## 2 Principle and Model
### Error Model Establishment and Analysis
parameters, such as structural parameters and optical parameters of the measuring device, is shown in Table 2.  
According to the principle of laser collimation, all the DOF motion errors can be obtained--except for the radial motion error along the \(x\) axis--with the two types of photoelectric detectors [12]. The angular positioning error is calculated using the formula:  
\[\varepsilon_{z}=-(Y_{\text{PSD1}}(\theta=0)-Y_{\text{PSD1}})/2f+\theta_{\text {encode}}-\theta_{\text{nominal}}, \tag{2}\]  
where \(Y_{\text{PSD1}}(\theta=0)=-f(-2\varepsilon_{z\alpha}+2\varepsilon_{zL})\). The tilt motion error around the \(y\) axis is calculated using the formula:  
\[\varepsilon_{y}=-(Z_{\text{PSD1}}-Z_{\text{PSD1}}(\theta=0))/2f-\varepsilon_ {xab}\sin\theta-\varepsilon_{yab}\cos\theta+\varepsilon_{yab}, \tag{3}\]  
where \(Z_{\text{PSD1}}(\theta=0)=-2f(\varepsilon_{yca}+\varepsilon_{yab}- \varepsilon_{ylL})\). The tilt motion error around the \(x\) axis is calculated using the formula:  
\[\varepsilon_{x}=-(\Delta-\Delta(\theta=0))/2D-\varepsilon_{xab}\cos\theta+ \varepsilon_{yab}\sin\theta+\varepsilon_{xab}, \tag{4}\]  
where \(\Delta=Z_{\text{QD1}}-Z_{\text{QD2}}\) and  
\[ \begin{array}{ll}\Delta(\theta=0)&=(Z_{\text{QD1}}-Z_{\text{QD2}})(\theta=0) =2D(-\varepsilon_{xca}-\varepsilon_{xab}+\varepsilon_{xL})+\big{(}-nQDC_{1x}^ {4}+nQDC_{2x}^{4}\big{)}\varepsilon_{yc}\\ &+\big{(}D-P_{1y}+P_{2y}\big{)}\varepsilon_{xc}\end{array} .\]  
According to Equation (2), the only influence on the angular positioning error is constant installation errors, which can be eliminated by subtracting the initial value in the first position of the rotary axis. As described in Equations (3) and (4), the two tilt motion errors are affected by the constant installation errors. The influence of the trigonometric form on the tilt motion errors is from the angle of the coaxiality deviation between the reference and target axes (\(\varepsilon_{xab}\) and \(\varepsilon_{yab}\)). This influence can be compensated for by function fitting. In addition, the manufacturing error (\(\varepsilon_{xc}\) and \(\varepsilon_{yc}\)) of the retro-reflector has a constant influence on the tilt error around the \(x\) axis, which can be eliminated by subtracting the initial value. The radial motion error along the \(y\) axis can be calculated using the equation:  
\[ \begin{array}{ll}\delta_{y}&=Y_{\text{QD}}/2-Y_{\text{PSD}}OC_{1x}/2 nf- \Delta(OC_{1x}+H_{a})/2D-\delta_{xab}\sin\theta-\delta_{yab}\cos\theta+\delta _{yab}\\ &-[Y_{\text{QD}}/2-Y_{\text{PSD}}OC_{1x}/2 nf-\Delta(OC_{1z}+H_{a})/2D](\theta=0 )\\ &+[\varepsilon_{zt}-\varepsilon_{zt}(\theta=0)](-\frac{OC_{x}}{n}+2P_{x}+QDC _{x})\\ &+\frac{(OC_{1x}+H_{a})}{2D}(-QDC_{1x}+QDC_{2x})\big{[}-\varepsilon_{yt}+ \varepsilon_{yt}(\theta=0)\big{]}\end{array}  \tag{5}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Naming** \\ \hline Height of the reference axis & \(H_{a}\) \\ Vertex coordinate of the retro-reflector (in \"3\") & \((OC_{1x},OC_{1y},OC_{1z})\) \\ Vertex distance of two retro-reflectors & \(D=OC_{2y}-OC_{1y}\) \\ Center point coordinate of the bottom edge of the measurement unit’s front surface (in \"0\") & \((P_{x},P_{y},P_{z})\) \\ The first laser output position coordinates on the measurement unit (in \"4\") & \((P_{1x},P_{1y},P_{1z})\) \\ Center coordinates of the QD1 photosensitive surface (in \"4\") & \((QDC_{1x}^{4},QDC_{1y}^{4},QDC_{1z}^{4})\) \\ Center coordinates of the PSD1 photosensitive surface (in \"4\") & \((PSD_{1x}^{4},PSD_{1y}^{4},PSD_{1z}^{4})\) \\ Focal length of the lens & \(f\) \\ Refractive index of the retro-reflector glass & \(n\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The naming of parameters. where  
\[