances, the value of \(k\) typically does not exceed 100. Therefore, this prime-based angular distribution strategy successfully eliminates harmonic suppression effects. Figure 5 demonstrates that the non-zero behavior of \(W_{k}\) at \(\varphi=37^{\circ}\) and \(\psi=313^{\circ}\) (two meticulously chosen prime angles) effectively validates the method's capability to avoid harmonic suppression.  
The experiment was performed in an environmentally controlled laboratory (\(\pm 0.1\)\({}^{\circ}\)C, VC-D vibration grade). The measurement system employed four Micro-Epsilon CSH02 high-precision capacitive displacement sensors featuring a \(\pm 100\)\(\upmu\)m measurement range with an exceptional 0.4 nm resolution. The sensors were connected to a dedicated Micro-Epsilon NCDT6530 data acquisition device functioning at a sampling frequency of 62.5 Hz. The precision rotary stage was driven at a constant speed of 10 rpm, achieving an angular sampling density of approximately one data point per degree. Data acquisition began following rotational stabilization, with the system performing continuous multi-revolution data capture cycles to provide an adequate dataset for subsequent signal processing and analysis.