# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 3 Multi-DOF Displacement Computation
### Solution Component Uniqueness Theory
As for the other condition that \(\text{rank}\left[\mathbf{p}_{1},\ldots\mathbf{p}_{N}\right]>\text{rank}\,\left[ \mathbf{p}_{1},\ldots\mathbf{p}_{n-1},\mathbf{p}_{n+1},\ldots,\mathbf{p}_{N}\right]\), the discussion is divided into the following two cases: For the case that \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{n-1},\mathbf{p}_{n+1},\ldots,\mathbf{p} _{N}\right]\) has full column rank, \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{N}\right]\) must have full column rank since its rank is larger than \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{p-1},\mathbf{p}_{n+1},\ldots,\mathbf{p} _{N}\right]\). Then, it follows from Lemma 1 that \(x_{n}\) is unique. For the other case that \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{n-1},\mathbf{p}_{n+1},\ldots,\mathbf{p} _{N}\right]\) has no full column rank, let \(\left[\mathbf{p}_{1},\ldots\mathbf{p}_{n-1},\mathbf{p}_{n+1},\ldots,\mathbf{p} _{N}\right]\)\(=\left[\mathbf{q}_{1},\ldots\mathbf{q}_{N}\right]\) after cutting down all the linearly dependent columns by taking the steps same as (14)-(17), where \(\left[\mathbf{q}_{1},\ldots\mathbf{q}_{N}\right]\) has full column rank and \(W<N-1\). Then, the new Jacobi matrix becomes \(\left[\mathbf{p}_{n},\mathbf{q}_{1},\ldots\mathbf{q}_{W}\right]\), which has full column rank. Hence, according to Lemma 1, \(x_{n}\) is unique. The proof of the second half of the statement is finished. From Theorem 2, the uniqueness of the solution component can be determined by computing whether the corresponding column in the Jacobi matrix is independent. Therefore, to verify the uniqueness of the displacement solution in the interferometer model, the columns dependence computation methods are discussed in the following. Although there are Gaussian elimination, determinant computation, and other numerical algorithms used to compute the dependence of vectors [23], since the numerical error exists in the Jacobi matrix computation, the dependence computation result might go wrong if the tolerance of the algorithms is improper (tolerance can only be selected according to empirical methods, without means of debugging or correction due to the lack of appropriate feedback). To make the dependence computation more accurate, a numerical algorithm for dependence computation based on principle component analysis (PCA) is proposed. For the given Jacobi matrix \(\mathbf{J}_{M\times N}=\left[\mathbf{p}_{1},\ldots\mathbf{p}_{N}\right]\), normalizing by  
\[\mathbf{A}=\left[\mathbf{p}_{1}-\text{mean}(\mathbf{p}_{1}),\ldots,\mathbf{ p}_{N}-\text{mean}(\mathbf{p}_{N})\right] \tag{19}\]  
The covariance matrix of \(\mathbf{A}\) is  
\[\mathbf{C}=\mathbf{A}^{T}\mathbf{A}/M \tag{20}\]  
Computing the eigenvalue decomposition  
\[\mathbf{C}=\mathbf{V}\mathbf{D}\mathbf{V}^{T} \tag{21}\]  
where \(\mathbf{V}=\left[\mathbf{v}_{1},\ldots,\mathbf{v}_{N}\right]\) is the eigenvector matrix with \(\mathbf{v}\in\mathbb{R}_{[-1,1]}^{N\times 1}\), \(\mathbf{D}=\mathbf{diag}(\sigma_{1},\ldots,\sigma_{N})\) is a diagonal matrix whose diagonal elements are eigenvalues in descending order. According to PCA, after setting the threshold \(\text{TH}_{\sigma}\) and \(r\) with that \(\sigma_{1}>\cdots>\sigma_{r}>\text{TH}_{\sigma}>\sigma_{r+1}>\cdots>\sigma_{N}\), we have  
\[\mathbf{J}_{\text{prin}}=\mathbf{J}\cdot\mathbf{V}_{r} \tag{22}\]  
for dimensionality reduction of \(\mathbf{J}\), where \(\mathbf{V}_{r}=\left[\mathbf{v}_{1},\ldots,\mathbf{v}_{r}\right]\). If \(\mathbf{p}_{n}\) in \(\mathbf{J}\) is strongly dependent, which indicates that it carries less information, we believe that it contributes less to \(\mathbf{J}_{\text{prim}}\) so that the weight of \(\mathbf{p}_{n}\) in Equation (22) is smaller. Suppose that \(\mathbf{V}_{r}=\left[\mathbf{w}_{1},\ldots,\mathbf{w}_{N}\right]^{T}\), \(\mathbf{w}\in\mathbb{R}^{r\times 1}\), then the weight of \(\mathbf{p}_{n}\) is \(\mathbf{w}_{n}\).