# Vertex-Weighted Consensus-Based Formation Control with Area Constraints and Collision Avoidance
## 3 Experimental Results
### Simulation Experiments  
In Figure 3, the hexagonal formation results using the conventional approach are reported. We can see how the virtual relabeling algorithm works, which can be beneficial in avoiding local stagnation. However, it is clear that we are not treating every agent as unique since the final formation has no order. Starting from robot 0 in the anti-clockwise direction, we have the robots 3, 1, 5, 4, and 2, which is not the desired order. In contrast, the hexagonal formation results using the proposed approach are reported in Figure 4. In this case, we achieved the same formation without the need for virtual relabeling. The final positions give us the desired order. Here, we have the following robot order--0, 1, 2, 3, 4, and 5--in anti-clockwise direction. As mentioned, the conventional approach suffers from local stagnation for some initial conditions of the robots. The performance of both the conventional and proposed approaches are illustrated in Figures 5 and 6, where the drawback of [conventional approach arose. When starting in a formation where all robots are lined up, the control in the conventional approach only actuates along this line, causing all agents to remain stuck in this region, as depicted in Figures 3b  
One of the benefits of adding the desired signed area is that the agents now have more knowledge about the final formation. In the proposed approach, agents need fewer connections to achieve their objective, which is in contrast to the conventional approach in which consensus-based communication is often required, especially in the case of collision avoidance and formation ambiguities. As can be seen in Figure 7, the conventional approach was unable to achieve the final formation, despite all robots being in their correct order, as suggested by the virtual relabeling algorithm. However, the proposed approach successfully achieved the desired formation. Figure 8 shows the square formation results starting from a line and using four non-holonomic robots. The drawbacks of the conventional approach arose due to the robots being initially arranged in a line formation. However, this inconvenience occurs when all of them are oriented in the same direction as the line. In contrast, the proposed approach successfully achieved the desired formation pattern.  
Figure 4: Hexagonal formation results using the proposed approach. The \(\star\) indicates the start position, and \(\bullet\) represents the final position. (**a**) Starting positions. (**b**) Achieved formation pattern.  
Figure 5: Rectangular formation results starting from a diagonal. The \(\star\) indicates the start position, and \(\bullet\) represents the final position. (**a**) Starting positions. (**b**) Achieved formation pattern using the conventional approach. (**c**) Achieved formation pattern using the proposed approach.  
Figure 6: Hexagonal formation results starting from a line. The \(\star\) indicates the start position, and \(\bullet\) denotes the final position. (**a**) Starting positions. (**b**) Achieved formation pattern using the conventional approach. (**c**) Achieved formation pattern using the proposed approach.  
Figure 9 shows the formation results of two equilateral triangles using four non-holonomic robots. In this case, the performance of both approaches is compared under situations where ambiguities arise. The formation produced by conventional control meets prescribed edge lengths but deviates from the desired configuration. To realize the desired formation with conventional control, additional inter-agent connections would be needed. For this assessment, we utilized the following formation values: \(k_{01}=k_{02}=k_{12}=k_{13}=k_{23}=0.852\) and \(z_{012}=z_{132}=0.433\).