Source localization using moving antenna S¸tefan Apostol EPFL - EDIC
[email protected] Semester project report 7.07.2014
Abstract With potential applications in victim rescue, we investigate how the source of an EM signal can be localized in 3D space using the apparent Doppler shifts recorded by a moving receiver equipped with an omnidirectional antenna. From the frequency measurements the receiver can infer the angle (relative to its moving direction) under which it sees the source. However, a single angle in 3D defines a cone rather than a direction and here we propose a parametric approach that transform the complex problem of intersecting cones into the much simpler one of intersecting spheres. This is done after an on-line linear regression algorithm smoothens the noisy information about the receiver’s position. In addition, we compare 4 variants of the parametric approach with the direct, naive method of finding the best point that fits the measurements to conclude that the naive approach over-performs any other analysed method.
1 Introduction Natural disasters such as earthquakes, floods and wildfires are unpredictable and may produce high damage to densely populated areas. In such an unfortunate scenario, when the communication infrastructure is partially damaged or unavailable, victims are unable to call for help and it is crucial to employ a wireless communication network that supports the rescuers in their mission and ensures emergency communication. One solution is to use a swarm of independent Micro Air Vehicles (MAVs) to construct a fast-deployable ad-hoc network with a reliable routing protocol, like the one proposed by the SMAVNET II project1 . Locating victims in traditional way is challenging as they might not be directly visible to the rescuers due to darkness, dust, smoke or fog. To overcome this difficulty the network can assist in victim localization and thus greatly reduce the rescue time especially in situations when the victim is unconscious. Nowadays mobile devices that are able to communicate with a wireless network are ubiquitous and almost everybody is in the close proximity of such a device (e.g. cellular phone, smart-phone, laptop, tablet). This eases the automatic localization problem in the absence of any other victim related tag and reduces it to the problem of locating a node in a wireless network. However, to be effective and practical, an automatic localization system must satisfy a series of requirements. First of all, because we assume an unaware victim, the system should not require any kind of active action from the wireless device. Next, because the situations when such a system is needed appear suddenly, it must be readily available and should not require any type of hardware modification to the widely used wireless devices. In addition, because the victim is passive any processing is done by the network and for a rapid localization the processing must be done in real time, imposing low complexity algorithms. Finally, accuracy is a desire but not a primary demand because localization errors in the order of meters are enough for a rescue team to rapidly find the victim. The capabilities of such a system have been recently proven by the team involved in SMAVNET II that used a MAV equipped with a GPS module and wireless dongle to localize an unaware device by sniffing the probe request frames that are regularly sent by its WiFi radio interface. Their experiments showed that, even in unfriendly conditions due to strong wind, the device is localized with a precision of under 100 meters by processing the signal strength information contained in the received frame. In this report we investigate how we can localize the source of an electromagnetic signal in 3D space by exploiting the properties of the signal received by a moving receiver, a scenario that overlaps with the set-up used in SMAVNET II. The novelty comes from the way the position is inferred by having access to measurements that only contain information about the angle under which the MAV sees the victim in different points in space. The remaining of this report is organized as follows: Section 2 briefly reviews the literature of victim 1
Swarming Micro Air Vehicle Network (SMAVNET II) is a joint project between The Intelligent Systems(LIS) and the Mobile Communication(LCM) laboratories, both from EPFL
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localization and presents the general approaches used to infer the position of a node in wireless networks. Section 3 formalizes the localization problem and proposes an algorithm for localization in 3D space using AoA in an ideal, noise-free, scenario. Section 4 generalizes the problem and analyses the case when the measurements are noisy whilst Section 5 presents and compares the results of the different methods proposed. Finally Section 6 concludes the report.
2 General Principles and Related Work In this section we start by presenting the general techniques used to locate the source of an EM signal, then we review the literature of victim localization and finally we present what techniques are generally used to perform localization in wireless networks. In a general set-up where a source S broadcasts a signal in free space, an arbitrary located receiver R can obtain some information about its relative position with respect to S from the properties of the received signal. If both S and R are fixed then the only information that R can extract is the distance between the two. This is achieved using a ToA (Time of Arrival) technique that is based on the fact that the signal travels at a known and constant velocity. Such an approach requires precise synchronization and loses any information about the direction of S relative to R because the source can be located anywhere on a sphere around the receiver. However, 4 receivers with known not coplanar positions are able to pinpoint the source using trilateration. This technique is used by the Global Positioning System [6] but it is not suitable for an ad-hoc network scenario because it demands precise synchronization and fixed positions for the receivers. The need of synchronization can be bypassed if the ToA is determined based on the time it takes for the S to respond to a request sent by R but this comes at the price of an aware and active S. A variation of ToA is based on measuring the power of the signal at R knowing the power emitted by S and a loss model for the wireless channel. Although packets used in WiFi carry information about the received signal strength inside RSSI (Received Signal Strength Indicator), this is not reliable for positioning purposes because there is no standardized relationship of RSSI to the power level [3, 9]. If instead of a single receiver we have an array of receivers then we can obtain information about the direction of S (relative to the direction of the array) using an AoA (Angle of Arrival) technique. If the array is 2D then we have have information about the direction of the source but not about the distance (exactly opposite to ToA) and we know that S lies on a straight line. Information from 2 non parallel 2D arrays is enough to pinpoint the source [4]. On the other hand if instead of an array of receivers we only have 2 receivers then we restrict the possible positions of S to a two-sheeted hyperboloid by measuring the difference between arrival times at the 2 receivers rather than measuring the absolute arrival times. This technique is called TDoA (Time Difference of Arrival) and has the advantage that, as in the case of AoA, only the receivers need to be synchronized [1]. If the receivers are positioned in the far field of S then the TDoA and AoA are equivalent [7]. Having information from multiple pairs of receivers (or from the same pair moved in different positions) we can recover the position of the source. Due to its importance, a lot of work has been done to design a system able to localize victims. For example the authors of [2] use a MAV to locate victims based on acoustic signals emitted by emergency whistles rather than EM signals. They process the signals from 4 microphones mounted on the MAV to extract information about the direction of arrival at different positions in space. The measurements are afterwards combined with the dynamics of the aerial platform in order to recover the position of the victim using a method based on particle filters. On the other hand the approach presented in [12] is based on active RFID tags placed by a paramedic on victims as part of triage. RFID readers placed at known positions use the unreliable RSSI values and estimates the location of the tagged victims using techniques such as tag calibration, tag averaging, time averaging and selective trilateration. They report an average error of under 4 meters when 5 readers are deployed to cover a square with the side of 30 meters, but their approach is limited to 2D. [13] presents the results of a field intensity based method that locates a GSM-compatible mobile phone hidden under smooth debris. They achieve this in a semi-automatic manner by jamming the GSM architecture using a custom base station that determines the phone to continuously send a known signal with maximum power. This allows the rescue team to localize the victim using a custom hand set device designed to measure the radiated field strength using a directional antenna. Also, a lot of research
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has been focused on localizing victims buried under snow in case of avalanches [8, 11, 10]. These methods rely on the fact that the victim carries an avalanche beacon (or even a more sophisticated wearable device) that transmits a magnetic field detectable by rescuers using special devices. The highly non-linear received signal is processed using an Extended Kalman Filter [11] or a Particle Filter [10] in order to infer the victim’s position relative to the rescue team. Regarding node localization in wireless networks, in general this is achieved by measuring the signal strength at some anchor nodes with known positions that cover the area of interest. If the anchor nodes have directive antennas then a node is localized using triangulation. If, on the other hand, the anchor nodes have omnidirectional antennas then the usual approach is to first perform measurements on a grid and assign each point on the grid a ”fingerprint” determined by the signals received by the anchor nodes from that point on the grid. Having a ”fingerprints-map” of the scene, the position of an unknown node is determined based on how similar is its ”fingerprint” to the ones kept at the anchor nodes. These methods are developed for 2D networks, where the geometry is simple in the case of directive anchors and the grid to be covered has a small number of points in the case of omnidirectional anchors, and cannot be easily generalized to 3D networks. In what follows we investigate how a source can be localized in 3D space using the Doppler shifts measured by a moving receiver which is equivalent to solving the localization problem in an AoA set-up with only one angle (elevation - defines a cone in 3D) instead of two (azimuth and elevation - define a line in 3D).
3 3D Source localization using AoA This section starts by presenting the general assumptions that we make and by formally defining the problem and afterwards presents how the problem can be solved in ideal conditions. The goal is to recover the unknown position of a source S that radiates a signal in free-space using the Doppler shifts measured at different points by a moving receiver R. We assume that both R and S lack directivity (they are equipped with omnidirectional antennas) and that the position of R is known at each measurement point. Let S, placed in S pSx , Sy , Sz qJ P R3 in an arbitrary coordinate system, continuously transmit a periodic signal with frequency fS . If Pptq P R3 and vptq P R3 are the position and velocity (in the same coordinate system) of R as a function of time t P R then the Doppler shift perceived by R has the form: f d ptq fS
fR
fS pS PptqqJ vptq c }S Pptq} ,
(1)
where fR is the frequency on which R is tuned (R and S are not synchronized and there is a frequency offset ∆f fS fR between them), c is the speed of light, } } is the standard euclidean norm and pqJ denotes the vector transpose. To be noted is that the assumption of unsynchronized devices introduces a new unknown (∆f ) which has to be found in order to recover S. An implementable digital system is not able to work with continuous signals and from now on we only consider discrete samples of the Doppler shifts. N measurements collected with uniform sampling period T from the perceived Doppler shifts will have the form: fid
f d piT q ∆f
fS pS Pi qJ vi c }S Pi} , i P t1, 2, . . . , N u
(2)
where Pi PpiT q and vi vpiT q. Because we only have access to fR , assuming that |∆f | ! min pfR , fS q, eq.(2) can be approximated by: fid
∆f
fR pS Pi qJ vi c }S Pi}
(3)
which in fact is the equation of a right circular cone (the surface C¯i defined below, in eq.(4)) with apex at
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Pi , axis parallel to vi and aperture 2 acos
} }
fid ∆f α vi
where α
fR c :
f d ∆f C¯i : pS Pi qJ vi i }S Pi}}vi} 0 α } vi }
(4)
Assuming that we know ∆f (which is equivalent to knowing the cones C¯i ), S will be found at the intersection of all the cones. Unfortunately, for arbitrary positioned cones, even if they are chosen such that to intersect in a single point, there is no closed form solution to recover S.
3.1 Naive Inverse Approach Because no closed form general solution can be found, the simplest way to recover S and ∆f is by solving a system of non-linear equations: F1 pδ, aq 0; where F1 i pδ, aq fid δ α
pa PiqJ vi @i }a Pi}
(5)
If we have at least 4 cones C¯i with non-collinear apexes then the system has only one solution. However, because we do not have a closed form solution, numerical methods designed for non-linear systems of equations must be used. The majority of numerical methods suited for this kind of problems are variations of the Newton’s method and it is helpful to also compute the Jacobian matrix JF1 pδ, aq associated to the system in eq.(5): JF1 pδ, aq r1N , α pvi b pa Pi q pa Pi q b vi qJ s,
(6)
where 1N is the all-ones vector of size N and b is the outer product defined as: u b v u vJ . The solution pδ, aq of eq.(5) correspond to the unknowns p∆f, Sq, so solving the system in eq.(5) is equivalent to finding the position of the source. Difficulties may appear if the system is underdetermined and thus admits multiple solutions but the ”correct” solution can be promoted by imposing constraints on the possible locations and frequency offset.
3.2 Parametric Approach If in the general case the naive approach may be the only feasible one, other methods exist if we impose that the trajectory of R is piece-wise linear, or at least rectilinear over a large fraction of the entire trajectory. This is not a strong constraint because in practice MAVs are programmed to follow long linear trajectories with constant air-speed. For this reason let us consider that M (say from 1 to M ) samples out of N (M N ) are measured while R follows a rectilinear and uniform trajectory. This means that vi v, @ 1 ¤ i ¤ M and that all Pi s lie on the same line which implies that the cones corresponding to these points have the apexes on the same (common) axis. After we rewrite eq.(4) for these M collinear points we get: f d ∆f C¯i : pS Pi qJ v vJ pS Pi q i
2
α2
pS P i qJ p S P i q 0
(7)
where this time, to avoid the euclidean norm we considered the two-sided cone for each measurement point. Intuitively, all these cones must intersect on a circle. In addition, due to the symmetry of the problem, the center of this circle must also lie on the common axis of the cones. With this intuition, let the circle of intersection be centered in P1 γv for some γ and have radius r. The center of intersection must clearly lie on all the cones. For the cone C¯i , the radius ri of the circle that lies on the cone and is centered on its axis, at P1 γv satisfies: ri2
pPi P1 γvqJ pPi P1 γvq
4
α2 v J v
fid ∆f
2
1
.
(8)
Finding the parameters γ and r of the intersecting circle (and the offset ∆f ) is equivalent to solving the following system of equations:
F2 pδ, r, γ q 0 where Fi2 pδ, r, γ q fid δ
2
J 2 J α 2v v pPi P1 γvJq pPi P1 γvq r pPi P1 γvq pPi P1 γvq
(9)
As for the naive approach, an analytical expression for the Jacobian matrix can be used by a rapidly convergent numerical method to solve the system of equations eq.(9). Having the circle for a rectilinear trajectory, to recover S we need to intersect at least two such circles (corresponding to different linear segments of the trajectory). However, while intersecting spheres in 3D is an easy task, intersecting circles in 3D is not straight-forward. To overcome this we consider each circle that corresponds to the intersection of cones along a rectilinear path to be obtained from the intersection of two spheres. The spheres are easy to determine because the circle centered in P1 γv and of radius r (obtained from a trajectory along v) is equivalent to the intersection of two spheres centered in P1 and P1 2γv both with radius squared equal to r2 γ 2 vJ v. As we will see in Section 4, both the naive and parametric approaches accurately localize the source and find the frequency offset in a noise-free scenario.
4 3D Source localization in the presence of noise In this section we consider measurements corrupted by noise (that affects both the recorded position and the perceived Doppler shifts) and analyse how it can be filtered in order to get good estimates of ∆f and S. Moving towards more realistic scenarios we consider that R is equipped with a GPS receiver that samples the positions at known moments of time. However the position given by the GPS module is affected by noise: i Pi nP where nP are assumed to be i.i.d. samples drawn from a zero-mean normal distribution with P i i 2: variance σP 2 nPi N 0, σP . We do not have any direct measurement of the velocity, but we assume that R moves on a piece-wise linear trajectory and then try to infer the velocity from the position measurements. Furthermore, the Doppler shifts measurements are also affected by white Gaussian noise with variance σf2 : d d f i fi
nfi where nfi
N
0, σf2 .
Trying to recover the unknowns from these noisy measurements is not an easy task. In each measurement point the frequency noise alters the aperture of the cone (nfi makes the cone to be wider or narrower) but this effect is negligible compared to the one caused by the position noise which completely deforms the cone by altering its apex, axis and aperture either directly (apex) or indirectly (axis, aperture) because the velocity is obtained from the (noisy) positions that randomly rotate the axis of the cone. To minimize its effect, the position noise is reduced in a preprocessing step that smoothens the trajectory and brings it as close as possible to a piecewise linear one. This smoothening step is basically an on-line i along with their associated measurement linear regression that takes as input the noisy measurements P c times ti and outputs a clean piecewise linear trajectory Pi . The algorithm that cleans up the trajectory is presented below and is based on the fact that if we have a bunch of points that originally lie on a straight line but are affected by i.i.d. Gaussian noise, then the normalized residual between the noisy points and the best fitted line (best in the sense of minimum residual) asymptotically decreases as the number of points increases. If at some moment the incoming points start to lie on a different line, then, as we consider more and more points, the normalized residual starts to increase and so we know where the trajectory makes a turn. In reality, the residual sometimes increases even if the points are noise-free and fluctuations in the normalized residual might appear. To bypass this effect, in practice only variations higher than 103 are recorded and a turn is decided only after the residual grows for T 10 consecutive steps. After the smoothening step for each measurement point we have the ”cleaned” position Pci and from it we obtain a ”clean” velocity vic both very close to the true Pi and vi if each linear segment of the trajectory contains a sufficient number of measurements (few tens).
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i ; Thresholds: T and Data: Noisy trajectory: ti , P Result: Clean, piecewise linear trajectory: pti , Pci q Initialization: rold Ð 0; count Ð 0; low Ð 1; idx Ð 1; r Ð r s; while idx ¤ N do perform linear regression on the block of points with indexes between low and idx to get the linearised trajectory between these points; assign the residual regression error to r pidxq; rnew r pidxq; if rnew rold ¡ then count ; else count Ð 0; end if count ¡ T then perform linear regression on the block of points with indexes between low and idx T ; assign to Pci the linearised trajectory; low Ð idx T 1; rlow Ð rpoldq; idx Ð low 1; count Ð 0; else rold Ð rnew ; idx ; end end Algorithm 1: Trajectory smoothing based on on-line linear regression
4.1 Naive Inverse Approach Having good approximations for position and velocity we recover S and ∆f as in the noise-free case but, instead of solving a non-linear system, we infer them by solving a minimization problem:
p∆f, Sq
J
d argmin F1 F1 where F1 i pδ, aq f i δα
pδ,aq
pa PciqJ vic . }a Pci}
(10)
Matlab’s optimization toolbox is equipped with the lsqnonlin function that can solve this type of optimization problems. Furthermore, the solution is found faster if we also compute the Jacobian matrix of F1 which has a form similar to the one in eq.(6).
4.2 Parametric Approach Instead of using the naive approach we can take advantage of the piecewise linearity of the trajectory, as in the ideal case, in order to recover a circle from each linear portion of trajectory. However we cannot use eq.(9) directly because it squares the (noisy) Doppler shifts and therefore the frequency noise acts in a strong non-linear manner that is hard to filter using standard techniques. We overcome this by using a slightly modified version of eq.(9) as new objective function for a minimization problem:
p∆f, γ, rq argmin F3JF3 where F3i pδ, r˜, γ˜q fid δ pδ,˜γ ,˜rq
αb r˜2
pPci Pc1 γ˜vcqJ vc . J c c c c c c pPi P1 γ˜v q pPi P1 γ˜v q
(11)
As for the naive approach, a rapidly convergent numerical method (like the one implemented in Matlab’s lsqnonlin function) can solve eq.(11) and thus find ∆f and the circle that best fits the Doppler shifts measured along a linear part of trajectory. Knowing the parameters of this circle we express it as the intersection of two spheres like in the ideal noise-free case and then S is recovered by finding the point that is closest to a bunch of spheres that results from different linear segments of R’s trajectory.
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Intersecting Spheres Here we briefly present the methods used to find the intersection point of at least 3 spheres. If we have n spheres then the intersection point (if exists) is at M1 pt 1n dq where t is a solution of
J M1 1n t2 1J nM
J M1 d 1 2t 1J nM
dJ MJ M1 d 0
M P Rn3 is the matrix that has on each row the centres of the spheres, d 21 diag M M J diag rrJ and r P Rn is the vector containing the radii of the spheres (If n ¡ 3 then M1 is the left inverse of M). Even if this closed-form solution is robust to small errors (it returns a point even if not all the spheres intersect in the same point) usually, when either the centres or the radii of the spheres are altered, this approach fails. If this happens we find the point that is the closest to all the spheres simultaneously by solving a minimization problem that promotes points which are closer to the majority of spheres. If we have
n spheres with centres Ci and radii Ri for i 1, n then the minimization problem can be posed as:
J
x ˆ argmin F4 F4 where F4 i pxq x
px CiqJ px Ciq 1, Ri2
(12)
a formulation similar to the ones in eq.(10) and eq.(11) and suitable for the same solver. Solving the Optimization Problems All the optimization problems presented above (eq.(10)-(12)) have a similar form; in addition the problem in eq.(5) can be formulated as an optimization problem similar to the one in eq.(10) and the solution to eq.(9) is the same as the solution to eq.(11) (if in eq.(11) the ”cleaned” variables - including the noisy perceived Doppler shifts - are replaced by the true, noise-free ones found in eq.(9)). Because all these optimization problems can be regarded as data-fitting problems, we solved them using Matlab’s lsqnonlin function which is specialized in solving non-linear least squares problem using the Trust-Region-Reflective algorithm2 . However, the solution to any of these problems (as well as the convergence time) is dependent on the starting point. A good approach is to start with a random guess and when new data is available, call the solver starting from the previous solution to reduce the convergence time. In addition, for more precise solutions, the parameters of the spheres corresponding to each linear segment of trajectory can be jointly estimated (rather (a) (b) than independently), imposing all the segments to have the same ∆f .
5 Results and discussion This section presents the results obtained using the framework described above. (c) (d) To analyse the localization capabilities of the proposed Figure 1: Test trajectories used in the experimethods we construct two test trajectories, presented ments: 1a,1b square planar trajectory; in blue in Fig.1. In the first case (Fig.1a,1b) the ideal 1c,1d rectangular trajectory with intrajectory of the receiver is a 900m 900m square placed creasing hight; on the left in 3D perat a constant height of 80m above the ground, whilst in spective; on the right seen from above the second case (Fig.1c,1d) the ideal trajectory of the receiver is a 600m 400m rectangle whose height varies linearly from 50m to 120m above the ground. In Fig.1 the left row presents a 3D perspective of the 2 cases and the right row presents them as seen from above. The red point represents the source that is randomly placed inside the projection of the trajectory on the ground at a height randomly chosen between 0m and 30m. 2
check http://www.mathworks.ch/ch/help/optim/ug/lsqnonlin.html for more details
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Also in red are depicted the noisy measurement points, which in these cases have been chosen to fluctuate randomly with σP 2m. In the rectangular case R performs 4 loops (16 linear segments) whilst in the case of square trajectory R performs 2 loops (8 linear segments). We test 5 methods presented in the previous sections and compare their performances: The naive inverse method presented in Section 4.2 - we will refer to it as ”Naive”; The parametric method presented in Section 4.3 combined with the analytical solution to get the intersection of the spheres. Parameter recovery is done independent after each linear segment - we will refer to it as ”OneLineExact”; A variation of the above method where parameter recovery is done jointly on all lines after each linear segment - we will refer to it as ”AllLineExact”; The parametric method presented in Section 4.3 combined with the numerical solution (result of optimization in eq.(12)) to get the intersection of the spheres. Parameter recovery is done independent after each linear segment - we will refer to it as ”OneLineOptim”; A variation of the above method where parameter recovery is done jointly on all lines after each linear segment - we will refer to it as ”AllLineOptim”;
We start by cleaning the trajectory as explained Section 4 and after each new linear segment covered by R we evaluate the Euclidean distance between the estimated position and the true position of S for all 5 methods.
Noiseless Measurements We first perform tests in ideal conditions. As expected, the preprocessing part yields the ideal trajectory and the all the methods perform well. Because in this case we have no type of noise, the analytical solution always returns the intersection of the spheres and the overall performance is ideal for all the methods, with estimation errors less than 109 m. In Fig.2a the ”OneLine” methods overlap perfectly on the corresponding ”AllLine” ones. In general, differences are negligible and appear due to rounding errors. The plots in Fig.2 present the minimum, mean and maximum estimation error obtained after performing 20 simulation in each case. These results validate all 5 localization methods in ideal conditions.
(a) Square trajectory
(b) Rectangular trajectory
Figure 2: Mininum, mean and maximum values of the distance between the estimated and the true position for 20 runs
Noisy Measurements Three experiments were performed to assess the performance of the five methods applied to noisy measurements. The first one consists of 20 runs on the square trajectory with σP 2m and σf 7Hz, the second and third one involve also 20 runs but on the rectangular trajectory with σP 2m and σf 10Hz respectively σP 3m and σf 5Hz. The mean values of the euclidean distance between the estimated locations and the true one are presented in Fig.3a-3c. We observe that for the square trajectory (Fig.3a) all the methods behave approximately the same, with a small advantage for the ”OneLine” ones while ”AllLineExact” performs the worst. The overall performance is poor, with average errors between 20 and
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(a)
(b)
(d)
(e)
(c)
(f)
Figure 3: Mean localization error after repeating each experiment 20 times. The left column presents the results of the first experiment, the middle one corresponds to the second experiment and the right column to the third experiment. The top row presents the average distance in 3D for every method tested whilst the bottom row shows the mean distance between the projections of the estimated positions and the true one on the ground 25m. However, a great improvement appears if we compare the projection of the estimated point with the projection of S on the z 0 plane (Fig.3d). In this case we observe a reversal in terms of performances, ”Naive” performing the best whereas ”OneLine” ones are the worst. The average errors ( 1m for ”Naive”) are much smaller in this case because the square trajectory has no variation along the z axis and therefore the resolution along this direction is very small. In the second experiment(Fig.3b-3e) the ”AllLine” methods perform better that the ”OneLine” ones, an intuitive result, because ∆f is optimized jointly, for all segments. Maybe not intuitively, the ”Optim” methods perform better than the ”Exact” ones and the reason for this is that in many cases the analytical approach fails to give a solution (when the spheres do not overlap). The ”Naive” approach performs again the best, yielding errors below 10m if we consider the 3D distance and below 2m if we only measure the distance between the projections on the ground. Finally, the general behaviour in the third experiment(Fig.3c-3f) is the same as in the second experiment, with an improved overall performance because in this case the frequency noise is less powerful. As a remark, the frequency noise is much more upsetting than the position one, because the latter is efficiently removed in the preprocessing step. Surprisingly, the best approach is the ”Naive” one but this is explainable if we recall that this method recovers all the unknowns in a single step by processing all the available points at once whilst all the other methods have some intermediate steps that accumulate the errors. The table in Fig.4 contains the numerical values resulted from the last experiment. Here, besides de mean values (that are plotted in Fig.3c-3f) we recorded also the variance of the estimation error and we see that, as the trajectory gets longer, the estimated points approximate better the solution and have smaller variations.
6 Conclusions This work investigated how the source of a signal can be localized in free-space by a moving receiver equipped with an omnidirectional antenna, a challenging problem with applications in victim rescue missions. The
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Figure 4: Mininum, mean and maximum values of the distance between the estimated and the true position for 20 runs of Experiment 3. receiver knows its position (affected by noise) and measures the Doppler shift of the received signal to infer information about the position of the source. Knowing the parameters of its trajectory, from the frequency measurements the receiver can infer the angle (relative to its moving direction) under which it sees the source. However, a single angle (elevation) defines a cone in 3D rather than a direction (for which we need both azimuth and elevation) and here we propose a parametric approach that transform the complex problem of intersecting cones into the much simpler one of intersecting spheres. While moving on a linear trajectory, the receiver reduces the possible positions of the source down to a circle (express as the intersection of two spheres) whose parameters are recovered by solving a non-linear least squares problem. Then the source is localized by solving another optimization problem to find the point which is closest to all recovered spheres. Four variants of this approach are tested and compared to the more naive, direct approach where the source is found as the point that best fits all the measurements. The results show that the source can be localize with small errors ( 10m while covering an area of 900m 900m and an error of under 2m if we measure the distance between the projections on the ground) and that the direct approach performs better than any parametric method. In addition, in order to increase the precision, the receiver must cover wide areas along all three dimensions (a receiver that moves parallel to the ground has a very poor resolution along the z axis). In all the cases, a first preprocessing step is required to reduce the noise on position because, if not smoothened, it makes the recovery process impossible. The smoothening is done by an on-line linear regression algorithm assuming that the receiver’s trajectory is piecewise linear. Future work towards a more realistic, implementable system impose the study of better models for GPS noise, wind speed and also Doppler shifts measurement of wideband signals.
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