# 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  
### Analysis of Conventional Three-Probe Measurement Method  
This section presents a concise analysis of the conventional three-probe measurement method and elucidates the correlation between the tilt error motions of the rotary stage corresponding to different probe installation angles. These findings lay the theoretical foundation for developing a multi-probe measurement method based on a reference plate.  
The conventional three-probe measurement method is primarily employed for assessing the radial error motion of the rotational axis. As illustrated in Figure 1, the measurement configuration comprises a spherical or cylindrical reference artifact concentrically affixed to the rotary stage alongside three displacement sensors arranged at varying installation angles \(\varphi\) and \(\psi\) within the identical measurement plane. The measurements recorded by the three displacement sensors are a summation of the artifact's roundness \(R(\theta)\), including a phase shift due to sensor location, and the \(D_{x}(\theta)\) and \(D_{y}(\theta)\) components of the rotary stage error motion [3, 23]. Here, \(\theta\) denotes the rotation angle of the rotary stage and serves as the independent variable for the function of error motion and form error. The roundness error of the artifact can be separated from sensor measurements through error separation, enabling the quantification of the rotary stage radial error motion.  
When the standard artifact exhibits an ideal spherical or cylindrical geometry (i.e., the roundness error \(R(\theta)=0\)), the sensor measurements exclusively reflect the superposition of the rotary stage radial error motions along the X and Y axes, denoted as \(D_{x}(\theta)\) and \(D_{y}(\theta)\), respectively, as mathematically expressed in Equations (1)-(3). The subscripts \(x\) and \(y\) denote the directions of the X and Y axes, respectively.  
\[m_{A}(\theta)=D_{x}(\theta) \tag{1}\]  
\[m_{B}(\theta)=D_{x}(\theta)\cos\varphi+D_{y}(\theta)\sin\varphi \tag{2}\]  
\[m_{C}(\theta)=D_{x}(\theta)\cos\psi+D_{y}(\theta)\sin\psi \tag{3}\]  
Actually, the measured radial error motion of the rotary stage in any measurement plane arises from the superposition of two distinct components: translational motion and axis tilting motion. Let \(T(\theta)\) signify the tilt error motion of the rotary stage axis and \(H\) represent the axial distance between the sensor measurement plane and the center of tilt motion. The radial error component generated by tilt is then expressed as the product \(T(\theta)\times H\). Consequently, the comprehensive radial error motion measurement \(m_{B}(\theta)\) at  
Figure 1: Schematic diagram of conventional three-probe measurement method.  
any given angular position, along with the X and Y directional radial error motion, can be described as follows [24]:  
\[m_{B}(\theta)=d(\theta)+T(\theta)H \tag{4}\]  
\[D_{x}(\theta)=d_{x}(\theta)+T_{x}(\theta)H \tag{5}\]  
\[D_{y}(\theta)=d_{y}(\theta)+T_{y}(\theta)H \tag{6}\]  
where \(d(\theta)\) denotes the radial error motion component arising from the translational motion of the rotary stage axis. Substituting Equations (4)-(6) into Equation (2) and integrating Equation (7) yields Equation (8).  
\[d(\theta)=d_{x}(\theta)\cos\varphi+d_{y}(\theta)\sin\varphi \tag{7}\]  
\[T(\theta)=T_{x}(\theta)\cos\varphi+T_{y}(\theta)\sin\varphi \tag{8}\]  
Equation (8) demonstrates that at arbitrary probe installation angle \(\varphi\) or \(\psi\), the tilt error motion \(T(\theta)\) of the rotary stage axis can be represented as a function of the orthogonal tilt error components \(T_{x}(\theta)\) and \(T_{y}(\theta)\) along the rotary stage X and Y axes. This analytical paradigm is also applicable to Equation (3), yielding identical functional relationships.