## 4 Performance Analysis
### Performance Analysis of the IKM Synthesized for the Hexapod's Leg  
Figure 10 compares the results obtained with the IKM synthesized for the hexapod's leg and those of the corresponding reference model. This figure shows that the calculated IKM successfully computes all the possible solutions inside the workspace of the hexapod's leg, while also finding all the singularities of the leg's mechanism, marked by the red circles (\"Singularities\"). These singularities are located in the axis defined by the intersection of the planes \(X=0\cap Y=0\).  
Figure 9: Trajectory tracking analysis for the PUMA’s IKM. The red dots represent the outputs of the reference model, for each of the PUMA’s DoFs, when it is supplied with the predetermined trajectory, while the continuous blue lines are the IKM’s outputs for that same trajectory. The upper left figure presents the computed outputs for the first DoF (\(q_{1}\)), the upper right one corresponds to the second (\(q_{2}\)), while the bottom figure are the ones for the third DoF (\(q_{3}\)). The data show that the IKM’s outputs follow the ones of the reference model, with high accuracy and a negligible error along all the desired trajectory, in the robot’s three DoFs.  
Figure 11 shows the hexapod's IKM performance following a trajectory. As shown, the outputs given by our IKM allow the hexapod's leg to follow any trajectory with a negligible positioning error.  
Table 7 contains a summary of the computation times required by the hexapod's IKM to follow the test trajectory (row identified as \"IKM\"), and compares them with the ones of the reference model (row named \"Ref\"). As it happened in the PUMA's case, these times are close enough, while also achieving negligible positioning error. Thus, it can be concluded that the hexapod's IKM, synthesized by the developed procedure, is equivalent to the reference model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mod** & **Min [s]** & **Avg [s]** & **Max [s]** \\ \hline Ref. & 0.98 \(\times\) 10\({}^{-4}\) & 1.23 \(\times\) 10\({}^{-4}\) & 1.50 \(\times\) 10\({}^{-3}\) \\ IKM & 2.19 \(\times\) 10\({}^{-4}\) & 2.68 \(\times\) 10\({}^{-4}\) & 3.69 \(\times\) 10\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Computation Times of the hexapod’s IKM and its reference model (“Ref.” row), for all the points in the predetermined trajectory. Column “Avg [s]” presents the average computation times obtained during the trajectory’s execution (in seconds), while “Min [s]” and “Max [s]” show the minimum and maximum registered times, respectively.  
Figure 10: Performance analysis of the synthesized IKM for the hexapod’s leg. The left side shows the isometric view of the leg’s workspace, while the right side presents the top, side, and front views of the same workspace. Marked in green (“Correctly calculated”) are all the the positions in which the IKM obtains the same amount of solutions as the reference model, with an RMS error lesser than 1 \(\times\) 10\({}^{-8}\). The red circles correspond to the singularities of the leg’s mechanism that are correctly identified by the synthesized IKM. The data show that all the positions of the analyzed workspace are correctly calculated.