## 3 Navigation Process
The swarm agents interaction is represented by graphs \((V,E)\) where \(V=\{1,2,\ldots,N\}\) is a set of nodes and \(E=\{(v_{i},v_{j}):i,j\in V,i\
eq j\}\) represents the communication and sensing topology between the \(i^{th}\) agent and each of the other \(j^{th}\) swarm agents as depicted in Appendix A. The dynamic model employed for the robot is a simple integrator model. Without loss of generality, since as exposed in [PERSON] (2009) it is possible to impose integrator dynamics for multiple robot structures employing kinematic control  
\[\dot{x}_{i}=u_{i}, \tag{1}\]  
where \(\dot{x}_{i}\) represents the linear speed of a robot and \(u\) is the control signal applied to the system in order to impose the desired behavior. The navigation approach in this work consists of the displacement of a robot swarm through an environment \(S\in\mathbb{R}^{2}\), in which, on the one hand, we have areas clear to move composed by the set \(S_{c}\) and on the other hand, \(S_{o}\) the areas with presence of obstacles that forbid the robots to move in it (Note that \((S=S_{c}\cup S_{o})\). Interaction between swarm agents:The interaction of the swarm agents is defined by the attraction and repulsion strategy allowing the navigation into the environment. This strategy allows the agents to keep a comfortable distance over the other swarm agents. The attraction parameter tries to maintain in close proximity every element of the swarm and thus gives the group a mechanism to remain grouped together. The attraction parameter is defined by (2)  
\[-k_{a}x_{ij} \tag{2}\]  
where \(k_{a}\) is the attraction force and \(x_{ij}\) is the distance between the agent \(i\) to \(j\), this mechanism can be local (with a restriction by the sensing range) or global (agents can move other agents from the group regardless of how far they are). The repulsion parameter allows the agent to keep a comfortable distance from the other members of the swarm. This avoids collision between agents when the swarm is moving. The repulsion mechanism can respond in two different ways. First one, when a comfortable distance is reached and its parameter is expressed by (3)  
\[[-k(\|x_{ij}\|-d)](x_{ij}) \tag{3}\]  
where \(\|x_{ij}\|=\sqrt[2]{(x_{i}-x_{j})^{T}(x_{i}-x_{j})}\), \(k>0\) is the repulsion magnitude and \(d\) is the comfortable distance between \(i^{th}\) agent and the \(j^{th}\) agent. On the other hand, if two agents are two close to each other, the parameter is represented by (4). \[k_{r}exp(\frac{-\frac{1}{2}\|x_{ij}\|^{2}}{r_{\rm s}^{2}})(x_{ij}) \tag{4}\]  
where \(k_{r}>0\), represents the repulsion magnitude and \(r_{\rm s}>0\), is the repulsion range. Interaction between agents and obstacles:The obstacles produce a repulsion force over the agents. This repulsion is modeled using (4) where \(k_{r}>0\), and \(r_{\rm s}\) is related to the size of the obstacle. Interaction between agents and victims:The victims exert an attraction force, where \(k_{r}<0\) and \(r_{\rm s}\) is proportional to the victims density in the near area. Sub-Swarm GenerationEvery agent is exposed to different attraction and repulsion forces, which help to keep the swarm coherent and at the same time pushes the swarm towards the goal. The victims generate an attraction force over close agents and this force can disturb the normal behavior of the swarm and reduce the velocity of the agents in proximity to the victims. The \(k_{r}\) magnitude was put in place so that it can stop the closest agents to the victims. When agents are navigating close to the victims, at least one agent must stop near the victims. Once at least one agent has stopped close to the victim and generated the alarm about victim detection, there is a relevant variation in the process that has to be exposed, which is the generation of a sub-swarm due to the localization of possible victim. The creation of a sub-swarm its generated by K-nearest neighbors approach. The K-nearest algorithm behaves as a classifier by selecting as his name indicates the \(k\) closest robots \(j\) to robot \(i\).