A Comprehensive Investigation on Range-free Localization Algorithms with Mobile Anchors at Different Altitudes
Francesco Betti Sorbelli, Sajal K. Das, Cristina M. Pinotti, Giulio Rigoni
AA Comprehensive Investigation on Range-free Localization Algorithmswith Mobile Anchors at Different Altitudes
Francesco Betti Sorbelli a , Sajal K. Das a , Cristina M. Pinotti b , Giulio Rigoni c a Dept. of Computer Science, Missouri University of Science and Technology, Rolla, MO, USA b Dept. of Computer Science and Math., University of Perugia, Italy c Dept. of Computer Science and Math., University of Florence, Italy
Abstract
In this work, the problem of localizing ground devices (GDs) is faced comparing the performanceof four range-free (RF) localization algorithms that use a mobile anchor (MA). All the investigatedalgorithms are based on the so-called heard/not-heard (HnH) method, which allows the GDs todetect the MA at the border of their antenna communication radius. Despite the simplicity ofthis method, its efficacy in terms of accuracy is poor because it relies on the antenna radiusthat continuously varies under different conditions. Usually, the antenna radius declared by themanufacturer does not fully characterize the actual antenna radiation pattern. In this paper,the radiation pattern of the commercial DecaWave DWM1001 Ultra-Wide-Band (UWB) antennasis observed in a real test-bed at different altitudes for collecting more information and insightson the antenna radius. The compared algorithms are then tested using both the observed andthe manufacturer radii. The experimental accuracy is close to the expected theoretical one onlywhen the antenna pattern is actually omnidirectional. However, typical antennas have strongpattern irregularities that vanish the accuracy. For improving the performance, we propose range-based (RB) variants of the compared algorithms in which, instead of using the observed or themanufacturer radii, the actual measured distances between the MA and the GD are used. Thelocalization accuracy tremendously improves confirming that the knowledge of the exact antennapattern is essential for any RF algorithm.
Keywords:
Drone, rover, localization, range-free, range-based, antenna model
1. Introduction
Considering the crescent interest in the Internet of Things (IoT), and in general in the WirelessSensor Networks (WSNs), the problem of localizing sparse wireless sensors is more and moreappealing to researchers. Due to their limited size and low cost, those sensors can be installedeverywhere for any specific task. For instance, localization can help to search and rescue peopleafter natural disasters, to monitor the structural health of building, or even to track people duringthe COVID-19 pandemic in order to decrease the circulation of the virus.Technically speaking, the localization problem aims at estimating the position of ground devices (GDs) deployed on the WSN. For this important task, special devices whose positions are knowna-priori, called anchors , are in charge of localizing the GDs. In the literature, localization can beclassified in respect to the type of anchors deployed, i.e., static anchors (SAs) or mobile anchors (MAs). The former scenario requires a discrete amount of SAs with fixed positions, while the latter
Preprint submitted to Journal of Pervasive and Mobile Computing February 12, 2021 a r X i v : . [ ee ss . SP ] F e b ne requires a single MA (e.g., rover, drone) with GPS capabilities (need of fresh coordinates whilemoving). Notice that the usage of SAs is not affordable in terms of costs when the WSN is large;analogously, such a fixed infrastructure cannot be quickly reused elsewhere. Hence, in this paperwe consider only the localization with a single MA. Localization algorithms can be also broadlycategorized as range-free (RF) or range-based (RB) approaches [1]. In the previous, the position isestimated only by discovering if the GD and MA are one in the range of the other. In the latter,the position of the GD is estimated by taking measurements (e.g., distance, angle, signal strength)between it and the MA. Usually, RB algorithms are known to be more accurate than RF ones butat the cost of additional specialized hardware. Finally, among the RF algorithms, the radius-based algorithms assume the knowledge of the transmission radius, while the radius-free ones do not.In this paper, we evaluate on a test-bed the accuracy of many state-of-the-art RF algorithms.They exploit the heard/not-heard (HnH) method, consisting of detecting two consecutive messagestransmitted by the MA, one heard and one not-heard from the GD, called endpoints . All thealgorithms, but one which is new, have been described in the literature under the ideal model .The ideal model assumes that both MA and GD are equipped with an isotropic antenna whoseradiation pattern is a perfect sphere. Consequently, the transmission and receiving areas, whichare given by the intersection of the sphere with the earth surface, are perfect circles. The idealmodel can be emulated in reality only with expensive hardware on noiseless areas, which are not theusual operating conditions of most localization applications that occur, for example, in precisionagriculture or search-and-rescue context. To reproduce the typical operating conditions, we adoptfor our test-bed a set of inexpensive DecaWave DWM1001 Ultra-Wide-Band (UWB) antennas andan off-the-shelf 3DR Solo drone as the flying MA. Our pursued goals can be resumed as follows: • Compare on a real scenario the performance of the investigated RF algorithms, with verylittle knowledge on the antenna radiation pattern; • Study how the MA’s altitude can affect the localization performance.The experimental results raised difficulties, doubts, and questions. First of all, collecting a large setof endpoints on the field has proven to be challenging. So, we search for the statistic distributionthat best fits the already observed endpoints to generate a large new set of synthetic endpoints onwhich comprehensively test the RF algorithms. To mitigate the poor observed performance in thereal scenario, we decide to exploit the DWM1001’s capability of ranging distance measurementsto gain awareness of the antenna radiation pattern. Despite the measurements transform thealgorithms from RF into RB, the original algorithmic rules are kept, and the awareness of theexact antenna radiation pattern, though incomplete, significantly improves the performance.The paper, which is an extension on [2], is structured as follows: Sec. 2 reviews the literature onlocalization. Sec. 3 presents the compared algorithms. Sec. 4 describes the test-bed. Sec. 5 analyzesthe antenna’s performance. Sec. 6 evaluates the algorithms under different scenarios. Sec. 7 showshow the algorithms can incorporate the distance measurements. Sec. 8 offers conclusions.
2. Related Work
In this section we cover the state-of-the-art about localization in WSNs, giving first a briefoverview about known techniques with SAs, and surveying then more in detail techniques using aMA. Finally, we describe a few works that implemented real test-beds about localization.2 he Static Anchors Scenario.
In literature, different algorithms exist to tackle the issue of localiza-tion of GDs using SAs. DV-HOP based techniques [3] and Amorphous algorithms [4] approximatethe distance between GDs using the number of hops between them and estimating the averagehop distance inside the WSN. Then, using the trilateration method, each GD computes its ownposition. The Centroid algorithm used in [5] is another technique where each SA broadcasts itsposition to the surrounding, and each GD computes its own position calculating the average ofall the coordinates of the SAs that it can hear. Other works propose solutions based on the Ap-proximate Point in Triangulation (APIT) [6] whose goal is to divide the area in triangles in whichunknown GDs reside. In each overlapped section (i.e., a polygon) of triangle area, the position ofthe unknown GD is computed calculating the centroid. The main issue for these solutions is therelative high number of SAs required for an acceptable localization error.
The Mobile Anchor Scenario.
All these algorithms emulate multiple SAs with a single MA thatcontinuously broadcasts its position. For localizing GDs, a MA has to plan a route (static path)in advance inside the WSN. On that route, the MA estimates the distance between itself and theGDs in range, and eventually the GDs’ positions are computed performing trilateration. In [7]three different 2D movement trajectories have been studied, i.e., SCAN, DOUBLE-SCAN, andHILBERT. The distance between two consecutive segments of the trajectories is defined as theresolution. The simplest algorithm is SCAN, in which the MA follows a path formed by verticalstraight lines interconnected by horizontal lines. Essentially, the MA sweeps the area along the y -axis. The main drawback is that it provides a large amount of collinear endpoints. In orderto resolve the collinearity problem, DOUBLE-SCAN sweeps the sensing area along both the x -axis and the y -axis. However, in this way the path length is doubled compared with SCAN.Finally, a level- n HILBERT curve divides the area into 4 n square cells and connects the centersof those cells using 4 n line segments, each of length equal to the length of the side of a squarecell. Generally, HILBERT provides more non-collinear endpoints but the path length can be verylong if the resolution increases. All the above techniques are based on straight lines and suffer ofcollinearity problem. In order to heavily reduce collinearity, S-CURVES [8] has been introduced,which is similar to SCAN except that it uses curves rather than straight lines. Even though thecollinearity problem is almost resolved, the main problem is that it does not properly cover thefour corners of the squared sensing area. A similar strategy to HILBERT is Z-CURVE [9]. Inconclusion, LMAT [10] is one of the best techniques. The main idea is to plan a series of MAs insuch a way as to form regular equilateral triangles, avoiding collinearity problems. Each GD insidea triangle is localized by a trilateration procedure using the three vertices.All the previous algorithms are RB. Less RF techniques have been proposed. Among them, theones proposed by Xiao [11] and Lee [12] rely on a ground MA, while the one proposed by Betti [13],relies on a flying MA. We will give more details and insights about those algorithms in Sec. 3. Implemented Test-beds.
At the best of our knowledge, only a few test-bed implementations havebeen done aimed at comparing the performance among RF algorithms with a MA. Recently in [14],a test-bed using inexpensive UWB antennas and a drone as MA evaluates the localization accu-racy of the RF
Drf algorithm proposed in [13]. Such an algorithm strictly relies on the goodquality of the antenna radiation pattern and requires very easy geometrical rules for estimatingthe GD’s position. Unfortunately, in practice, the experimental localization error obtained by theimplemented
Drf algorithm is large. Another recent study in [15] shows that the irregularitiesof the hardware antenna radiation pattern can heavily affect the air-to-ground (A2G) link quality3etween a MA (drone) and a GD. The authors study the A2G link quality of BroadSpec UWBantennas from Time Domain Inc by observing the Received Signal Strength Indication (RSSI).They show a dependency on the link quality from the antennas’ orientations, their elevation, andtheir distance. The authors in [16] report the UWB A2G propagation channel measurements in anopen field using a drone. Three scenarios are considered at a different type of obstructions while adrone was orbiting above a GD at different altitudes and ground distances. Also, different antennaorientations are considered for the drone. Experimental results show that the received power ishighly dependent on the antenna gain of the line-of-sight (LoS) component in the elevation planewhen the antennas are aligned (same orientation). In a work proposed in [17], authors investigate atime difference of arrival (TDoA) based approach for localizing drones taking into account a simpleA2G 3D antenna radiation pattern. Experimental results show that accounting for antenna effectsmakes a significant difference and reveals many important relationships between the localizationaccuracy and the altitude of the drone. Most importantly, they finally show that the localizationperformance varies in a non-monotonic pattern with respect to the drone altitude.
Motivations.
Considering that the RF algorithms have been around for a long time, that thealgorithm tested in [14] on the field uses a drone (i.e., flying MA) brought bad performance, andthat the irregularities showed in [15, 16, 17] are for A2G links, we start thinking that altitudecould be one of the main cause of poor results for the algorithm tested in [14] and in general forRF algorithms. Therefore, these considerations motivate us to investigate more concerning theaccuracy of RF algorithms at different altitudes and the quality of antennas.
3. The Range-free Algorithms
In this section, we describe the RF algorithms that we compare. We start defining a rover as a MA that moves on the ground, and a drone as a MA that flies on the sky. During thelocalization procedure, called mission , the MA visits specific points in its path, called waypoints .Along this path, the MA continuously transmits a beacon that can be used by any GD withinthe communication range for estimating its position. The beacon includes the current MA’s GPSposition and it is sent at regular intervals of time. The distance among any two consecutivetransmitted beacons is called inter-waypoint distance I w and depends on the MA’s current speed.All the algorithms that we study exploit the HnH method relying on the detection of endpoints.Performing HnH, the GD learns that the MA is currently transmitting at its transmission areaborder. Moreover, the GD learns that the last (first) not-heard beacon is at a distance I w fromthe first (last) heard beacon (endpoint), and hence at a distance no more I w from the edge of thetransmission area. In general, three applications of HnH are required for localizing the GD.In the following, we describe three RF algorithms that we evaluate, namely, Drf [13],
Xiao [11],and
Lee [12]. We also introduce a new variant of
Drf , called
DrfE . In these algorithms, the MAtravels along a static path for localizing the GD. The static path is formed by a sequence ofconsecutive straight segments in which the MA regularly broadcasts message beacons (its currentposition) that the GD is listening to. Once the GD has collected enough information, it can finallycompute and estimate its position according to the performed algorithm.
Drf
Algorithm
Drf [13] is a lightweight RF radius-free algorithm designed for drones. This algorithm isbased on the notion of chord . In geometry, the perpendicular bisector of any circle’s chord passes4 w OA A A A B (a) Drf . OI w A A A A P P P P (b) Xiao . r − I w OA A rP P P P (c) Lee . Figure 1: The
Drf , Xiao , and
Lee localization algorithms. In
Xiao e Lee there are two symmetric intersectionareas: a third point (not illustrated) is required to find and disambiguate the intersection area where GD resides. through the center O of the circle itself. So, the bisector of another non-parallel chord and theprevious one intersect at O point. In Fig. 1(a), the GD is located at the point O , while initiallythe MA travels along the segment that intersects the points in sequence A first and A then. Theradio receiving area of the GD is identified by the circle centered at O , so if the MA transmits amessage outside that circle (e.g., in A ), the GD cannot hear any message. However, when the MAcrosses such a circle and transmits in A , the GD can now receive and record messages, becausethe relative distance between them is less than or equal to the transmitting/receiving radius. Thesame reasoning can be applied for the points A (heard) and A (not heard). Accordingly, the firstchord is denoted by the segment with endpoints A A . It is easy to understand that when the MAcrosses the GD’s receiving area along another segment (e.g., the one that intersects the point B ),another endpoint is detected, and eventually two chords are identified by the pairs A A and A B .Finally, the GD starts to estimate its position once it has detected these two chords computingfirst the associated perpendicular bisectors (dashed lines) and then their intersection point (whichcan be different from point O ). The detection of chords incurs several problems that eventuallyaffect the localization accuracy. Recalling that the MA regularly broadcasts its current position(waypoint) at discrete intervals of time and that two consecutive waypoints are at distance I w , theendpoints of the chords may not exactly fall on the circumference of the receiving disk, even if thereceiving disk is a perfect circle (e.g., A and A ). Xiao
Algorithm
Xiao [11] is a RF radius-based localization algorithm initially developed for ground MAs. Like
Drf , the
Xiao algorithm exploits the HnH method in order to detect special points used forbuilding a constrained area that bounds the GD’s position. Unlike
Drf , Xiao also uses the valueof the communication radius r . In Fig. 1(b), the GD is located at the point O while the MA travelsalong the segment that intersects first the point A and then A . Once applied the HnH method,the GD initially detects the first pair of heard endpoints A and A . However, since the segment lieson a straight line and the value of I w is known, the GD can also compute two additional non-heardbeacons, i.e., A and A , associated with A and A , called pre-arrival and post-departure. Then,four circles of radius r centered at each of these four points are drawn. Those circles create twosymmetrical intersection areas (e.g., the first one is bounded by the points P , P , P , P ) wherethe GD may reside. Hence, the GD’s position can be at the “center” of one of the two intersectionareas. In order to disambiguate in which intersection area the GD resides, a third HnH beacon isrequired, and the final estimated position is the one which has the closest distance, from that third5oint, to the radius. The definition of center varies depending on whether the intersection area isdelimited by four or five vertices [11]. Lee
Algorithm
Lee [12] is a RF radius-based algorithm very similar to
Xiao . It builds a constrained areausing the HnH method and the knowledge of both r and I w , similarly to Xiao . In Fig. 1(c), oncethe GD has detected the two extreme endpoints ( A and A ), it traces two circles of radius r and r − I w on both the points, which create two annuli, intersecting in two distinct and symmetricalintersection areas (e.g., one is bounded by the points P , P , P , P ). Finally, GD resides at thecenter of one of such areas, and a third endpoint is used to disambiguate the correct one. DrfE
Algorithm
Now we present a new RF algorithm, called
DrfE , that shares the “chord” idea with the
Drf algorithm and the radius information r with Xiao and
Lee . In the special case that thetwo endpoints A and A that delimit the chord exactly lay on the circumference, the GD resideson the point O of the perpendicular bisector that is distant r from the two endpoints. Precisely,there are two O ’s points, one on the left and one on the right of the chord. Usually, using a thirdendpoint non-collinear with A and A , it is possible to disambiguate the correct intersection. I w I w OA A A A P P P rr Figure 2: The points P , P , and P of the DrfE algorithm.
In general, since the MA’s path is sampled with discrete beacons at distance I w among them, A and A may not lay on the circumference and thus GD may not exactly reside on the perpendicularbisector of the chord A A , but in its vicinity. So, to find the GD’s position, DrfE repeats theabove construction for the three chords A A , A A , and A A , where A and A are the twonon-heard beacons associated with the endpoints A and A , as illustrated in Fig. 2. From thethree chords A A , A A , and A A , three intersection points P , P , and P are obtained atdistance r from the endpoints of their chord. In other words, P , P , and P form three isoscelestriangles (cid:52) ( A A P ), (cid:52) ( A A P ), and (cid:52) ( A A P ) with two oblique sides of equal length r . Asbefore, this construction finds three vertices on the left of the chord and three vertices on the rightthat will be disambiguated using another non-collinear endpoint (not illustrated). Let us supposethat GD is on the right of the chord A A . DrfE places GD at the centroid of P , P , and P onthe right of the chord A A . Since the RF radius-based algorithms require the knowledge of thetransmission radius, in the next next we describe how to collect such information in a real test-bed.
4. Test-bed Setup
In this section, we describe the test-bed and the used hardware. We fix a Cartesian coordinatesystem with origin at the special position
Home (0 , , h ), with h = 1 m. At Home , we place the6D’s antenna at the top of a tripod of height h . Also, the MA is equipped by an antenna. Whenwe set h = 0 m, we refer to a rover mission where the MA and the GD are placed at h , while whenwe set h > h + h . h sr PtWh P . (0 , x , y )( x , y ) ( x , y )( x , y )( x , y ) ( x , y ) (b) The MA’s random path. VV HV (c) VV and VH.
Figure 3: The test-bed setup.
In our test-bed, we employ different hardware components: a few DecaWave DWM1001 UWBantennas [18], a Raspberry Pi, and a 3DR Solo drone [19]. According to DecaWave, the transmissionradius is 60 m and hence, in this paper, we set the manufacturer radius to r = 60 m. The maincomponent that pilots the MA and sends commands to the GD is the Raspberry Pi. The Raspberrycan be used for the experiments on the ground using a rover, or together with the drone for theaerial ones. In the former case, i.e., MA as a rover, we simulate the rover’s behavior just walkingin the field at a regular walking speed (about 3 km / h as measured by a smart-watch) keeping theRaspberry on the hands at h = 1 m above the ground. Moreover, since the rover has to send GPSpositions, we rely on a cheap USB GPS module connected to the Raspberry. Finally, in the lattercase, i.e., MA as a drone, we use the 3DR Solo drone which is able to fly up to 25 min [19].A mission is a static path Π that consists of n segments S i , i = 0 , . . . , n −
1. Such a list of n segments is made by generating n + 1 random points in the deployment area. Each segment isdelimited by two random points, and any two consecutive segments share one random point bysetting the waypoints’ coordinates ( x W i , y W i , h ) for each W i , i = 0 , . . . , n − x, y ) positions. This process continues until the MA reaches thelast waypoint of Π. When the mission is accomplished, the MA comes back to Home .Concerning the GD’s side, in our experiments, we set the antenna on the tripod placed at
Home , laying on two different sides, as sketched in Fig. 3(c). In the left case, the antenna layson the short side, i.e., its xy side is parallel to the ground. In the right case, the antenna lays onthe long side, i.e., its yz side is parallel to the ground. The drone’s antenna is always verticallyplaced with the short side parallel to the drone’s body keeping the UWB transceiver at the bottomfor guaranteeing the most available free space. We indicate the first configuration where the twoantennas lay on the same side, but opposite direction, with vertical-vertical (VV); whereas we referto the other configuration with vertical-horizontal (VH).
5. Antenna Analysis
In this section, we recap the UWB technology, report the DecaWave’s technical datasheetinformation, and analyze the experimental data.7 .1. The Ultra Wide Band Technology
UWB is a promising radio technology that can use a very low energy level for short-range,high-bandwidth communications over a large portion of the radio spectrum. Nowadays, its primarypurpose is in the field of location discovery and device ranging. Differently from both Wi-Fi andBluetooth, UWB is natively more precise and accurate, uses less power and, as production of UWBchips blows up over time, holds the promise of a lower price point. Moreover, UWB offers relativeimmunity to multipath fading.In this paper we rely on a kit of DWM1001 UWB antennas produced by DecaWave. Accordingto DecaWave’s datasheet document [18], those antennas provide 10 cm accuracy for the measure-ments. Moreover, those chips have a 6 . − . / MHz, and the typical receiversensitivity is −
93 dBm / Fig. 4 shows the antenna radiation patterns of the UWB antennas according to DecaWave’smanufacturer document [18] for different configurations. The solid dark line of Fig. 4(a) showsthat it is possible to obtain the same gain in all the directions in the xy -plane when an antennavertically placed (i.e., on xy -plane) is observed by another antenna which shares the same vertical orientation (Φ polarization ), i.e., they are concordant. We recreate this situation by implementingthe antennas as VV. Thus, we expect that the VV configuration experiences the same gain atdifferent angles, at least when the two antennas are at the same height.
270 1800 90ΘΦ 90180270 -30-20-1000 V (a) xy -plane. 0 90180270 Φ Θ 90180270 -30-20-1000 H (b) yz -plane. Figure 4: The DWM1001 radio pattern: dBm vs angle.
Both the dashed lines of Fig. 4(a) and Fig. 4(b) refer to the VH configuration because they showthe gain when an antenna is observed by another antenna perpendicularly oriented (Θ polarization ),i.e., they are discordant. Both the dashed lines have nulls at certain angles that can limit the gainand can introduce “holes” and “bubbles” in the pattern. Thus, we expect that the VH configurationexperiences different gains at different angles, and we also expect a relevant variability given that thedashed lines of Fig. 4(b) and Fig. 4(a) are different, although the relative position of the antennasseems to be the same. Analyzing these technical data, it seems that the gain is omnidirectional atleast when the two antennas are placed as VV.We have not found, for DWM1001, any data which correlates the gain and the polar angle.However, in a document of a former antenna model called DW1000 [18], DecaWave gives the gainvalues (reported in Tab. 1) for an antenna vertically placed (as in Fig. 4(a)) in an anechoic chamber.8 able 1: The DW1000 elevation gains (in dBm) at 6 . Θ (discordant) Φ (concordant) yz -plane peak 0 .
30 2 . − . − . xz -plane peak 0 .
26 1 . − . − . We report these values just to confirm the not negligible impact of the elevation : the 3D radiationpattern is far from being a sphere, with the same gain in all the directions. A A tO (a) xy -plane. OA th A (cid:48) A (cid:48) A xz -plane. Figure 5: The ideal (dashed) and actual (solid) antenna radiation profile in xy -plane and xz -plane. Then, we conjecture that, when the MA is a drone, the 3D antenna pattern is highly irregularand it can be sketched as a nibbled apple (Fig. 5). Therefore, when such an antenna shape isprojected on the ground, holes and bubbles can be found. We speculate that the altitude is themain cause of the pattern irregularity. So, we conjecture that when the two antennas are both onthe ground (rover as MA) in VV configuration, the gain is the same in almost all the directions.
In our experiments, we wish to characterize the 2D antenna pattern observing the range ofvalues of its radius. The MA starts at
Home in (0 ,
0) at different altitudes h = { , , } m. Asexplained in Sec. 4, the MA traverses the deployment area with n segments that aim to cross thereceiving shape of the GD. Moving along each random segment, the MA continuously broadcastsits current ( x, y ) position, and the GD registers the first and the last heard endpoints sent by theMA. Since we know in advance the Home position of the GD, we can compute the actual 2D radiusfor each detected endpoint. Note that we observe the radius on the ground. It is important to recallthat the beacons are sent at regular intervals of time, and so the observed radii have an intrinsicerror of at most I w . In this paper, we fix I w to 0 .
40 m since we have experimentally observed thisvalue, which clearly depends on the speed of the MA. From the collected endpoints during thesame experiment, we compute the mean µ and the standard deviation σ of the set of observedradii. Then, we apply the goodness-of-fit method in order to assess whether a given distribution issuitable to the built data-set . We repeat the experiments for the two antenna configurations (VVand VH) and for different altitudes.We start reporting in Tab. 2 the results of the first experiment with the rover (i.e., h = 0 m)and VV configuration. According to DecaWave’s datasheet (solid line in Fig. 4(a)), we expect Note also that the 3D antenna pattern is completely defined if we know its behavior in 3 planes: xy , xz , and yz . Fixed a distribution and a set of categories, we determine if there is a significant difference between the expectedand observed frequencies in one or more categories by using the chi-squared test.
9n almost uniform radius in the experiments, at least when the two antennas are at the sameheight. We observed 38 different endpoints, with mean µ = 97 .
10 m and σ = 39 .
74 m, and alsowith min = 32 .
14 m and max = 162 .
40 m. According to the Pearson χ -test, at h = 0 m, the radiiof the VV configuration have a uniform distribution. As we will see, this is the only configurationwith uniform distribution of the radii. Therefore, we agree with DecaWave that this configurationis somehow special. However, the radius cannot be considered really constant. Table 2: h = 0 m, VV class radii (in m) frequencies U ) normal ( N )1 32.14 58.34 8 0.08 3.142 58.34 84.54 7 0.01 0.133 84.54 110.74 8 0.08 0.344 110.74 136.94 5 0.69 1.065 136.94 163.14 10 1.06 8.15likelihood 0.75 0.01 Then, in Tab. 3 we report a second experiment at altitude h = 10 m, with VH configuration.We observed 28 different endpoints out of 38, thus confirming several null angles. The observedradii have mean µ = 66 .
34 m, σ = 22 .
81 m, min = 16 .
52 m, and max = 121 .
66 m. According to thePearson χ -test, at h = 10 m, the radii of the VH configuration have a normal distribution. Table 3: h = 10 m, VH class radii (in m) frequencies U ) normal ( N )1 16.52 41.52 5 1.68 0.682 41.52 66.52 9 0.00 0.143 66.52 91.52 11 0.52 0.074 91.52 141.52 3 12.23 0.15likelihood 0.01 0.90 Finally, in Tab. 4 we summarize the statistic distributions that fit the observed experimentalradii. For each distribution, i.e., Uniform ( U ) and Normal ( N ), we give the observed µ and σ , andthe likelihood. Except for the rover in VV, all the experiments show that the radius most likelyfollows a normal distribution, but with a large σ . It is worthy to note that increasing h , the meanof the radii decreases. The mean decreases faster with VV while with VH it remains quite stable(see Tab. 4). The values of the radii are generally more concentrate with VH than VV. Table 4: The radii (in m) distribution with its parameters D ( µ, σ ) and its likelihood p . VV VH h = 0 m U (97 . , . . N (63 . , . . h = 10 m N (84 . , . . N (66 . , . . h = 20 m N (62 . , . . N (57 . , . . We conclude that, oppositely to our conjecture, VH seems better than VV and the radii obtainedwith a drone seem more concentrated than those obtained with a rover. Marginally, let us pointout that organizing a localization mission is easier with a drone than with a rover because the droneis faster and less attention has to be paid to the terrain. Although the results are different fromwhat we expected, we continue our investigation in localization algorithms accuracy. Thus, we usethe results reported in Tab. 4 to generate a large synthetic set of endpoints that fit the estimatedparameters of the radii distributions for testing the different algorithms surveyed in Sec. 3.10rom now on, we refer to the radius reported in Tab. 4 as the observed radius r = µ , while tothe DecaWave declared radius as the manufacturer radius r = r .
6. Range-free Comparative Evaluation
In this section, we compare all the RF localization algorithms using first the set of syntheticendpoints, and then the set of real endpoints collected during the experiments.Our goal is to analyze the localization error and the percentage of unsuccessful localizationsof
Drf , Xiao , Lee , and
DrfE . From now on, with height h = { , , } m and antenna con-figurations { VV, VH } we refer to a particular scenario. For each simulated scenario, we run 200localizations generating at random three endpoints, B , B , and B , with the distribution and theparameters of the simulated scenario derived in Sec. 5.3 from the observed endpoints (see Tab. 4).For each triple, we invoke the four algorithms using either the observed radius r = µ used to gener-ate the endpoints or the manufacturer radius r = r . In the former case, we test the performanceof algorithms when they receive in input the actual radius, but still, the endpoints can be affectedby the antenna irregularity (i.e., σ ). In the latter case, we test the performance of algorithms whenthey receive in input a completely different radius from the actual one.Reinterpreting the constraints to improve the accuracy given in [20], the three selected end-points, i.e., B , B , and B we use for localizing the GD should satisfy two constraints: theminimum distance r min = 60 m and the minimum angle α min = 20 deg between them. The con-straint r min means that the distances d ( B , B ), d ( B , B ), and d ( B , B ), must be at least aslong as r min . The α min constraint means that the three angles α = ∠ B B B , α = ∠ B B B ,and α = ∠ B B B , must be at least as large as α min . From a geometrical point of view, thesetwo constraints guarantee that the three selected endpoints are sufficiently apart each other, thusavoiding the construction of degenerated triangles. Therefore, in the experiments, we discard anytriple of endpoints that does not satisfy r min = 60 m and α min = 20 deg. We repeat the endpointextraction until we find three suitable endpoints.We compare the RF algorithms under two metrics; the localization error , defined as the Eu-clidean distance between the actual GD’s position and the estimated one outputted by the algo-rithms, and the percentage of unlocalized . Concerning the first metric, we report the localizationerror resumed into a boxplot that highlights the median (horizontal line), the average value (solidcircle), and the data between the first Q and the third quartile Q (box). Additionally, the ex-tremes of the whiskers represent the Q − . Q + 1 . Q − Q . Lastly, about the second metric, an unsuc-cessful localization is an application of the algorithm which does not return any constrained area orin general any geometrical intersection where the GD can reside, i.e., the GD remains unlocalized.This mainly happens for the RF radius-based algorithms when the radius is under-estimated. In this section, we evaluate the performance of the RF algorithms. We start considering asynthetic set of endpoints with an average radius equal to the observed radius µ , but very smallstandard deviation, emulating an ideal model. This experiment is to support the observationthat a high accuracy is possible when the antenna is almost isotropic. Moreover, we discuss theperformance of the RF algorithms on the synthetic set of endpoints generated according to theobserved distributions in Tab. 4. These experiments, as those that use the manufacturer radius,11how poor accuracy. In all the experiments, the impact of using a rover or a drone is considered.Finally, we report the comparison between the performance of Drf and
DrfE . Under the Ideal Model.
Let us start considering the nearly ideal model in Fig. 6 in which theendpoints are generated with the observed radius µ given in Tab. 4 for the VV configuration,but selecting σ equal to 1. We evaluate the algorithms simulating an almost omnidirectionalantenna. Since the dispersion is low, the radius can be considered almost constant. As expected,all the radius-based algorithms perform well when they use the observed radius r = µ , which isthe same radius used to generate the endpoints. In such a case, the localization error is on theorder of a couple of meters or less for all the algorithms . Drf is slightly better than the otheralgorithms at any altitude, while
DrfE is worse than
Drf . The number of unlocalized GDs is verysmall. However, the performance of the radius-based algorithms drastically drops down when thealgorithms run using the manufacturer radius r = 60 m, while the endpoints have been generatedwith the observed radius. The percentage of unsuccessful localizations is extremely large. DRF XIAO LEE DRFEAlgorithms01234 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (a) Error: r = µ . DRF XIAO LEE DRFEAlgorithms0204060 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (b) Error: r = r . DRF XIAO LEE DRFEAlgorithms020406080100 U n l o c a li ze d ( % ) h = 0 m h = 10 m h = 20 m (c) Unlocalized: r = µ . DRF XIAO LEE DRFEAlgorithms020406080100 U n l o c a li ze d ( % ) h = 0 m h = 10 m h = 20 m (d) Unlocalized: r = r . Figure 6: The impact of the radius on the ideal model.
These results confirm what we said in Sec. 3: to obtain an accurate localization, not only theantenna must be of good quality (i.e., with small σ ), but also the observed radius µ must be exactlyknown by the algorithm. The results show that the error due to the use of the radius r decreaseswhen h increases because decreases the difference between the observed µ and the manufacturer r radii. Alongside, note that since Drf is radius-free, the performance of
Drf is not influencedby the radius selection, as shown in Fig. 6(b).
Drf only requires an omnidirectional antenna.
Using the Synthetic Endpoints Set.
Figs. 7(a), 7(b), 7(e), and 7(f) show the results when both theendpoints and the algorithms use the observed radius in Tab. 4. Figs. 7(c), 7(d), 7(g), and 7(h)instead show the results when the endpoints are generated using the observed radius and thealgorithms run with the manufacturer radius. This is what happens in practice whenever theend-user implements the RF radius-based algorithms using the DecaWave datasheet radius on ourtest-bed. The errors of the algorithms are compared at different altitudes, antenna configurations,and radii. For all the algorithms, the average error is large. The worst error occurs with the VVconfiguration of the antennas and h = 0 m: in such a scenario, the endpoints follow a uniformdistribution. In general, the VH makes an error smaller than the VV probably because the radiusthat generated the endpoints for VH is less dispersed (i.e., σ is smaller) than that for VV. The Please note that the scale of y axis in Fig. 6(a) is zoomed with respect to the scale of y axis in Fig. 6(b) Xiao , Lee , and
DrfE , exhibit the same performance. They areinspired by slightly different ideas, but they actually act the same.
DRF XIAO LEE DRFEAlgorithms04080120 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (a) Error: VV, r = µ . DRF XIAO LEE DRFEAlgorithms04080120 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (b) Error: VH, r = µ . DRF XIAO LEE DRFEAlgorithms04080120 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (c) Error: VV, r = r . DRF XIAO LEE DRFEAlgorithms04080120 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (d) Error: VH, r = r . DRF XIAO LEE DRFEAlgorithms020406080100 U n l o c a li ze d ( % ) h = 0 m h = 10 m h = 20 m (e) Unlocalized: VV, r = µ . DRF XIAO LEE DRFEAlgorithms020406080100 U n l o c a li ze d ( % ) h = 0 m h = 10 m h = 20 m (f) Unlocalized: VH, r = µ . DRF XIAO LEE DRFEAlgorithms020406080100 U n l o c a li ze d ( % ) h = 0 m h = 10 m h = 20 m (g) Unlocalized: VV, r = r . DRF XIAO LEE DRFEAlgorithms020406080100 U n l o c a li ze d ( % ) h = 0 m h = 10 m h = 20 m (h) Unlocalized: VH, r = r . Figure 7: Comparisons between all the algorithms in the studied scenarios.
The
Drf algorithm experiences the worst average error, but the error is only slightly morethan that of the radius-based algorithms. The whisker of the largest error of
Drf is the longestwhisker among all the algorithms probably because
Drf finds a localization also in extreme caseswhen other algorithms return an unsuccessful localization. The error in
Drf decreases when thealtitude increases, moreover the error of the VV configuration is worse than that of the VH one.The localization error is almost the same regardless of the adopted radius (observed or man-ufacturer). As witnessed by comparing the whiskers of the boxplots in Figs. 7(a) and 7(c) (resp.,by Figs. 7(b) and 7(d)), the localization error of the radius-based algorithms, when they use themanufacturer radius r , is slightly worse than when they use the observed radius µ . Instead, theknowledge of µ reduces percentage of unlocalized GDs as illustrated in Figs. 7(e) and 7(g) for VV.The improvement in Figs. 7(f) and 7(h) is weaker for VH than for VV because for VH the differencebetween the observed radii µ = { . , . , . } m and r = 60 m is smaller than VV.In conclusion, the average localization error for VV and VH is 50 m and 30 m, respectively,regardless of if the algorithm knows the true radius used to generate the endpoints or not. Theknowledge of the radius used to generate the endpoints by the algorithm only matters when theradius dispersion is small. The take-away lesson here is that to apply a RF radius-based algorithmthe antenna must be omnidirectional and its radius must be known by the algorithm. Between
Drf and
DrfE . In Fig. 8, we compare the performance of
Drf , DrfE µ ( DrfE withradius µ ), and DrfE r ( DrfE with radius r ). In general, DrfE µ performs better than DrfE r ,but DrfE µ has always an error much greater than the error of the ideal model (see Fig. 6).13 RF DRFE µ DRFE r Algorithms04080120160 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (a) Error: VV, Drf vs DrfE . DRF DRFE µ DRFE r Algorithms04080120160 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (b) Error: VH, Drf vs DrfE . Figure 8:
Drf vs DrfE . Using Real Endpoints.
In Fig. 9, we show the results using the real endpoints collected during ourtest-bed. For the radius-based algorithms, in VV, the results of the localization error seem to follow
DRF XIAO LEE DRFEAlgorithms020406080100 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (a) Error: VV, µ . DRF XIAO LEE DRFEAlgorithms020406080100 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (b) Error: VH, µ . DRF XIAO LEE DRFEAlgorithms020406080100 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (c) Error: VV, r . DRF XIAO LEE DRFEAlgorithms020406080100 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (d) Error: VH, r . Figure 9: Error using real data. the trend already seen on the synthetic generated set. In general, VV has a larger error comparedto that of VH. In VH, the results are slightly better than that obtained on the synthetic generatedset, although the number of valid triples is small. Indeed, the average error at h = 20 m is about15 m although σ is greater than 20 m when the radius µ is used. The average error increases to20 m when the radius r is used. This can be explained with the fact that there is a much strongercorrelation between the endpoints that are not fully captured by the two constraints r min and α min that we imposed for the selection of the synthetic endpoints. Finally, the error of Drf is worsewith real endpoints than with synthetic endpoints, especially for the VV configuration.So far, we learned the performance of all the RF algorithms strongly relies on the radiationantenna pattern. The simplest way to discover the radiation antenna pattern is to measure foreach direction up to which distance the MA and the GD are one in the range of the other. So,in the next section, we significantly improve the localization accuracy exploiting the capability ofthe UWB antennas used in our test-bed of taking distance measurements. Even though there arebubbles and holes in the antenna pattern, it is always possible to discover them if the distancefrom the GD and the MA is taken (see Fig. 10).14 A Or r Figure 10: The two radii measure by the GD when the antenna pattern is irregular.
7. Range-based Comparative Evaluation
Our interest in RF algorithms lay in the fact that they are simple to implement, do not requirespecialized hardware, and are scalable with respect to the number of GDs. Also, they are immune toproblems that come from the measurement of the 3D distance. Indeed, distance measurements areaffected by several errors that depend on the adopted technology and GPS. Technologies like WiFior Bluetooth are much less accurate than UWB and may incur in measurement errors of the orderof tens of meters, whereas GPS and barometer inaccuracies together with bad weather conditionscan seriously impact the drone’s position. Such combined 3D slant errors are then reflected on theground, leading to 2D ground errors. Although range measurements come with their troubles, giventhe results of the previous section, we decided to include distance measurements in our algorithms.Since in our test-bed both the MA and GD are equipped with UWB antennas that are able to takedistance measurements, we do not need to heavily modify our test-bed. OA A A A r r (a) RB Xiao . r − I w r r r − I w OA A (b) RB Lee . OA A A A r r P P P (c) RB DrfE . Figure 11: The
Xiao , Lee , and
DrfE localization algorithms using two radii.
In the algorithm variants that we are going to propose, the actual measured distance betweenMA and GD is used to run the algorithms instead of using the observed or manufacturer radius.This adaption makes the RF algorithms actually RB algorithms, but we maintain the originalalgorithmic rules (e.g., intersection of annuli or circumferences). It is worth noting that we canapply modifications only to the RF algorithms that actually make use of the radius in their local-ization rules, and hence only for the radius-based ones, i.e.,
Xiao , Lee , and
DrfE . Accordingly,the simplest radius-free algorithm
Drf cannot be adapted to measurements.
Range-based version of
Xiao . Fig. 11(a) shows how the
Xiao algorithm is modified. When theGD hears for the first time the beacon in A , the GD measures the distance r (cid:48) = d ( A , O ) betweenits position O and A . Whenever the MA is inside the receiving area of the GD, the GD still takesdistance measurements neglecting the intermediate measurements. When the last beacon A isheard, the GD saves the last distance r (cid:48) = d ( A , O ) between its position O and A . As in the15riginal version of Xiao , the GD computes the other two points A and A knowing the line thatinterconnects A and A and the value of I w . Then, the algorithm proceeds as before, except thatit draws two circumferences of radius r = r (cid:48) + I w centered in A and A , and two circumferencesof radius r = r (cid:48) + I w centered in A and A . That is, the predefined observed or manufacturerradii are substituted by the distances from the endpoints and the GD. Note that the radius r and r are used instead of r (cid:48) = d ( A , O ) and r (cid:48) = d ( A , O ) to avoid that O falls outside the intersectionarea in case of measurement error that underestimated the distances (see Fig. 11(a)). Range-based version of
Lee . As seen for
Xiao , also for
Lee , the two distances r (cid:48) and r (cid:48) betweenthe endpoints and the GD are ranged. The algorithm proceeds by drawing two annuli centeredat A and A , with outer radius equal to r = r (cid:48) + I w and r = r (cid:48) + I w , and inner radius equalto r − I w and r − I w , respectively. The main difference with the previous version is that theradii are replaced by the distances between the GD and the endpoints. As in Xiao , we use theradii r = r (cid:48) + I w and r = r (cid:48) + I w instead of r (cid:48) or r (cid:48) to limit the risk that, due to measurementinaccuracy, O falls outside the intersection area. The width I w of the annuli is preserved. Range-based version of
DrfE . We replace r with r = r (cid:48) = d ( A , O ) and r = r (cid:48) = d ( A , O ). Theintersection of the two circles of radius r and r centered, respectively, at A and A , returns P which coincides with O (assuming no measurement errors). To repeat the construction of the RF DrfE , the points P and P are drawn in a way similarly to P . Precisely, P is at the intersectionbetween a circumference of radius r centered in A and a circumference of radius r centered in A . P is at the intersection between a circumference of radius r centered in A and a circumference ofradius r centered in A . As a result, differently from the original version where there were threeisosceles triangles lying on the same line passing through A and A , here there are three scalenetriangles, i.e., (cid:52) ( A A P ), (cid:52) ( A A P ), and (cid:52) ( A A P ). Eventually, the centroid resulting fromthe vertices P , P , and P is selected as the estimation point, as shown in Fig. 11(c). In this section, we argue concerning the localization error that can be obtained once the originalRF versions of the algorithms have been adapted to be actually RB. We focus on
Xiao , but a verysimilar analysis applies to
Lee . Some words will be finally spent for
DrfE .Without loss of generality, we assume that the MA moves along a straight line along the x -axis(see Fig. 11(a)). Let O = ( x O , y O ) be the actual GD’s location with respect to Cartesian coordinatesystem with origin in A = (0 , P = ( x P , y P ) be the estimated GD’s location by thealgorithm. We fix A and A so as r = d ( A , O ) ≤ d ( A , O ) = r and let A = ( I w ,
0) and A = ( kI w , k is an integer number that represents the number of times the MA has sentthe beacons from A to A . Finally, let A = (( k + 1) I w , Xiao relies on two different radii, i.e., r applied to A and A , and r applied to A and A , for constructing four circumferences, i.e., (a) x + y > r , centered in A , (b) ( x − I w ) + y ≤ r , centered in A , (c) ( x − kI w ) + y > r , centered in A , and (d) ( x − ( k + 1) I w ) + y ≤ r ,centered in A . We also denote as L the lune delimited by Eqs. (a)–(b), and as L the lunedelimited by Eqs. (c)–(d), also in accordance with the original Xiao algorithm.Note that r and r are affected by the measurement errors [20] and for this reason we do notdirectly conclude that the estimated position P is at the intersection of Eqs. (b)–(c). Moreover, In plane geometry, a lune is the concave-convex region bounded by two circular arcs. kI w = d ( A , A ) is influenced by the MA’s speed, and hence it cannot be consideredexact as well. Therefore, we prefer to select P inside the intersection among L and L .The correctness of the RB Xiao algorithm is completely different from that of the original
Xiao . Although our implementation only memorizes the first and the last measured radius, i.e., r and r , the RB Xiao algorithm performs measuring, at each i th beacon A i = ( iI w , r ( i ) = d ( A i , O ). Therefore, for each intermediate beacon A i and correspondingmeasurement r ( i ), with 1 ≤ i ≤ k , RB Xiao knows that O ∈ ( L ∩ L ( i )), i.e., where L ( i ) isthe lune created by the two circumferences of radius r ( i ) centered in A i and A i +1 , respectively.Hence, although we do not implement this feature in our algorithm, after each measurement r ( i )RB Xiao could stop. As we will see, the best moment to stop would be when r ( i ) is minimum.This is the main reason that makes RB Xiao robust to hole and bubbles: whenever it stops, theintersection area is limited. However, a discussion about the intersection of lunes while A i variesalong the x -axis is needed to complete the error analysis.Assuming an omnidirectional antenna pattern, as MA moves along the x -axis, it crosses theGD’s receiving area disk. The distances d ( A i , O ) decrease until MA reaches the closest positionto O in A k ∗ = ( k ∗ I w ,
0) (precisely, x O ≤ k ∗ I w < x O + I w ). Then, d ( A k ∗ , O ) ≈ y O . After A k ∗ ,the distances d ( A i , O ) start to increase again up to A = (( k + 1) I w , A = ( kI w ,
0) is d ( A , O ) = r . The observed distances while MA moves in A i , with i = 1 , . . . , k ∗ , . . . , k , form aunimodal sequence R with minimum in k ∗ I w . As long as the radii in R decrease, i.e., x A i < x A k ∗ ,the lunes L and L ( i ) have the same curvature (see Fig. 12(a)), while when the radii in R increase,i.e., x A i > x A k ∗ , L and L ( i ) have opposite curvature (see Fig. 12(c)). In A k ∗ , the tangent to thecircumferences of the lune L ( k ∗ ) is parallel to the x -axis (see Fig. 12(b)). D BC A (a) Tangent concordant.
D BCA (b) Parallel to x -axis. D BCA (c) Tangent discordant.
D BA (d) When C is undefined. Figure 12: The possible intersection areas depending on the lunes: the solid line is the x -axis. In a general scenario in presence of holes and bubbles, the MA stops at A in one of the threecases depicted in Fig. 12: (i) the curvatures of A and A are concordant, i.e., 1 < k < k ∗ or k ∗ < < k ; (ii) the tangent at A is parallel to x -axis, i.e., 1 < k = k ∗ ; (iii) the curvature A and A are discordant, i.e., 1 < k ∗ < k . To evaluate the size of the intersection among L and L in these three cases, let observe that they cross each other making at most four intersections, i.e., A is the intersection among Eqs. (b)–(c), B among Eqs. (b)–(d), C among Eqs. (a)–(d), and D among Eqs. (a)–(c) (see Fig. 12), where: The curvature is the sign of the tangent to the curve. = Ö r − r k − I w + k + 12 I w , Ã r − Ç r − r + ( k − I w k − I w å è B = Ö r − r kI w + k + 22 I w , Ã r − Ç r − r + k I w kI w å è C = Ö r − r k + 1) I w + k + 12 I w , Ã r − Ç r − r + ( k + 1) I w k + 1) I w å è D = Ö r − r kI w + k I w , Ã r − Ç r − r + k I w kI w å è Note that A is stable when A moves because it is at the intersection of the two measurements of O . If there were no errors, O ≡ A . It is worth noting that y B = y D and x B − x D = I w = d ( D, B )regardless of k . Now we are in the position of clarifying the three previous cases. Recalling that A = ( I w , A = ( kI w , Case (i): The lunes have the same curvature, as illustrated in Fig. 12(a). The area where P can be selected has width I w and height y C − y A . In particular, if k is very small, r → r , andthus y C → r . If y A →
0, the error y C − y A ≈ r . As k approaches k ∗ , y C − y A quickly decreases.A similar error occurs when k ∗ < < k . Case (ii): By intersecting the coordinates x C = x A , and all the four points A , B , C , and D are very close. The area where P can be selected is very small. Case (iii): Observe that x C can be re-written as x C ≈ ( x A + I w ) k − k +1 for k ≥
2. Thus, x C < x A + I w for any k . Due to this and the fact that the curvatures of L and L are opposite,their intersection area is quite limited. Unfortunately, we could not find a simple formula fordescribing | y A − y C | here, but it can be computed by approximating the curves with their tangentin A . Hence, the intersection of the two lunes can be inscribed in a rectangle with two sides parallelto the x -axis of length I w and two vertical sides whose length depends on the angular coefficientof the tangents in A . We can only add that their lengths decreases below 2 I w when arctan y A x A > P can be selected only depends on I w . When arctan y A x A ≤
1, the trivial bound is y A , which cannot be very large however. So, the size of the area where P can be selected dependson I w and y A . When r ≤ x A < r + I w , C is undefined (see Fig. 12(d)), Hence, the vertical sideof the rectangle that contain P has length at most y A ≤ » r − ( r − I w ) . This leads to the sameerror described in Xiao when the lunes have only 3 intersection points [11].In presence of irregularities, RB
Xiao , as in our implementation, stops at the last heard beacon.The error can be large only when A and A fall on the same side with respect to x O , and thus theassociated lunes have concordant curvatures. In order to limit the occurrences of Case (i), in ourimplementation we have forced B , B , and B , i.e., the three selected endpoints, to respect the r min and α min constraints, putting them sufficiently apart, so as the two lunes will have oppositecurvature. Moreover, in Cases (ii) and (iii) the error is quite limited and only depends on I w .Without irregularities, RB Xiao always falls in Cases (ii) or (iii). The same happens for theoriginal
Xiao which stops when r = r and thus the curvatures are opposite.To conclude, the above analysis also applies for Lee . In
Lee , the GD belongs to the intersectionof two annuli of different radii but with the same width I w , and the error depends on the curvatureof the lunes created by the intersection of annuli. As regard to DrfE , P , P , and P are theintersections of pair of circumferences, and here the lunes coincide with portions of circumferences.Differently from Xiao and
Lee , the position uncertainty depends only on the measurement errorsand on I w , and it is bounded according to [20]. 18 .2. Range-based Versions Results In this section, we evaluate the performance of the RB variants, and we compare these resultswith those of the RF original implementation.In reality, in our previous experiments, since the goal was to investigate the performance of RFalgorithms, we did not take any distance measurement even though our UWB antennas were ableto range measurements with good accuracy. Indeed, we only knew in advance the GD’s
Home position (0 ,
0) and the MA’s position with respect to
Home . Instead, the new RB variants relyon the 3D slant distances between the MA and the GD, which are then converted in 2D grounddistances. However, it is important to recall that, due to the pandemic COVID-19, it is forbiddento take new distance measurements in the open field. According to those restrictions and based onour previous research experience, we decided to estimate the 2D ground distances (see Fig. 3(a))between the MA and the GD just calculating the Euclidean distance from the endpoints to the
Home in (0 ,
0) plus a random error (overestimation or underestimation) computed as proposedin [20] . In other words, we perturb any Euclidean distance r adding the error E r : E r = γ d + hr γ h + e s · h r = 1 . hr . e s · h r (1)where h is the drone’s altitude, γ d = 1 . γ h = 0 . e s is a random number in the range [ − (cid:15) s , + (cid:15) s ], where (cid:15) s = 0 . r = 30 m, the maximum ground error E r isapproximately 1 . . . h is 0 m, 10 m, and 20 m, respectively. First, we comparethe accuracy of the new RB variant algorithms toward that of their RF implementation. Then, wemake remarks on the accuracy of the new RB variants of the three RB algorithms. Range-free vs Range-based.
In Tab. 5 we report the localization accuracy of
Xiao , Lee , and
DrfE showing the comparison between the original RF and adapted RB versions of them. Tab. 5 alsoreports the number of times (in %) that an algorithm does not localize the GD. The experimentalresults take into account the constraints r min = 60 m and α min = 20 deg, and only VV. In particular,for the original RF versions we report the average error for both the radii, i.e., the observed radius r = µ and the manufacturer radius r = r . For the adapted RB versions we report the averageerror for the measured radius, denoted as r = d ( A i , O ), where A i is an endpoint. Table 5: Error (in m) and unlocalized (in %) between RF and RB algorithms in VV.
RF RB r = µ r = r r = d ( A i , O )Algorithm h error unloc error unloc error unloc Xiao
Lee . . . DrfE . . . Our decision is supported by our previous experience in converting slant from/to ground measurements [20, 21].
19t is clear that the average error, using measurements, is one order of magnitude smaller.We have seen that the knowledge of the exact distance significantly improves the accuracy andpractically clear the number of unlocalized devices. This also enforces our original statement thatthe knowledge of the radiation pattern is fundamental for the success of the localization. Bubblesand holes in the antenna radiation pattern are no longer a problem because we range the effectivedistances between the endpoints and the GD. So to avoid measurements, any manufacturer shouldgive as much information as possible on the antenna radiation pattern as possible.
Range-based Comparison.
Here, we present performance-wise the RB versions of the
Xiao , Lee ,and
DrfE algorithms. As previously said,
Drf is not present because it is not radius-based. Forthis simulation evaluation, we consider different cases varying the constraints of minimum distance r min = { , } m and minimum angle α min = { , } deg, as explained in Sec. 6. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (a) Error: VV, r min = 0 m. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (b) Error: VH, r min = 0 m. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (c) Error: VV, r min = 60 m. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (d) Error: VH, r min = 60 m. Figure 13: Error using distance measurements when α min = 20 deg and r min varies. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (a) Error: VV, r min = 0 m. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (b) Error: VH, r min = 0 m. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (c) Error: VV, r min = 60 m. XIAO LEE DRFEAlgorithms0246810 E rr o r ( m ) h = 0 m h = 10 m h = 20 m (d) Error: VH, r min = 60 m. Figure 14: Error using distance measurements when α min = 40 deg and r min varies. In Fig. 13 and Fig. 14, we evaluate the performance of the RB algorithms. In particular, inFig. 13 we compare the algorithms fixing the minimum angle α min = 20 deg, while in Fig. 14 suchconstraint is kept to α min = 40 deg. Overall, we can see that DrfE works better than the othertwo algorithms with an average error smaller across all the tests, and that
Xiao works better than
Lee with those settings. Also, a better localization accuracy can be obtained for the experimentscarried on the ground at h = 0 m, exception made for Fig. 13(c) in which, oddly, the error is smallerat h = 20 m. More specifically, in both Fig. 13 and Fig. 14 is also evident that VH benefits slightlymore from the r min constraint of 60 m, while it is the opposite for VV which behaves better showing20maller error with no r min constraint. Finally, there is very little improvement while testing withlarger α min . It is extremely important to note that using measurements, all the selected endpointsare suitable for localizing the GD in O . This means that the number of unlocalized GDs is zero,but also this leads to the growth of the outliers number. This can be seen by the presence oflong whiskers and a considerable difference between the average (i.e., solid circle) and median (i.e.,horizontal line) values. However, even though such whiskers are pretty long, they are really shortwith respect to the ones seen in the previous experiments in Sec. 6, Fig. 9.
8. Conclusion
In this paper, we compared the performance of four RF algorithms based on HnH on a realtest-bed using the DecaWave DWM1001 UWB antennas as MA and GDs. We implemented andsimulated the algorithms on a large data-set of endpoints collected in the field. We analyzedthe antenna radiation pattern of the GD at different altitudes and configurations, i.e., VV andVH. We have shown how such algorithms actually perform assuming (i) first the datasheet radiusof the antenna equal to that released by the manufacturer, (ii) then the experimental observedradius, and (iii) finally the actual radius obtained via distance measurements. The manufacturerdatasheet radius poorly performs because usually it does not characterize the antenna radiationpattern very well. The observed radius can help only if it is almost constant in all the directions.When the antenna is irregular, only the knowledge of the distances between the MA and the GDcan alleviate the localization error. We conclude that the RF algorithms are simple and elegant,but they can be very inaccurate and can leave a high percentage of unlocalized GDs if the antennais not omnidirectional and measurements are not allowed. However, the exact knowledge of theirregular antenna can make accurate the RF algorithms.
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