# Decentralization of Virtual Linkage in Formation Control of Multi-Agents via Consensus Strategies
## 5 Comparison with Hierarchical Virtual Linkage Approach
### Scalability  
As described in Section 3.1, each agent can completely decide its movement and only exchange states with its local neighbors. To verify the scalability of the proposed approach, five agents are required to align in a line formation with a uniform distance of 0.1 m and perform the same simulation task in Section 4.2 using the proposed decentralized virtual linkage approach.  
To perform such a task, a virtual linkage which also consists of two virtual links is designed as follows:  
\[Virtuallinkage\_B=\left\{\begin{array}{l}virtuallink1=\{agent1,agent2,agent3\} \\ virtuallink2=\{agent3,agent4,agent5\}\end{array}\right. \tag{19}\]  
\[\begin{array}{l}{}^{v1}p_{1}^{Int}=[0,0],{}^{v1}p_{2}^{Int}=[0.1,0],{}^{v1} p_{3}^{Int}=[0.2,0],\\ {}^{v2}p_{3}^{Int}=[0,0],{}^{v2}p_{4}^{Int}=[0.1,0],{}^{v2}p_{5}^{Int}=[0.2,0] \end{array} \tag{20}\]  
The extract/expansion rate is set as \(\lambda_{d}=[\lambda_{1x},\lambda_{1y},\lambda_{2x},\lambda_{2y}]=[1,1,1,1]\). Figure 13 shows the designed virtual linkage and communication topologies used for this simulation. Figure 14 shows the snapshot during the simulation.  
Figure 12: Detailed architecture of the hierarchical virtual linkage approach for simulation.  
Figure 13: Schematic of designed virtual linkage and communication topologies.  
Recall that in Figure 3, each agent exchanges its own understanding of virtual linkage states \(\xi_{i}=(r_{i},q_{i},\Theta_{i},\lambda_{i})\) and the individual tracking error \({}^{w}p_{i}-{}^{w}p_{i}^{d}\) with its neighbors. Supposing \(6K\) bits are required to encode each instantiation of virtual linkage \(\xi_{i}\), \(2K\) bits for individual tracking error \({}^{w}p_{i}-{}^{w}p_{i}^{d}\), then each agent only needs exchange these \(8K\) bits message with its neighbors. If an agent has \(M\) neighbors and communicates with them for every period of time \(T\) and then the required bandwidth in units of bits per second is:  
\[BW=8K\cdot M/T \tag{21}\]  
The maximum communication bandwidth \(BW_{1}\) in this five agents involved simulation occurs in agent 2, since it has three neighbors (See Figure 13b):  
\[BW_{1}=8K\cdot 3/T=24K/T \tag{22}\]  
Meanwhile, nine agents are involved in the simulation in Section 4.2, the maximum communication bandwidth also occurs in agent 2:  
\[BW_{2}=8K\cdot 3/T=24K/T=BW_{1} \tag{23}\]  
It is worth noting that, the maximum communication bandwidth in the team is determined by the maximum number of agents' neighbors and does not increase with the number of agents in the group.  
In contrast to the decentralized virtual linkage approach, the hierarchical virtual linkage approach does not scale well with the number of agents. As illustrated in Reference [18], the maximum communication bandwidth increases with the number of agents in the team. The results indicate that the proposed decentralized virtual linkage approach is more suitable for situations when a large number of agents and/or limited communication are involved.  
Figure 14: Snapshot of the formation moving of five agents.