# Decentralization of Virtual Linkage in Formation Control of Multi-Agents via Consensus Strategies
## 1 Introduction  
Formation control is one of the most leading research areas in robotics. It has been extensively studied by researchers around the world on different platforms: mobile robots, aerial robots, spacecraft, and autonomous surface and underwater vehicles [1; 2; 3; 4; 5; 6; 7; 8]. In the literature, formation control approaches can be classified into three basic strategies: leader-following, behavior-based, and virtual structure. In the leader-follower approach [9; 10; 11], some agents are considered as leaders, while others act as followers which track the leaders with predefined offset. However, the leader's motion is independent of the followers. When a follower fails, the leader will keep on moving as predefined and results in the break of the formation shape. In the behavior approach [12; 13; 14], several reactive behaviors are prescribed (e.g., move-to-goal, avoid-robot, avoid-static-obstacles, and maintain-formation). The action of each agent is derived by a weighted sum of all the behaviors. The main problem with this approach is that it is difficult to formalize the group mathematically and the team of agents is not guaranteed to converge to the desired formation configuration. The virtual linkage approach considers the entire formation as a single rigid body and is able to maintain the formation shape in high precision during manoeuvers [15; 16; 17]. Perhaps themain criticism of the virtual structure approach is that it has poor reconfiguration ability and needs to refresh the relative positions of all the team members when a different formation pattern changes. Although these three approaches have been used in many applications, they focus more on maintaining a specified formation pattern throughout a task and few studies address the effects of formation reconfiguration. However, situations also exist where different formation patterns are needed, for example, a group of agents might need to reconfigure into different patterns to go through a gallery. A reconfigurable formation control method named virtual linkage is proposed by the authors in Reference [18]. Instead of treating the whole formation as a single rigid body, as in the virtual structure, the virtual linkage approach considers the formation as a mechanical linkage which is a collection of rigid bodies connected by joints. A virtual linkage is defined as an assembly of virtual structures (named \"virtual link\") connected by virtual joints. By coordinating the value of each virtual joint, the virtual linkage approach is able to reconfigure a group of agents into different formation shapes. Currently, the virtual linkage approach uses hierarchical architecture. In detail, the states of the virtual linkage are implemented in a virtual linkage server and broadcast to all the virtual link servers. Each virtual link's state is in turn calculated in the corresponding virtual link server and transmits to all its virtual link members. The principal limitation of this hierarchical architecture is that it does not scale well with the number of agents in the team. In addition, due to the communication range limitations, the virtual linkage server might lose communication with some virtual link servers when the agent group covers a large area. A possible way to solve these drawbacks is to use decentralized architecture in which each agent runs a consensus strategy and totally decides its moving action with communication with parts of the members. The concept of consensus is an important idea in control and information theory, and it has been applied to the formation control of multiple agents [2; 16; 19; 20; 21]. The basic idea of a consensus algorithm is that each agent updates its state's information only based on its neighbors' state's information and finally enable the convergence of all the agent's state's information. The main contribution of this paper is the decentralization of the virtual linkage approach via consensus strategies.