# Decentralization of Virtual Linkage in Formation Control of Multi-Agents via Consensus Strategies
## 5 Comparison with Hierarchical Virtual Linkage Approach
### Reconfiguration Ability  
As compared to the virtual structure approach, an important property of the hierarchical virtual linkage approach is that it can reconfigure the group of agents into different formation patterns by coordinating the value of virtual joint angles. To evaluate the reconfiguration ability of the virtual linkage approach, the definition of reconfiguration space is proposed:  
**Definition 3** (Reconfiguration Space): _In virtual linkage approach, the reconfiguration space is defined as the set of formation configuration that can be realized by the designed virtual linkage model. In other words, it corresponds to the formation patterns to which the designed virtual linkage can be reconfigured._  
To compare the reconfiguration ability between the original and proposed approach, the reconfiguration spaces of these two approaches are evaluated. In the proposed approach, instead of using \(\xi^{d}=(r^{d},q^{d},\Theta^{d})\) to control the designed behavior of the team like the hierarchical virtual linkage approach (See Figure 12), \(\xi^{d}=(r^{d},q^{d},\Theta^{d},\lambda^{d})\) is used as the coordinate variable. Meanwhile, each agent \(i\) record its relative percentage position \({}^{vj}p_{i}^{ini}\) in the corresponding virtual link \(j\) rather than the relative position \({}^{vj}p_{i}\). The benefit of this is that the virtual linkage is now able to extract/expansion the virtual link using Equation (6), which facilitates the reconfiguration ability of the proposed virtual linkage.  
In detail, the reconfiguration spaces of the hierarchical and decentralized approaches are denoted as \(RS_{hier}\) and \(RS_{dec}\) respectively. Recall that \({}^{d}r,{}^{d}q\) are the predefined trajectory and attitude of the virtual linkage, the formation pattern is only determined by the vectors \({}^{d}\Theta~{}=~{}(\theta_{1},\ldots,\theta_{n})\) and \(\lambda_{d}=[\lambda_{1x},\lambda_{1y},\ldots,\lambda_{nx},\lambda_{ny}]\). Then the \(RS_{hier}\) and \(RS_{dec}\) can be expressed as vector spaces:  
\[RS_{hier}=\left({}^{d}\Theta,{}^{d}\lambda_{const}\right)=(\theta_{1},\ldots \theta_{n},1,1,\ldots 1,1) \tag{24}\]  
\[RS_{dec}=\left({}^{d}\Theta,{}^{d}\lambda\right)=\left(\theta_{1},\ldots \theta_{n},\lambda_{1x},\lambda_{1y},\ldots,\lambda_{nx},\lambda_{ny}\right) \tag{25}\]  
It is worth noting that \(RS_{hier}\) can be seen a special case of \(RS_{dec}\) when  
\[\lambda_{d}=[\lambda_{1x},\lambda_{1y},\ldots,\lambda_{nx},\lambda_{ny}]=[1, 1,\ldots,1,1] \tag{26}\]  
and the decentralized approach has stronger reconfiguration ability than the hierarchical approach since  
\[RS_{hier}\subset RS_{dec} \tag{27}\]  
Figure 15 shows an intuitive way of illustrating the stronger reconfiguration ability by introducing the scale factor \(\lambda_{d}\). In the original approach, the designed virtual linkage corresponds to a mechanical linkage which has fixed length and width and is only able to reconfigure into different shapes by coordinating the joint angles. However, proposed approach enables each virtual link to extract/expansion its length and width by introducing the scale factor \(\lambda_{d}\), which allows the mechanical linkage to reconfigure into more shapes by extracting/expanding each link and changing the joint angles simultaneously.