## 2 Multi-Robot Cooperative Formation Control Based on Group Formation
### Formation Control Principle  
The distributed control structure in the leader-follower formation is adopted in this article. The formation control is realized by using the consistency algorithm given below on the basis of the previous graph theory. The consistency is defined as one or some states of all robots in a multi-robot system that tend to be consistent with the evolution of time. The leader-follower formation is a classical formation form. The main idea of the first-order formation algorithm based on consistency is to control the relative position and speed between the leader and the follower. When these quantities reach a certain stable state, the formation is realized, in which the leader robot plays a vital role. The first-order continuous system model of the formation system is Equation (8) as follows.  
Then  
\[\dot{x}_{i}=u_{i}, \tag{8}\]  
where \(x_{i}\) and \(u_{i}\) represent the state quantity and input quantity of node \(i\) respectively, and \(x_{i}\), \(u_{i}\)\(\in\)\(\mathbf{R}_{n}\), \(n\) represent the dimension of state quantity. Ideally, consider the following forms of control input. Then  
\[u_{i}=\sum_{j\in N_{i}}a_{ij}\big{(}x_{j}-x_{i}\big{)}, \tag{9}\]where \(a_{ij}\) is the formation adjacency matrix element and \(N_{i}\) is the neighbor set of member \(i\). By substituting Equation (9) into Equation (8) and introducing the global state vector \(\mathbf{x}=\left(x_{1},\ \ldots,\ x_{n}\right)^{\mathrm{T}}\in\)\(R_{n}\), the global dynamic relationship can be obtained as follows in Equation (10). Then  
\[\dot{x}=-\mathbf{D}x+\mathbf{A}x=-\mathbf{L}x, \tag{10}\]  
Based on Equation (10), it can be concluded that the closed-loop dynamic characteristics of the formation depend on the matrix \(\mathbf{L}\). In addition, if and only if the topological graph \(\mathbf{G}\) of the formation has a spanning tree, \(-\mathbf{L}\) eigenvalues are located in the left half plane of the complex plane. Hence, Equation (10) can ensure the consistency of the system. The final state value is shown as follows.  
Then  
\[c=\sum_{i=1}^{N}p_{i}x_{i}(0), \tag{11}\]  
In discrete time, Equation (8) can be reduced to Equation (12). Then  
\[x_{i}(k+1)=x_{i}(k)+u_{i}(k), \tag{12}\]  
In this subject, it is assumed that a scene is a scene in which the leader leads other robots to complete a certain distance and move from the starting point to a target point. Ideally, the algorithm of the follower in the formation adopts the following form. Then  
\[u_{i}(k)=\varepsilon\sum_{j\in N_{i}}a_{ij}\big{(}x_{j}(k)-x_{i}(k)-r_{ij}(k) \big{)}, \tag{13}\]  
Among \(\varepsilon>0\), \(r_{ij}\) represents the relative position between robot \(i\) and robot \(j\), which is related to the distance between them, while the control algorithm of leader \(\mathbf{N}\) is Equation (14). Then  
\[u_{N}(k)=m+k\mathbf{D}(k)+\sum_{i\in N_{i}}a_{Ni}r_{Ni}(k). \tag{14}\]  
where \(r_{Ni}\) represents the relative position between the leader and other robots, \(m\) and \(k\) are constants, \(\mathbf{D}(k)\) represents the distance from the leader to the target point at time \(k\), and the speed of the leader is affected by the adjacent nodes and the distance from the leader to the target point.