## 2 The Mathematical Modeling and Simulation  
In order to eliminate the Abbe measurement error, we designed a mathematical model of SMLI which can synthesize the three laser light paths into the one measured center line of the step gauge, as shown in Figure 1. The P\({}_{1}\), P\({}_{2}\) and P\({}_{3}\) are the installation points of three reflector mirrors, and the O is the measurement point of the step gauge. The three reflector mirrors and the step gauge are installed in the same moveable platform. The \(\overline{\text{P}_{1}\text{P}_{1}^{\prime}}\) is the first laser measurement path during the measurement that is marked as l\({}_{1}\)\(u_{1}\) (l\({}_{1}\) is the measurement value of the first interferometer, and \(u_{1}\) is its unit directional vector), the \(\overline{\text{P}_{2}\text{P}_{2}^{\prime}}\) is the second laser measurement path that is marked as l\({}_{2}\)\(u_{2}\) (l\({}_{2}\) is the measurement value of the second interferometer, and \(u_{2}\) is its unit directional vector), the \(\overline{\text{P}_{3}\text{P}_{3}^{\prime}}\) is the third laser measurement path that is marked as l\({}_{3}\)\(u_{3}\) (l\({}_{3}\) is the measurement value of the third interferometer, and \(u_{3}\) is its unit directional vector), and the O O, (the O point is the first measurement position and the O' is the nth measurement position) is the synthesized measurement path of the measured step gauge that is marked as r (r is the only unknown quantity). The \(\beta_{\text{i}}\) is the angle between the direction of OPi and X axis of coordinate system, which is known and definite. The \(\theta_{\text{i}}\) is the angle between the direction of PiPi' (i = 1,2,3)and the Z axis of the coordinate system, which represents the change of pitch angle and yaw angle during the measurement and is measured by the auto-collimator. According to Figure 1, the mathematical model based on the closed-loop vector principle [15; 16; 17] is as follows. \[\overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{e}_{i}}+l_{i}\overrightarrow{\mathbf{u}_{ i}}-\overrightarrow{\mathbf{e}_{i}},\ i=1,2,3, \tag{1}\]  
where \(\overrightarrow{\mathbf{e}_{i}}=e_{i}\Big{(}\ \cos\beta_{i}\ \ \ \sin\beta_{i}\ \ \ 0\ \ \Big{)}^{T}\ (i=1,2,3)\),\(\|\overrightarrow{OP_{i}}\|=\|\overrightarrow{\mathbf{e}_{i}}\|_{i}\)it is known. where \(\overrightarrow{\mathbf{u}_{i}}=(\sin\theta_{i}\cos\beta_{i}\ \sin\theta_{i}\sin\beta_{i}\ \cos\theta_{i})^{T}\),it is known. Then, \(l_{i}\) is also known. Changing Equation (1) to Equation (2):  
\[(\mathbf{r}-\mathbf{e}_{i}-l_{i}\mathbf{u}_{i})^{T}(\mathbf{r}-\mathbf{e}_{i}-l_{i}\mathbf{u}_{i})=e_{ i}^{2}. \tag{2}\]  
Expanding Equation (2) to Equation (3) is done as follows. \[\Big{[}(\mathbf{r}-\mathbf{e}_{i})^{T}-(l_{i}\mathbf{u}_{i})^{T}\Big{]}[(\mathbf{r}-\mathbf{e}_{i} )-(l_{i}\mathbf{u}_{i})]-e_{i}^{2}=0. \tag{3}\]  
Then, Equation (3) is expanded as follows. \[\Rightarrow(\mathbf{r}-\mathbf{e}_{i})^{T}(\mathbf{r}-\mathbf{e}_{i})-(\mathbf{r}-\bm {e}_{i})^{T}(l_{i}\mathbf{u}_{i})-(l_{i}\mathbf{u}_{i})^{T}(\mathbf{r}-\mathbf{e}_{i})+l_{i}^{ 2}-e_{i}^{2} =0,\] \[\Rightarrow(\mathbf{r}-\mathbf{e}_{i})^{T}(\mathbf{r}-\mathbf{e}_{i})+l_{i}^{2}-e _{i}^{2}-(\mathbf{r}-\mathbf{e}_{i})^{T}(l_{i}\mathbf{u}_{i})-(l_{i}\mathbf{u}_{i})^{T}(\mathbf{r} -\mathbf{e}_{i}) =0.\]  
Suppose the coordinate of point O is (0,0,0), and the coordinate of point O' is (x,y,z), then:  
\[G_{i}-l_{i}\mathbf{r}^{T}\mathbf{u}_{i}+l_{i}\mathbf{e}_{i}^{T}\mathbf{u}_{i}-l_{i}\mathbf{u}_{i}^{ T}\mathbf{r}+l_{i}\mathbf{u}_{i}^{T}\mathbf{e}_{i}=0, \tag{4}\]  
Figure 1: Mathematical model of Synthesis Technology of Multi-Path Laser Interferometers (SMLI). where:  
\[G_{i}=\left(\mathbf{r}-\mathbf{e}_{i}\right)^{T}\left(\mathbf{r}-\mathbf{e}_{i} \right)+l_{i}^{2}-e_{i}^{2}=\mathbf{r}^{T}\mathbf{r}-\mathbf{r}^{T}\mathbf{e}_{i}-\mathbf{e}_{i}^{T} \mathbf{r}+\mathbf{e}_{i}^{T}\mathbf{e}_{i}+l_{i}^{2}-e_{i}^{2}\] \[=x^{2}+y^{2}+z^{2}-\left( \begin{array}{cccc}x&y&z\end{array}  \right)\left( \begin{array}{cccc}e_{i}\cos\beta_{i}&e_{i}\sin\beta_{i}&0\\ 0\end{array}