# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 2 Interferometer Measurement Model
### 3-DOF Displacement Measurement Model
\begin{array}{l}V_{n1}=\frac{2\cos(\theta_{n}-\alpha_{1})}{\cos(2\beta_{ 2}-2\alpha_{1})}[(x_{n}-x_{Al})\cos\theta_{n}+(y_{n}-y_{Al})\sin\theta_{n}-b/2 ]+\frac{2\cos\alpha_{1}}{\cos 2\alpha_{1}}(x_{Al}+b/2)\\ V_{n2}=\frac{2\cos(\theta_{n}-\alpha_{1})}{\cos(2\beta_{2}-2\alpha_{1})}[(x_{n} -x_{Al})\cos\theta_{n}+(y_{n}-y_{Al})\sin\theta_{n}-b/2]+\frac{2\cos\alpha_{1} }{\cos 2\alpha_{1}}(x_{Al}+b/2)\\ V_{n3}=\frac{2\cos(\theta_{n}-\alpha_{3})}{\cos(2\beta_{2}-2\alpha_{1})}[(x_{n} -x_{Al})\cos\theta_{n}+(y_{n}-y_{Al})\sin\theta_{n}-b/2]+\frac{2\cos\alpha_{1} }{\cos 2\alpha_{1}}(x_{Al}+b/2)\\ V_{n4}=\frac{2\cos(\theta_{n}-\alpha_{4})}{\cos(2\beta_{2}-2\alpha_{1})}[(y_{n }-y_{Al})\cos\theta_{n}+(x_{n}-x_{Al})\sin\theta_{n}-h/2]+\frac{2\cos\alpha_{1 }}{\cos 2\alpha_{1}}(y_{Al}+h/2)\\ V_{n5}=\frac{2\cos(\theta_{n}-\alpha_{5})}{\cos(2\beta_{2}-2\alpha_{1})}[(y_{n }-y_{Al})\cos\theta_{n}+(x_{n}-x_{Al})\sin\theta_{n}-h/2]+\frac{2\cos\alpha_{1 }}{\cos 2\alpha_{1}}(y_{Al}+h/2)\\ V_{n6}=\frac{2\cos(\theta_{n}-\alpha_{4})}{\cos(2\beta_{2}-2\alpha_{1})}[(y_{ n}-y_{Al})\cos\theta_{n}+(x_{n}-x_{Al})\sin\theta_{n}-h/2]+\frac{2\cos\alpha_{1 }}{\cos 2\alpha_{1}}(y_{Al}+h/2)\\ V_{n6}=\frac{2\cos(\theta_{n}-\alpha_{4})}{\cos(2\beta_{2}-2\alpha_{1})}[(y_{ n}-y_{Al})\cos\theta_{n}+(x_{n}-x_{Al})\sin\theta_{n}-h/2]+\frac{2\cos\alpha_{1 }}{\cos 2\alpha_{1}}(y_{Al}+h/2)\\ \end{array} \right. \tag{4}\]  
where \(V_{ni}\) is the reading of the \(i\)th interferometer at the \(n\)th measurement position, \(n=1,\ 2,\dots,\ 10\), \(i=1,\ 2,\dots,\ 6\). The equation system (4) is the proposed 3-DOF displacement measurement model. As we can see, the number of equations is 60, which is larger than the number of unknowns (48 in total, including \((x_{n},y_{n},\theta_{n}),n=1,\ 2,\dots,\ 10\) and \((x_{di},y_{di},\alpha_{i}),i=1,\ 2,\dots,\ 6\)). The 3-DOF displacement can be computed from Equation (4) by using the nonlinear equations system algorithm, which is discussed in Section 3. From Equation (4), it should be noted that since the installation positions and angles \((x_{di},y_{di},\alpha_{i})\) are unknowns, the installation errors will not bring errors to the displacement computation. In addition, there is no linear approximation in Equation (4), which also makes the displacement computation result accurate. It should also be noted that the number of interferometers and the number of measurement positions in the measurement model can be varied, and their selection rule is as follows: let the number of interferometers be \(M\) and the number of measurement positions be \(N\) (in Equation (4), \(M=6\) and \(N=10\)). According to the above analysis, there are \(MN\) equations and \((3M+3N)\) unknowns in the model. To make the number of unknowns not greater than the number of equations, there is  
\[3M+3N\leq MN \tag{5}\]  
It can be seen from Equation (5) that \(M\) is supposed to be larger than 3, otherwise the number of equations will never catch up with the number of unknowns. Hence, Equation (5) can be rewritten as  
\[N\geq 3+\frac{9}{M-3} \tag{6}\]  
Equation (6) shows that \(N\) can be smaller with larger \(M\). Under the restriction of Equations (5) and (6), less \(MN\) means less computational complexity, but large \(MN\) provides enough redundant information to solve the unknowns. Therefore, it is selected that \(M=6\) and \(N=10\) in this section.