## 2 Multi-Robot Cooperative Formation Control Based on Group Formation  
### Graph Theory Description  
The control cooperative multi-robot dynamic system is a dynamic problem related to each other through a communication diagram, which shows the information flow between each node. The goal of cooperative control is to design control protocols for each node and to ensure the state synchronization behavior of all nodes in a specific sense. In a cooperative system, any control protocol must carry out the distributed control according to its specified graph topology. The control protocol of each node is only allowed to rely on the information about the node and its neighbor nodes in the graph. The communication restrictions imposed by the topology may seriously limit the work. This work is completed by the local distributed control protocol on each node. In multi-robot cooperative formation, this article focuses on the behavior and interaction of dynamic systems. These dynamic systems are interconnected through the link of a communication network. The communication network is modeled as a graph with directed edges corresponding to the allowed information flow between systems. Each robot in the formation is modeled as a node in the graph. The most basic problem is how the graph topology interacts with the local feedback control protocol of nodes to generate the overall behavior of interconnected nodes. This problem can be solved by the graph theory. The following shows some basic graph theory concepts that are essential in the study of multi-robot cooperative formation. The topological graph can be represented as \(\mathbf{G}=(\mathbf{V},\mathbf{E})\), where \(\mathbf{V}=\{v_{1},\ \ldots\,\ v_{N}\}\) represents the set of \(\mathbf{N}\) nodes, and \(\mathbf{E}\) represents the set of \(\mathbf{E}\) edges. Assuming that the graph is a simple directed graph, that is \((v_{i},\ v_{j})\
otin\mathbf{E}\). At this time, there are no multiple edges between the same pair of nodes. The elements \((v_{i},\ v_{j})\) in the set \(\mathbf{E}\) represent the edges from node \(v_{j}\) to node \(v_{i}\). In the formation, it represents that the information flows from node \(v_{j}\) robot to node \(v_{i}\) robot. The out degree of node \(v_{i}\) robot \(d_{i}^{o}\) represents the number of edges with it as the data sending end. The in degree \(d_{i}\) represents the number of edges with it as the data receiving end. The structure and properties of graphs can be studied by studying the properties of some matrices related to graphs, which is algebraic graph theory. After the weight \(a_{ij}\) of the edge is given, the graph can be represented by an adjacency matrix or connected matrix \(\mathbf{A}=(a_{ij})\), where when \((v_{i},\ v_{j})\
otin\mathbf{E}\), \(a_{ij}=1\), otherwise \(a_{ij}=0\). The weighted penetration of point \(i\) at this time can be expressed as the sum of all values in row \(i\) of matrix \(\mathbf{A}\), as shown in Equation (1). Then  
\[d_{i}=\sum_{j=1}^{N}a_{ij}, \tag{1}\]  
Then, the weighted output can be expressed as Equation (2). Then  
\[d_{i}^{o}=\sum_{j=1}^{N}a_{ji}. \tag{2}\]  
Both in degree and out degree are local properties of the graph. In formation topology communication, the out degree can reflect the impact of the node on other robots in the formation, that is, the greater the out degree, the greater the impact of the node robot on the formation consistency. The diagonal penetration matrix and graph Laplacian matrix are defined as \(\mathbf{D}=diag\ d_{i}\) and \(\mathbf{L}=\mathbf{D}-\mathbf{A}\), respectively. Many properties of a graph can be studied as the properties of graph Laplacian matrix. It is of great significance in the study of graph theory of dynamic multi-node systems.