# Vertex-Weighted Consensus-Based Formation Control with Area Constraints and Collision Avoidance
## 1 Introduction  
Formation control is a very well-studied problem due to its potential applications in collaborative and cooperative tasks [1]. Formation control aims to drive a multi-agent system to achieve prescribed patterns. Commonly, formation control is solved as a consensus problem [2, 3, 4], where the state variables of all agents converge to a common value that defines the centroid of the formation. A remarkable advantage of consensus algorithms is that a consensus agreement is reached even in the presence of delays in communication [5, 6]. Moreover, the various problems for consensus formation control have been addressed, like formation control with collision avoidance [7, 8, 9], formation control with communication delays [10, 11], noise tolerance formation control [12], fuzzy formation control with input saturation [13], PID formation control with disturbances [14], and formation control with leader-follower networks [15, 16, 17]. Since collision among agents can occur during the formation control task, artificial potential field approaches are commonly used to deal with collisions [18, 19, 20]. Repulsive potential fields are combined with consensus-based formation control to include collision avoidance. However, incorporating repulsive fields does not guarantee system stability in the presence of delays [21]. Edge-weighted consensus-based formation control guarantees collision avoidance among agents [22, 23]. These approaches exploit the properties of weighted graphs, allowing the formation to rotate and adapt its shape to avoid collision among robots. However,the desired formation cannot always be preserved due to local minima. Local minima may occur for some particular agent positions, especially when agents are aligned. Moreover, a connection between agents is required to avoid a collision between agents. Then, when more agents are involved in a network, more connections are required to avoid collisions. Thus, to guarantee formation and collision capabilities, fully consensus-based communication is always required. In [6], a virtual relabeling strategy, along with a broadcasting algorithm, is proposed to avoid local minima while ensuring that agents converge to the desired formation. The virtual relabeling strategy assigns a virtual label to each agent. Local minimum stagnation is avoided by modifying agents' labels following an anti-clockwise order concerning the formation's center of mass. However, the broadcasting algorithm assigns the appropriate label to an agent based on its own position and the position of all other agents, losing the capability of performing a decentralized formation control. Moreover, some particular initial agents' positions still provoke local minima stagnation. Additionally, the virtual relabeling strategy required fully consensus-based communication. In [24; 25], formation control is achieved based on a gradient descent control law, which incorporates a signed area term to mitigate ambiguities within the formation. When attempting to attain a specific formation, ambiguities may arise in the absence of sufficient inter-agent connections. In the context of \(n\) agents that are not fully connected, multiple formations may exist that meet the desired inter-agent distance criteria [25]. This approach enables formation to remain translation- and rotation-independent. However, the proposed control law does not inherently ensure collision avoidance among agents. Additionally, the methodology in [25] is structured around a hierarchical approach, implying that control strategies employed differ among agents within the information. In this work, vertex-weighted consensus-based formation control is addressed, where a control law is proposed that incorporates a vertex-tension function along with signed area constraints. Inspired by the edge-tension function from [6], a vertex-tension function is proposed, which maintains the capabilities of collision avoidance among agents.