# Multi-Probe Measurement Method for Error Motion of Precision Rotary Stage Based on Reference Plate
## 4 Experimental Results and Discussion  
The error motion experiment was conducted according to the configurations outlined in Section 3 and the error separation algorithm established in Section 2, based on the analytical results presented in Figure 5 at the angular locations \(\varphi=37^{\circ}\) and \(\psi=313^{\circ}\). Figure 6 illustrates the measured error motion curves of the rotary stage, with the first column displaying the comprehensive error motion along with the synchronous component (shown by the red curve) and the second column depicting the asynchronous error components. The first two rows represent the tilt error motions along the X and Y axes (\(T_{x}(\theta)\) and \(T_{y}(\theta)\), respectively), while the third row exhibits the axial error motion \(Z(\theta)\). Notably,  
Figure 5: Behavior of determinant \(W_{k}\) (\(\varphi=37^{\circ}\), \(\psi=313^{\circ}\)). Under current angular configuration, \(W_{k}\) does not equal zero for any value of \(k\) (except for \(k=1\)), indicating absence of harmonic suppression. the synchronous error motion features distinct third-order harmonic characteristics, as indicated by three periodic peaks per entire revolution. The error motion measurement curves acquired at \(\varphi=240^{\circ}\) and \(\psi=313^{\circ}\) are presented in Figure 7, and the synchronous error motion demonstrates identical third-order harmonic characteristics to those observed in Figure 6. The extensive experimental data compiled in Table 1 includes (1) comparative results from these two measurement sets, (2) supplementary measurements obtained at varying probe installation angles, and (3) the ensemble average of all measurements. Statistical analysis reveals significant uniformity across all datasets, with the maximum relative deviation of synchronous errors not exceeding 4.69% in relation to the mean value. The repeatability of the synchronous error during the error separation procedure reflects the stability of the measuring method and necessitates careful consideration [3]. The experimental evidence conclusively demonstrates that the reference plate-based multi-probe methodology attains significant measurement stability, enabling precise measuring of both tilt error motions and axial error motion in precision rotary stages. Theoretically, the measurement results of rotary stage error motion should remain consistent regardless of variations in probe installation angles. The observed discrepancies in Table 1 primarily stem from the following factors: asynchronous rotating axis inaccuracy, probe mounting precision, and variances in sensor performance. Figure 6: Curves of measured error motion (\(\varphi=37^{\circ}\), \(\psi=313^{\circ}\)). Figure 8b illustrates the curve of the residual form error \(P_{r}(\theta)\) of the reference plate derived from the error separation method detailed in Section 2. The measured curve exhibits a peak-to-valley (PV) value of 54.9 nm, indicating a 5.67% relative deviation from the direct measurements acquired with the interferometer shown in Figure 8a. Notably, both profiles display remarkably similar variation tendencies throughout the circumference, characterized by an alternating pattern of two raised and two depressed regions. The discrepancy in the observed PV value arises due to the exclusion of the first-order form  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Components**} & **Tx(\(\theta\)).Sync** & **Tx(\(\theta\)).Async** & **Ty(\(\theta\)).Sync** & **Ty(\(\theta\)).Async** & **Z(\(\theta\)).Sync** & **Z(\(\theta\)).Async** \\  & \(\mu\)rad & /\(\mu\)rad & /\(\mu\)rad & /\(\mu\)rad & /\(\mu\)rad & /\(\mu\)m & /\(\mu\)nm \\ \hline \([0^{\circ}\),\  
error component (fundamental form error \(P_{f}(\theta)\)) from the multi-probe measurement. These findings collectively confirm that the multi-probe error separation method successfully separates the reference plate form error from the rotary stage error motion, hence further demonstrating the accuracy of the multi-probe measurement method.