# Vertex-Weighted Consensus-Based Formation Control with Area Constraints and Collision Avoidance
## 5 Conclusions  
This paper delves into the _xy-plane_ formation problem, aiming to create formations with a minimum of three agents. The results show that incorporating the signed area term, coupled with collision avoidance control, consistently enables us to achieve the desired formations under various conditions.  
The incorporation of the signed area in the control law allows us to address the formation problem with fewer connections, a crucial advantage when dealing with substantial distances between agents. This is illustrated in Figure 7, where, as previously discussed, the compared formation control method would necessitate a connection between agent one and agent four. However, this link would be the longest in the formation, and in real-world scenarios, measuring such a distance might be impractical.  
Regardless of the initial agent's positions, the proposal shows that local minima stagnation is avoided using area constraints based on triplets, a drawback of conventional edge-weighted consensus-based formation control. Moreover, the proposal avoids collision among agents thanks to the vertex-tension function, which is not considered in conventional formation control with distance and area constraints.  
The proposed approach requires providing formation patterns based on triples. Then, the minimum number of agents to consider in a formation problem is three. However, defining triplets in a network of a bunch of agents can take time and effort. The signed area can be extended to use other geometric figures, such as squares, pentagons, and hexagons, to easily represent connection among agents.  
The proposed control approach offers a decentralized solution to the formation problem, applying the same control law for all agents without requiring knowledge of the full network, in contrast to the conventional control method with the virtual relabeling algorithm. This characteristic facilitates easier implementation in real-world experiments.  
Looking ahead, our proposed control approach can be extended to the _xyz-space_, utilizing the concept of the signed volume of a tetrahedron. In this context, the analysis should focus on _quartets_ rather than _triplets_.  
Conceptualization, U.H.-V. and J.H.-B.; data curation, A.Y.A. and U.H.-V.; funding acquisition, A.Y.A., J.H.-B., I.G.J. and G.M.-S.; investigation, U.H.-V. and J.H.-B.; methodology, U.H.-V. and J.H.-B.; project administration, A.Y.A., I.G.J. and G.M.-S.; software, U.H.-V. and J.H.-B.; validation, A.Y.A., U.H.-V. and J.H.-B.; visualization, A.Y.A., U.H.-V. and J.H.-B.; writing--original draft, A.Y.A., U.H.-V. and J.H.-B.; writing--review and editing, I.G.J. and G.M.-S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by CONAHCYT Mexico through Project PCC-2022-319619.  
The data analyzed are available from the authors upon request.  
The authors also thank Universidad de Guadalajara for their support in this research.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
ROS Robot operating system