## 4 Results
### Performance Evaluation  
Table 4 shows a comparison of metrics' average performance: distance, number of points on the path, maximum jerk, acceleration deviation, and computing time for 100 scenarios, shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Method**} & **Distance** & **Points on** & **Maximum Jerk** & **Accel.** & **Computing** \\  & **[cm]** & **the Path** & **[Degree/s\({}^{3}\)]** & **Deviation** & **Time [s]** \\ \hline PRM & 43.03 \(\pm\) 6.87 & 13.80 \(\pm\) 2.05 & 10.91 \(\pm\) 2.81 & 3.35 \(\pm\) 0.57 & **2.26 \(\pm\) 0.82** \\ TPP* & 45.90 \(\pm\) 6.48 & 14.52 \(\pm\) 1.91 & 9.05 \(\pm\) 4.01 & 2.87 \(\pm\) 0.55 & 2.75 \(\pm\) 0.52 \\ TPP & **42.86 \(\pm\) 6.29** & **13.42 \(\pm\) 1.82** & **7.91 \(\pm\) 3.22** & **2.69 \(\pm\) 0.57** & 5.37 \(\pm\) 1.82 \\ \hline \end{tabular}
\end{table}
Table 4: Comparisons of metrics of methods for path planning shown in Figure 8. The average for 100 scenarios.  
Figure 7: Examples of PRM, TPP* and TPP, respectively: (**a–c**) the nodes, (**d–f**) the roadmps and (**g–i**) the final shortest path.  
As shown in Table 4, we highlight that the TPP presented better results in relation to the PRM and TPP* in the distance criteria, which were the smallest in relation to the PRM and TPP*. Regarding the number of points on the path, the TPP also presented lower results when compared to the PRM and TPP*. The paths shown in Figure 7 illustrate that the TPP path tends to be shorter, thus containing fewer points. The maximum jerk for TPP was the lowest among the others. Regarding the standard deviation of acceleration, the TPP also presented a lower result compared to the PRM and TPP*. Figure 8 shows how smooth the TPP path tends to be, considering the method's intrinsic topological restriction. The TPP has an average processing time that is twice that required by the PRM. However, TPP* has no statistically significant difference to PRM in this criterion. This means that, depending on the application's needs, TPP* can be used instead of TPP, ensuring all the advantages of TPP over PRM.  
Figure 9 shows details of the performance metrics. It can be highlighted here, according to Figure 9 and Table 4, that the number of points on the path is similar for the three methods. However, for TPP, the acceleration deviation value is smaller than the other methods. This is important for the application requirements and is explained because the trajectories of the manipulator generated by TPP are smoother.  
Although the PRM is clearly faster, due to the randomness of PRM points initialization, some distributions may result in areas of scarcity, thus impeding the shortest path algorithm to find a solution, since it depends on the neighbor distance parameter. Using greater values for the neighbor distance can help to solve this problem.  
Table 5 presents the average energy consumption of joint motors for 100 scenarios. The energy consumption values were calculated using Equations (8) and (9) and with the torque values calculated from the dynamic model (Equations (6) and (7)). Notice that TPP's  
Figure 8: Results of (**a**) PRM, (**b**) TPP*, and (**c**) TPP for three different scenarios, same scenario per row.  
trajectory presents a similar average energy consumption for the application presented in this research when compared to the other methods.