Transponder positioning within monitoring area divided into multiple polygonal segments

Abstract

The invention relates to positioning transponders within a monitoring area of a positioning system, in which each receiver (R1-R8) measures the signal strength of a periodic signal from each beacon (B1-B6) and calculates a probability density function (700, 715, 717, 718) of the signal strength for each beacon. Based on the probability density function, a confidence interval (71-76) of the signal strength within each segment is determined, and a signal strength signature (10, 20, 30, 40, 50, 60, 70) is calculated accordingly. Further, a signal strength fingerprint (500) is generated from measurements of the transponder, and a section (104, 107) having a signal strength signature (40, 70) that matches the signal strength fingerprint is selected to determine the position of the transponder. The position of the transponder is determined from the candidate positions, and the position has a maximum match between the measurement of the transponder and the probability density function.

Description

TECHNICAL FIELD

[0001] The present invention relates to a method for positioning at least one transponder within a monitoring area, in particular an indoor area, of a positioning system, a positioning system for carrying out the method, a processing unit for such a positioning system and a mobile object comprising such a processing unit. BACKGROUND

[0002] Positioning devices can be based on, for example, RFID tags (radio frequency identification), which are attached to objects to be identified or positioned. The RFID tags comprise a storage unit, the content of which can be read using an RFID reader and can also be changed by means of the RFID reader. In each RFID tag, at least one identifier is usually stored. To read the information stored in the RFID tags, the RFID reader sends out a request signal by generating an alternating electromagnetic field. On the one hand, this electromagnetic alternating field is used, inter alia, to supply energy to passive RFID tags, which have no own energy source. On the other hand, the electromagnetic alternating field is modulated by the RFID tags to transmit a response signal, for example, by load modulation or by changing its antenna impedance.

[0003] Positioning systems for industrial automation systems must meet special requirements in terms of reliable data transmission, authenticity of the transmitted data and robustness against potential sources of interference. Since industrial automation systems are used for monitoring and controlling technical processes, in particular in factory automation systems, process automation systems and building automation systems, a malfunctioning, tampered or incorrectly configured positioning system can have serious consequences, in the worst case the automation system is shut down.

[0004] In the earlier international patent application WO 2024 / 013096 Al, a method for mobile object positioning in a wireless indoor transmission system is described, which has an improved accuracy. By utilizing machine learning techniques for processing wireless channel condition values, which are finally transformed into corresponding spatial distributions of estimated radio coverage and uncertainty, an improved spatial accuracy is achieved. The wireless channel condition values can be determined by signal strength or RSSI (Received Signal Strength Indication), latency, throughput measurements or channel impulse response. Furthermore, the calculation of an uncertainty metric is introduced. This metric is beneficial for improving the accuracy of the positioning as well as for reporting a confidence level of the predicted position. Using this uncertainty metric advantageously eliminates the need for explicit information about the boundaries of the scene, as these are automatically taken into account by high uncertainty values on and outside the boundaries of the scene.

[0005] Xue, Weixing et al., "Improved Wi-Fi RSSI measurement for indoor localization", IEEE Sensors Journal, vol. 17, no. 7 (2017), pages 2224-2230, propose and evaluate a technique to obtain indoor localization by means of WiFi signals. Its main feature is a smoothing algorithm that filters out low-power measurements corresponding to non-line-of-sight conditions, which deteriorate the overall results. However, the conversion between signal power and position is quite basic, since it is based on range estimation, and the evaluation is performed in a simplified scenario.

[0006] In Hoang, Minh Tu et al., "Recurrent neural networks for accurate RSSI indoor localization", IEEE Internet of Things Journal, vol. 6, no. 6 (2019), pages 10639-10651, a use of WiFi Received Signal Strength Indication (RSSI) fingerprinting is described. RSSI fingerprinting is used in combination with recurrent neural networks, such as long short-term memory algorithms, to accurately predict the position of a mobile device along a trajectory. However, it cannot be directly used to predict the position of a receiver with an unknown trajectory.

[0007] Talvitie, Jukka, et al. "Distance-Based Interpolation and Extrapolation Methods for RSS-Based Localization With Indoor Wireless Signals." IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, vol. 64, no. 4, April 2015, pp. 1340-1353, describes a wireless local area network (WLAN)-based fingerprinting using received signal strength (RSS) as an indoor positioning solution. In particular, the collection of correct fingerprints and their maintenance in a fingerprint database is discussed, and an incomplete fingerprint database with realistic coverage gaps is presented. More specifically, the accuracy of interpolation and extrapolation methods is investigated by artificially removing fingerprints from the database using a randomization procedure and by comparing the estimated fingerprints with the original ones. It has been shown that the correct interpolation and extrapolation methods can be utilized to improve both horizontal positioning accuracy and inter-floor detection probability.

[0008] In Alonazi, Abdullah, "Delaunay Triangulation Based Interpolation for Radio Map Construction with Reduced Calibration," 2017 9th IEEE-GCC Conference and Exhibition (GCCCE), fingerprint-based indoor positioning techniques are discussed as a method to provide high-accuracy information about the location of mobile users. In particular, a novel Delaunay triangulation-based interpolation method is presented to construct a radio map with reduced calibration effort, where a measured grid is triangulated on a less calibrated radio map according to the principles of Delaunay triangulation. A complete radio map is constructed with cubic interpolation inside each triangle.

[0009] In many industrial use cases, indoor positioning is a much desired functionality as it enables enhanced monitoring and task automation, leading to high production efficiency. However, existing positioning deployments based on signal strength measurements have so far failed to provide good positioning accuracy as signal strength measurements are typically highly variable. The goal of received signal strength indication (RSSI) fingerprinting is to mitigate the high variability of signal strength measurements and at the same time maintain low deployment cost, while achieving high positioning accuracy. In particular, it is a challenging task to adequately process the information contained in RSSI fingerprints. SUMMARY

[0010] It is an object of the present invention to create a method of positioning at least one transponder within a surveillance area of a positioning system, which allows for an economic and accurate positioning based on signal strength measurements and which aims at providing suitable means for implementing the method.

[0011] According to the invention, the problem is solved by means of a method having the features specified in claim 1, by means of a processing unit having the features specified in claim 12, by means of a positioning system having the features specified in claim 13 and by means of a mobile object having the features specified in claim 14. Advantageous embodiments of the invention are specified in the dependent patent claims.

[0012] According to the invention, a method of positioning at least one transponder within a surveillance area of a positioning system is provided. The positioning system comprises a plurality of beacons and receivers located at known positions. The surveillance area is divided into a plurality of polygonal sections, wherein a receiver is located at a respective corner of the polygonal sections. The method comprises the steps of measuring, by each receiver, a signal strength of a periodic signal from each beacon and, for each beacon, calculating a probability density function of the signal strengths measured by the receivers. Furthermore, the method comprises determining, for each beacon, a confidence interval of the signal strength within each section from the probability density function and calculating a signal strength signature of each section from the confidence interval.

[0013] Preferably, the polygonal segments are triangles and the signal strength signatures are barycentric signatures computed for each triangle from probability density functions of signal strengths measured by the respective receivers at each corner of the triangle. Advantageously, the probability density functions are computed for each receiver and each beacon, wherein the probability density functions associated with the position of the receiver are interpolated within the triangle using barycentric interpolation. In particular, when using barycentric interpolation of the probability density functions for a point inside the triangle, a weighted combination of the probability density functions can be computed based on the barycentric coordinates of the point. Thus, the barycentric signature can provide a confidence limit for any point within the triangle. The barycentric signatures can be easily computed and are less sensitive to the effects of distant objects, e.g. obstacles positioned or moving within the monitored area, and to fast changing radio propagation conditions due to the presence of objects when localizing the transponder. These effects can frequently occur in indoor scenarios, in particular within factory or process automation systems.

[0014] The method according to the present application further comprises the step of performing, by the transponder to be localized, a measurement of the signal strength of the periodic signal from each beacon. Furthermore, the method comprises generating a signal strength fingerprint from the measurements of the transponder and selecting segments having signal strength signatures matching the signal strength fingerprint. Moreover, the method comprises determining, within each selected segment, candidate positions at which the transponder is most likely located from the measurements of the transponder and the respective probability density functions. Finally, the method comprises the step of determining, from the candidate positions, the position having the largest match between the measurements of the transponder and the probability density functions.

[0015] The present application improves the efficiency of finding high probability candidate positions of the transponder by checking the signal strength fingerprint, in particular the RSSI fingerprint, measured by the transponder to be localized against the signal strength signatures, preferably barycentric signatures. By doing so, a large number of segments of the monitored area can be discarded before running an optimization algorithm to determine candidate positions only within a limited number of selected segments. Thus, the present application allows using low-cost radio beacons as source for signal strength fingerprinting based on accurate localization information.

[0016] According to a preferred embodiment of the present application, the receivers periodically update their respective measurements of the signal strength of the periodic signal from the beacons, wherein the receivers comprise stationary receivers and / or mobile devices having a known position and / or moving along a known trajectory. The receivers periodically updating the measurements reduces the effects of other objects within the monitored area on the propagation path of the periodic signal from the beacons and further enhances the accuracy when determining the position of the transponder.

[0017] In a further advantageous embodiment of the present application, the probability density functions are calculated and updated by a processing unit, which can be integrated in the transponder or implemented as a significant instance of a positioning system serving all transponders or mobile devices. In particular, the confidence interval of the signal strength within each segment is determined by the processing unit considering each receiver in the respective segment, wherein the confidence interval comprises the minimum and maximum of the signal strength over all receivers in the respective segment. Advantageously, the monitoring area is divided into a plurality of polygonal segments by the processing unit based on the locations of the receivers. Furthermore, preferably, the signal strength fingerprint is generated by the processing unit from the signal strengths measured by the transponder for each beacon. This allows a fast and efficient signal processing, in particular in scenarios where time is a critical factor.

[0018] According to a further advantageous embodiment of the present application, the segments with a signal strength signature matching the signal strength fingerprint are selected by the processing unit, wherein the selected segments are candidate areas where the transponder is likely to be located. In particular, the candidate locations are determined by the processing unit from the measurements of the transponder for each beacon and the respective probability density functions. On this basis, the processing unit can determine one candidate location for each candidate area, thereby allowing an efficient determination of the relevant candidate locations.

[0019] The positioning system according to the present application is designed for locating at least one transponder within a monitoring area and comprises a plurality of beacons and receivers located at known positions. The beacons are configured to transmit periodic signals, while the receivers are configured to measure the signal strength of the periodic signals from each beacon. Furthermore, the transponder is configured to perform measurements of the signal strength of the periodic signals from each beacon. Moreover, the positioning system comprises a processing unit according to the present application for calculating, for each beacon, a probability density function of the signal strength measured by the respective receivers and for dividing the monitoring area into a plurality of polygonal segments, wherein the receivers are located in the respective corners of the polygonal segments.

[0020] In particular, the processing unit according to the present application is configured for determining, for each beacon, from the probability density functions, a confidence interval of the signal strength within each segment, and for calculating, from the confidence intervals, a signal strength signature for each segment, and for generating a signal strength fingerprint from the measurements of the transponder and selecting the segments with a signal strength signature matching the signal strength fingerprint. Finally, the processing unit is configured for determining, within each selected segment, from the measurements of the transponder and the respective probability density functions, a candidate location where the transponder is most likely to be located, and for determining, from the candidate locations, the location with the greatest match between the measurements of the transponder and the probability density functions. BRIEF DESCRIPTION OF DRAWINGS

[0021] The present application will be explained in more detail with reference to the exemplary embodiments illustrated in the drawings, in which

[0022] Figure 1 A schematic diagram of a positioning system comprising a plurality of beacons and receivers at known locations and at least one transponder to be located within a monitored area is shown;

[0023] Figure 2 A flowchart of a method according to the present application is shown. Figure 1 A block diagram of a processing unit of the positioning system depicted in

[0024] Figure 3 A representation of a confidence interval of a signal strength of a periodic signal of a selected beacon within a selected triangular section of the positioning system depicted in Figure 1

[0025] Figure 4 A representation of a barycentric signature of a selected triangular section computed from confidence intervals of signal strengths of periodic signals from all beacons is shown.

[0026] Figure 5 A representation of a selection of a triangular section having a barycentric signature matching a signal strength fingerprint generated from measurements of the transponder is shown.

[0027] Figure 6 A flowchart of method steps according to a preferred embodiment of the present application is shown. DETAILED DESCRIPTION

[0028] Figure 1 The positioning system shown in Fig. 1 is designed for locating at least one transponder T1 within a monitored area 100, in particular an indoor area, e.g. within a factory. The positioning system comprises a plurality of beacons B1-B6, such as Bluetooth, UWB, WiFi or 5G access points, at known locations and receivers R1-R8. The beacons B1-B6 are configured to transmit periodic radio signals, while the receivers R1-R8 are configured to measure a received signal strength indication (RSSI) of the periodic signal from each beacon B1-B6. Furthermore, the transponder T1 is also configured to perform measurements of the RSSI of the periodic signal from each beacon B1-B6. In general, the receivers can comprise stationary receivers and mobile devices having known locations or moving along known trajectories, respectively. The transponder T1 can be implemented as a client having a Bluetooth, UWB, WiFi or 5G radio unit attached to, e.g., an automated guided vehicle (AGV).

[0029] ​The monitoring area 100 is divided into multiple triangular segments 101-107, with receivers R1-R8 located at the corresponding corners of segments 101-107. To efficiently calculate the position of transponder T1 (as will be explained in more detail later), the positions of receivers R1-R8 and transponder T1 are expressed using centroid coordinates. Figure 1 In the example shown, receiver R7, associated with triangular segment 106, has barycentric coordinates (0, 0), while the other two receivers R5 and R8, associated with the same triangular segment, have barycentric coordinates (1, 0) and (0, 1), respectively. Assume transponder T1 is also located within triangular segment 107 and has barycentric coordinates (0.45, 0.31), as per... Figure 1 The example is illustrated in the diagram.

[0030] In addition, the positioning system includes a processing unit 200, such as Figure 2 As shown, the processing unit is connected to receivers R1-R8 on one side and to transponder T1 on the other. The RSSI measurements performed by receivers R1-R8 and transponder T1 are provided to the input component 201 of the processing unit 200 and then passed to the CPU component 202 and the RSSI database 203. Receivers R1-R8 periodically update their respective RSSI measurements. These updates are stored accordingly in the RSSI database 203 for retrieval by the CPU component 202 to determine the location of transponder T1. According to this exemplary embodiment, the processing unit 200 also performs the step of dividing the monitoring area 100 into triangular segments 101-107 based on the locations of receivers R1-R8. The processing unit 200 may be a component of the transponder 200, or it may be implemented as a separate dedicated unit, particularly as the central unit constituting the positioning system.

[0031] With the aid of CPU component 202, processing unit 200 calculates and updates the probability density function (PDF) of RSSI measured by each receiver R1-R8 for each beacon B1-B6. The PDF is calculated for each receiver R1-R8 and each beacon B1-B6. In particular, using centroid interpolation, the PDF associated with the position of receiver R1-R8 is interpolated within triangle segments 101-107 to provide the PDF for any point within triangle segments 101-107. This is done with respect to beacon B1 and triangle segment 107 using... Figure 3The processing unit 200 is configured to determine, from the PDFs 717, 715, 718, a confidence interval 71 of RSSIs within the triangular section 107 for the beacon Bl. The confidence interval 71 is determined by the processing unit 200 considering each receiver R7, R5, R8 in the triangular section 107 and comprises the minimum and maximum RSSI values on all receivers R7, R5, R8 in the triangular section 107 for the beacon Bl. This is done accordingly for each of the beacons B1-B6 within each triangular section 101-107.

[0032] As Figure 4 indicated in the middle, the processing unit 200 computes, based on the PDFs 700, an RSSI signature 70 for the triangular section 107 from the confidence intervals 71-76 for each beacon B1-B6 and for each receiver R5, R7, R8 associated with the triangular section 107. The RSSI signature 70 is a barycentric signature computed for the triangular section 107 from the PDFs 717, 715, 718 of RSSIs measured by the receivers R7, R5, R8 at the corners of the triangular section 107. Such an RSSI signature 10, 20, 30, 40, 50, 60, 70 is computed for each triangular section 101-107 (see Figure 5 ).

[0033] Furthermore, the processing unit 200 is configured to generate an RSSI fingerprint 500 from the RSSI measurements of the transponder T1 to be located and to compare the RSSI fingerprint 500 with the RSSI signatures 10, 20, 30, 40, 50, 60, 70 of the triangular sections 101-107. In particular, the RSSI fingerprint 500 is generated by the processing unit 200 from the RSSIs measured by the transponder T1 for each beacon B1-B6.

[0034] According to Figure 5 , the processing unit 200 selects the triangular sections 104, 107 having an RSSI signature 40, 70 matching the RSSI fingerprint 500. The selected triangular sections 104, 107 constitute candidate areas where the transponder T1 is likely to be located.

[0035] Furthermore, the processing unit 200 determines, within each selected triangular section 104, 107, a candidate location where the transponder T1 is most likely to be located from the RSSI measurements of the transponder T1 and the respective PDF. Thus, the processing unit 200 determines one candidate location for each candidate area. Finally, the processing unit 200 is configured to determine, from these candidate locations, the location having the greatest match between the RSSI measurements of the transponder T1 and the PDF.

[0036] In the following, the method of localizing a transponder T1 within the monitoring area 100 will be described in more detail. According to Figure 1 The present exemplary embodiment is based on a setup of B = 6 beacons B1-B6 and R = 8 stationary or fixed receivers R1-R8 according to the scenario shown in Fig. 1. The RSSI fingerprint generated from the RSSI measurements of the transponder to be localized results in an array of B received RSSIs. Due to radio propagation characteristics (e.g. due to obstacles within the monitoring area), some of the RSSI measurements can be missing if the respective received radio power is below the sensitivity threshold of the respective receiver or transponder.

[0037] Naive Bayes classification is an accurate method of assigning an RSSI fingerprint to one of the locations when the PDFs of the RSSI measurements of all receivers or transmitters at known locations are available. This classification technique works by computing the joint probability of obtaining all measured RSSI values in the fingerprint of all known RSSI distributions in all known locations. Usually, the true PDF of the measurable RSSI values at all points in space cannot be obtained. An approximate solution to this problem is to measure the PDF at discrete points and use them for PDF interpolation at the desired locations, e.g. by using the so-called barycentric interpolation. Then, a global optimization method can be used to find the point with the highest probability given the generated RSSI fingerprint. Usually, this would be a computationally intensive task if the above optimization method would include all triangular segments, because interpolation would have to be performed for a large number of unknown points.

[0038] The present method for localizing a transponder within a monitoring area improves the efficiency of finding a high probability point by checking the RSSI fingerprint against the so-called barycentric signature, which can be used to discard large parts of the monitoring area before running the optimization algorithm. The main functional steps are summarized in the following and illustrated by means of Figure 6

[0039] 1. In a process called triangulation, the area bounded by the convex hull of a set of fixed receivers (hereafter called the inner area) is divided into non-overlapping triangular areas (step 601). There are various techniques for generating this triangulation (such as Delaunay triangulation). For convenience, the resulting triangulation can be modeled as a triangulation matrix T = [t i,j ] of dimension T x 3, where T is the number of resulting triangles. The elements in the i-th row of the matrix T represent the indices of the fixed receivers in the corners of triangle i. For example, the triangulation matrix in the Fehler! Verweisquelle konnte nicht gefundenwerden.(German: Error! Reference source not found.) scenario is:

[0040] ​

[0041] The position of a point within triangle i in the inner region can be represented by means of its barycentric coordinates with respect to one of the triangle's corners, i.e. to the position of one of the fixed receivers. These barycentric coordinates of an arbitrary point are henceforth referred to as (a, β). These coordinates satisfy the properties a < 1, β < 1 and a + β < 1 for all points within the triangle, such that the points (0, 0), (0, 1) and (1, 0) correspond to the fixed receivers in the corners of the triangle. This is depicted in Fehler! Verweisquelle konnte nicht gefunden werden. (German: Error! Reference source not found).

[0042] 2. Each fixed receiver r listens to the periodic pilot signals sent by the radio transmitters b and generates an updated estimate of their RSSI X b,r PDF, denoted as (Step 602). There are multiple state-of-the-art methods for performing this estimation, such as kernel density estimation or distribution fitting. From this PDF, the quantile function of a given probability p can also be computed directly.

[0043] 3. After selecting an appropriate confidence level e, the valid RSSI interval V b,i is defined for each transmitter b and triangle i (Step 603):

[0044]

[0045] This interval contains all arbitrary interpolated PDFs of RSSI that can be obtained by interpolating from the barycentric interpolation of the fixed receivers' estimated PDFs, to RSSI values that are achieved with a probability higher or equal to e (see also Figure 3 Fehler! Verweisquelle konnte nicht gefunden werden. (German: Error! Reference source not found).

[0046] 4. The valid RSSI ranges of all transmitters of a given rectangle i are collected (Step 604) and these ranges are summarized in the triangle's barycentric signature S i = <V 1,i ,..., V B,i> , because this can be used to rule out the possibility of a given RSSI value set measured within the triangle (see also Figure 4 ).

[0047] 5. A device at an unknown position measures a set of RSSI values from all transmitters and collects the RSSI fingerprint H = <hi,..., hi>b ,…,h B >, where h b The RSSI value is received from transmitter b and reported to the processing unit.

[0048] 6. The processing unit compares the RSSI fingerprint H with the centroid signatures of all triangles (step 605). If a signature mismatch occurs (step 606), where any value h of the fingerprint... b All are outside the valid RSSI range V b,i Then triangle i is discarded (step 607) and ignored in subsequent steps (see also...). Figure 5 This saves computational workload and can help increase positioning accuracy.

[0049] 7. For each of the remaining triangles (in (The set of other triangle indices), the RSSI value X measured from transmitter b at position l = (α, β). b PDF The angle t of the triangle can be calculated using its centroid coordinates. i,1 t i,2 and t i,3 X b The distribution is a combination of interpolations, i.e., barycentric interpolation. The simplest interpolation will have the following form:

[0050]

[0051] By continuously shifting the mean of the interpolation elements to obtain a smoother and finer interpolation, we get:

[0052]

[0053] Where μ b,r X is measured at receiver r. b,r The mean.

[0054] 8. At this point, it can be calculated that And a set of triangles where RSSI fingerprints may have been measured. To find the location where an RSSI fingerprint is most likely to have been measured, the naive joint probability of obtaining such a fingerprint can be maximized for each possible triangle, similar to known Naive Bayes classification. This means maximizing the product of the probability densities of each possible triangle i (step 608):

[0055]

[0056] There are various global or local optimization methods for obtaining (near-) optimal solutions to the above problem, such as gradient descent, evolutionary algorithms, particle swarm optimization, simulated annealing, etc.

[0057] 9. Finally, the most likely position is selected as the position with the highest prior probability for all remaining triangles (Step 609).

Claims

1. A method of locating at least one transponder within a surveillance area of a positioning system, wherein, The positioning system comprises a plurality of beacons and receivers located at known positions, the method comprising the steps of: - measuring, by each receiver (R1-R8), the signal strength of a periodic signal from each beacon (B1-B6) and computing, for each beacon, a probability density function of the signal strength measured by the receivers (700, 715, 717, 718); - dividing the monitored area (100) into a plurality of polygonal sections (101-107), the receivers being located at respective corners of the polygonal sections; - determining, for each beacon, a confidence interval of the signal strength within each section from the probability density function (71-76) and computing a signal strength signature of each section from the confidence intervals (10, 20, 30, 40, 50, 60, 70); - performing, by a transponder (T1) to be located, measurements of the signal strength of the periodic signal from each beacon; - generating a signal strength fingerprint (500) from the transponder's measurements and selecting sections (104, 107) having signal strength signatures (40, 70) matching the signal strength fingerprint; - determining, within each selected section, a candidate position where the transponder is most likely located from the transponder's measurements and the respective probability density function; and - determining, from the candidate positions, a position having the greatest match between the transponder's measurements and the probability density function.

2. The method of claim 1, wherein, The polygonal sections (101-107) are triangles, and wherein the signal strength signatures (10, 20, 30, 40, 50, 60, 70) are barycentric signatures computed for each triangle from the probability density functions of the signal strength measured at each corner of the triangle by the respective receivers (R1-R8).

3. The method of claim 2, wherein, The probability density functions (700, 715, 717, 718) are computed and updated for each receiver (R1-R8) and each beacon (B1-B6), wherein the probability density functions associated with the receivers' positions are interpolated within the triangles using barycentric interpolation.

4. The method of any one of claims 1 to 3, wherein, The receivers (R1-R8) periodically update their respective measurements of the signal strength of the periodic signal from the beacons (B1-B6).

5. The method of any one of claims 1 to 4, wherein, The receivers comprise stationary receivers and / or mobile devices having known positions and / or moving along known trajectories.

6. The method of any one of claims 1 to 5, wherein, The probability density functions (700, 715, 717, 718) are computed and updated by a processing unit (200).

7. The method of claim 6, wherein, The confidence intervals of the signal strength within each section (101-107) are determined by the processing unit (200) considering each receiver (R1-R8) in the respective section, and wherein the confidence intervals comprise a minimum and a maximum of the signal strength over all receivers in the respective section.

8. The method of any one of claims 6 or 7, wherein, The monitored area (100) is divided by the processing unit (200) into a plurality of polygonal sections (101-107) based on the receivers' (R1-R8) positions.

9. The method of any one of claims 6-8, wherein, generating, by the processing unit (200), the signal strength fingerprint (500) from the signal strengths measured by the transponder (T1) for each beacon (B1-B6).

10. The method of any one of claims 6-9, wherein, selecting, by the processing unit (200), the segments (104, 107) having a signal strength signature (40, 70) matching the signal strength fingerprint (500), wherein the selected segments are candidate areas where the transponder (T1) is likely to be located, and wherein the candidate locations are determined by the processing unit from the transponder's measurements for each beacon and the corresponding probability density functions.

11. The method of claim 10, wherein, The processing unit determines one candidate location for each candidate area.

12. A processing unit for a positioning system, the processing unit being configured for: - computing, for each beacon (B1-B6) of the positioning system transmitting a periodic signal, a probability density function (700, 715, 717, 718) of the signal strength of the periodic signal measured by each receiver (R1-R8) of the positioning system located at a known location; - dividing a monitoring area (100) into a plurality of polygonal segments (101-107), the receivers being located at respective corners of the polygonal segments; - determining, for each beacon, a confidence interval (71-76) of the signal strength within each segment from the probability density function, and computing a signal strength signature (10, 20, 30, 40, 50, 60, 70) of each segment from the confidence interval; - generating a signal strength fingerprint (500) from measurements of the signal strength of the periodic signal from each beacon performed by a transponder to be located, and selecting the segments (104, 107) having a signal strength signature (40, 70) matching the signal strength fingerprint; - determining, within each selected segment, a candidate location where the transponder is most likely to be located from the transponder's measurements and the corresponding probability density function; and - determining, from the candidate locations, a location having a maximum match between the transponder's measurements and the probability density function.

13. A positioning system for locating at least one transponder within a monitored area, the positioning system comprising a plurality of beacons (B1-B6) located at known positions and receivers (R1-R8), wherein, The beacons are configured to transmit a periodic signal, wherein the receivers are configured to measure the signal strength of the periodic signal from each beacon, and wherein the transponder (T1) is configured to perform measurements of the signal strength of the periodic signal from each beacon, the positioning system further comprising the processing unit according to claim 12.

14. A mobile object comprising the processing unit according to claim 12 and a transponder configured to perform measurements of the signal strength of a periodic signal from each beacon within a positioning system.

15. A computer program product comprising computer-executable instructions for performing the method according to claim 13 when the program is executed on a computer.

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