# Orbital-Based Automatic Multi-Layer Multi-Pass Welding Equipment for Small Assembly Plates
## 3 Equipment Debugging and Simulation Analysis
### Simulation Analysis of the Robotic Arm
#### 3.2.1 Kinematic Analysis of the Robotic Arm
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Serial Number** & \(\theta_{1}\)/\({}^{\circ}\) & \(\theta_{2}\)/\({}^{\circ}\) & \(\theta_{3}\)/\({}^{\circ}\) & \(\theta_{4}\)/\({}^{\circ}\) & \(\theta_{5}\)/\({}^{\circ}\) & \(\theta_{6}\)/\({}^{\circ}\) \\ \hline
1 & 30 & 30 & 45 & 45 & 60 & 60 \\
2 & 30 & \(-60\) & \(-45\) & \(-135\) & \(-60\) & \(-120\) \\
3 & 30 & \(-109.6\) & 45 & 76.84 & 38.96 & \(13.29\) \\
4 & 30 & \(-109.6\) & 45 & \(-103.16\) & \(-38.96\) & \(-166.7\) \\
5 & \(-150\) & \(-250.3\) & \(-45\) & 76.84 & \(-38.96\) & \(-166.7\) \\
6 & \(-150\) & \(-250.3\) & \(-45\) & \(-103.16\) & \(-38.96\) & \(13.29\) \\
7 & \(-150\) & 60 & 45 & 45 & \(-60\) & \(-120\) \\
8 & \(-150\) & 60 & 45 & \(-135\) & 60 & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Inverse kinematics solution details. The above method uses the [PERSON] iterative formula to solve the inverse kinematics of the robotic arm; alternatively, it can be solved using the SVD method, which is a computational method for finding the pseudoinverse of a non-square matrix, as shown in Equation (32):  
\[ \begin{array}{l}f_{\theta_{d}}=\left[p_{x_{4}},p_{y_{4}},p_{z_{4}},\left(r_{1},r_{2},r_{3},r_{4},r_{5},r_{6},r_{7},r_{8},r_{9}\right)_{d}\right]^{T}\\ f_{\theta_{n-1}}=\left[p_{x_{n-1}},p_{y_{n-1}},p_{z_{n-1}},\left(r_{1},r_{2},r_ {3},r_{4},r_{5},r_{6},r_{7},r_{8},r_{9}\right)_{n-1}\right]^{T}\\ \theta_{n}=\theta_{n-1}+SVD(Jaco(\theta_{n-1}))(f_{\theta_{d}}-f_{\theta_{n-1}} )\end{array}  \tag{32}\]  
After the kinematic solution of the robotic arm is completed, MATLAB Robotics Toolbox 10.4 is used to verify the robotic arm's postures. Four sets of experimental postures are selected. The joint variable values are \([pi/4,pi/4,pi/3,pi/3,pi/6,pi/6]\), \([pi/2,pi/4,pi/2,pi/4,pi/3,pi/3]\), \([0,pi/2,pi/6,0,pi/2,pi/6]\). The corresponding forward kinematic end-effector pose matrices T1, T2, T3, T4 and the inverse kinematic solutions with corresponding joint angle values and pose matrices Ta, Tb, Tc, Td are obtained as shown below, where the fourth set of posture is the DH model diagram of the initial posture when the rail-mounted robotic arm starts, as seen in Figure 26.  
\[ \begin{array}{l}T_{1}=\left[\begin{array}{cccc}-0.9212&-0.3066&0.2396&487.6 2\\ 0.3509&-0.3887&0.8519&391.79\\ -0.1681&0.8689&0.4656&537.32\\ 0&0&0&1\end{array}\right]\\ T_{3}=\left[\begin{array}{cccc}0.75&0.433&0.5&358.842\\ 0.5&-0.866&0&0\\ 0.433&0.25&-0.866&484.967\\ 0&0&0&1\end{array}\right]\\ T_{4}=\left[\begin{array}{cccc}1&0&0&-383.5\\ 0&1&0&0\\ 0&0&1&386.9\\ 0&0&0&1\end{array}\right]\\ \end{array}  \tag{33}\]  
\[ \begin{array}{l}T_{4}:\theta_{1}\sim\theta_{6}:(45^{\circ},45^{\circ},60^{ \circ},60^{\circ},30^{\circ},30^{\circ})\\ T_{c}:\theta_{1}\sim\theta_{6}:(0^{\circ},90^{\circ},30^{\circ},0^{\circ},90^{ \circ},30^{\circ})\\ \end{array} \]  
At this point, the calculation of experimental data within the effective variable ranges of joint angles for four sets of six-axis welding robotic arms has been completed. The output corresponding postures and joint angles have been verified, further confirming the correctness of the model and calculations, providing support for subsequent simulation work [23]. Figure 26: Robotic arm attitude verification diagram.