# Reconfigurable Formation Control of Multi-Agents Using Virtual Linkage Approach
## 3 Formation Control: Virtual Linkage Approach
### Architecture of the Virtual Linkage  
As defined in Section 2.1, a virtual linkage consists of a set of virtual links, and each virtual link in turns consists of a set of agents. Hence, it is natural to apply the hierarchical architecture to the control of a virtual linkage. In this architecture, each virtual link oversees the performance of the other virtual links. Each agent of a virtual link in turn oversees the other agents in the same links.  
The architecture block diagram of the virtual linkage is shown in Figure 6. The module \(Vl_{i}\) is the \(i\)th virtual link in the virtual linkage, with the coordination input \(\xi\), and output vector \(v_{i}\) representing the state information of the \(i\)th virtual link. The system \(F\) represents the formation control in the virtual linkage. The inputs to \(F\) are the performance of each virtual link \(z_{uli}\), and the outputs \(y_{G}\) of the supervisor. Also, a formation feedback from \(F\) to \(G\) can be induced by the performance measure \(z_{F}\). In reality, the module \(G\) and \(F\) can be implemented in a specified agent or a server computer outside of the agent.  
Figure 5: Bi-direction flow control of the virtual linkage.  
Figure 6: Architecture of the virtual linkage.  
The second layer of the architecture is the coordination of each virtual link module \(Vl_{i}\). Here, the coordination architecture proposed in [27] is directly used because each \(Vl_{i}\) is a small virtual structure, as stated in the Definition 3. The systems \(S_{i}\) and \(K_{i}\) are the agent and corresponding local controller. Meanwhile, \(u_{i}\) and \(y_{i}\) represent the control variable to the agent and measurable outputs, respectively. The rest of the modules and variables are much the same as the first layer. In practical applications, a specified agent is responsible for running the \(G_{\mathit{v}il}\) and \(F_{\mathit{v}il}\) module, including receiving the state of the agents that participate in the same link, handling the control algorithm and sending desired information \(\xi_{\mathit{v}il}\) to the agents.  
Figure 7 shows the scheme of the centralized architecture based on the virtual structure approach [28]. In contrast to the centralized formation control [19; 24; 29], the virtual linkage architecture has several advantages. Firstly, the hierarchical architecture does not need to exchange all agents' state information to a central location, which reduces the burden of communication bandwidth, and scales much better than the centralized approaches. Secondly, the heavy computation of fitting agents into the virtual linkage also has been distributed to several low-level controllers \(F_{\mathit{v}il}\) and can be calculated simultaneously. This makes the proposed algorithm faster and more suitable for real-time control. Finally, by varying virtual joint angle \(\theta_{d}\) in the coordination input \(\xi=\left(r^{d},q^{d},\theta^{d}\right)\) different formation patterns can be achieved without the need to re-plan the whole formation procedure. In other words, the virtual linkage approach provides the formation reconfiguration ability that is not provided by the virtual structure approach.