# A Neural Network Based Approach to Inverse Kinematics Problem for General Six-Axis Robots
## 5 Results and Discussion
### Test with Random Pose
Error in this magnitude is sufficient for some collaborative robots, such as those for goods delivery. However, the percentage falls to 34.27% when the standard rises to 1 mm and 0.1\({}^{\circ}\), and further reaches 1.96% at 0.1 mm and 0.1\({}^{\circ}\). Apart from these, under certain circumstances, the error can reach up to 1.5 centimeters and 4.6\({}^{\circ}\). This poses substantial challenges in tasks that require significantly high precision, such as surgery. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{e_{x}}\) **(m)** & \(\mathbf{e_{y}}\) **(m)** & \(\mathbf{e_{z}}\) **(m)** \\ \hline mean absolute error & 0.0004 & 0.0005 & 0.0005 \\ minimum absolute error & 6 \(\times\) 10\({}^{-8}\) & 3.6 \(\times\) 10\({}^{-7}\) & 1.9 \(\times\) 10\({}^{-7}\) \\ maximum absolute error & 0.014059 & 0.007147 & 0.007430 \\ \hline  & \(\mathbf{e_{\phi}}\) **(*)** & \(\mathbf{e_{\theta}}\) **(*)** & \(\mathbf{e_{\psi}}\) **(*)** \\ \hline mean absolute error & 0.1344 & 0.0812 & 0.1687 \\ minimum absolute error & 6.241 \(\times\) 10\({}^{-5}\) & 1.8 \(\times\) 10\({}^{-5}\) & 3.946 \(\times\) 10\({}^{-5}\) \\ maximum absolute error & 4.5682 & 2.0621 & 4.3986 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Error indicators for initial solution. Figure 6: Absolute errors distribution of initial solutions in each direction. Figure 7: Absolute errors distribution of optimized solutions in each direction. Thus, further optimization is required. By contrast, with Newton-Raphson based error minimization procedure, all the final solutions meet our presupposed requirements. More than expected, even setting a small iterative criterion as 0.001 radians (0.0573\({}^{\circ}\)), all solutions have errors less than 0.01\({}^{\circ}\) for orientation. One interesting thing is that, although the classification system fails to categorize all the test samples correctly, the system still manages to give reliable predictions all the time. These incorrectly labeled cases are further checked and details are sorted in Table 9. Comparing the initial solution norm with respective \(e_{max}\), it is surprised to find the influence of utilizing MLP from mismatched subspace was negligible since the former is much more smaller. An explanation for this is the IK function under some subspaces are extremely similar and the classifiers are thus prone to make mistakes in these cases. However, similar IK functions also result in analogical weights of the prediction system for these subspaces, making the impact insignificant as a result. The existence of this phenomenon definitely makes our algorithm more efficient than it appears to be. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(e_{x}\) **(m)** & \(e_{y}\) **(m)** & \(e_{z}\) **(m)** \\ \hline mean absolute error & 5.461\(\times 10^{-7}\) & 5.462\(\times 10^{-7}\) & 5.884\(\times 10^{-7}\) \\ min absolute error & 1.612\(\times 10^{-10}\) & 5.875\(\times 10^{-11}\) & 3.483\(\times 10^{-10}\) \\ max absolute error & 9.998\(\times 10^{-7}\) & 9.994\(\times 10^{-7}\) & 9.998\(\times 10^{-7}\) \\ \hline  & \(e_{\phi}\) **(\({}^{\circ}\))** & \(e_{\theta}\) **(\({}^{\circ}\))** & \(e_{\phi}\) **(\({}^{\circ}\))** \\ \hline mean absolute error & 1.868\(\times 10^{-4}\) & 1.233\(\times 10^{-4}\) & 2.572\(\times 10^{-4}\) \\ min absolute error & 1.209\(\times 10^{-7}\) & 2.025\(\times 10^{-8}\) & 1.059\(\times 10^{-8}\) \\ max absolute error & 5.7\(\times 10^{-3}\) & 2.1\(\times 10^{-3}\) & 6.3\(\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Error indicators for optimized solutions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample No. ** & **True Label** & \ \begin{tabular}{c} **Classification** \\ **System Output** \\ \end{tabular}  & \begin{tabular}{c} **Initial Solution** \\ **Error (\(L_{2}\) Norm)** \\ \end{tabular}