## 3 Navigation Process  
In nature, there are several species that display swarm behavior, such as bacteria, insects, birds, fish, lions, hyenas, and horses among others. As can be expected, these groups are composed of individuals, who possess specific distinct capabilities and behaviors, but when they all get together as one group and operate as such, they will behave differently in order to work as a group. One of the more notable examples of this group behavior is a swarm of honey bees choosing a place to build a new honeycomb, this phenomenon was presented by [PERSON] et. al. in [37], whose paper showed that this swarm behavior, which takes place during the spring and summer, it is best exemplified when the colony outgrows its honeycomb and then proceeds to divide itself to find new grounds and then gets back together as a swarm once it finds the correct location to build a bigger hive. To be more specific, The new site selection begins with several hundred scout bees, who leave the colony in search of potential new sites, once the scouts find an appropriate candidate, they return to the hive in order to report with a wiggle dance where the potential new place is located. Once the scouts select the correct location from the pre-selected sites, they will steer the rest of the swarm to the new location by chemical stimulation. After the selection step is completed, the division process of the hive begins, and the old queen with nearly half of the colony leaves the hive to build the new one, while a younger queen stays with the other half of the colony in the old honeycomb. It is also important to highlight that the ability of the swarm to navigate can be affected by a set of critical factors, such as collisions between themselves, large obstacles in the route, pheromone communication errors, erroneous scout indications and even poor sense of direction. We can represent the swarm of bees using graph theory with the application of the concepts worked on by [PERSON] in [PERSON] (2018). One can describe the system by defining \(N\) agents representing each member of the bee swarm, or in the case of our application a robot, and a connection representing the communication and transmission of information between the agents, either bees or robots. As mentioned by [PERSON], we first need to define the agents of the system \((V,E)\), where \(V\) is the set of nodes in the system and \(E\) stands for the topology of the system, we also need to define \(N(i)\) as the neighborhood or adjacent nodes to node \(i\). Regarding the links between the agents, we can describe them as a dynamic system. It is advantageous to apply graph theory to control swarm behavior, because it allows us to apply concepts such as the Laplacian matrix, which represents the system and allows the consensus analysis, connectivity grade calculation, and applying a control law for the whole system. In addition to that, we assume that the system has moderated sensor noise, links are bidirectional given the capability for any robot to share and receive information with the agents within own communication range. The design of the swarm navigation algorithm focuses on the generation of simple navigation rules that are applied to each of the agents individually. However, the sum of the individual decisions of the agents generates a collective behavior typical of a swarm. The applied criteria to the simple navigation algorithm of the swarm agents are based on Reynolds rules [PERSON] (1964). These rules settle down the base conduct for the generation of swarm behavior, such as (1) avoid collisions with neighbors, (2) match speed and direction with neighbors, and (3) stay close to neighbors. Before establishing the interaction policy to fit the [PERSON] rules, it is necessary to define the agent concept and the kinematic model of the robotics agent used in the current work. Each agent is described as a point in the space with position and velocity. We assumed that each agent can detect the position and velocity of the rest of the swarm agents.