## 6 Conclusions  
This paper presented an approach based on an image processing algorithm to perform path planning with collision avoidance. Our method, named topological path planning (TPP), uses morphological operations on a map of obstacles to create a topological path that is collision-free. This path is used to guide an SCARA manipulator and make it move around the obstacles from start to goal points. Because of the topological constraint, TPP provides smoother paths than the PRM, a desired characteristic to create the manipulator trajectories. However, because the number of points in TPP is higher than for TPP*, the latter is faster.  
We highlight that the TPP presented better results in relation to the PRM and TPP* in the distance criteria, which were the smallest, 42.86 cm, in relation to the PRM and TPP*'s 43.03 cm and 45.90 cm, respectively. Regarding the number of points on the path, the TPP also presented lower results, 13.42, while they were 13.80 for the PRM and 14.52 for the TPP*. The maximum jerk for TPP was the lowest among the others, which was 7.91 degree/s\({}^{3}\); for PRM, it was 10.91 degree/s\({}^{3}\), and for TPP*, it was 9.05 degree/s\({}^{3}\). Regarding the standard deviation of acceleration, the TPP also presented a lower result, 2.69, while that of the PRM was 3.35 and that of the TPP* was 2.87. In the computation time criterion, PRM presented the best result, 2.26 s, TPP* spent 2.75 s, and TPP spent 5.37 s, which was the longest time. From the five criteria compared, TPP presented the best result in four criteria, being below the others only in relation to computing time. With this, it can also be concluded that TPP can be used as a collision-free trajectory planning method for industrial manipulators.  
The performance of TPP was compared to the PRM, a fast method that is also based on the previous knowledge of the static obstacle map. The results show overall advantages to using TPP since it provides better manipulator control, considering the smaller variations in the joint velocities. Furthermore, TPP is a non-stochastic method that provides a collision avoidance path planning solution with repeatability and predictability, since, for a given scenario, the solution is always the same. Evaluating TPP only for 2D trajectory planning and the use of only two SCARA axes are the main shortcomings of this study. Furthermore, TPP is often slower than the PRM because the script generated by TPP contains more points than necessary, making it more difficult to find the shortest path. In future works we intend to optimize TPP and use the full SCARA model to evaluate it in pick-and-place tasks, considering 3D obstacles. Implementing TPP on our SCARA platform is not straightforward, requiring PLC programming to interface a computer system that generates the maps and the paths. These implementations will be carried out in future work.  
Conceptualization, J.G.B. and G.L.B.R.; methodology, J.G.B. and G.L.B.R.; software, J.G.B., G.L.B.R. and D.S.F.; validation, J.G.B., G.L.B.R. and D.S.F.; formal analysis, G.L.B.R.; investigation, J.G.B. and G.L.B.R.; resources, J.G.B., G.L.B.R. and D.S.F.; data curation, M.A.T. and A.L.O.; writing--original draft preparation, J.G.B., G.L.B.R., M.A.T. and A.L.O.; writing--review and editing, J.G.B., G.L.B.R. and D.S.F.; visualization, J.G.B., D.S.F. and G.L.B.R.; supervision, G.L.B.R.; project administration, J.G.B., D.S.F. and G.L.B.R. All authors have read and agreed to the published version of the manuscript.  
This research was funded by PROINFRA/PRPI/IFCE 13/2022 grant number 13208.  
Not applicable.  
The authors thank FUNCAP for the financial support of this work.  
The authors declare no conflict of interest.