# Vertex-Weighted Consensus-Based Formation Control with Area Constraints and Collision Avoidance
## 3 Experimental Results
### Real-World Experiments  
In real-world experimentation, four Turtlebot3(r) Waffle Pi robots are considered. The Turtlebot3(r) Waffle Pi is an ROS-based differential-drive mobile robot used for ed  
Figure 8: Square formation results starting from a line and using four non-holonomic robots. 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 7: Hexagonal formation results starting from a rectangular one. The \(\star\) indicates the start position, and \(\bullet\) marks the final position. (**a**) Starting positions. (**b**) Achieved formation pattern using the conventional approach. (**c**) Achieved formation pattern using the proposed approach. Figure 9: Square formation results of two equilateral triangles and using four non-holonomic robots. 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. ucation, research, and product development applications [27] (see Figure 10). The ROS (ROS is a registered trademark of Open Robotics, USA) packages of Turtlebot3(r) provide access to send velocity control inputs and read its odometry. In this section, we perform two experiments, which we call Experiment 1 and Experiment 2. In Experiment 1, we will address the initial line position to achieve the square formation depicted in Figure 1(c) with the values listed in Table 3. In Experiment 2, we will tackle a specific ambiguity case where the desired formation is two equilateral triangles. Thus, the formation patterns are defined as \(k_{01}=k_{02}=k_{12}=k_{13}=k_{23}=0.852\), and the desired signed areas are \(Z^{*}_{012}=Z^{*}_{132}=0.433\). In the case of Experiment 1, Figure 11 illustrates the performance of the conventional approach without virtual relabeling. We have intentionally omitted the use of a virtual relabeling algorithm to highlight its significance within this control law. To assess the performance of the proposed approach, please refer to Figure 12. It is reported that the line starting position does not pose a challenge for the proposed approach, regardless of whether it is applied to holonomic or non-holonomic robots. The achieved formation patterns are depicted in Figures 10(b) and 11(b). In the case of Experiment 2, Figure 13 reports the performance of the conventional approach with virtual relabeling under the presence of ambiguities. Moreover, the performance of the proposed approach is illustrated in Figure 14. Notably, even with the use of the virtual relabeling algorithm, the conventional approach fails to attain the desired formation pattern, in stark contrast to the proposed approach, which correctly achieves the desired formation. The incorporation of the signed area term allows for achieving the desired formation pattern, even in the presence of ambiguities. Figure 11: Formation control results of the conventional approach for Experiment 1. The \(\star\) indicates the start position, and \(\bullet\) represents the final position. (**a**) Achieved formation pattern with trajectories. (**b**) Achieved formation pattern with Turtlebot3\({}^{\text{\textregistered}}\) robots. (**c**) Linear velocity of each robot. (**d**) Angular velocities of each robot. Figure 12: Formation control results of the proposed approach for Experiment 1. The \(\star\) indicates the start position, and \(\bullet\) represents the final position. (**a**) Achieved formation pattern with trajectories. (**b**) Achieved formation pattern with Turtlebot3\({}^{\text{\textregistered}}\) robots. (**c**) Linear velocity of each robot. (**d**) Angular velocities of each robot. Figure 14: Formation control results of the proposed approach for Experiment 2.