## 3 Cooperative Algorithms for Mobile Sensors
### Area Coverage Performance and Algorithm Limitations  
In this section the area-coverage performance of the algorithm are shown with respect to optimal geometrical solutions. The total sonar coverage depends on the sonar detection range \(R_{D}\) and on the maximum distance allowed \(R_{M}\) between two vehicles. Figure 6 shows the result obtained applying the proposed method using three vehicles. The agents place themselves in order to position the asset at the center of the detection area. The final configuration reached allows the vehicles to completely protect a circle of radius \(R_{M}\) around the asset with a minimum overlap \((R_{M}-R_{D})\) of the vehicle sonars, and with a detection level always greater than the minimum desired. The final agent positions coincide with the analytical solution of the area coverage problem of a circle of radius \(R_{M}\), with the vehicles placed on the circumference, as vertices of the equilateral triangle inscribed in the circle itself. The configuration reached has three axes of symmetry, one per each vehicle, and along these axes the detection coverage is more effective (detection directivity). It is interesting to note that the stability of the final configuration reached is related to the information communicated among the team members. In particular, if each vehicle communicates to the remainder of the team its foreseen location at its next communication step after applying the proposed motion  
coordination algorithm, then a circular motion around the asset is automatically stimulated and the vehicles periodically scan the whole area around the asset. Note in fact that, for symmetry reasons, all the configurations with the asset placed at the center of mass of the team detection area are all optimal. Increasing the number of vehicles, the area that can be protected becomes larger and different solutions can be obtained depending on the relative weight given to the two tasks. Simulations are performed for a four agents case and for a more complex scenario with ten vehicles. Figure 7 shows three configurations reached using four agents. In the first case (Figure 7(a)) the configuration is symmetric with the asset placed at the center of mass of the vehicles' detection area. In this case (obtained setting \(N=20\) in \(h_{A}\), \(q=1\) in \(h_{C}\)) the agents are placed as vertices of a square around the asset, which is however located inside an area characterized by a lower level of detection (_i.e._, asset further than \(R_{M}\)). Increasing the interest of the vehicles in moving towards the asset (_i.e._, increasing the priority of task 1 with respect to task 2: \(N=10\) in \(h_{A}\)) the configuration of Figure 7(b) can be reached: one of the vehicles is located directly on the high value asset to protect while the remainder of the agents reach a final stable configuration around it. In this case, the locations of the external agents in the final configuration depend on their initial conditions (all the points around the vehicle in the middle are equally attractive). Increasing the priority of task 2 (\(q=100\) in \(h_{C}\)) yet another configuration can be obtained. The vehicles are taken to an asymmetric final configuration characterized by detection directivity. As shown in Figure 7(c), the asset is always placed at the center of the sonar detection area, and the vehicle positions coincide with the vertices of a regular polygon around the asset to be defended. In Figure 7(c), agents \(V_{1}\) and \(V_{4}\) are further away from the asset than vehicles \(V_{2}\) and \(V_{3}\) providing the increase in directivity. Again, as in the three vehicles case, the stability of the final configuration is related to the information communicated among the team members.