## 3 Simulations and System Validation  
Tests were conducted virtually to evaluate and refine the performance of the robots in a controlled environment. The first step involved validating the initial scenario, which entailed transporting an object from a specific point in the working area of Robot 1 to an endpoint in the working area of Robot 2. This scenario is crucial for measuring the efficiency and coordination of robots when transferring objects between virtual work areas. To address this scenario, three specific cases representing different levels of complexity were proposed using obstacles with dimensions of 0.8 m \(\times\) 0.03 m \(\times\) 0.4 m.  
First case: Obstacles are present only in the working area of Robot 1. Robot 1 must detect and avoid these obstacles while transporting the object to the transfer zone. This challenge tests the navigation and adaptability of the robot in a partially obstructed virtual environment. The relevant positions and dimensions in this case are presented in Table 4.  
In this virtual scenario, Robot 1 demonstrates its ability to efficiently maneuver around an obstacle located in its working area, thereby ensuring that the object is transferred without incident to the transfer point. See Figure 11.  
Second case: In this scenario, obstacles are present only in the working area of Robot 2. Robot 1 transports the object to the transfer zone without any issues; however, Robot 2 must navigate around obstacles when receiving and moving the object to the final point. This case evaluated Robot 2's ability to operate efficiently in a restricted virtual environment. The positions of the objects and obstacles are presented in Table 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Position in X (m)** & **Position in Y (m)** & **Position in Z (m)** \\ \hline Piobj & 0.1406 & 1.1435 & 0.33 \\ PFobj & 0.5494 & \(-\)2.98 & 0.33 \\ Ptrans & 0.1756 & \(-\)1.1227 & 0.33 \\ Center of the Obstacle & 0.6046 & 0.4783 & 0.4 \\ \hline \hline \end{tabular} where Piobj: initial position of the object; Pfobj: final position of the object; and Ptrans : location of the transfer point.  
\end{table} 
Table 4: Initial, transfer, and final positions of the object to be transported in the first case. In the virtual environment shown in Figure 12, Robot 2 demonstrated its ability to navigate around two obstacles located in its working area, ensuring that the object was moved safely and efficiently to its final location. Third case: In this scenario, obstacles are present in the working areas for Robots 1 and 2. This is the most complex scenario because both robots must demonstrate their detection, navigation, and cooperation skills to overcome obstacles and successfully complete a task. Coordination between robots is crucial for avoiding collisions and optimizing transport time. The positions of the objects and obstacles are presented in Table 6.  
Figure 11: Transfer of an object from one place to another with obstacles in Robot 1’s working area: (**a**) initial position of the robots; (**b**) Robot 1 transfers the object; (**c**) Robot 2 transfers the object; and (**d**) task completion. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Position in X (m)** & **Position in Y (m)** & **Position in Z (m)** \\ \hline Piobj & 0.4560 & 0.8165 & 0.33 \\ PGbj & 0.4560 & \(-\)2.98 & 0.33 \\ Ptrans & 0.19 & \(-\)0.8657 & 0.33 \\ Center of Obstacle 1 & 0.8282 & \(-\)1.4745 & 0.4 \\ Center of Obstacle 2 & 1.2083 & \(-\)1.9217 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Positions of the object to be transported and the obstacles in the second case. In this virtual environment, the robots must work together efficiently. Robot 1 must navigate and avoid obstacles in its area while transporting an object to a transfer point. Once there, Robot 2 must receive the object and, in turn, avoid obstacles in its own environment to carry the object to the final location.