## 4 Victim Detection
### Formation Control  
Taking into account, that sub-swarms were generated in the navigation process due to some robots from the robot network detect a possible victim through their sensors and decide to leave the main swarm to create a new sub-swarm. Consequently, an algorithm capable of relocating all the robots in the sub-swarm around to the possible victim location is carried out in order to improve measurements from the sensor network on the detection of a victim. Giving to the sensor network the ability to increase their probability of well-classifying and avoid that a far robot introduces noise in the consensus. The algorithm has the capability to bring robots closer to the victims using the rendezvous algorithm and then take agents to the perimeter of a circle making certain modifications to the rendezvous consensus algorithm exposed in [38]. In which the Laplacian matrix and the state vector are used, giving rise to the consensus equation that models the dynamics of the system, expressed in (5), and explained with more detail in Appendix A. Where the state vector contains the dynamical information of the agents. \[\dot{x}=-Lx \tag{5}\]  
According to [38] the consensus equation shown above will have a space of agreement \(A=\{X\in R^{n}\mid x_{i}=x_{j}\forall i,j\}\) in which, when \(\dot{x}_{i}=0\) there will be a consensus. On the other hand, in the time domain, we have \(x(t)=e^{-Lt}x_{0}\) in which, when the limit when \(t\) tends to infinity is evaluated and performing an expansion in Taylor Series is possible to note that except for the first term all the others are zero. The first term turns out to be \(x(t)=e^{-\lambda_{1}t}((u_{1})^{T}x_{0})u_{1}\) letting that the first eigenvalue is zero because the whole exponential part tends to 1 which causes the following expression \(((u_{1})^{T}x_{0})u_{1}\). Extending it to  
Figure 1: Sub-swarm generation. all the agents in the graph, we have that the consensus of a non-directed graph is the average of the initial conditions of the system, this means that the consensus can be written as follows  
\[consensus=\sum\frac{x_{0i}}{n}. \tag{6}\]  
Taking into account that this rendezvous consensus allows all agents to reach the same position in the space, the next step is to guarantee the connection between the agents while the signal control is modified to reach the goal position for each agent based on the approach shown in [11] which in order to solve the problem of maintenance communication, artificial potential functions was performed used to guarantee connectivity and keeping a distance among agents with the aim to avoid collisions. The artificial potential functions are of the form,  
\[\psi_{ij}=\frac{1}{\rho_{2}^{2}-||x_{ij}||^{2}}-\frac{1}{||x_{ij}||^{2}-\rho_{ 1}^{2}}, \tag{7}\]  
where \(\psi_{ij}\) is the artificial potential function between the agents \(i\) and \(j\), \(\rho_{2}\) corresponds to the radius of connectivity of the agents and \(x_{ij}\) corresponds to the distance between the agents \(i\) and \(j\). From these functions, the dynamics of the agents is imposed by \(\dot{x_{i}}=-\sum_{j\in N_{i}^{\sigma}}\
abla_{x_{i}}\psi_{ij}\), note that this function tends to infinite when the distance between the agents tends to \(\rho_{2}\). In fact, the more the distance between agents increases, the more the intensity of the attraction among the robots increments to avoid communication breaks. The artificial potential functions approach demonstrates connectivity maintenance by evaluating the total energy of the graph connections given by  
\[\psi=\frac{1}{2}\sum_{i=1}^{n}\psi_{i}=\frac{1}{2}\sum_{i=1}^{n}\sum_{j\in N_ {i}^{\sigma}}\psi_{ij}, \tag{8}\]  
where \(\psi\) is the total energy of all the links and \(\psi_{i}\) corresponds to the energy of all the links concerning the agent \(i\).