## 4 Simulation Experiments and Optimization
### Simulation Experiments
Figure 15: The precision for the solution of Nadam + ELU. The obtained experimental results after optimization of the Nadam optimizer, the training time required by the corresponding method to achieve the corresponding accuracy is significantly reduced compared with that before optimization. The results show that the proposed method can greatly improve the accuracy and efficiency of the robot kinematics solution. The optimized time results and efficiency are shown in Figure 7:0  
1. The relationship between joint angle and terminal position and orientation can be understood as nonlinear regression of \(x\) and \(y\), the nonlinear fitting is carried out by the full connection of layer to layer of DNN, represented as \(y=f(x)\). After running the program, the predicted result of terminal position and orientation can be obtained, and then compared with the theoretical analytical value, the absolute mean error of orientation forward kinematics solution is about 0.02\({}^{\circ}\), the absolute mean error of position forward kinematics solution is about 0.3 mm, they are all within the margin of error. It shows that there is little difference between the predicted value and the analytical solution, which can meet the precision requirement of engineering applications. The comparison curves between the analytic solution and the predicted solution of forward kinematics are shown in Figure 17, the selected sample data is shown in Figure 7:1
2. Similarly, the relationship between terminal position and orientation and joint angle can be understood as nonlinear regression of \(y\) and \(x\). The deep neural network is combined with the inverse kinematics solving program, and the unique inverse kinematics solution of the robot is determined by filtering and limiting conditions. After running the program, the predicted result of joint angle can be obtained and then compared with the theoretical analytical value, the absolute mean error of orientation inverse kinematics solution is about 10\({}^{-5}\), and it is almost close to 0. It shows that there is little difference between the predicted value and the analytical solution (negligible), which can meet the precision requirement of engineering applications. The comparison curves between the analytic solution and the predicted solution of inverse kinematics are shown in Figure 18, the selected sample data is shown in Figure 7:2 \begin{table}
\begin{tabular}{c c c c c} \hline
**Method** & **Time Required for 80\%** & **Time Required for** & **Time Required for** & **Time Required for** \\ **+ Accuracy** & **Accuracy** & **85\% Accuracy** & **90\% Accuracy** & **95\% Accuracy** \\ \hline SGD + Tanh & About 30 s & N/A & N/A & N/A \\ Nadam + Tanh & About 20 s & About 30 s & N/A & N/A \\ Nadam + ELU & About 20 s & About 30 s & About 45 s & About 10 min \\ \hline \end{tabular} (1)  
The relationship between joint angle and terminal position and orientation can be understood as nonlinear regression of \(x\) and \(y\), the nonlinear fitting is carried out by the full connection of layer to layer of DNN, represented as \(y=f(x)\). After running the program, the predicted result of terminal position and orientation can be obtained, and then compared with the theoretical analytical value, the absolute mean error of orientation forward kinematics solution is about 0.02\({}^{\circ}\), the absolute mean error of position forward kinematics solution is about 0.3 mm, they are all within the margin of error. It shows that there is little difference between the predicted value and the analytical solution, which can meet the precision requirement of engineering applications. The comparison curves between the analytic solution and the predicted solution of forward kinematics are shown in Figure 17, the selected sample data is shown in Table 8.  
\end{table} 
Table 7: The optimized time results and efficiency. Figure 17: Comparison curves of analytic and predictive solutions of forward kinematics.