# Decentralization of Virtual Linkage in Formation Control of Multi-Agents via Consensus Strategies
## 4 Simulation and Results
### Formation Reconfiguration Using Decentralized Virtual Linkage Approach  
In this part, formation reconfiguration simulation is performed to show the virtual linkage approach's reconfiguration ability by coordinating the desired virtual joint angle \(\Theta_{d}\) and expansion/contraction vector \(\lambda_{d}\).  
In the original virtual linkage approach [18], the designed virtual linkage is able to present different formation shapes by coordinating the desired virtual joint angle \(\Theta_{d}\). Moreover, an expansion/contraction vector \(\lambda_{d}\) is introduced in this decentralized virtual linkage approach to enable the designed virtual linkage to be reconfigured into more formation patterns, as compared to the hierarchical virtual linkage approach. In this simulation, the group of agents is designed as the virtual linkage defined in Equations (9) and (10) and move along the trajectory in Equations (13) and (14). The \(\Theta_{d}\) and \(\lambda_{d}\) are specified as Equations (17) and (18) to illustrate the formation reconfiguration ability.  
\[\Theta_{d}=\theta_{1}=\left\{\begin{array}{cc}\frac{\pi}{100}t&t\leq 50\\ -\frac{\pi}{100}t+\frac{\pi}{2}&50<t<100\end{array}\right. \tag{17}\]  
\[\lambda_{d}=[\lambda_{1x},\lambda_{1y},\lambda_{2x},\lambda_{2y}]=[0.4-0.002t,1,0.4-0.002t,1] \tag{18}\]  
Figure 10 shows the snapshots during the simulation for 100 s. As can be seen, the team of agents move around a circle with varying formation patterns. It is worth noting that the two virtual links  
Figure 9: Simulation results of nine agents move around a circle in line formation.  
have different lengths at each moment during the simulation, which indicates that the introduction of \(\lambda_{d}\) has provided a stronger reconfiguration ability to the virtual linkage approach.  
Figure 11a,b report the reference state and desired trajectory of the virtual linkage defined in Equations (13), (14), (17) and (18). The individual elements of each \(\xi_{i}\) are plotted in Figure 11c-h, respectively. As can be seen from the figures, the nine random initialized \(\xi_{i}\) finally converge into the same value, which indicates that the nine agents finally form a specified virtual linkage and move in desired formation shapes along the specified path. Notice that \(\lambda_{1x},\lambda_{2x}\) and \(\Theta\) of each \(\xi_{i}\) track well with the varying function. Recall that a virtual linkage can reconfigure into different formation patterns by changing the joint anglesand extracting/expanding each link with the scale factor \(\lambda_{d}\). Thus, the simulation results indicate the effectiveness of formation reconfiguration using the proposed decentralized virtual linkage approach. What is important for us to recognize here, is the coordinate variable \(\xi_{r}=(r_{d},q_{d},\Theta_{d},\lambda_{d})\) in the proposed approach can be arbitrarily set, which provides great potential to perform complicated tasks. For example, when a group of agents is required to go through a gallery, the desired trajectory and varying formation shapes (\(\xi_{r}=(r_{d},q_{d},\Theta_{d},\lambda_{d})\)) can be solved by plan method.  
Figure 10: Snapshot of the formation reconfiguration using virtual linkage.