## 3 Establishment and Simulation of the Error Model
### Ray Tracing and Error Representation
#### 3.2.5 The Axial Motion Error along Z Axis  
The outgoing light \(L^{8}\) of the LDS3 along the Z direction can be expressed as this:  
\[\left[ \begin{array}{cccc}1&0&0&-V_{y}^{8}\\ 0&1&0&-V_{y}^{8}\\ \end{array} \right]\left[ \begin{array}{c}x^{8}\\ y^{8}\\ z^{8}\\ 1\\ \end{array} \right] \tag{14}\]  
The top surface matrix of the polyhedral prism is \(\left[ \begin{array}{cccc}0&0&1&0\\ 0&0&1&-H_{b}\\ \end{array} \right]\); \(H_{b}\) is the sum of the thickness of the polyhedral prism and the plane reflector. Considering the parallel error of the polyhedral prism, the bottom surface of the plane reflector can be expressed as this:  
\[\left[ \begin{array}{cccc}0&0&1&-H_{b}\\ \end{array} \right]\left[ \begin{array}{cccc}1&0&\varepsilon_{yb}&0\\ 0&1&-\varepsilon_{xb}&0\\ -\varepsilon_{yb}&\varepsilon_{xb}&1&0\\ 0&0&0&1\\ \end{array} \right]=\left[ \begin{array}{cccc}-\varepsilon_{yb}&\varepsilon_ {xb}&1&-H_{b}\\ \end{array} \right] \tag{15}\]  
The \(\varepsilon_{xb}\) and \(\varepsilon_{yb}\) are the parallel errors between the top surface and the bottom surface. In CS8, the reflection surface of the plane reflector can be expressed as this:  
\[\left[ \begin{array}{cccc}&-\varepsilon_{yb}&\varepsilon_{xb}&1&-H_{b}\\ \end{array} \right]\left[ \begin{array}{c}X^{2}\\ Y^{2}\\ Z^{2}\\ 1\\ \end{array} \right]=\left[ \begin{array}{cccc}&-\varepsilon_{yb}&\varepsilon_ {xb}&1&-H_{b}\\ \end{array} \right]T_{2}^{1}T_{1}^{0}T_{1}^{8}\left[ \begin{array}{c}X^{8}\\ Y^{8}\\ Z^{8}\\ 1\\ \end{array} \right] \tag{16}\]  
The coordinates of the reflection points that can be obtained from the incident light and the reflection surface are as follows:  
\[ \begin{array}{c}X^{8}=V_{z}^{8},\ Y^{8}=V_{y}^{8},Z^{8}\\ \hskip 14.226378 pt=V_{z}^{8}\left(-\varepsilon_{yb}cos\theta-\varepsilon_{xb} sin\theta+\varepsilon_{y}-\varepsilon_{yb}\right)+V_{y}^{8}\left(-\varepsilon_{yb} sin\theta+\varepsilon_{xb}cos\theta+\varepsilon_{x}+\varepsilon_{x}\right)\\ \hskip 14.226378 pt-\varepsilon_{yb}\left(cos\theta V_{x}+sin\theta V_{y} \right)+\varepsilon_{xb}(-sin\theta V_{x}+\varepsilon_{yb}\theta V_{x}- \varepsilon_{x}V_{y}+V_{z}-\delta_{x}-H_{b}\\ \end{array} \right)\end{array} \tag{17}\]  
The measurement value of the LDS3 is \(\mathrm{LZ}_{lds}\). We can use the spatial geometric position relationship to obtain \(\mathrm{LZ}_{lds}=\sqrt{Z^{8}}^{2}+\left(X^{8}-V_{y}^{8}\right)^{2}+\left(Y^{8} -V_{y}^{8}\right)^{2}\approx Z^{8}\). Because the valuesof \(X^{8}-V_{x}^{8}\) and \(Y^{8}-V_{y}^{8}\) are first order infinitesimal, \(\text{LZ}_{lds}\approx Z^{8}\). Some system errors can be eliminated by reducing the initial value. The axial motion error along the Z direction is this:  
\[ \begin{split}&\delta_{z}=\text{LZ}_{lds}-\text{LZ}_{lds(t=0)}- \left(V_{y}+V_{y}^{8}\right)\varepsilon_{x}+\left(V_{x}+V_{x}^{8}\right) \varepsilon_{y}\\ -&(cos\theta-1)\left[\left(V_{x}+V_{x}^{8}\right) \left(\varepsilon_{yb}-\varepsilon_{ybc}\right)+\left(V_{y}+V_{y}^{8}\right) \left(\varepsilon_{xbc}-\varepsilon_{xb}\right)\right]\\ &-sin\theta\left[\left(V_{x}-V_{x}^{8}\right)\left(\varepsilon_{ xb}-\varepsilon_{xbc}\right)+\left(V_{y}+V_{y}^{8}\right)\left(\varepsilon_{ yb}-\varepsilon_{ybc}\right)\right]\end{split}  \tag{18}\]  
The main factors affecting the axial motion error along Z axis include the following: the parallel errors \(\varepsilon_{xb}\) and \(\varepsilon_{yb}\); the perpendicularity error \(\varepsilon_{py}\); the working angle manufacturing error \(\varepsilon_{pz}\); the position coordinates \(V_{x}\) and \(V_{y}\) of the origin of CS8 in CS0. The position of LDS3 outgoing light is the coordinates \(V_{x}^{8}\) and \(V_{y}^{8}\) in CS8. The error compensation method is the same as the radial motion error along X axis. The effects of the polyhedral prism's manufacturing errors on the measurement of the six DOF motion errors are shown in Table 1.  
The above model includes forty-six errors: the six DOF motion errors of the rotary axis, the installation errors of each component, and the manufacturing errors.