# Reconfigurable Formation Control of Multi-Agents Using Virtual Linkage Approach
## 1 Introduction  
Formation control is an important and fundamental problem in the coordinated control of multi-agent systems. In many applications, a group of agents is expected to move around while maintaining a specified spatial pattern. For example, aircraft flying in V-shaped formation could reduce the fuel consumption required for propulsion [1]. As for the box pushing problem, formation control of a team of identical robots provides the ability to move heavy boxes, which is difficult for a single robot [2]. In military and security patrol missions, formations allow each member of the team to monitor only a small part of the environment for full area coverage and surveillance [3; 4; 5; 6]. In the literature, there are roughly three formation control approaches, namely leader-following, behavioral, and virtual structure approaches. In the leader-following approaches [7; 8; 9; 10], some agents are considered as leaders, while others act as followers. The formation control problem is converted into two tracking problems: the leaders track the predefined trajectories, and the followers track the transformed coordinates of the leader with some prescribed offsets. The primary advantage of this approach is that the formation stability can be analyzed as tracking stabilities through standard control theories. In addition, the analysis is easy to understand and implement using a vision system equipped on the robot [11; 12]. The disadvantage is that the leader has no information about the followers' states. If a follower fails, the leader will still move as predefined, and the formation cannot be maintained. In the behavioral approaches [13; 14; 15; 16], a set of reactive behaviors are designed for the individual agent, such as move-to-goal, avoid-static-obstacle, avoid-robot and maintain-formation. The control action of each agent is generated as a weighted sum of all the behaviors accordingly. One advantage ofthis approach is that it provides a natural solution for each agent when it has to maintain formation and avoid obstacles. Learning-based methods can also be used to determine agents' behavior to improve the formation performance [17; 18]. Furthermore, the formation feedback is implicitly induced through control action, which partly depends on the states of neighboring agents. However, the mathematical analysis of this approach is difficult, and the group stability is not guaranteed. In the virtual structures approach [19; 20; 21], the entire formation is treated as a single rigid body. The desired position of each agent in the team is derived from the virtual particles embedded in the virtual structure. One advantage of the virtual structure approach is that it is straightforward to prescribe the desired behavior for the whole group. Another advantage of this approach is that the group is able to keep a geometric pattern in high precision during movement. Nevertheless, most approaches based on the virtual structure are designed to hold the same shape during the whole task [22; 23; 24; 25], and tend to have poor performance when the formation shape needs to be reconfigured frequently. When talking about formation control, it is desirable to have a group of agents moving in formation with high precision when a specified shape needs to be preserved. Nevertheless, the formation stability is not guaranteed using behavioral approaches. In leader-following approaches, the formation shape also cannot be maintained when a follower fails. Therefore, it is a suitable choice to have all the agents move as a single rigid body by using a virtual structure, out of the three approaches mentioned above. However, the virtual structure approach has limited formation reconfiguration ability for the group of agents. The main aim of this paper is to provide high precision formation control with some reconfiguration ability for a group of agents. Motivated by the advantages and disadvantages of the virtual structure approach, a novel idea named virtual linkage (VL) is proposed in this paper.