## 2 Principle and Model
### Error Model Establishment and Analysis
\begin{array}{l}[Y_{\text{QD}}-OC_{1x}Y_{\text{PSD}}/nf-\Delta(OC_{1z}+H_{a})/D]( \theta=0)\\ =2\delta_{yca}+2\varepsilon_{zL}P_{x}-2\varepsilon_{xL}P_{z}+2\delta_{yab}\ +2H_{a} \varepsilon_{xca}+(-OC_{1z}-H_{a}-P_{z}+P_{1z})\varepsilon_{xc}\\ -\varepsilon_{yc}\big{(}-nQDC_{1x}^{4}+nQDC_{2x}^{4}\big{)}(OC_{1z}+H_{a})/D\\ +\big{(}nP_{x}-OC_{1x}+n\ QDC_{1x}^{4}\big{)}\varepsilon_{zc}-\big{(}D-P_{1y}+ P_{2y}\big{)}\varepsilon_{xc}(OC_{1z}+H_{a})/D\end{array} .\]  
The axial motion error can be calculated using the equation:  
\[ \begin{array}{l}\delta_{z}=Z_{\text{QD}}/2-OC_{1x}Z_{\text{PSD}}/2 nf+OC_{1y }\Delta/2D-\big{[}Z_{\text{QD}}/2-OC_{1x}Z_{\text{PSD}}/2 nf+OC_{1y}\Delta/2D \big{]}(\theta=0)\\ -\frac{\varepsilon_{yt}-\varepsilon_{yt}(\theta=0)}{2}\Big{(}-\frac{OC_{x}}{ D}QDC_{1x}+\frac{OC_{y}}{D}QDC_{2x}-QDC_{x}-2P_{x}+\frac{OC_{x}}{n}\Big{)}\end{array}  \tag{6}\]  
where  
\[ \begin{array}{l}\big{[}Z_{\text{QD}}/2-OC_{1x}Z_{\text{PSD}}/2 nf+OC_{1y} \Delta/2D\big{]}(\theta=0)=2\delta_{zca}+2\delta_{zab}+2\varepsilon_{xL}P_{y} -2\varepsilon_{yl}P_{x}\\ +\big{(}-nP_{x}-nQDC_{1x}^{4}+OC_{1x}\big{)}\varepsilon_{yc}\ +\big{(}-P_{y}+OC_{1y}-P_{1y}\big{)} \varepsilon_{xc}\\ +\varepsilon_{yc}(-nQDC_{1x}^{4}+nQDC_{2x}^{4})OC_{1y}/D\ +\big{(}D-P_{1y}+ P_{2y}\big{)}\varepsilon_{xc}OC_{1y}/D\end{array} .\]  
As described in Equations (5) and (6), the manufacturing errors and some of the installation errors have constant influences on the radial and axial motion errors. These influences can be eliminated by subtracting the initial value. The measurement result of the radial motion error along the \(y\) axis is affected by the offset of the coaxiality deviation of the two rotary axes. This effect--the trigonometric terms in equations--can be compensated for using function fitting. In Equation (5), \(Y_{\text{PSD}}/2f\) is related to the angular positioning error, and \(\Delta/2D\) is related to the tilt motion error around the \(x\) axis. Thus, these two motion errors have influences on the measurement of the radial motion error along the \(y\) axis, which is the error crosstalk. Similarly, in Equation (6), two tilt motion errors have influences on the measurement of the axial motion error. All of the error crosstalk can be compensated for by the above model. To measure the radial motion error along the \(x\) axis, the interference principle is used. Therefore, accurate analysis of the total optical path of the interference part is the key to describing the radial motion error along the \(x\) axis. The total optical path is divided into three parts: the optical path of the output light (\(L_{o}\), yellow), the optical path in RR3 (\(L_{i}\), red), and the optical path of the reflected light (\(L_{r}\), blue), as shown in Figure 4. According to the equations of the output and reflected light, the incident and emitted positions of the lights on the measurement unit or the retro-reflector can be obtained, and the total optical path of the interference can be calculated. The optical path in RR3 is affected by the angular changes of the incident light, and the influences are second- and higher-order items, as shown in Equation (7). \(h\) is the height of the retro-reflector and \(i\) is the angle of incidence. \[L_{i}=2 nh\Big{(}1-\sin^{2}i/n^{2}\Big{)}^{-1/2} \tag{7}\]  
Figure 4: Optical path of the interference. The angular motion errors have larger influences on the optical paths of the output light and reflected light. The influences include the first-order items. After ignoring the second- and higher-order items, the optical paths of the output light \(L_{o}\) and the reflected light \(L_{r}\) are:  
\[