# Improved Design of LNG Marine Loading Arm Docking Method Based on TRIZ Theory
## 4 Innovative Design of Problem-Solving Method Based on TRIZ
### Design of Automatic Control of LNG Marine Loading Arm  
As depicted in Figure 11, the LNG marine loading arm appears to be a three-axis mechanical arm. To establish the relationship between adjacent rods through the coordinate system and express it in the form of a matrix, the \(D-H\) method was employed. This method facilitates the establishment of the kinematics equation of the marine loading arm [24]. The homogeneous matrix of each adjacent member can be represented as follows:  
\[\overset{i-1}{i}T=\begin{bmatrix}c\theta_{i}&-s\theta_{i}&0&\alpha_{i-1}\\ s\theta_{i}ca_{i-1}&c\theta_{i}ca_{i-1}&-s\alpha_{i-1}&-d_{i}s\alpha_{i-1}\\ s\theta_{i}sa_{i-1}&c\theta_{i}sa_{i-1}&ca_{i-1}&d_{i}c\alpha_{i-1}\\ 0&0&0&1\end{bmatrix} \tag{6}\]  
where \(c\theta_{n}=cos\theta_{n}\), and \(s\theta_{n}=sin\theta_{n}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Serial Number**} & **Distance** & **Distance** & **Absolute Error** & \multirow{2}{*}{**Relative Error**} \\  & **Measurement** & **Calculation** & & \\ \cline{2-5}  & **(mm)** & **(mm)** & & \\ \hline
1 & 497 & 497.2 & 0.2 & 0.03\% \\
2 & 601 & 598.5 & 2.5 & 0.41\% \\
3 & 698 & 696.9 & 1.1 & 0.16\% \\
4 & 797 & 794.7 & 2.3 & 0.29\% \\
5 & 900 & 894.4 & 5.6 & 0.63\% \\
6 & 1000 & 990.1 & 9.9 & 0.99\% \\
7 & 1103 & 1097.0 & 6.0 & 0.54\% \\
8 & 1202 & 1193.6 & 8.4 & 0.69\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the results.  
Figure 11: Coordinate system diagram of LNG marine loading arm link.  
The parameters listed in Table 4 are substituted into Equation (6) in order to derive the transformation matrix for each link.  
\[\begin{array}{c}{}_{1}^{0}T=\begin{bmatrix}c\theta_{1}&-s\theta_{1}&0&0\\ s\theta_{1}&c\theta_{1}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\end{array} \tag{7}\]  
\[\begin{array}{c}{}_{2}^{1}T=\begin{bmatrix}c\theta_{2}&-s\theta_{2}&0&0\\ 0&0&-1&0\\ s\theta_{2}&c\theta_{2}&0&0\\ 0&0&0&1\end{bmatrix}\end{array} \tag{8}\]  
\[\begin{array}{c}{}_{3}^{2}T=\begin{bmatrix}c\theta_{3}&-s\theta_{3}&0&a_{2}\\ s\theta_{3}&c\theta_{3}&0&d_{3}\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\end{array} \tag{9}\]  
Multiplying the transformation matrices of each link, the expression for the position of the end of the LNG marine loading arm relative to the base can be obtained.  
\[\begin{array}{c}{}_{3}^{0}T=\begin{smallmatrix}0\\ 1\end{smallmatrix}T(\theta_{1})_{2}^{1}T(\theta_{2})_{3}^{2}T(\theta_{3})\end{array} \tag{10}\]  
According to the expression, it is possible to obtain all possible solutions for the variables \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\), followed by screening out the solutions that meet the specified requirements. This process is known as finding the inverse solution for the manipulator. To control the LNG marine loading arm, the solution program that satisfies the requirements can be programmed into the microcontroller [25]. In order to verify the feasibility of the proposed solution, a model of the LNG marine loading arm was printed using 3D printing technology and a motor was installed. The inverse solution was calculated and written in C language and then burned onto a microcontroller. Ultimately, the LNG marine loading arm can be controlled freely to move to a specified coordinate position. As shown in Figure 12, only by inputting the coordinates of the target into the microcontroller can the LNG marine loading arm model accurately move in front of the target.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Link (\(\theta\))** & \(\mathbf{a}_{i-1}\)/\({}^{(\circ)}\) & \(\mathbf{a}_{i-1}\) & \(\mathbf{d}_{i}\) & \(\theta_{i}\)/\({}^{(\circ)}\) \\ \hline
1 & 0 & 0 & 0 & \(\theta_{1}\) \\
2 & 90 & 0 & 0 & \(\theta_{2}\) \\
3 & 0 & \(a_{2}\) & \(d_{3}\) & \(\theta_{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: D–H parameter table.  
Figure 12: LNG marine loading arm model.