# Multi-Probe Measurement Method for Error Motion of Precision Rotary Stage Based on Reference Plate
## 2 Multi-Probe Measurement Method for Error Motion Based on Reference Plate
### Multi-Probe Measurement Method Based on Reference Plate
Subsequently, we  
Figure 2: Schematic diagram of multi-probe measurement method based on reference plate. substitute Equations (11) and (12) and subtract Equation (16) from Equations (13)-(15), resulting in the following:  
\[{m^{\prime}}_{1}(\theta)=P_{r}(\theta)+F_{x}(\theta) \tag{17}\]  
\[{m^{\prime}}_{2}(\theta)=P_{r}(\theta-\varphi)+F_{x}(\theta)\cos\varphi+F_{y}( \theta)\sin\varphi \tag{18}\]  
\[{m^{\prime}}_{3}(\theta)=P_{r}(\theta-\psi)+F_{x}(\theta)\cos\psi+F_{y}( \theta)\sin\psi. \tag{19}\]  
To separate \(P_{r}(\theta)\) from \(F_{x}(\theta)\) and \(F_{y}(\theta)\), a weighted function \(M(\theta)\) for the three measurements is constructed as follows:  
\[ \begin{array}{ll}M(\theta)&={m^{\prime}}_{1}(\theta)+{am^{\prime}}_{2}( \theta)+{bm^{\prime}}_{3}(\theta)\\ &=P_{r}(\theta)+aP_{r}(\theta-\varphi)+bP_{r}(\theta-\psi)\end{array} . \tag{20}\]  
Thus, the function \(M(\theta)\) can be expressed as a weighted linear combination of the reference plate's residual form error \(P_{r}(\theta)\), including a phase shift due to probe location. The weighted coefficients \(a\) and \(b\) are derived by solving Equations (17)-(20), yielding the following:  
\[\left\{ \begin{array}{l}a=\frac{-\sin\psi}{\sin(\varphi-\varphi)}\\ b=\frac{\sin\varphi}{\sin(\varphi-\varphi)}\end{array} .\right. \tag{21}\]  
By using Fourier transformation, \(P_{r}(\theta)\) and \(M(\theta)\) can be decomposed as follows:  
\[P_{r}(\theta)=\sum_{k=2}^{\infty}A_{k}\cos k\theta+B_{k}\sin k\theta \tag{22}\]  
\[M(\theta)=\sum_{k=2}^{\infty}F_{k}\cos k\theta+G_{k}\sin k\theta. \tag{23}\]  
The substitution of Equations (22) and (23) into Equation (20), followed by simplification, leads to the following:  
\[\sum_{k=1}^{\infty}F_{k}\cos k\theta+G_{k}\sin k\theta=\sum_{k=1}^{\infty} \left(A_{k}\alpha_{k}-B_{k}\beta_{k}\right)\cos k\theta+\left(A_{k}\beta_{k}+ B_{k}\alpha_{k}\right)\sin k\theta \tag{24}\]  
where  
\[\left\{ \begin{array}{l}\alpha_{k}=1+a\cos k\varphi+b\cos k\psi\\ \beta_{k}=a\sin k\varphi+b\sin k\psi\end{array} \right.. \tag{25}\]  
Equation (24) can be reformulated in the following matrix format:  
\[ \begin{bmatrix}\alpha_{k}&-\beta_{k}\\ \beta_{k}&\alpha_{k}\end{bmatrix}\begin{bmatrix}A_{k}\\ B_{k}\end{bmatrix} = \begin{bmatrix}F_{k}\\ G_{k}\end{bmatrix} . \tag{26}\]  
According to Equations (23) and (25), once the probe installation angles \(\varphi\) and \(\psi\) are determined, the respective \(F_{k}\), \(G_{k}\), \(\alpha_{k}\), and \(\beta_{k}\) may be calculated for each integer \(k\). The matrix Equation (26) establishes the correlation between the unknown variables (\(A_{k}\), \(B_{k}\)) and the known quantities (\(F_{k}\), \(G_{k}\)). Solving this matrix equation allows us to determine \(A_{k}\) and \(B_{k}\), which subsequently enables the calculation of \(P_{r}(\theta)\) through Equation (22). Then we may obtain \(F_{x}(\theta)\) and \(F_{y}(\theta)\) using Equations (17) and (18), ultimately yielding the final solutions \(T_{x}(\theta)\) and \(T_{y}(\theta)\). The measured signals \(Z(\theta)\), \(T_{x}(\theta)\), and \(T_{y}(\theta)\) encompass both synchronous and asynchronous error components. The synchronous error components, which occur at integer multiples of the rotational frequency, can be derived by averaging multiple revolutions of the probe signals:  
\[\left\{ \begin{array}{l}Z_{s}(\theta)=\frac{1}{l}\sum\limits_{i=0}^{l-1}Z_{i}( \theta+2\pi i)\\ T_{xs}(\theta)=\frac{1}{l}\sum\limits_{i=0}^{l-1}T_{xi}(\theta+2\pi i)\\ T_{ys}(\theta)=\frac{1}{l}\sum\limits_{i=0}^{l-1}T_{yi}(\theta+2\pi i)\\ \end{array} \right. \tag{27}\]  
where \(l\) represents the number of consecutive revolutions recorded. The asynchronous error components occur at non-integer multiples of the rotational frequency. The asynchronous error in a given single revolution can be determined by subtracting the synchronous error components from the measurement results:  
\[\left\{ \begin{array}{l}Z_{a}(\theta)=Z_{i}(\theta)-Z_{s}(\theta)\\ T_{xa}(\theta)=T_{xi}(\theta)-T_{xs}(\theta)\\ T_{ya}(\theta)=T_{yi}(\theta)-T_{ys}(\theta)\\ \end{array}