## 5 Discussion  
In this section, we present a quantitative and qualitative analysis of each algorithm regarding the path planning with collision avoidance and the manipulator dynamics. Figure 10 shows an overall comparison between the TPP path, features PRM, and TPP*. PRM is very fast for a small number of random nodes. However, this reduces its chance of finding a solution and the randomness provides different solutions for the same scenario, as shown in Figure 11. Indeed, in the worst case, for the same scenario, PRM can find a valid solution for one execution, but not find a valid solution for the next execution. This unpredictability and inconsistency are not desired in collaborative robotics. Because the TPP roadmap is built to preserve the topology of the free space, it constrains the resulting path at an intermediate distance between obstacles. The path is short, minimizes undesired detours, and ultimately prevents collisions with the obstacles, as shown in Figure 10. TPP finds solutions for any number of nodes, and every solution is feasible. Obviously, the greater the number of nodes, the greater its execution time. Despite TPP always finding the same solution for the same scenario, an alternative solution can be chosen if the prior must be discarded for any reason. Figure 10d illustrates an alternative solution obtained using TPP. Thus, TPP results highlight two main general advantages of its collision-free path: it is predictable and smooth. Although it provides a better single solution per scenario, it does not always deliver the shortest path and the greater number of nodes of TPP compared to TPP* makes the prior slower. Another relevant property is the fact that the TPP has only one parameter to adjust. This aspect makes the algorithm less dependent on external factors, thus forcing context independence and contributing to simplicity. Finally, TPP always generates collision-free paths, which makes it safer and more efficient. It is also fundamental in the context of real-time environments that require a predictable solution. \begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Energy Consumption [kWh]** \\ \hline PRM & 0.028 \(\pm\) 0.0084 \\ TPP* & 0.030 \(\pm\) 0.0095 \\ TPP & 0.029 \(\pm\) 0.0092 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average energy consumption of joint motors for 100 scenarios for PRM, TPP*, and TPP methods. Figure 9: Evaluation metrics for 100 random scenarios. Figure 12 shows an SCARA manipulator trajectory where the manipulator is positioned at the start, middle, and goal points for two scenarios. Figure 12 also shows that all paths are feasible. We can state that the paths generated by the methods are executable by the manipulator, without having problems with singularities. Another important issue that we can highlight is that the manipulator's movements are carried out within its workspace. The values of \(\theta\) for the PRM, TPP\({}^{*}\), and TPP, presented in Figure 13 were found from the inverse kinematic model presented in Equations (3) and (4). From the values of \(\theta\), the velocities, accelerations, and jerks can be found. It should also be noted here that the values of \(\theta\) will be sent to the joint controllers to carry out their displacements; that is, they are the controllers' set points. Figure 11: Example of PRM, showing two different solutions for the same scenario. Figure 10: Best features of TPP path (magenta) in comparison to PRM (blue) and TPP\({}^{*}\) (green). TPP delivers (**a**) shorter, (**b**) less conservative (fewer detours), (**c**) smother, and (**d**) alternative paths. Figure 12: SCARA manipulator trajectories for (**a**) PRM, (**b**) TPP*, and (**c**) TPP. Manipulator positioned at start (green), middle (black), and goal (red) points for two different scenarios (top and bottom rows). Figure 13: Results comparison for same scenario. (**a,b**) PRM path and thetas. (**c,d**) TPP* path and thetas. (**e,f**) TPP path and thetas.