# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 3 Multi-DOF Displacement Computation
### Principle of Displacement Computation  
In this Subsection, the algorithm of the displacement computation of the interferometer measurement model is proposed, and the high accuracy of the displacement computation result is verified. Since the classic Gauss-Newton method is not applicable to SNES, the Newton method with MP inverse is proposed to solve the displacement from the interferometer model (4). The linear expansion approximation of the equation system (12) at the given iteration \(\mathbf{X}^{(k)}\) is  
\[\mathbf{F}(\mathbf{X}^{(k)})+\mathbf{J}(\mathbf{X}^{(k)})(\mathbf{X}-\mathbf{ X}^{(k)})=\mathbf{0} \tag{25}\]  
In the Newton method, the solution of \(\mathbf{X}\) in (25) is taken as the next step \(\mathbf{X}^{(k+1)}\) in the iteration. In the case that \(\mathbf{J}\left(\mathbf{X}^{(k)}\right)\) is not a square matrix, the Gauss-Newton method solves \(\mathbf{X}^{(k+1)}\) by multiplying Equation (25) by \(\left[\mathbf{J}^{T}\left(\mathbf{X}^{(k)}\right)\mathbf{J}\left(\mathbf{X}^{ (k)}\right)\right]^{-1}\mathbf{J}^{T}\left(\mathbf{X}^{(k)}\right)\), which failswhen \(\mathbf{J}^{\mathsf{T}}\left(\mathbf{X}^{(k)}\right)\mathbf{J}\left(\mathbf{X}^{( k)}\right)\) is singular. In this paper, let Equation (25) be the inconsistent linear equation system of \(\left(\mathbf{X}-\mathbf{X}^{(k)}\right)\), then the minimum 2-norm solution is  
\[\mathbf{X}-\mathbf{X}^{(k)}=\mathbf{J}^{\mathsf{T}}(\mathbf{X}^{(k)})[-\mathbf{ F}(\mathbf{X}^{(k)})] \tag{26}\]  
where \(\mathbf{J}^{\mathsf{T}}\left(\mathbf{X}^{(k)}\right)\) is the MP inverse of \(\mathbf{J}\left(\mathbf{X}^{(k)}\right)\), which is calculated by singular value decomposition  
\[\mathbf{J}^{\mathsf{T}}(\mathbf{X}^{(k)})=\mathbf{V}_{1}\mathbf{S}_{1}^{-1} \mathbf{U}_{1}^{\mathsf{T}} \tag{27}\]  
where \(\mathbf{V}_{1},\mathbf{S}_{1},\mathbf{U}_{1}\) satisfy  
\[\mathbf{J}(\mathbf{X}^{(k)})=\mathbf{U}\mathbf{S}\mathbf{V}^{\mathsf{T}}= \begin{bmatrix}\mathbf{U}_{1}\\ \mathbf{U}_{2}\end{bmatrix} ^{\mathsf{T}} \begin{bmatrix}\mathbf{S}_{1}&\mathbf{0 }\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{V}_{1}\\ \mathbf{V}_{2}\end{bmatrix}  \tag{28}\]  
where \(\mathbf{S}\) is the diagonal matrix, whose diagonal elements are singular values in descending order. Therefore, according to Equation (26), the iteration formula is  
\[\mathbf{X}^{(k+1)}=\mathbf{X}^{(k)}+\mathbf{J}^{\mathsf{T}}(\mathbf{X}^{(k)})[ -\mathbf{F}(\mathbf{X}^{(k)})] \tag{29}\]  
The brief process of the proposed Newton method with MP inverse is summarized as follows. Since the convergence of the Newton method with MP inverse is discussed [22], Algorithm 2 can be used to solve the interferometer model (4). However, the solution component uniqueness needs to be verified. ```
1:Initialization: Given \(k_{\text{max}}\), \(\mathbf{X}^{(0)},\ k{:=}\ 0\)
2:While\(k<k_{\text{max}}\)do
3:Singular Value Decomposition \(\mathbf{J}\left(\mathbf{X}^{(k)}\right)=\mathbf{U}\mathbf{S}\mathbf{V}^{ \mathsf{T}}\)
4:Find \(\mathbf{V}_{1},\mathbf{S}_{1},\mathbf{U}_{1}\) satisfy (28)
5:\(\mathbf{J}^{\mathsf{T}}\left(\mathbf{X}^{(k)}\right)=\mathbf{V}_{1}\mathbf{S} _{1}^{-1}\mathbf{U}_{1}^{\mathsf{T}}\)
6:\(\mathbf{X}^{(k+1)}=\mathbf{X}^{(k)}+\mathbf{J}^{\mathsf{T}}\left(\mathbf{X} ^{(k)}\right)\left[-\mathbf{F}\left(\mathbf{X}^{(k)}\right)\right]\)
7:\(k=k+1\)
8:endwhile
```  
**Algorithm 2**Newton method with MP inverse. Algorithm 1 is used to compute the solution uniqueness of the 48 unknowns in Equation (4). Number the 48 unknowns as  
\[ \begin{array}{rl}\mathbf{X}=&\begin{bmatrix}x_{d1},x_{d2},\ldots,x_{d6},x_{ 1},x_{2},\ldots,x_{10},\\ &y_{d1},y_{d2},\ldots,y_{d6},y_{1},y_{2},\ldots,y_{10},\\ &\alpha_{1},\alpha_{2},\ldots,\alpha_{6},\theta_{1},\theta_{2},\ldots,\theta_ {10}]\end{array}  \tag{30}\]  
Then, the eigenvalues \([\sigma_{1},\ldots,\sigma_{48}]\) in Equation (21) are shown in Figure 6(a), and the IDFs of the 48 vectors in the Jacobi matrix are shown in Figure 6(b).