## 5 Discussion
The joint's torques were obtained from the dynamic model and Equations (6) and (7) of the manipulator, and are shown in Figure 14. A scenario is presented for each method: PRM, TPP*, and TPP. Torques were calculated by taking the trajectories, calculated by inverse kinematics (Equations (3) and (4)), in addition to the values of speeds and accelerations of the joints. An important contribution to this case study can be highlighted by the analysis carried out using the standard deviation of acceleration (see Table 4 and Figure 9), which is a measure that indicates how uniform the acceleration values are, which is why, in this research, this value was calculated to be analyzed. The idea of this check was to analyze how smooth the trajectories of the manipulator are; that is, how the joints behave. It is also important to highlight that the algorithms used are generating paths for a manipulator and not for a mobile robot. In view of this, some observations must be taken into consideration, as a manipulator will be transporting some material trapped in its claw. Another point that must be analyzed is the joint jerks, which represent how quickly a joint will move from one point to another. This must also be analyzed very carefully, as the manipulator may suffer damage that could damage them and cause problems in couplings and mechanical and electrical systems. Figure 14: SCARA manipulator joint’s torques calculated from trajectories, velocities, and accelerations for PRM, TPP*, and TPP results for same scenario. (**a,b**) PRM path and torques. (**c,d**) TPP* path and torques. (**e,f**) TPP path and torques. We highlight that TPP presents smoother trajectories than the PRM, and overall shorter paths than TPP*, as can be seen in Figure 10. Using TPP, the manipulator will have a better performance for the proposed application, though TPP* is faster. We also highlight here that TPP has some advantages over classical methods, such as the PRM and APF. TPP will always generate consistent paths; that is, it does not have probabilistic routes like the PRM, nor is it very sensitive to parameter adjustment as is the case with APF. APF adjustments of the three parameters are necessary, such as the gain in the attraction force of the endpoint, the gains in repulsion from the obstacles, and the initial position repulsion. In the case of TPP, as stated here, it will always generate the same path for the same adjustment and its single parameter is very easy to tune. The TPP also generated smooth trajectories for the displacements of the manipulator's joints, as shown in Figure 13.  
TPP is based on the framework of mathematical morphology that was developed for application in robotic manipulators and mobile robotics for path planning and collision avoidance applications. TPP produces a roadmap in only one step without using additional optimization to prune nodes and has only one parameter to adjust \(\in[0,\inf]\), which can be used to tune the compromise between processing time and trajectory quality (considering the application requirements) without changing the success rate to achieve the goal. Table 6 presents a summary comparison of the characteristics and each method mentioned in the references of this work and in comparison with the TPP, the proposed method. The method proposed in this paper will be used for pick-and-place tasks. Thus, the manipulator must perform slow movements, as it is moving with a load held in the gripper. According to our applications, for manipulation tasks, the speeds that we want to apply are in the range of 0.1 to 1 m/s (linear speeds) and 5 to 50 degrees/s (angular speeds) [35]. For inspection tasks, the speed of the camera (mounted on the end-effector) will be in the range of 0.10 to 0.30 m/s [40].