# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 3 Multi-DOF Displacement Computation
### Solution Component Uniqueness Theory  
As shown in Figure 6, the convergence of each variable of the nonlinear equation system with a singular Jacobi matrix is different, which is proven to be related to the uniqueness of the solution component in this section. To prove the solvability of displacement in the interferometer measurement model, the solution component uniqueness theory is proposed. Without loss of generality, consider the nonlinear equation system  
\[\mathbf{F}(\mathbf{X})=\mathbf{0} \tag{12}\]  
where \(\mathbf{X}\in\mathbb{R}^{N}\), \(\mathbf{F}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{M}\) is a continuously differentiable nonlinear function. Equation (12) is said to be a singular nonlinear equation system (SNES) if it has a singular Jacobi matrix. Let \(\mathbf{X}=[x_{1},\dots,x_{N}]^{T}\) be any one in the solution set of Equation (12), the solution component \(x_{n}\) is said to be unique or with uniqueness if there is only one value for it, where \(n\in\mathbb{N}\), \(1\leq n\leq N\). Let \(\mathbf{X}^{*}= \begin{bmatrix}x_{1}^{*},\dots,x_{N}^{*}\end{bmatrix} ^{T}\) be the true value of the unknowns in Equation (12). If we look at the interferometer model (4), \(\mathbf{X}^{*}\) is the true value of the 48 unknowns. Let \(\hat{\mathbf{X}}=[\hat{x}_{1},\dots,\hat{x}_{N}]^{T}\) be the solution computed by any improved algorithm. For the case that Equation (12) is SNES, as mentioned in Section 2, \(\hat{\mathbf{X}}\) varies with the initial iterate, and the solution of Equation (12) is a set with at least one dimension. \(\mathbf{X}^{*}\) and \(\hat{\mathbf{X}}\) both belong to the solution set, but there is no evidence that \(\mathbf{X}^{*}=\hat{\mathbf{X}}\) holds. The following Theorem 1 shows that the distance between \(x_{n}^{*}\) and \(\hat{x}_{n}\) depends on the solution component uniqueness. **Theorem 1**. : _For \(n\in\mathbb{N},1\leq n\leq N\), \(\hat{x}_{n}=x_{n}^{*}\) holds if \(\hat{x}_{n}\) is unique. Otherwise, \(\hat{x}_{n}=x_{n}^{*}\) might not hold._  
Proof. : For the condition that \(\hat{\mathbf{x}}_{n}\) is unique, there is only one value for the solution of \(\mathbf{x}_{n}\). Since both \(\mathbf{X}^{*}\) and \(\hat{\mathbf{X}}\) are the solution of (12), it is easy to check that \(\hat{\mathbf{x}}_{n}=x_{n}^{*}\) holds. In contrast, for the condition that \(\hat{\mathbf{x}}_{n}\) is not unique, there are two distinct solutions of (12) set as \(\mathbf{X}_{\rm A}\) and \(\mathbf{X}_{\rm B}\) with two distinct \(n^{\rm th}\) components set as \(\mathbf{x}_{\rm nA}\) and \(\mathbf{x}_{\rm nB}\). Without loss of generality,  
Figure 6: (**a**) Detail of iteration error of \(x_{1}\). (**b**) Detail of iteration error of \(x_{d1}\). suppose that \(\mathbf{X}^{*}=\mathbf{X}_{\mathrm{A}}\) and \(\hat{\mathbf{X}}=\mathbf{X}_{\mathrm{B}}\), then we have \(\mathbf{x}_{\mathrm{n}}^{*}=\mathbf{x}_{\mathrm{nA}}\) and \(\hat{\mathbf{x}}_{\mathrm{n}}=\mathbf{x}_{\mathrm{nB}}\), which indicate that \(\hat{\mathbf{x}}_{\mathrm{n}}\
eq\mathbf{x}_{\mathrm{n}}^{*}\) holds. Thus, the proof is finished. Theorem 1 implies that only for the solution component with uniqueness, the result computed by the algorithm always converges to the true value. Therefore, for the aim to solve the 30 displacement unknowns in Equation (4), the uniqueness of the corresponding 30 solution components must be verified. The following Theorem 2 provides the basis to help verify the solution component uniqueness. **Lemma 1**. : _All the components of solution \(\mathbf{X}\) are unique if \(\mathbf{J}\left(\mathbf{X}^{*}\right)\) has full column rank._  
Proof.