# Research Trends in Collaborative Drones
## 6 Risk and Obstacle Avoidance
### Approach and Methodology  
There are many ways to measure the risk when a given region is traversed by the drone. For example, when the source of the risk is not known, it could be measured by the length of the portion of the path traversed by the drone that is inside the given region (see Figure 5), which we also refer to as the exposure. In general, the hazardous region will be enclosed by the perimeter of a complicated domain (represented as oval in Figure 5). However, if the source of the risk is also known (something we refer to as hotspot), then sometimes, it may be possible to measure the risk from either the distance from the hotspot or an area covered by the drone during the traversal (e.g., see Figure 6.) This is of course an idealized scenario, but it can be useful for making computational estimates on the complexity of the problem. More generally, consider regions in the plane representing risky areas, which were initiated from various sources (which may or may not be known) of pollution, viral infection, population density, etc. As depicted in Figure 6, a drone \(M\) is traversing a straight-line trajectory in a given terrain. The goal of the drone is to navigate the terrain so as to find a path that minimizes its risk of exposure. In many instances, one might be able to measure this quantity precisely. Measurements can be done in various ways. For example, associated to each region, the authors of [51] consider only disks which are generated by hotspots located at the respective centers of the disks. The critical parameter is a distance \(r>0\) that represents how far the hotspots can reach. The value of \(r\) depends on the hotspot's intensity to cause harm, so that the bigger the \(r\), the higher the degree of harm that the hotspot can cause. Moreover, beyond \(r\), no harm is possible. A drone in proximity to the hotspot may get harmed depending on its distance from it. The amount and intensity of harm depends on the distance between the drone and the hotspot (see [51] for additional details). More specifically, the metric introduced in [51] is defined by the area of the circular sector delimited by the points \(A,E,B,D\); it is easy to see that this satisfies \(Area(AEBD)=\arccos\left(\frac{h}{r}\right)\cdot r^{2}-h\cdot\sqrt{r^{2}-h^{2}}\) (see Figure 7). Figure 5: A drone \(M\) is traversing a straight line trajectory. Its exposure during the trajectory may be measured by the length of the path inside the risky region. Figure 6: A drone \(M\) is traversing a straight line trajectory. Its exposure during the trajectory is measured by the sum of the exposures from each of the regions (depicted as ovals) within its range. For the metric of harm proposed here, one may consider the length of the portion of the trajectory of the mobile inside the disk. For this metric, we observe the following. Assume a drone is traversing a straight line trajectory at distance \(h\) from a hotspot (e.g., an infected station) with critical distance \(r\), where \(r\geq h\geq 0\). The total amount \(H(r,h)\) of harm that the drone \(M\) incurs for its entire trajectory within the hotspot's range is equal to  
\[H(r,h)=2r\sqrt{1-h^{2}/r^{2}} \tag{3}\]  
(see Figure 7 where a drone \(M\) is traversing a straight line trajectory at distance \(h\) from the source \(C\) entering at \(A\) and exiting at \(B\)). Similar estimates may also be possible to derive for other types of regions. There are various kinds of hotspots, for example, for the case of an infection, the function \(H(\cdot,\cdot)\) may be the well-known viral load which is studied in epidemics, but in general, it does not have to be limited to this. More generally, if a drone traverses a path \(P\) consisting of \(k+1\) vertices \(u_{0},u_{1},\ldots,u_{k}\) with respective sensitivity distances \(r_{0},r_{1},\ldots,r_{k}\), then the total harm accumulated will be equal to  
\[\sum_{i=0}^{k}H(r_{i},h_{i}), \tag{4}\]  
where \(h_{i}\) is the distance of the drone from node \(u_{i}\).