# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 3 Multi-DOF Displacement Computation
### Solution Component Uniqueness Theory
: For the \(\mathbf{J}\left(\mathbf{X}^{*}\right)\) with full column rank, it follows from the continuity of \(\mathbf{F}(\mathbf{X})\) that there exists the neighborhood of \(\mathbf{X}^{*}\) as \(\mathrm{B}\left(\mathbf{X}^{*},\mathbf{r}\right)\) with \(\mathbf{r}>0\) such that \(\mathrm{rank}[\mathbf{J}(\mathbf{X})]=\mathrm{rank}\left[\mathbf{J}\left( \mathbf{X}^{*}\right)\right]\) holds for all the \(\mathbf{X}\in\mathrm{B}\left(\mathbf{X}^{*},\mathbf{r}\right),\) i.e., for all the \(\mathbf{X}\in\mathrm{B}\left(\mathbf{X}^{*},\mathbf{r}\right)\), \(\mathbf{J}(\mathbf{X})\) has full column rank. Thus, it follows from the Newton method [22] that \(\mathbf{X}^{(k)}\) converges to the unique solution \(\mathbf{X}^{*}\). Therefore, all the components of \(\mathbf{X}\) converge to the unique solution, the proof is finished. **Theorem 2**. : _Let \(\mathbf{J}\left(\mathbf{X}^{*}\right)=[\mathbf{p}_{1},\ldots\mathbf{p}_{ \mathrm{N}}]\), where \(\mathbf{p}_{\mathrm{n}}\in\mathbb{R}^{M\times 1}\) for \(n=1,2,\ldots,N.\) Then, \(x_{n}\) is not unique if \(\mathrm{rank}\left[\mathbf{p}_{1},\ldots\mathbf{p}_{\mathrm{N}}\right]= \mathrm{rank}\left[\mathbf{p}_{1},\ldots\mathbf{p}_{\mathrm{n-1}},\mathbf{p}_{ \mathrm{n+1}},\ldots\mathbf{p}_{\mathrm{N}}\right]\) holds. Otherwise, \(x_{n}\) is unique._  
Proof. : According to the definition of the Jacobi matrix, the total differential of (12) at \(\mathbf{X}^{*}\) is  
\[\sum_{j=1}^{N}\mathbf{p}_{j}\Delta x_{j} = \mathbf{0} \tag{13}\] \[\text{If }\mathrm{rank}[\mathbf{p}_{1},\ldots\mathbf{p}_{ \mathrm{N}}]=\mathrm{rank}\left[\mathbf{p}_{1},\ldots\mathbf{p}_{\mathrm{n-1} },\mathbf{p}_{\mathrm{n+1}},\ldots\mathbf{p}_{\mathrm{N}}\right]\text{ holds, there is}\] \[\mathbf{p}_{n}=\sum_{j=1,j\
eq n}^{N}\alpha_{i}\mathbf{p}_{j} \tag{14}\]  
where \(\alpha_{\mathrm{j}}\in\mathbb{R}\). Substituting Equation (14) into Equation (13), then we have  
\[\sum_{j=1,j\
eq n}^{N}\mathbf{p}_{j}\big{(}\Delta x_{j}+\alpha_{j}\Delta x_{n} \big{)} = \mathbf{0} \tag{15}\]  
Let  
\[y_{l}=\left\{ \begin{array}{c}x_{l}+\alpha_{l}x_{n},1\leq l<n\\ x_{l+1}+\alpha_{l+1}x_{n},n\leq l\leq N-1\end{array} \right. \tag{16}\]  
Substituting Equation (16) into Equation (15), there is  
\[\left[\mathbf{p}_{1},\ldots,\mathbf{p}_{n-1},\mathbf{p}_{n+1},\ldots,\mathbf{ p}_{N}\right]\cdot\left[\Delta y_{1},\ldots,\Delta y_{N-1}\right]^{T} = \mathbf{0} \tag{17}\]  
Taking (17) as the total differential of the equation system, \(\mathbf{F}(\mathbf{Y})=\mathbf{0}\) where \(\mathbf{Y}=\left[y_{1},\ldots,y_{\mathrm{N-1}}\right]^{\mathrm{T}}\), for which the Jacobi matrix is \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{\mathrm{n-1}},\mathbf{p}_{\mathrm{n+1} },\ldots\mathbf{p}_{\mathrm{N}}\right]\). Without loss of generality, suppose that \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{\mathrm{n-1}},\mathbf{p}_{\mathrm{n+1} },\ldots\mathbf{p}_{\mathrm{N}}\right]\) has full rank, otherwise we can repeat the above process to cut down the column elements. Then, it followsfrom Lemma 1 that all the solutions of \(y_{1}\) are unique. Let \(y_{1}^{*}\) be the unique solution of \(y_{1}\), it follows from (16) that  
\[ \begin{pmatrix}1&&\alpha_{1}&&\\ &\ddots&&\vdots&\\ &&1&\alpha_{n-1}&&\\ &&&\alpha_{n+1}&1&\\ &&&\vdots&\ddots&\\ &&&\alpha_{N}&&&1\end{pmatrix} _{(N-1)\times N}\quad \begin{pmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{N}\end{pmatrix} \!=\! \begin{pmatrix}y_{1}^{*}\\ y_{2}^{*}\\ \vdots\\ y_{N-1}^{*}\end{pmatrix}  \tag{18}\]  
holds, which is the linear equation system of \(\left[x_{1},\ldots,x_{N}\right]^{T}\). Notice that the blanks in the matrix in (18) are all 0. It is easy to check that \(x_{n}\) in (18) has infinitely many solutions, which means \(x_{n}\) is not unique. The proof of the first half of the statement is finished.