# A New Method for Finding the Shape of the Main Cable in the Special Cable Plane
## 2 Force Analysis of Spatial Special-Shaped Cable Plane
### Modified Least Squares Method  
The Gauss-Newton iteration method is one of the least squares methods, which has unique advantages in solving nonlinear overdetermined equations [26,27]. This paper introduces the modified least squares method to the calculation of finding the main cable shape of spatial special-shaped cable planes. In view of the difficulty in calculating the Jacobian matrix, this paper uses the influence matrix to replace the Jacobian matrix approximately. In order to avoid generating singular matrices in the calculation process, the Levenberg-Marquardt algorithm was used to modify the least squares method. In order to ensure that the objective function of each iteration is lowered, the step size is searched in three dimensions.  
The specific process of the modified least squares method is as follows:  
1. Determination of objective function  
The objective function of the least squares method depends on the calculation of the control node. Referring to the calculation function of the least squares method of the space cable plane proposed by [PERSON] [10], this paper proposes an objective function suitable for the calculation model of four coordinate constraints \(\phi(x)\):  
\[\phi(x)=\sum_{i=1}^{4}f_{i}^{2}(x)\]  
where \(f(x)\) is calculated according to the calculation model (Equation (4)) proposed in this paper.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(x_{i}\) & \(b_{i}\) & \(f(x)\geq f(b)\) & \(\Delta\theta_{i}\) & \multicolumn{6}{c}{\(s(\alpha_{i}\beta_{i})\)} \\ \cline{5-8}  & & & & \(\alpha_{i}\beta_{i}>0\) & \(\alpha_{i}\beta_{i}<0\) & \(\alpha_{i}=0\) & \(\beta_{i}=0\) \\ \hline
0 & 0 & False & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & True & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & False & \(\varepsilon\) & \(+1  
The calculation of the Jacobian matrix is the key to conduct the modified least squares method. Because there is no explicit expression for the calculation model of the space main cable, it is difficult to solve the partial derivative. This paper refers to the influence matrix method and uses the influence matrix to replace the Jacobian matrix approximately. The specific calculation of the influence matrix method is as follows:  
Take \(H_{x0}=H_{x0}+1\), \(H_{z0}=H_{z0}\), \(V_{0}=V_{0}\); \(H_{x0}=H_{x0}\), \(H_{z0}=H_{z0}+1\), \(V_{0}=V_{0}\); \(H_{x0}=H_{x0}\), \(H_{z0}=H_{z0}\), \(V_{0}=V_{0}+1\);  
Substitute them into Equation (4) and calculate the corresponding calculation error to obtain the influence matrix \(A\)  
\[A=\begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\\ A_{41}&A_{42}&A_{43}\end{bmatrix}\]
3. Calculate the best step size In order to avoid the problem of convergence failure due to the singularity of ATkAk in the calculation process, this paper uses the Levenberg-Marquardt algorithm to modify it.  
\[d^{(k)}=-\left(A_{k}^{T}A_{k}+\alpha I\right)^{-1}A_{k}^{T}f^{k}\]  
In order to ensure that the value of the objective function decreases in each iteration, a one-dimensional search is conducted from \(x(k)\) to obtain the best step size \(\lambda d^{(k)}\)  
\[\phi(x^{(k)}+\lambda_{k}d^{(k)})=\underset{\lambda}{\text{min}}\phi(x^{(k)}+ \lambda d^{(k)})\]
4. Correct the initial value \[x^{(k+1)}=x^{(k)}+\lambda_{k}d^{(k)}\]  
The initial value shall be corrected continuously, and the iteration shall be repeated until convergence.