## 2 Multi-Robot Cooperative Formation Control Based on Group Formation
### Implementation of Group Formation Control Algorithm Based on Consistency for Nonholonomic Systems  
The basic idea of the consistency algorithm is that each robot uses the robot network to transmit information. It designs an appropriate distributed control algorithm. It couples the robot dynamics with the robot communication topology to form a complex system, so as to realize the continuous or synchronization of the state.  
The goal of this section is to design a unified grouping consistent formation (GCF) scheme for a system that can be divided into \(k\) groups and composed of \(N\) incomplete mobile multi-robots to meet the large-scale multi-robot system formation. \(\mathbf{V}=\{v_{1},\ \ldots,\ v_{N}\}\) represents a collection of \(N\) nodes, \(\mathbf{V}^{k}\) represents the \(k\)-th group, and \(\mathbf{V}^{k}\in\)\(\mathbf{V}\), this article adopts the multi-group formation mode to realize a large-scale multi-mobile robot system, as shown in the following figure. \(\mathrm{R}_{k}{}^{L}\) represents the leading robot of the \(k\)-th group, and \(\mathrm{R}_{k-s}\) represents the \(s\)-th follower robot of the \(k\)-th group, as shown in Figure 4. While the leader robot \(\mathrm{R}_{k}{}^{L}\) forms a formation with the inside of the group, it also forms a formation with the leader robots of other groups. The leader robot only has a communication topology diagram with the robots in the group and the leader robots in each group. The group formation greatly increases the size of the formation, and also reduces the amount of information processing of the navigator in the multi-robot system. It is of great significance to a large number of robot systems.  
The corresponding communication topology matrix is shown as follows in Equation (15). Then  
\[\left\{\begin{array}{l}A_{k}=\left\{a_{k-ij}\right\}\\ A^{m\times m}=\left\{A_{k}\right\}\end{array}\right., \tag{15}\]  
In order to verify the formation consistency algorithm, five robots are used to form an experimental robot formation in the form of leader-follower to find a stable communication topology matrix \(A^{\prime}\) as shown in Equation (16). Then  
\[A^{\prime}=\left(\begin{array}{ccccc}0&1&1&1&1\\ 0&0&0&0&1\\ 0&0&0&1&1\\ 0&0&1&0&1\\ 0&0&0&0&0\end{array}\right), \tag{16}\]  
After using the communication topology matrix to make a simulation experiment in Section 4, the formation and its communication topology are shown in Figure 5. The arrow indicates the direction of information flow. In Figure 5, it can be seen that the outgoing degree of node 5 is the largest. So, node 5 is the key node. That is the leader robot. The outgoing degree of node 1 is 0, which has the least impact on the formation. Both in and out degrees are local properties of the graph. In formation topology communication, their out and in degrees can reflect the influence of nodes on other robots in the formation. The larger the exit degree, the greater the influence of the node robot on the formation consistency. When the penetration is larger, the influence of the node robot on the formation consistency is smaller.  
Figure 4: Communication topology of grouping consistent formation.