# Reconfigurable Formation Control of Multi-Agents Using Virtual Linkage Approach
## 4 Simulation and Results
### Comparison with Virtual Structure Approach (\(k=1\))  
According to the Definition 3, a virtual linkage degenerates into a virtual structure because a virtual linkage is a collection of virtual structures (virtual links). Therefore, the centralized virtual structure approach can be seen as a special case of the virtual linkage approach, where the number of the virtual links \(k=1\). In this section, the same simulation task in Section 4.2 is conducted using the virtual structure approach [19]. Then, the comparison between the centralized virtual structure and the proposed virtual linkage approach is discussed. Since virtual structure approaches treat each formation as a single rigid body, the whole virtual structure should be redesigned once the formation shape changes. Therefore, a different virtual structure should be redesigned each time to achieve the formation shape defined in Equation (15). Figure 27 shows some virtual structures when different formation shape involves. Each time the nine desired place holders should be refreshed by communication. Figure 28 shows simulation solution to this problem using virtual structure approach [19]. Although the formation reconfiguration simulation can be successfully performed, the virtual structure has been redesigned during the whole simulation. Figure 29 shows that, in the centralized virtual structure, the position information of all the nine agents is sent into the single virtual structure server to implement the formation control algorithm. In contrast, in the virtual linkage defined in Equations (11) and (12), there is no need to transmit all the agents' position information into a single localization. The virtual link1 server only needs to receive the position information of agents 1, 2, 3, 4, 5. The virtual link2 server also only needs to communicate with part of the agents, namely agents 5, 6, 7, 8, 9. The significance of this property is that the largest bandwidth of communication equipment in the system is reduced. Figure 27: **(a)** Real and desired attitude of the virtual linkage; **(b)** error between real and desired attitude. Another advantage of virtual linkage lies in the time complexity as compared to the centralized virtual structure approach. Using the virtual structure approach [19], the optimal position and attitude of the virtual structure \(X_{v}^{*}\) is calculated using Equation (1) with nine agents' positions. Nevertheless, the whole fitting process presented in Section 3.3.1 is broken into individual parts. The \(X_{v1}^{*}\), \(X_{v2}^{*}\) can be simultaneously calculated using only five corresponding agents' position on the two virtual link severs. This allows the virtual linkage approach can be implemented faster and to be more suitable for real-time control. Roughly speaking, if a virtual linkage consists of \(k\) virtual links and each virtual link has the same number of agents, the virtual linkage approach runs \(k\) times faster than centralized virtual structure. It is worth mentioning that this particular task shows the novelty of the virtual linkage approach. A virtual linkage is able to be reconfigured as different formation shapes by changing the virtual joint angle \(\theta_{d}\). Using Equations (8) and (9), it can be seen that the proposed virtual linkage only needs a communication bandwidth of  
\[BW_{2}=K\cdot k/T=\frac{2K}{T} \tag{23}\]  
bits per second to realize formation reconfiguration. In contrast, for the virtual structure approach, a bandwidth of  
\[BW_{1}=3\cdot N\cdot K/T=3\cdot 9\cdot K/T=\frac{27K}{T} \tag{24}\]  
bits per second is needed. The bandwidth ratio is as follows:  
\[\frac{BW_{2}}{BW_{1}}=\frac{k}{3\cdot N}=\frac{2}{27}\approx 7.4\% \tag{25}\]  
Figure 28: Formation reconfiguration using virtual structure approach. Figure 29: Comparison between (**a**) centralized virtual structure and (**b**) virtual linkage approach. This implies that only 7.4% communication bandwidth is required to reconfigure the formation shape using the virtual linkage approach.