## 2 Principle and Model
### Error Model Establishment and Analysis
\begin{array}{ll}L_{o}=&-B^{3}+\big{(}P_{y}+P_{1y}\big{)}(\varepsilon_{zca}- \varepsilon_{zL}+\varepsilon_{z})\\ &-(H_{a}-P_{z}-P_{1z})\big{(}-\varepsilon_{yca}-\varepsilon_{y}+\varepsilon_{ yL}-\varepsilon_{yab}\cos\theta-\varepsilon_{xab}\sin\theta\big{)}\\ L_{r}=&-B^{3}-\big{(}P_{y}+P_{1y}-20\mathcal{C}_{y}\big{)}(\varepsilon_{zca}- \varepsilon_{zL}+\varepsilon_{z})\\ &-(P_{z}+P_{1z}-H_{a}-20\mathcal{C}_{z})\Big{(}-\varepsilon_{yca}-\varepsilon_ {y}+\varepsilon_{yL}-\varepsilon_{xab}\sin\theta-\varepsilon_{yab}\cos\theta \Big{)}\end{array}  \tag{8}\]  
where  
\[B^{3}=\delta_{\text{sca}}+\delta_{x}+\delta_{x}\cos\theta-\delta_{yab}\sin \theta+H_{a}\Big{(}\varepsilon_{yab}\cos\theta+\varepsilon_{xab}\sin\theta+ \varepsilon_{y}-\varepsilon_{yL}\Big{)}-P_{x}-P_{y}\varepsilon_{zL}+P_{z} \varepsilon_{yL}+L_{s}\]  
The total optical path can be expressed as:  
\[ \begin{array}{ll}L=&L_{o}+L_{i}+L_{r}=\\ &-2\Big{(}\delta_{xca}+\delta_{x}+\delta_{xab}\cos\theta-\delta_{yab}\sin \theta\Big{)}+\mathcal{O}\mathcal{C}_{y}\mathcal{Y}_{\text{PSD}}/f+(H_{a}+ \mathcal{O}\mathcal{C}_{z})Z_{\text{PSD}}/f\\ &-2\big{(}-P_{y}\varepsilon_{zL}+P_{z}\varepsilon_{yL}-H_{a}\varepsilon_{yca} \big{)}-2(-P_{x}+L_{S}-n\mathcal{O}\mathcal{C}_{x})+\mathcal{O}\mathcal{C}_{y }\varepsilon_{zt}-(H_{a}+\mathcal{O}\mathcal{C}_{z})\varepsilon_{yt}\end{array} . \tag{9}\]  
Therefore, the radial motion error along the \(x\) axis is:  
\[ \begin{array}{ll}\delta_{x}=&-(L-L(\theta=0))/2+\mathcal{O}\mathcal{C}_{y}( Y_{\text{PSD}}-Y_{\text{PSD}}(\theta=0))/2f\\ &+(H_{a}+\mathcal{O}\mathcal{C}_{z})(Z_{\text{PSD}}-Z_{\text{PSD}}(\theta=0))/ 2f-\Big{(}\delta_{xab}\cos\theta-\delta_{yab}\sin\theta-\delta_{xab}\Big{)} \\ &+\mathcal{O}\mathcal{C}_{y}(\varepsilon_{zt}-\varepsilon_{zt}(\theta=0))/2-(H _{a}+\mathcal{O}\mathcal{C}_{z})\big{(}\varepsilon_{yt}-\varepsilon_{yt}( \theta=0)\big{)}/2\end{array} , \tag{10}\]  
where  
\[ \begin{array}{ll}L(\theta=0)=-2(\delta_{xca}+\delta_{xab})+\mathcal{O} \mathcal{C}_{y}Y_{\text{PSD}}(\theta=0)/f+(H_{a}+\mathcal{O}\mathcal{C}_{z}) Z_{\text{PSD}}(\theta=0)/f\\ &-2\big{(}-P_{y}\varepsilon_{zL}+P_{z}\varepsilon_{yL}-H_{a}\varepsilon_{yca} \big{)}-2(-P_{x}+L_{s}-n\mathcal{O}\mathcal{C}_{x})+\mathcal{O}\mathcal{C}_{y} \varepsilon_{zt}(\theta=0)-(H_{a}+\mathcal{O}\mathcal{C}_{z})\varepsilon_{yt} (\theta=0)\end{array} . \tag{11}\]  
\(L_{s}\) is the distance between the laser emission surface and the front surface of the measurement unit. The constant influence of some installation errors can be eliminated by subtracting the initial value in the first position of the rotary axis. The offset of the coaxiality deviation (\(\delta_{xab}\) and \(\delta_{yab}\)) has an influence on the measurement of the radial motion error. The trigonometric items can be compensated for by function fitting. In Equation (10), two angular motion errors also influence the measurement of the radial motion error. This kind of error crosstalk can be compensated for by using Equation (10). The complete theoretical analysis shows that the measurement results of the six DOF motion errors are affected by several errors. The compensation methods are given in the above theoretical model, which can effectively compensate for error crosstalk, installation errors, and manufacturing errors and thus improve measurement accuracy.