# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 3 Multi-DOF Displacement Computation
### Problem of Solving Nonlinear Equation System  
Rewriting the nonlinear equation system (4) as  
\[\mathbf{F(X)=0} \tag{9}\]  
Figure 4: Scheme of 6-DOF displacement measurement.  
where **X** is the variable consisting of the 48 unknowns; \(\textbf{F}(\cdot)\) is the equation function of Equation (4).  
The classical algorithm of the nonlinear equation system (9) is the Gauss-Newton method [16], in which the iteration formula is  
\[\textbf{X}^{(k+1)}=\textbf{X}^{(k)}-\left[\textbf{J}^{T}\Big{(}\textbf{X}^{(k)} \Big{)}\textbf{J}\Big{(}\textbf{X}^{(k)}\Big{)}\right]^{-1}\textbf{J}^{T}\Big{(} \textbf{X}^{(k)}\Big{)}\textbf{F}\Big{(}\textbf{X}^{(k)}\Big{)} \tag{10}\]  
where \(k\) is the number of iterations and \(\textbf{J}(\cdot)\) is the Jacobi matrix of \(\textbf{F}(\cdot)\).  
The problem is that in the computation process of solving (4) in the form of Equation (9) by using Equation (10), \(\textbf{J}\Big{(}\textbf{X}^{(k)}\Big{)}\) is always going to be singular and \(\textbf{J}^{T}\Big{(}\textbf{X}^{(k)}\Big{)}\textbf{J}\Big{(}\textbf{X}^{(k)} \Big{)}\) is not invertible due to its huge condition number, which makes it impossible to continue the iteration. In fact, for a nonlinear equation system with a singular Jacobi matrix, it is proven that the solution is not unique, and both the Newton method and Gauss-Newton method lose their convergence [16]. Hence, the improved algorithm needs to be applied. As it is proven that the solution of a nonlinear equation system with a singular Jacobi matrix is a set with at least one dimension and the Levenberg-Marquardt method (LM method) can be used to converge to a solution on the solution set [20], the iteration formula is  
\[\textbf{X}^{(k+1)}=\textbf{X}^{(k)}-\left[\textbf{J}^{T}\Big{(}\textbf{X}^{(k) }\Big{)}\textbf{J}\Big{(}\textbf{X}^{(k)}\Big{)}+\mu\textbf{I}\right]^{-1} \textbf{J}^{T}\Big{(}\textbf{X}^{(k)}\Big{)}\textbf{F}\Big{(}\textbf{X}^{(k) }\Big{)} \tag{11}\]  
where **I** is the identity matrix and \(\mu\) is the steepest descent factor.  
However, although there are improved algorithms such as the LM method and Newton method based on the MP inverse mentioned in [21] for solving nonlinear equations with a singular Jacobi matrix, and since the solution of Equation (4) is not unique and the iteration varies with the initial iterate, there is still a deviation between the computation result of the 48 unknowns and their true value. As an illustration, the computation of solving Equation (4) with the Newton method is simulated: a set of true values of unknowns is given, and 10 different initial values are randomly generated. In contrast, the iterations of \(x_{1}\) and \(x_{d1}\) using the 10 initial values are shown in Figure 5. The detail of Figure 5 is drawn as Figure 6 to observe the convergence results.  
Figure 5: (a) Iteration error of \(x_{1}\). (b) Iteration error of \(x_{d1}\).  
From Figure 5, the iterations of \(x_{1}\) and \(x_{d1}\) both converge; however, from Figure 6, the convergence of \(x_{d1}\) changes with initial values. Hence, the convergence accuracy of \(x_{1}\) and \(x_{d1}\) are different, which means that the accuracy of each component of the equation system solution is different. Since we only care about the solution of displacement \((x_{n},y_{n},\theta_{n}),n=1,\;2,\dots,\;10\), can it be proven that all the displacement unknowns can be accurately solved? The answer is yes, and the proof is given in the following sections.