## 3 Establishment and Simulation of the Error Model
### Matrix Description of Relative Motion  
The origin of CS0 is located at the center of the top surface of the target rotary axis measured at the initial timepoint, which represents the position-pose of the initial timepoint and without errors of the target rotary axis. The origin of CS1 is the center of the top surface of axis C, which coincides with CS0 at the initial timepoint. In the movement process of the axis, there are two tilt errors, \(\varepsilon_{x}\) and \(\varepsilon_{y}\), one angular positioning error, \(\varepsilon_{z}\), two radial motion errors, \(\delta_{x}\), \(\delta_{y}\), and one axial motion error, \(\delta_{z}\). The theoretical angle of rotation is \(\theta\), and the coordinate transformation matrix when CS1 is reached is this:  
\[T_{0}^{1}=\left[ \begin{array}{ccc}\cos(\theta+\varepsilon_{z})&-\sin(\theta +\varepsilon_{z})&\varepsilon_{y}&\delta_{x}\\ \sin(\theta+\varepsilon_{z})&\cos(\theta+\varepsilon_{z})&-\varepsilon_{x}& \delta_{y}\\ -\varepsilon_{y}\cos(\theta+\varepsilon_{z})+\varepsilon_{x}\sin(\theta+ \varepsilon_{z})&\varepsilon_{y}sin(\theta+\varepsilon_{z})+\varepsilon_{x} \cos(\theta+\varepsilon_{z})&1&\delta_{z}\\ 0&0&0&1\end{array} \right] \tag{1}\]  
CS2 is fixed at the center of the bottom surface of the polyhedral prism and moves with that. There are five installation errors between the initial coordinate system and CS1, excluding the positioning error. The installation error between them is expressed in  
Figure 2: Schematic diagram of experimental device and coordinate system. pairwise lowercase letters. For example, \(\delta_{zkc}\) represents the position error between B and C in the Z direction. Therefore, the transformation matrix from CS2 to CS1 is this:  
\[T_{1}^{2}=\left[ \begin{array}{cccc}1&0&\varepsilon_{ybc}&\delta_{xbc}\\ 0&1&-\varepsilon_{xbc}&\delta_{ybc}\\ -\varepsilon_{ybc}&\varepsilon_{xbc}&1&\delta_{zbc}\\ 0&0&0&1\end{array} \right] \tag{2}\]  
CS3 and CS2 have the same origin of coordinates, and its \(X\)-axis direction is constantly perpendicular to the current working surface of the polyhedral prism. Therefore, CS3 and CS2 have an angular difference, \(\theta+\varepsilon_{pz}\), around the Z axis (manufacturing errors of the working angle of the polyhedral prism) and angular error \(\varepsilon_{py}\) around the Y axis or angular error \(\varepsilon_{px}\) around the X axis (\(\varepsilon_{px}\) and \(\varepsilon_{py}\) are perpendicularity errors between the working surface and datum plane, and change with the \(\theta\) angle). When the target rotary axis rotates \(\theta\), the transformation matrix from CS3 to CS2 is as follows: (This matrix is for measuring the error \(\varepsilon_{y}\). For measuring the error \(\varepsilon_{x}\), the variables in the matrix should be changed accordingly). \[T_{2}^{3}=\left[ \begin{array}{cccc}\cos(\theta+\varepsilon_{pz})&\sin( \theta+\varepsilon_{pz})&\varepsilon_{py}&0\\ -\sin(\theta+\varepsilon_{pz})&\cos(\theta+\varepsilon_{pz})&0&0\\ -\varepsilon_{py}\cos(\theta+\varepsilon_{pz})&-\varepsilon_{py}\sin(\theta+ \varepsilon_{pz})&1&0\\ 0&0&0&1\end{array} \right] \tag{3}\]  
The original point P of CS4 is at the center of the front surface of autocollimator AC1, and the P point in CS0 is \(\left( \begin{array}{cccc}P_{x}&P_{y}&P_{z}&1\end{array} \right)^{T}\). The angle installation errors between CS4 and CS0 are expressed as \(\varepsilon_{xl}\), \(\varepsilon_{yl}\), and \(\varepsilon_{zl}\). The transformation matrix from CS4 to CS0 is this:  
\[T_{0}^{4}=\left[ \begin{array}{cccc}1&-\varepsilon_{zl}&\varepsilon_{yl}&P_{ x}\\ \varepsilon_{zl}&1&-\varepsilon_{xl}&P_{y}\\ -\varepsilon_{yl}&\varepsilon_{xl}&1&P_{z}\\ 0&0&0&1\end{array} \right] \tag{4}\]  
The position of the exit light \(P^{4}\) is \(\left( \begin{array}{cccc}0&P_{y}^{4}&P_{z}^{4}&1\end{array} \right)^{T}\) and the direction is parallel to the X axis. The light vector is \(\left[ \begin{array}{cccc}x&0&0&0\end{array} \right]^{T}\). Therefore, the ray equation \(L^{4}\) is this:  
\[\left[