A Novel Multi-Layer Framework for BVLoS Drone Operation: A Preliminary Study

Obstacles such as buildings or trees occupy some cubes in $B$, where obviously drones cannot fly. However, drones are free to fly over obstacles at a sufficient height, and hence this layer is defined at multiple heights. In general, the higher the drone altitude, the fewer obstacles will be encountered. The cubes that include obstacles are not considered in $B$, and therefore the number of cubes in $B$ is at most $nmh$.

No Fly-zones are areas where drone flight is forbidden, e.g., close to airports and military warehouses. We assume that no-fly zones are effective at any height. As before, the cubes that include no-fly zones are removed from $B$.

We assume that in the environment $B$ there is already a deployed wireless network infrastructure that comprises a set of cellular towers. These towers provide Internet connectivity (e.g., 4/5G) to ground users. Specifically, let $T$ be a set of towers with a given discretized position $t=(x_{t},y_{t},z_{t})\in B$. For simplicity, we assume that each tower $t\in T$ is at ground level, so $z_{t}=1$. We also assume that there is no more than one tower on each cube.

Drones can connect to these antennas if some physical constraints are met at the same time. We have to distinguish between the transmitter device, i.e., the tower, and the receiver device, i.e., the drone. In this paper, we use the Friis transmission equation (see Eq. (1)) in order to determine the quality of the BVLoS communication link if a drone in cell $c$ establishes a connection to a tower $t$, i.e., by evaluating:

where $P_{W}(c,t)$ is the receiving power at the drone, $P_{W}(t)$ is the transmitting power of the tower, $G(t)$ and $G(c)$ are the antenna gains of the transmitting and receiving devices, respectively, $\lambda$ is the wavelength representing the effective aperture area of the receiving antenna, and $d_{S}=\|c-t\|_{2}$ is the Euclidean slant distance that separates the antennas.

Having said that, the wireless infrastructure layer is defined at multiple heights, for multiple reasons. First, at different altitudes, the distance $d_{S}$ in Eq. (1) varies, which in turn affects the received power at the drone. Another aspect to take into account is the probability of being on the Radio Line of Sight (briefly, LoS) which increases if the height of the drone increases. Namely, the chances of being LoS increase with the relative elevation angle between the drone and the tower. Not only the drone altitude but also the presence of obstacles impact the LoS. So, when dealing with obstacles, the chances of being LoS decrease if the density of obstacles increases.

To wrap up, given two parameters $\alpha$ and $\beta$¹¹1The $\alpha$ and $\beta$ parameters are called here the S-curve parameters, i.e., a modified Sigmoid function. that model the surrounding environment (e.g., urban, rural) by assuming that the drone is flying at the cell $c=(x_{c},y_{c},z_{c})$, the probability to be LoS with the tower $t=(x_{t},y_{t},1)$ is defined as [19]:

So, the higher the altitude, the longer the distance $d_{S}$, the less will be the received power $P_{W}(c,t)$ by Eq. (1), and the less will be the quality of the BVLoS link. But, at a higher altitude, a drone can potentially detect more towers due to its increased elevation. Finally, given both $c$ and $t$, let

be the link-reliability function that returns the product among the probability of being LoS, and the normalized received power at the drone. Potentially, there is an edge between $(c,t)$ as long as $\mathcal{L}(c,t)>0$.

For a given cell $c=(x_{c},y_{c},z_{c})\in B$, let $s(z_{c})=1+2z_{c}$. To simplify our multi-layer model, from now on we assume that the probability of LoS, i.e., $\mathcal{P}_{\text{LoS}}(c,t)$, is null or negligible for all towers outside a square of size $s(z_{c})\times s(z_{c})$. Thus, when $z_{c}=1$, $s(z_{c})=3$; while when $z_{c}=2$, $s(z_{c})=5$. In general, the square sides depend on the $\alpha$ and $\beta$ parameters, but currently, we omit the dependency. Formally, let $f_{T}(c)$ be the subset of towers that the drone can see when it flies inside the cell $c$, i.e., the towers that belong to the square of size $s(z_{c})$ centered in $c$, as represented in Figure 4. Precisely

where $T$ consists of cells $(x_{t},y_{t},1)$ such that $x_{c}-z_{c}\leq x_{t}\leq x_{c}+z_{c}$ and $y_{c}-z_{c}\leq y_{t}\leq y_{c}+z_{c}$. Under the above simplification, we assume that a drone in cell $c$ can communicate with any tower $t\in f_{T}(c)$ with a receiving power $P_{W}(c,t)$ determined according to Eq. (1). For any $t\not\in f_{T}(c)$, there is no connection between the drone and the tower. Note that $f_{T}(c)$ increases with the cell height. Moreover, we normalize the value $P_{W}(c,t)$ in the interval $[0,1]$ where $P_{W}(c,t)=1$ represents the maximum received power at the drone, i.e., when the drone and the tower share the same position $c=t$. Notice that $0\leq\mathcal{L}(c,t)\leq 1$.

The risk-map layer models the potential risks that unexpected events can cause to ground people. In this paper, we do not consider air risk, i.e., the collision with other flying vehicles (manned or unmanned). The ground risk-map layer specifically evaluates the risk on the ground depending on a concatenation of different probabilities. Given a cell $c\in B$, the risk [12] is defined as follows:

where $\mathcal{P}_{\text{event}}$ is the probability that the drone crashes on the ground; $\mathcal{P}_{\text{impact}}$ is the probability to impact a person when the drone crash on the ground; and $\mathcal{P}_{\text{fatality}}$ is the probability to produce fatal injuries after a person has been impacted.

In the next section, exploiting the presented multi-layer framework, we will create a suitable graph-based data structure used to solve our problem.

After we have presented the multi-layer framework, which is at the basis of our BVLoS flight of drones, and before constructing the graph, we need to introduce some fundamental prerequisites. The graph we will create has vertices and edges. The vertices represent “locations” or “locations associated with a tower”, while the edges represent “movements between adjacent cells” or “tower handovers”. The graph is also weighted, where the weight of each edge represents its dependability. In this paper, we combine the information from the multi-layer framework that forms the path dependability. More formally, the path dependability jointly takes into account the ground-safeness, the link-reliability, and the handover-success-rate. For us, the term dependability means the quality of being trustworthy and reliable, and therefore we measure it through a probability, i.e., the larger the probability, the more the path dependability.

So, ground-safeness, link-reliability, and handover-success-rate are individual probabilities to combine in order to obtain the path dependability. Consider an edge that enters the cell $c^{\prime}$ associated with the tower $t$ and exits the cell $c$ associated with the tower $t$. Intuitively, the ground-safeness represents the dependability of the link with respect to the ground and the possible related safety risks. It is defined as $\mathcal{P}_{\text{GS}}(c^{\prime},t)=1-\mathcal{R}(c^{\prime})$, where $\mathcal{P}_{\text{GS}}(c^{\prime},t)=0$ represents the highest risk, while $\mathcal{P}_{\text{GS}}(c^{\prime},t)=1$ does not represent risk. The link-reliability is simply $\mathcal{P}_{\text{LR}}(c^{\prime},t)=\mathcal{L}(c^{\prime},t)$, which combines the probability to be in LoS, and the normalized received power at the drone. To summarize, the edge that represents a movement between the two cells has weight $\mathcal{P}=\mathcal{P}_{\text{GS}}(c^{\prime},t)\ \mathcal{P}_{\text{LR}}(c^{\prime},t)$, i.e., the product among the ground-safeness, and the link-reliability of the destination cell.

Finally, we assume that the Handover-Success-Rate (HSR) is a value $\mathcal{P}_{\text{HSR}}\in\{\frac{1}{2},1\}$, where $\mathcal{P}_{\text{HSR}}=1$ represents the fact that the drone does not change the tower, continuing communication with the same tower $t$, while $\mathcal{P}_{\text{HSR}}\ =\frac{1}{2}$ represents the scenario when the drone changes the connection with another nearby tower $t^{\prime}$. In this latter case, we assume that something can go wrong, and hence the HRS is fixed to $\frac{1}{2}$. A more complex HSR probability can be utilized in our model, which depends on the $\mathcal{P}_{\text{LoS}}$ and $\mathcal{P}_{\text{LR}}$ parameters. Therefore, the edge that represents a tower handover has weight $\mathcal{P}=\mathcal{P}_{\text{HSR}}$.

During a drone mission, the drone moves from a source to a destination and traverses a path consisting of several edges. The path dependability is given by the product of the weights of the traversed edges.

Let $G=(V,E)$ be the weighted directed graph which is defined by a set $V$ of vertices and a set $E$ of edges, which is built from the box $B$. Recall that the box $B$ does not include either obstacle or no-fly zones. We build the graph $G$ as follows. For each cell $c\in B$, we create a vertex called $v_{c}$ that represents the cell $c$, plus $|f_{T}(c)|$ additional vertices to represent the towers to which it is possible to connect from $c$, denoted as $v_{c}^{t}$. So, the set of vertices is $V=\{v_{c}:c\in B\}\cup\{v_{c}^{t}:t\in f_{T}(c)\ \forall c\in B\}$, while the set of edges $E$ is a bit more complicated. In general, there are two types of edges: intra-edges (i.e., tower handover) and inter-edges (i.e., moves between adjacent cells). An intra-edge is in the form of $(v_{c},v_{c}^{t})\in E$, while an extra-edge is in the form of $(v_{c}^{t},v_{c^{\prime}}^{t})\in E$.

With regard to the intra-edges, these are used to connect a drone located at a given cell to all the possible visible towers in that cell. For each $v_{c}^{t}\in V$ we add a directed edge $(v_{c}^{t},v_{c})\in E$ with cost $\mathcal{P}_{\text{HSR}}=1$, and a directed edge $(v_{c},v_{c}^{t})\in E$ with cost $\mathcal{P}_{\text{HSR}}=\frac{1}{2}$ (Algorithm 1, Line 1). Crossing this latter edge represents the drone changing the connection to a different tower. Note that we could have used the opposite manner, but the graph construction guarantees that the handover is only paid once. The pseudocode of the graph construction is reported in Algorithm 1.

With regard to the inter-edges, these are used when the drone moves between adjacent locations. Given $c=(x_{c},y_{c},z_{c})\in B$, let $Adj(c)$ be the set of adjacent cells of $c$ where $Adj(c):=\{c^{\prime}=(x_{c^{\prime}},y_{c^{\prime}},z_{c^{\prime}})\neq c:x_{c^{\prime}}=x_{c}\pm\{0,1\},y_{c^{\prime}}=y_{c}\pm\{0,1\},z_{c^{\prime}}=z_{c}\pm\{0,1\}\}$. So, as illustrated in Figure 3, $\max|Adj(c)|=26$. For each neighbor of $c\in B$, i.e., $\forall c^{\prime}\in Adj(c)$, we connect pairs of vertices that represent the same tower $t$ in the two different cells $c$ and $c^{\prime}$, i.e., the towers in $t\in f_{T}(c)\cap f_{T}(c^{\prime})$. For each of these pairs, we add a directed edge $(v_{c}^{t},v_{c^{\prime}}^{t})\in E$ with cost $\mathcal{P}_{\text{LR}}(c^{\prime},t)\ \mathcal{P}_{\text{GS}}(c^{\prime})$, and a directed edge $(v_{c^{\prime}}^{t},v_{c}^{t})\in E$ with cost $\mathcal{P}_{\text{LR}}(c,t)\ \mathcal{P}_{\text{GS}}(c)$ (Line 1). Crossing these edges represents the drone moving, e.g., from the cell $c$ to $c^{\prime}$, thus considering the ground-safeness and the link-reliability of the coming cell.

An example of the graph construction is reported in Figure 4. The yellow vertex in position $(2,3)$ has $8$ neighbors, i.e., the ones inside the green highlighted square. Moreover, there are $3$ towers, i.e., $t_{a}$, $t_{b}$, and $t_{c}$. Among these, only $t_{a}$ and $t_{b}$ are in range with position $(2,3)$, and therefore intra-edges from $v_{2,3}$ are only created between $v_{2,3}^{a}$ and $v_{2,3}^{b}$. The blue cell is in position $(3,3)$ which is in range with all the $3$ towers. As we can see, both $v_{2,3}$ and $v_{3,3}$ share the towers $t_{a}$ and $t_{b}$, and hence inter-edges between $v_{2,3}^{a}$ and $v_{3,3}^{a}$, and $v_{2,3}^{b}$ and $v_{3,3}^{b}$, are created. The same construction can be repeated for the red cell in $(3,2)$.

Note that by decoupling the tower handover from the movements between adjacent cells, the number of edges decreases. Recall that at level $1\leq i\leq h$, the drone can see at most $|T_{i}|=(1+2i)^{2}$ towers and $\max|Adj(c)|=26$. Also, let $T_{\max}=\max|T_{i}|$. In our construction, instead of preparing an edge ad hoc for each pair $v_{c}^{t}=(c,t)$ to $(v_{c^{\prime}}^{t^{\prime}}=(c^{\prime},t^{\prime})$, we first move from $(c,t)$ to $(c,t^{\prime})$, and then from $(c,t^{\prime})$ to $(c^{\prime},t^{\prime})$. Without the decoupling, $(c,t)$ would have had $\mathcal{O}(26\times T_{\max})$ outgoing edges. With the proposed decoupling, $(c,t)$ has $\mathcal{O}(T_{\max})$ intra-edges plus $\mathcal{O}(26)$ inter-edges. Thus, the graph construction has been strongly improved by decoupling the tower handover and the cell movement.

With our assumptions, the box $B$ has $\mathcal{O}(nmh)$ cells and $\mathcal{O}(nmh)$ overall inter-edges. So, the number $|V|$ of vertices of the graph $G$ is upper-bounded by $|V|\leq nmh+\sum_{i=1}^{h}{|T_{i}|nm}=\mathcal{O}(nmh^{3})$. With regard to the edges, we can upper bound $|E|\leq 2\sum_{i=1}^{h}{T_{i}nm}+2\sum_{i=1}^{h}{26T_{i}nm}$, where the first summation refers to the intra-edges, while the second summation refers to the inter-edges (note the multiplicative factor $2$ that creates the two directed edges). In conclusion, $|V|\in\mathcal{O}(nmh^{3})$, and $|E|\in\mathcal{O}(nmh^{3})$.

The constructed graph represents the multi-layer framework and models any drone mission by a path whose weight is its dependability.

Given a box $B$, the multi-layer framework, a starting and a destination cell $s,g\in B$ respectively, and built the weighted dependability graph $G=(V,E)$, we are in position for solving the Maximum Path Dependability Problem (MPDP) whose objective is to find a drone’s path $\pi^{*}$ that starts in $s$ and finishes in $g$ such that the dependability of $\pi=\{(s,\cdot),\ldots,(\cdot,g)\}$ is maximized. Formally,

Thus, the original objective to find the maximum probability path is equivalent to finding the minimum shortest path where each edge weight (probability) is replaced by the absolute value of its logarithm, and the path cost is the sum of the edge weights. We remark that the above graph construction enables standard path planning algorithms, such as Dijkstra’s algorithm, to retrieve the optimal path for the drone to traverse in the area prioritizing both safety and communication reliability, and thus enabling the BVLoS operation.

So our problem is polynomially solvable in $|E|$. Moreover, our model is flexible because by changing the path dependability we can find paths that optimize different criteria. For example, focusing on the reliability of the communication and posing all $\mathcal{P}_{\text{GS}}=1$, MPDP find the path that maximizes the connectivity issues. Similarly, fixing $\mathcal{P}_{\text{LR}}=\mathcal{P}_{\text{HSR}}=1$, MPDP will find the path with minimum ground risk. Or, just preserving $\mathcal{P}_{\text{HSR}}$ (and $\mathcal{P}_{\text{LR}}=\mathcal{P}_{\text{GS}}=1$), $\pi^{*}$ optimizes the number of handovers.