# 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  
Similar to radial error motion, the measured face error motion of a rotary stage arises from translational and tilting motions of its axis. Specifically, the face error motion component \(F(r,\theta)\) induced by the tilting motion at radius \(r\) is expressed as [15]. \[F(r,\theta)=T(\theta)r \tag{9}\]  
Multiplying both sides of Equation (8) by radius \(r\) and substituting the result into Equation (9) yields the following expression:  
\[F(r,\theta)=F_{x}(r,\theta)\cos\varphi+F_{y}(r,\theta)\sin\varphi. \tag{10}\]  
Equation (10) demonstrates that the face error motion \(F(r,\theta)\), associated with any probe installation angle (\(\varphi\) or \(\psi\)), can be represented as a function of the face error motions \(F_{x}(r,\theta)\) and \(F_{y}(r,\theta)\) along the X and Y axes at a certain radius \(r\). The subscripts \(x\) and \(y\) denote the X and Y axes, respectively. Building upon these findings, the tilt error motion of the rotary stage axis can be quantitatively assessed through appropriate error separation techniques, provided that the face error motion of the rotary stage is accurately measured. Figure 2 illustrates the principle of our proposed multi-probe measurement system employing a reference standard artifact. The measurement setup consists of (1) a precision reference plate affixed to the rotary stage being tested and (2) an array of four displacement sensors positioned above the reference plate. The measurement configuration comprises three probes arranged along the circumference of radius \(r\): one aligned with the X axis of the measurement plane and the other two positioned at angular offsets \(\varphi\) and \(\psi\) relative to the X axis. The fourth probe is precisely aligned with the rotational axis of the rotary stage, serving as an essential reference for detecting axial motion. Using an analysis similar to that presented in [4; 15], we define the following: \begin{tabular}{l l} \(Z(\theta)\): & axial error motion of the axis of rotation, coaxial to the axis; \\ \(Z_{f_{\rm s}}(\theta)\): & fundamental synchronous axial error motion of the axis of rotation; \\ \(Z_{\rm rs}(\theta)\): & residual synchronous axial error motion of the axis of rotation; \\ \(Z_{\rm a}(\theta)\): & asynchronous axial error motion of the axis of rotation; \\ \(Z_{\rm a}(\theta)\): & form error of the reference plate as a function of the radial (\(r\)) and angular \\ \(P(r,\theta)\): & (\(\theta\)) positions; \\ \(P_{f}(r,\theta)\): & fundamental form error of the reference plate; \\ \(P_{r}(r,\theta)\): & residual form error of the reference plate. \\ \end{tabular}   
Their interrelationship is as follows:  
\[Z(\theta)=Z_{f_{\rm s}}(\theta)+Z_{\rm rs}(\theta)+Z_{a}(\theta) \tag{11}\]  
\[P(r,\theta)=P_{f}(r,\theta)+P_{r}(r,\theta). \tag{12}\]  
For the sake of notation, we assume that \(r\) is an arbitrarily chosen value. Accordingly, \(P(r,\theta)\), \(P_{f}(r,\theta)\), and \(P_{r}(r,\theta)\) are abbreviated as \(P(\theta)\), \(P_{f}(\theta)\), and \(P_{r}(\theta)\), respectively, while \(F(r,\theta)\), \(F_{x}(r,\theta)\), and \(F_{y}(r,\theta)\) are shortened as \(F(\theta)\), \(F_{x}(\theta)\), and \(F_{y}(\theta)\), respectively. The experimental data acquired by the four displacement sensors can be represented by Equations (13)-(16):  
\[m_{1}(\theta)=P(\theta)+Z(\theta)+F_{x}(\theta)+V_{1}(\theta) \tag{13}\]  
\[m_{2}(\theta)=P(\theta-\varphi)+Z(\theta)+F_{x}(\theta)\cos\varphi+F_{y}( \theta)\sin\varphi+V_{2}(\theta) \tag{14}\]  
\[m_{3}(\theta)=P(\theta-\psi)+Z(\theta)+F_{x}(\theta)\cos\psi+F_{y}(\theta)\sin \psi+V_{3}(\theta) \tag{15}\]  
\[m_{4}(\theta)=Z(\theta) \tag{16}\]  
where \(V_{1}(\theta)\) to \(V_{3}(\theta)\) represent the deficiency of squareness in mounting the reference plate relative to the axis of rotation. We remove the first-order harmonic components from the measurement data to rectify inaccuracies generated by misalignment.