In the last years, mega-wild res have become routine news. For the re ghters, it is crucial to rely on systems able to prevent and detect those disastrous events. Wireless Sensor Networks (WSNs) become an important issue such as environmental and forest monitoring. An important requirement for the WSNs is the ability to precisely localize their sensors for augmenting the collected data with their position. In this paper, we survey three previous presented algorithms, called Short, Dir, and Omni, for localizing terrestrial sensors using a drone. We compare their performance under a variety of di erent conditions: we evaluate the path length, we compare the localization performance in terms of precision and error, and we analyze the impact of using multilateration instead of trilateration during each localization. Our study supports the importance of the geometry in selecting the waypoints from which a sensor is measured. The common belief that multilateration leads to a more precise localization is not completely correct. Indeed, we have shown by simulations that, when the waypoints follow a well pondered geometry, multilateration is not useful. Whereas, the multilateration can improve the precision for algorithms that select the waypoints according to a best-e ort policy.
It is widely know that, during the dry season, the risk to have a forest re is very high. Every year, such situations create huge damages in terms of money and even in terms of human lives. Nowadays, we are used to watch on television re ghters that try to extinguish res where the majority of which are caused from humans fraudulently. Therefore, it is appropriate to build robust solutions for re ghting in order to prevent and detect such emergencies. As an example, the Mediterranean basin, which is a very large area, is continuously a ected by res in the summer. Hence, it is required to cover and precisely monitor these potential hazardous areas by sensors.
In such scenario, a Wireless Sensor Network (WSN) can be built inside a forest for monitoring tasks. Deploying
sensors close to dangerous places is essential in order to detect potential clues, for example, unexpected
temperature increases in a very dry zone. Once the detected data from the terrestrial sensors are delivered to the master
gateway node, proper and adequate solutions are taken. To build the WSN, sensors can be randomly deployed
in the target area and a mobile vehicle can be used to be the contact point between the WSN and the external
world. Therefore, the random deployment results in sensors initially unaware of their location. Localization is
one the most important task in a WSN, since it provides fundamental support for many location-aware protocols
and applications. Due to limitations in form factor, cost per unit, and energy budget, individual sensors are not
expected to be GPS-enabled. Many existing localization algorithms in the literature require a large number of
xed anchor points, i.e., sensors whose positions are known a-priori [
In order to decrease the setup costs of WSNs, we propose to replace xed anchor sensors with a single mobile
anchor sensor, such as an Unmanned Aerial Vehicle (UAV) [
In this paper, we survey three previous presented algorithms: Short [
The three algorithms make di erent assumptions of the available hardware at the drone for communicating with the terrestrial sensors. Dir assumes that the drone is equipped with directional antennas, which can adjust their beamwidth according to the required precision. Short and Omni assume only a regular omnidirectional antenna. All the solutions adopt the Impulse-Radio Ultra-Wide-Band (IR-UWB) technology for the antennas. Dir and Omni are based on a static path for the drone over the deployment area, and guarantee the localization precision required by the nal user, while Short produces a very short static path, but it cannot guarantee any a-priori localization precision. Again, Dir and Omni prefer precision to path length, whereas Short sacri ces precision for being able to accomplish an entire mission without recharging the battery.
It is very important to stress that all the three algorithms locate all the nodes independently of their number and their actual position. While traversing the path, the drone calculates the distances with the sensors at special positions, called waypoints: each waypoint acts as a xed anchor node. Once a sensor has received three measurements, it can calculate its position by applying the trilateration procedure. Moreover, in this paper, to localize themselves, the sensor can either perform a multilateration using six di erent measurements or the usual trilateration procedure.
The main contribution of this work is to present an extensive comparison of the three algorithms that studies, in particular, whether using a greater number of measurements for each sensor is possible to increase the localization precision. Our ndings contradict the common belief that multilateration always help.
The rest of paper is organized as follows. Sec. 2 revises the current literature on localization algorithms using drones. Speci cally, in Sec. 2.1, we survey Short, in Sec. 2.2, we survey Dir, while in Sec. 2.3, we survey Omni. The experimental results are discussed in Sec. 3, while Sec. 4 concludes the paper. 2
We consider a network of n sensors deployed in a rectangular area Q of size Qx Qy. From now on, sensor P is short form for the sensor that resides at a point P in Q. Since, in principle any point P in Q is candidate to contain a sensor, with a little abuse of notation, we denote P indistinctly as the point P or the sensor P . Moreover, we use the notation P P 0 to denote the ground distance between two waypoints/sensors P and P 0.
Our mobile anchor is a drone. It ies at a xed altitude h [
Assuming that the path is used for localizing objects, the drone will take the measurements at prede ned points along its trajectory. The projection on the ground of a measurement point will be called waypoint. (a) Scan (b) Double Scan (c) Hilbert (d) S-Curves (e) Lmat
In [
It is worthy to note that Dir and Omni adopt a Scan 2D trajectory. Short, instead, starts with a trajectory similar to Lmat, which is then modi ed by solving an instance of the Traveling Salesman Problem (TSP).
In all the three algorithms compared in this work, i.e., Short, Dir, and Omni, during the mission the drone takes measurements at pre-established points of the path called waypoints.
We assume that the drone, using the IR-UWB antenna, measures its distance from the sensors by the roundtrip time of messages exchanged with them. To take a measurement, the drone acts as follows. At each waypoint wi, the drone sends a beacon signal with a unique identi er which depends on the coordinates (xwi ; ywi ) of the waypoint and the current timestamp. Each sensor on the ground that can hear a beacon replies to the drone with an ack message that contains its ID, the current timestamp, and the identi er of the beacon received. The drone computes the distance between itself and the sensor using the round-trip time of the beacon message and then sends a message with the computed distance to the sensor. Once a sensor has collected three measurements, it can locally apply the trilateration algorithm (Eq. (1)) to discover its position P . Hence, given three ground measures, the estimated position of P is the point (xP ; yP ) that minimizes the sum of the least squares, that is: min s.t. p(xwi xP )2 + (ywi yP )2 + i = wiP for i = 1; 2; 3
The multilateration (in our case, a six-lateration) procedure can be considered as the generalization of trilateration, as reported in Eq. (2), when six measurements are considered: min
P6 2 i i s.t. p(xwi xP )2 + (ywi yP )2 + i = wiP for i = 1; : : : ; 6 (1) In other words, slant distances when projected on the ground cannot be larger than dmax. This dmax constraint holds for all the three surveyed algorithms.
Slant distances are a ected by errors that depend on the adopted technology [
1 cos( )
r = es 1 + h2 d2 where is the angle of incidence of the slant distance to the ground, d is the distance on the ground between the drone and the sensor, h is the altitude, and es the actual measured slant distance error (see Fig. 2).
The drone and the sensor have communication ranges, rdrone and rsensor, respectively. Since a message can be exchanged between the drone and the sensor only if they can hear each other, throughout this paper, the communication range will be r = min frdrone; rsensorg.
The drone measures the (3D) slant distance s, which is de ned as the line-of-sight between itself and the measured sensor. Clearly, s r. By simple geometric argument, since we assume negligible error in altitude, it is easy to see that any ground measured distance d satis es: d = mP2aQx ed(P ) = s s 1 +
h2 dmin2
From Eq. (4), the ground error increases when the ground distance decreases. Since the slant error is maximum
when the distance on the ground is minimum, xed dmin as the minimum allowed ground distance, the maximum
possible ground error is [
However, only Dir and Omni are interested in bounding the localization error so, the dmin constraint is required only for them.
Moreover, to minimize the localization error eL(P ) due to the trilateration, the three waypoints from which
a sensor is measured cannot be collinear neither among them nor collinear with the sensor [
L = max eL(P ) =
P 2Q
sin
d
min
2
where min is the minimum angle obtained during the localization. The best case occurs when min = =3 [
For the same reason mentioned above, the min constraint is satis ed only for Dir and Omni since they need to bound the localization error. min is not actually satis ed in Short because the built geometry by triangles does not allow to bound the minimum angle min. For example, if a generic point P resides in the midpoint of a triangle's side, dmin is respected but min = 0.
To summarize, the aim of the precise Dir and Omni algorithms is to obtain an L-precise localization for each point, where the maximum localization error L is determined as a function of dmin and min. In conclusion, the sensor P in Q is L-precisely localized, where L is given by Eq. (6), if the drone chooses three ranging waypoints w1(P ), w2(P ) and w3(P ) for P such that they satisfy the following constraints: 1. dmax: which controls the reachability for each point P in Q: wi(P )P dmax for i = 1; 2; 3; 2. dmin: which controls the ground precision d for P in Q: wi(P )P dmin for i = 1; 2; 3;
min: which controls the collinearity of the drone with P : (P ) min; 4. non-linearity : which controls the collinearity among waypoints: w1(P ), w2(P ), and w3(P ) cannot belong to the same straight line.
As regard to Short, it only respects the reachability and collinearity constraints.
We nally resume the constraints for each algorithm in Tab. 1. Short assumes the drone to be equipped with a regular omnidirectional antenna. The aim of this algorithm is to nd a static path as short as possible for localizing all the sensors, but any a-priori bound on the localization error is accepted. Short operates in two phases: waypoint grid construction and waypoint ordering.
As depicted in Fig. 3, in the rst phase, Short builds a set of waypoints forming a grid of isosceles triangles (dashed lines) which covers the whole deployment area Q.
Ty
S
Tx
Qy
Qx
The waypoints reside at the vertices of each triangle (gray dots). The triangles have the same base Tx and the same height Ty. Isosceles triangles are more exible than equilateral ones, because they can be resized along both dimensions. This allows us to cover the whole deployment area as tightly as possible, without wasting path length for covering external areas uselessly. The base and the height of the triangles must be as long as possible, without violating the maximal triangle side dmax. Given a base Tx, the longest feasible height is (Tx=2)2. Short computes the base and the height of the triangles as follows:
Qx dmax Tx = Qx=
Ty = Qy=
Qy Tymax
Finally, in the waypoint ordering phase, Short executes the TSP solver to connect all the waypoints in order to drastically reduce the path length. The TSP solver takes in input all the waypoints' locations and gives in output the static path S . An example of S is depicted as a dashed line in Fig. 4, where the gray dots represent the waypoints, while the black dots are the sensors inside Q.
In our solution the directional antennas cover six di erent sectors, as illustrated in Fig. 5. The drone can
send at the same time the beacons in all six orientations using an array of directional antennas [
The static path D is depicted as a dashed line in Fig. 6. D is uniquely determined posing Fx = dmin=2, Fy = 0, F = ( Fx; Fy), and E = (xE ; yE ), where xE = Qx + Fx and yE can be either 0 or Qy depending on the parity of the number of vertical scans. The waypoints reside only on the vertical scans. The inter-waypoint distance between two consecutive waypoints is denoted by Iw.
Fx H Dir assumes the drone to be equipped with directional antennas. The drone is equipped with an array of six directional antennas. Each antenna is assumed to transmit the beacon in a circular sector, centered at the antenna position, of radius r, beamwidth 2 , where = arctan dIwm=in2 , and orientation .
From each vertical scan , the points of Q that we can precisely measure using any type of orientation are those at distance at least dmin=2 and at most (dmax 2Iw)=2 from the scan . In fact, although the antennas of type hor can precisely measure the points at distance from dmin up to dmax from , the most stringent limits for the precise measurements come from the antennas of type up and dw. Therefore, any two consecutive vertical scans are xed at distance no greater than at distance greater than (dmax dmin 2Iw)=2. Having xed the rst and the last scan respectively at Fx and Qx + Fx, the length of H is xed in order to evenly distribute Qx + 2Fx in stripes of width as tight as possible to the maximum value. In this way, the whole area is covered without wasting path length. Finally, observe that the left and right stripes of Q of width dmin=2 adjacent to each vertical scan are not measured by to satisfy the dmin constraint. The left and right stripes adjacent to are then measured by the scan that precedes and the scan that follows .
During the ight, for each waypoint, the drone sends the beacon's message along the six di erent orientations of beamwidth: up+, up , dw+, dw , hor+, and hor . When a sensor receives the drone's message from orientation , for = d =3 and d = 0; : : : ; 5, it rst checks whether the d-th location of its register R is empty or not. In the rst case, the sensor is receiving for the rst time the beacon from the orientation d. In the second case, since the sensor has already heard that beacon from orientation d, the message is ignored. When the sensor receives one orientation for the rst time, it replies to the drone with an ack message. The drone infers the distance from the time of the round-trip message, and sends to the sensor the measure. The sensor stores the measure in the local register R. The trilateration is performed by the sensor when it has received three measurements.
As explained previously, the localization error depends on the position of the three waypoints from which
the point is ranged. From Eq. (6), the error is minimum when for each sensor min = =3. In our localization
technique, the minimum angle at P is =3 if and only if the sensor P resides at the intersection of the orientations
of the three sectors centered at the waypoints w1(P ), w2(P ) and w3(P ). Although it is not possible to achieve
for every point P that the minimum angle is =3, we can prove that the algorithm Dir provides [
Considering Eq. (7), we are now in a position to rewrite Eq. (6) as a function that depends on h, dmin and Iw.
Inverting this expression we can ensure the precision required by the nal user. In conclusion, according to [
Now we present Omni, which does not require specialized hardware such as directional antennas. Actually, Omni is based on a standard omnidirectional antenna.
The omnidirectional antenna sends beacons with the drone's position isotropically. Since the sensors cannot distinguish from which direction they receive the beacon, it is di cult to select the waypoints that guarantee the desired geometry. For this reason, Omni adopts a two phase approach. In the rst phase, the sensors obtain a rough estimation of their position. In the second phase they use this estimation to pick among all the available waypoints the best ones, called precise waypoints, that equally divide the turn angle around the sensor itself to perform a trilateration and obtain a precise localization.
The static path O is depicted in Fig. 7. The drone's mission starts at F = ( Fx H; Fy), where Fx = dmin=2, Fy = p3(dmax Iw)=2 and nishes at E = (xE ; yE ), where xE = Qx + Fx and yE can be either 0 or Qy. The inter-scan distance is set to H = (dmax dmin 2Iw)=2. O consists of two vertical scans outside Q that are used to measure the sensors in the leftmost stripe of Q close to the vertical border, while the last vertical scan is at distance no larger than dmax2 2Iw from the rightmost border.
The behavior of the drone under Omni is the same as during algorithm Dir. However, the behavior of the sensors is quite di erent. According to Omni, until the rst trilateration, the sensor stores all the measurements that it receives which are above the dmin ground distance. As soon as the sensor has collected three distance measurements which belong to two di erent scans and are greater than the minimum ground distance, the sensor performs the rst trilateration to compute a rough estimation of its position P^ = (xP^ ; yP^ ). min 3
2 s L = s 1 +
h2 dmin2 r 2 1 1 + p3
Iw=2 2 dmin Iw=2 dmin Iw
F
O
Fy Fx
Omni logically tessellates the area by diamonds (see Fig. 8). Then, from P^, the sensor locates the closest vertical scan on its left. Then, conceptually, the sensor derives the six rays that equally divide the turn angle around its current position. The intersections of such rays with the scan on the left of P^ are then the most desirable positions as regard to the waypoint geometry (i.e., the min constraint) from where taking the nal measurements. However, such points may not coincide with waypoints because the drone takes measurements at regular distance and may also not satisfy the dmin constraint. So, to nd the three precise waypoints obeying all the constraints, the sensor locally computes the three precise waypoints w1(P^); w2(P^); w3(P^) on the scans on its left. Selected w1(P^); w2(P^); w3(P^), the sensor continues to hear to the drone until it has collected all the three precise waypoints and discards the measurements which are useless. Finally, the sensor computes the second precise trilateration using the precise waypoints, and the localization process is nished. (9) (10) H w3 dmdinmin 2 A
C B
D w2
The localization error strictly depends on the position of the three precise waypoints w1 and w2, and w3 from
which the point P is ranged. It has been proved in [
Considering Eq. (9) and from Eq. (6) rewritten in terms of h, dmin and Iw, the localization precision in Omni
can be expressed as [
We have implemented the Short, Dir, and Omni localization algorithms in MATLAB programming language in order to compare their performance under a variety of di erent conditions.
Let rst observe that Omni and Dir belong to the class of the precise algorithms because they aim to guarantee the user-de ned localization precision. Instead, Short belongs to the class of the best-e ort algorithms that cannot ensure any a-priori guaranteed localization precision L, but they return a very short static path. Consequently, we cannot use L as the main metric to compare the performance of the three algorithms. Namely, Short does not respect dmin and min, and we cannot bound a-priori the localization error.
Considering the two di erent classes to which the three algorithms belong, in order to compare the goodness of them, we can only evaluate the experimental position error, expressed as the distance P P 0 between the original position P and the estimated position P 0. For each drone's mission, we record the worst experimental error and then we compute the error bound L as the average value of the worst bound on the position error in all the missions. We also compute the 95%-con dence interval of the data reported in the experiments. Moreover, we will continue to evaluate the ability of Dir and Omni of guaranteeing a-priori localization L given in input to the algorithm. We compare and evaluate the three di erent algorithms under the metrics:
2. we compare localization performance in terms of precision and error, and 3. we analyze the di erence of using multilateration instead of trilateration.
The goal of the experiment shown in Fig. 9, is to compare the length of the static paths S , D, and O of the Short, Dir, and Omni algorithms having xed Iw = 5 m and L = 0:30 m for Dir and Omni. Dir and Omni have more or less the same number of vertical scans, but each vertical scan in Omni goes well beyond the deployment area. Precisely, in Omni, each vertical scan has length 2Fy + Qy, where Fy = (dmax Iw)p3=2, whereas Fy = 0 for Dir. Fig. 9 shows that D is better than O. The increase on the path length of the static path of Omni is the paid price for not using specialized hardware. Note that the path length in Omni is not monotonic decreasing because we use a xed inter-scan value without resizing it as we do in Dir. Obviously, Short is the shortest static path regardless of the length of the communication radius r, but one has to remember that Short requires to solve o -line an instance of TSP before the drone can start its mission. Dir is twice as long as Short. Short is the shortest path but it does not have neither dmin nor min constraints, and thus its precision is not guaranteed (see Fig. 10 for the analysis of the precision of Short).
14 12 10 h tg 8 n e l 6 4
Note that dmax, and thus the communication range r, is the parameter that mostly in uences the length of the path in Dir and Omni. For this reason, we omitted to report the value of dmin adopted in Fig. 9 for Dir and Omni.
In the following, we compare in Fig. 10 the localization performance in terms of user-de ned precision and error.
In the rst experiment shown in Fig. 10(a) we compare the user-de ned precision L for the precise algorithms Dir and Omni. Speci cally, we want to experimentally verify that our bound holds, and thus L < L. In Dir and Omni we set Iw = f5; 10g m and L = 0:30 m. Given the localization precision L, we select dmin for Dir and Omni by applying Eq. (8) and Eq. (10), respectively, and compare it with the results from the experiments. It is worthy to note that the error bound in Fig. 10(a) is always smaller than the user-de ned localization precision L = 0:30 m. The Omni algorithm is slightly more precise than Dir because Omni uses values of dmin larger than Dir, and the min angle is larger in Omni than Dir. Moreover, Omni uses two trilaterations.
In the second experiment, reported in Fig. 10(b), we compare the localization error among all the three algorithms, considering also the path length. Dir and Omni run using the values of dmin and Iw that guarantee a maximum error of L = 1 m. For Short, instead, we just use for each sensor the three waypoints that form the vertices of the triangle that better circumscribes the sensor itself. Although in Short the three waypoints surround the point to be localized, neither minimal ground distance dmin nor minimum angle min constraints are respected in a prede ned way. Thus, the error for Short increases in Fig. 10(b). In fact, even if the sensor is inside the triangle formed by the three waypoints, it is su cient that it is close to one of the vertices to break the dmin constraint. Hence, it is easy to see from Eq. (8) that, when the ground distance tends to 0, the error blows up. Moreover, the reported error and the con dence interval of Short is higher when n is large than when n is small because the probability of encountering sensors in the worst position increases. Contrary, Dir and Omni have a good level of precision since all the four constraints dmax, dmin, min, and non-linearity , are satis ed and as a consequence of the setting of the parameters Iw and dmin, they guarantee an a-priori localization error of L = 1 m. It is important to note that the path length of Short is 3:2 km, which is less than half the path length of Dir (6 km) and Omni (8:2 km).
In the third experiment depicted in Fig. 10(c), we have relaxed the dmin constraint accepting any measurement, i.e., dmin = 0. With this experiment we want to evaluate if, paying a higher localization error, we can considerably shorten the path of the precise algorithms. First, note that the dmin constraints cannot be relaxed for Dir, since = arctan dIwm=in2 . Hence, in Dir, dmin must be positive. In omni we can set dmin = 0. The error increases, but it remains smaller than that of Short. Nonetheless, the path length of Omni is still the double of that of Short. In conclusion, Omni and Dir have been mainly designed to be precise, and reducing the precision does not reduce the path length. So, precision cannot be traded for path length in Omni and Dir.
Finally, we compare our algorithms when multilateration is used instead of trilateration. A crucial nding of our experiments is that multilateration helps when the algorithm is best-e ort, but it does not help when the algorithms are precise.
OMNI Iw = 5; dmin = 0:00
SHORT DIR DIR6 max PP0 : h = 30; r = 150
OMNI OMNI6 100 200 300 400 500 600
nodes (a) Short vs Short6 100 200 300 400 500 600
nodes (b) Dir vs Dir6 100 200 300 400 500 600
nodes (c) Omni vs Omni6
In the experiment shown in Fig. 11 we want to analyze whether an optimal geometry of three waypoints is better than six generic waypoints, or not. Speci cally, Short6, Dir6, and Omni6 are, respectively, the modi ed version of Short, Dir, and Omni, which localize each sensor exploiting the rst (in order of time) six noncollinear waypoints that a sensor hears from the drone. Fig. 11(b) and Fig. 11(c) show that a good geometry, is better than a blind multilateration. It is not su cient to increase the number of waypoints for obtaining a smaller error. Namely, since Dir6 and Omni6 choose the rst three waypoints from a vertical scan and the other three from the subsequent one, min can be very small. Although Omni6 (see Fig. 11(c)) is less precise than Omni, the localization degrades less than between Dir and Dir6 (see Fig. 11(b)). We believe that this is due to the fact that in general Omni and Omni6, according to the regular omnidirectional antenna, can take measurements at distances greater than compared to those taken in Dir6 with directional antennas. On the other hand, Fig. 11(a) highlights that using the rst six waypoints for Short6 the precision improves. Although no geometry has been set in Short6, the experimental error is smaller than that in the original algorithm Short which aims to build a weak geometry formed by isosceles triangles. Short performs as the common belief: more waypoints help for improving the localization. Therefore, multilateration helps precision for the best-e ort algorithm Short. max PP0 : h = 30; r = 150 SHORT SHORT6OPT max PP0 : h = 30; r = 150
DIR DIR6OPT max PP0 : h = 30; r = 150
OMNI OMNI6OPT 1.2 1 0.8
In the experiment shown in Fig. 12 we study whether multilateration improves the localization when the geometry constraints are respected also by the six measurements. Speci cally, be Short6OPT the modi ed version of Short which considers the initial three associated vertices of the triangle that better circumscribes the sensor plus the three midpoints along each triangle's side. Dir6OPT is the modi ed version of Dir which localizes all the sensors exploiting all the di erent possible six orientations that a sensor can hear. Moreover, Omni6OPT is the modi ed version of Omni which uses for each sensor the optimal triple of waypoints plus another triple, clone of the rst one. Fig. 12(b) and Fig. 12(c) show that the geometry of the waypoints, for any algorithm belonging to the precise class, is essential to localize the sensors with a small error. In practice for Dir and Omni, a non-blind multilateration works almost the same than as the simple trilateration. However, since the precision is almost the same for trilateration and multilateration when both satisfy the geometry constraints, there is no reason to pay an extra computational cost. Even in this experiment, in Fig. 12(a), Short behaves di erently from Dir and Omni: with Short6OPT the precision deteriorates. This shows that it is di cult to de ne which is a good geometry for an algorithm without rigid constraints for precision.
In summary we believe that it is di cult to achieve small localization error without a careful design of the localization algorithms. On the other hand, when we have a well de ned geometry that guarantees precision, trilateration is good enough. 4
In this paper, we surveyed three localization algorithms that replace multiple xed anchor sensors with a ying drone. We investigated the impact of multilateration for the three di erent algorithms. For Dir and Omni, which are precise algorithms that measure the sensors obeying to a rigid geometry, a multilateration that just increases the number of waypoints is not useful to improve the precision of the localization. A multilateration that increases the number of waypoints respecting the geometry improves localization in a negligible way. So we can conclude that when the waypoints for each sensor follow a well pondered geometry, multilateration is not useful. Whereas, the precision of the Short algorithm, which follows a best-e ort policy in selecting the waypoints for each sensor, bene ts of taking more measurements. For the best-e ort algorithms, the design of sophisticated multilateration techniques that preserve geometry requires further investigation.
In our future works, in addition to further look for short drone's paths that preserve precision, we intend to analyze the energy consumption of the localization algorithms on both the drone and the sensors side, considering also the number of the waypoints. We also wish to perform a test-bed with a drone of our localization algorithms.