Positioning system

Abstract

To provide a positioning system that is able to measure a target position highly accurately while restricting the amount of calculation.SOLUTION: A positioning system according to the present embodiment includes: a plurality of RDFs 10-1 to 10-6 each of which estimates a direction of arrival (DOA) of a radio signal transmitted from an aircraft 40 that is a target of three-dimensional positioning; and a positioning unit 20 that positions a position of the aircraft 40, based on the DOA estimated by each of the plurality of RDFs and a position of each of the plurality of RDFs. The positioning unit 20 has a factor graph structure formed using a plurality of nodes.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a positioning system for determining the position of a target. [Background technology]

[0002] In the current air traffic control system, to ensure safe and smooth flight operations, communication is carried out between ground air traffic controllers and aircraft pilots, and pilots navigate according to the controller's instructions. Communication between the controller and pilots is carried out using analog voice AM radio, and the controller gives instructions through verbal communication.

[0003] There are multiple aircraft flying in the airspace that an air traffic controller is responsible for, and it is stipulated that these aircraft use the same frequency. However, in the current air traffic control system using voice radio communication, the controller cannot know which pilot he is communicating with, except for communication such as confirming call signs. This system restriction has led to incidents due to controller confusion and pilot recognition errors.

[0004] In order to alleviate this problem, in Europe, as shown in FIG. 11, multiple RDFs (Radio Direction Finders) 110 are used to measure the direction of arrival (hereinafter, DOA: Direction Of Arrival) φ of a voice radio signal transmitted from an aircraft 140. m (m is the RDF number). In the RDF system in FIG. 11, six RDFs, RDF110-1 to 110-6, are installed. The DOA φ estimated by each RDF110 is m is transferred to the positioning unit 120. The positioning unit 120 estimates that the aircraft 140 is located at the intersection of the direction of arrival lines. The positioning result by the positioning unit 120 is transferred to the radar console 130 and marked on the radar console 130. Since the position of the aircraft itself is known by another radar system, the marking by this RDF system allows the controller to identify the aircraft currently in communication, thereby alleviating the above problem. [Prior art documents] [Non-patent literature]

[0005] [Non-Patent Document 1] Meng Cheng,et al.,“A DOA-Based Factor Graph Technique for 3D Multi-Target Geolocation”,IEEE Access,vol7(2019) Summary of the Invention [Problem to be solved by the invention]

[0006] In the positioning unit 120 of the above RDF system, the DOA φ m The aircraft position (x, y) is calculated based on the above. As this positioning method, for example, the LSE (Least Square Estimation) method or the GN (Gauss-Newton) method can be applied.

[0007] LSE performs calculations based on a standard that minimizes the squared positioning error, but since the amount of calculations is proportional to the cube of the number of RDFs, the amount of calculations increases exponentially as the number of RDFs increases. The display on the radar console 130 must be displayed within several hundred milliseconds after receiving the communication radio waves, and an increase in the amount of calculations leads to a delay in the radar console display. In addition, the positioning accuracy of LSE is lower than that of GN, making it difficult to apply it to aircraft positioning systems that require high positioning accuracy.

[0008] On the other hand, GN can obtain good positioning accuracy by providing appropriate initial values ​​and performing iterative calculations. However, if the initial values ​​are not provided appropriately, it may converge to an incorrect positioning result. In addition, GN also has the problem of a large amount of calculations, just like LSE.

[0009] The present invention has been made in consideration of the above-mentioned conventional circumstances, and has an object to provide a positioning system capable of measuring a target position with high accuracy while suppressing the amount of calculation. [Means for solving the problem]

[0010] In order to achieve the above object, in the present invention, a positioning system is configured as follows. That is, the positioning system of the present invention comprises a plurality of sensors that each estimates the direction of arrival of a radio signal emitted from a target, and a positioning unit that locates the position of the target based on the direction of arrival estimated by each of the plurality of sensors and the respective positions of the plurality of sensors, and is characterized in that the positioning unit has a factor graph structure formed using a plurality of nodes.

[0011] Here, the positioning unit may be configured to have a sub-factor graph node of a factor graph structure provided for each sensor, which calculates the position of the target based on the direction of arrival estimated by the sensor and the position of the sensor, and a target position node which locates the position of the target based on the calculation result by each sub-factor graph node.

[0012] In addition, each sensor may estimate the azimuth angle and elevation angle of the target as seen from the sensor as the direction of arrival, and each sub-factor graph node of the positioning unit may calculate the average and variance of the azimuth angle and elevation angle estimated by the sensor corresponding to the sub-factor graph node, calculate the relative position of the target as seen from the sensor based on the approximate average and approximate variance of a trigonometric function obtained using these averages and variances, and convert the relative position into an absolute position and output the result of the calculation to the target position node.

[0013] Also, each sub-factor graph node of the positioning unit may be configured to feed back the positioning result by the target position node to the calculation of the relative position of the target. Effect of the Invention

[0014] According to the present invention, it is possible to provide a positioning system capable of measuring a target position with high accuracy while suppressing the amount of calculation. [Brief description of the drawings]

[0015] [Figure 1] 1 is a diagram illustrating an example of the configuration of a positioning system according to an embodiment of the present invention. [Diagram 2] 2 is a diagram illustrating an example of the configuration of a positioning unit in the positioning system of FIG. 1; [Diagram 3] 3 is a diagram illustrating an example of the configuration of a sub-FG in the positioning unit of FIG. 2. [Figure 4] FIG. 13 is a diagram showing message updates at trigonometric function node F. [Diagram 5] FIG. 13 is a diagram showing message update (two variables) for FA →Δx. [Figure 6] FIG. 13 is a diagram showing message update (three variables) for FB →Δx. [Figure 7] FIG. 1 is a diagram showing the mean and variance of the product of two probability density functions. [Figure 8] FIG. 13 is a diagram showing the update of Δx→RA. [Figure 9] FIG. 1 illustrates a product of multiple random variables. [Figure 10] 13 is a diagram showing the message update of x→RA,m. [Figure 11] FIG. 1 is a diagram illustrating an example of the configuration of an RDF system according to a conventional example. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0016] A positioning system according to an embodiment of the present invention will be described with reference to the drawings. Fig. 1 is a diagram showing an example of the configuration of a positioning system according to an embodiment of the present invention. The positioning system in Fig. 1 uses six RDFs 10-1 to 10-6, which are examples of sensors according to the present invention, to measure the three-dimensional position of an aircraft 40, which is an example of a target according to the present invention.

[0017] Before describing the positioning system according to the present invention, the definition of the geometry shown in Fig. 1 will be described. The position of the aircraft 40 is (x, y, z), and the position of the m-th RDF 10 is (x m ,Y m ,Z m ) and the aircraft position (x,y,z) and the RDF position (X m ,Y m ,Z m ) relative distance to (Δx m ,Δy m ,Δz m ), the relationship between these is (Equation 1). (Δx m ,Δy m ,Δz m ) is also the position of the aircraft 40 relative to the RDF.

[0018]

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[0019] Each of the RDFs 10-1 to 10-6 receives a communication radio wave from the aircraft 40 and estimates the direction of arrival (DOA) of the radio wave by using a direction of arrival estimation algorithm such as MUSIC (Multiple Signal Classification). m and the elevation angle θ m The subscript m indicates a number that identifies each RDF10, and m=1 to M. In the example in Figure 1, M=6.

[0020] In the above explanation, an example of estimating the DOA from a received wireless signal has been described, but the DOA may be estimated using a signal other than a wireless signal. For example, a method of estimating the azimuth angle and elevation angle from an image captured by a camera may be used. The present invention estimates the angle (φ m ,θ m ) and is independent of the angle estimation technique.

[0021] Relative position (Δx m ,Δy m ,Δz m ) and DOA(φ m ,θ m ) The relationship is expressed as (Equation 2).

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[0022] DOA estimation results obtained with RDF10-1 to 10-6 (φ m ,θ m ) are input to the positioning unit 20. An example of the configuration of the positioning unit 20 is shown in FIG. 2. The positioning unit 20 includes a plurality of sub-FGs 21-1 to 21-M corresponding to the respective RDFs 10, and a target position node P22. The positioning unit 20 has a graph structure called a factor graph (hereinafter, "FG"), and further, the sub-FGs 21 to 21-M also have an FG structure. The FG is formed using a plurality of nodes that propagate messages.

[0023] An example of the configuration of sub-FGs 21-1 to 21-M is shown in Figure 3. In the FG, messages are propagated in the direction of the arrow. The propagated message is expressed using the mean and variance, and its probability density is assumed to be a Gaussian distribution. Each node in the FG propagates the message (mean and variance) while updating it. As explained below, by repeating message propagation in the downward direction (hereinafter "Forward") and upward direction (hereinafter "Backward") of Figures 2 and 3, it is possible to achieve high positioning accuracy with a small amount of calculation.

[0024] Sub-FG21-1 to 21-M show the angles (φ m ,θ m ) is input as a message. The message from each of the sub FGs 21-1 to 21-M is propagated to the target position node P 22 (Forward). The message from the target position node P 22 is not only output to the radar console 30 as a positioning result, but is also propagated to each of the sub FGs 21-1 to 21-M (Backward).

[0025] First, we will explain the message propagation in the forward direction of FG21-1 to 21-M. m ,θ m ) is input to observation node D(1D,2D). Observation node D(1D,2D) is connected to the observation node D(1D,2D) with the azimuth angle φ m The average μ φm and variance σ φm 2 , elevation angle θ m The average μ θm and variance σ θm 2 The average and variance may be calculated at the observation node D(1D,2D) as in this example, or may be calculated in advance in RDF.

[0026] Azimuth angle φ obtained at observation node D (1D, 2D) m Mean·variance (μ φm ,σ φm 2 ) and elevation angle θ m Mean·variance (μ θm ,σ θm 2 ) is input to the DOA variable node N(1N,2N). The DOA variable node N(1N,2N) uses the azimuth angle φ m Mean·variance (μ φm ,σ φm 2 ) and elevation angle θ m Mean·variance (μ θm ,σ θm 2 ) to the trigonometric function node F(1F, 2F, 3F).

[0027] Trigonometric function node F(1F,2F,3F) updates the message according to (Equation 2) and propagates it to the subsequent nodes. Here, linearity must be ensured for message propagation in FG. However, (Equation 2) uses trigonometric functions and is not linear. Therefore, before explaining message updates in trigonometric function node F(1F,2F,3F), we will explain linear approximation of trigonometric functions using Taylor expansion.

[0028] The first-order Taylor expansion is as follows (Equation 3), where α = μ α It is possible to linearly approximate it with

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[0029] Therefore, the average μ f(α) and variance σ f(α) 2 The linear approximation of is given by (Equation 4).

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[0030] If we use this (Equation 4) to show a specific example of tan(α) used in the first equation of (Equation 2), the linear approximation using Taylor expansion becomes (Equation 5).

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[0031] Table 1 shows the mean μ and variance σ, which are linear approximations of the trigonometric functions described in Equation 2. 2 Summarize. [Table 1]

[0032] The linear approximation described above provides the conditions for message updating in FG. In the configuration shown in Figure 3, the message update in trigonometric function node F (1F, 2F, 3F) is performed by (φ m,θ m ) and (Δx m ,Δy m ,Δz m ) and can be classified into two cases, two variables (a,c) and three variables (a,b,c), as shown in Fig. 4. In Fig. 4, (a) and (b) correspond to the DOA node N, (c) and (Δ) correspond to the relative position variable node Δ, and [F] in the center corresponds to the trigonometric function node F. The direction of the arrow indicates the direction in which the message is propagated.

[0033] For the product of two (a,c) or three (a,b,c) independent random variables shown in Figure 4, the mean and variance of the result are summarized in (Table 2). Although details are omitted, (Table 2) can be derived from the definition of the mean and variance. [Table 2]

[0034] The message updates in these two-variable and three-variable trigonometric function nodes F (1F, 2F, 3F) are explained below using examples. First, the two-variable trigonometric function node F A As an example of message updating in (1F), A →Δx will be explained using Figure 5. As mentioned above, in FG message propagation, the mean and variance are propagated. Here, F A →The propagation of Δx is Δx shown in (Equation 2) m =Δy m tan(φ m ) (first equation). Here, the average μ tan(φ) and variance σ tan(φ) 2 According to (Table 1), the approximate average tan(μ φ ) and approximate variance σ φ 2 sec 4 (μ φ ) can be converted into the mean and variance corresponding to the c nodes. Δy→FA , σ Δy→FA 2Applying these values ​​to the two variables in Table 2 gives us Equation 6.

[0035]

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[0036] Similarly, as an example of message update in a three-variable trigonometric function node F(2F,3F), F B →Δx will be explained using Figure 6. In the case of three variables, the mean and variance are propagated in the same way as in the case of two variables. Here, F B →The propagation of Δx is Δx shown in (Equation 2) m = Δz m ·sin(φ m ) tan(θ m ) (the second equation). The average μ sin(φ) and variance σ sin(φ) 2 According to (Table 1), the approximate mean sin(μ φ ) and approximate variance σ φ 2 cos 2 (μ φ ) and the average μ tan(φ) and variance σ tan(φ) 2 is the approximate average tan(μ φ ) and approximate variance σ φ 2 sec 4 (μ φ ) can be converted into the mean and variance corresponding to the c nodes. ΔZ→FB , σ ΔZ→FB 2 Applying these values ​​to the three variables in Table 2 gives us Equation 7.

[0037]

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[0038] Below is the message update formula for all nodes in F → Δ. (Equation 8) is F A →Δx is the message update formula. (Equation 9) is F A →Δy is the message update formula. (Equation 10) is F B →Δx is the message update formula. (Equation 11) is F B →Δz is the message update formula. (Equation 12) is F C →Δy is the message update formula. (Equation 13) is F C →This is the message update equation for Δz. Note that (Equation 8) and (Equation 10) are the same as (Equation 6) and (Equation 7) described above, and are merely repeated here for convenience of explanation.

[0039]

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[0040]

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[0041]

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[0042]

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[0043]

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[0044]

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[0045] From the above explanation, the message update (μ F→Δ ,σF→Δ 2 ) was shown.

[0046] Next, the message update (μ Δ→R ,σ Δ→R 2 ) will be explained. The message update at the relative position function node R (1R, 2R, 3R) corresponds to Figure 7. Figure 7 shows the average and variance by the product of two probability density functions. In the message update at the relative position function node R (1R, 2R, 3R), the update is performed according to (Table 3).

[0047] [Table 3]

[0048] As a concrete example, Δx→R A The update of is explained with reference to Fig. 8. According to the update formula in (Table 3), (Formula 14) is obtained.

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[0049] Below is the update formula for all nodes from the relative position variable node Δ(1d, 2d, 3d) to the relative position function node R(1R, 2R, 3R). (Formula 15) is Δx→R A is the message update formula. (Equation 16) is Δy→R B is the message update formula. (Equation 17) is Δz→R C This is the message update formula. Note that (Formula 15) is the same as (Formula 14) described above, and is merely repeated here for convenience of explanation.

[0050]

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[0051]

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[0052]

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[0053] In message propagation between the relative position function node R (1R, 2R, 3R) in each of the sub-FGs 21-1 to 21-M and the target position node P22, the relative position (Δx m ,Δy m ,Δz m ) to the absolute position (x, y, z) of the target (aircraft 40), forward message propagation to combine messages from other sub-FGs, and backward message propagation from the absolute position (x, y, z) to the relative position function node R (1R, 2R, 3R) in each sub-FG.

[0054] Relative position (Δx m ,Δy m ,Δz m ) to the target's absolute position (x,y,z), the average μ R→P becomes (Equation 18), and since the variance does not depend on the position, it is kept as it is and becomes (Equation 19).

[0055]

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[0056]

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[0057] Next, the messages (μ R→P ,σ R→P 2 ) at the target location node P22, and the message (μ P ,σ P 2In (Equation 14), the message is propagated by the product of two probability density functions, but in the target location node P22, the message is propagated by M variables as shown in FIG. 9, and this is shown in (Equation 20).

[0058]

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[0059] By combining the messages calculated by each of the sub-FGs 21-1 to 21-M using (Equation 20), the positioning accuracy can be improved. A,i The update formula for absolute position x (R A →x) is shown in (Equation 21).

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[0060] Similarly, the R of each sub-FG21-1 to 21-M B,i The update formula for absolute position y (R B →y) is shown in (Equation 22), and R C,i The update formula for absolute position z (R C →z) is shown in (Equation 23).

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[0061]

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[0062] The above-described process is the message propagation in the forward direction of FG. X ,μ Y ,μ Z ) is the estimated target position, and (σ X 2 ,σ Y 2 ,σ Z 2These values ​​are the variances of the positioning results by the positioning unit 20, which are transferred to the radar console 30 and marked on the radar console 30.

[0063] Next, the backward message propagation in the FG will be described. The absolute position of the target calculated at the target position node P22 (μ P ,σ P 2 ) in the backward direction. The relative position function node R of the m-th sub-FG21-m from the target position node P22 m Message to (μ P→Rm ,σ P→Rm 2 ) For example, x → R A,m The message update is shown in (Equation 24).

[0064]

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[0065] This is the relative position function node R of the m-th sub-FG21-m from the target position node P22. A,m This is a backward update to the node. In (Equation 24), both the mean and variance are calculated from other sub-FGs excluding the node itself (the mth). A,m The backward update of is shown.

[0066] Similarly, y → R B,m The message update of z → R is shown in (Eq. 25). C,m The message update is shown in (Equation 26).

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[0067]

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[0068] Next, the backward update (μ R→Δ ,σ R→Δ 2 From (Equation 1), the average μ R→Δ is expressed as (Equation 27), and the variance σ R→Δ 2 Since it does not depend on the position, it is kept unchanged and becomes (Equation 28).

[0069]

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[0070]

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[0071] The above-described process is the backward message propagation of FG. This completes one iterative calculation process.

[0072] The second and subsequent iterative calculations of FG will now be described. As mentioned above, in the first iterative calculation, the trigonometric function node F (1F, 2F, 3F) is updated by messages from the DOA variable node N (1N, 2N) and the relative position variable node Δ (1d, 2d, 3d). The message of the DOA variable node N (1N, 2N) does not change due to the iterative calculation, but the message of the relative position variable node Δ (1d, 2d, 3d) is updated by processing in the backward direction. Therefore, by updating the trigonometric function node F (1F, 2F, 3F) again, it is possible to realize message updates with higher accuracy than the first time. In this way, by repeating the iteration, the positioning accuracy is improved, and high positioning performance can be realized.

[0073] As described above, the positioning system of this example comprises a plurality of RDFs 10-1 to 10-6 (sensors according to the present invention) that each estimate the direction of arrival (DOA) of a radio signal transmitted from aircraft 40, which is the target of three-dimensional positioning, and a positioning unit 20 (positioning unit according to the present invention) that locates the position of aircraft 40 based on the DOA estimated in each of the plurality of RDFs and the respective positions of the plurality of RDFs, and the positioning unit 20 has a factor graph structure formed using a plurality of nodes.

[0074] More specifically, the positioning unit 20 has sub-FGs 21-1 to 21-M (sub-factor graph node units according to the present invention) having a factor graph structure, which are provided for each RDF and calculate the position of the aircraft 40 based on the DOA estimated in that RDF and the position of the sensor, and a target position node P22 (target position node according to the present invention) which locates the position of the target based on the calculation results by each of the sub-FGs 21-1 to 21-M.

[0075] Here, the RDFs 10-1 to 10-6 estimate the azimuth angle and elevation angle of the aircraft 40 as seen from themselves as the DOA, and the sub-FGs 21-1 to 21-M of the positioning unit 40 calculate the average and variance of the azimuth angle and elevation angle estimated in the RDF corresponding to themselves, calculate the relative position of the aircraft 40 as seen from the RDF based on the approximate average and approximate variance of a trigonometric function obtained using these averages and variances, and output the relative position converted into an absolute position as the calculation result to the target position node P 22. In addition, the sub-FGs 21-1 to 21-M of the positioning unit 40 feed back the positioning result by the target position node P 22 to the calculation of the relative position of the aircraft 40.

[0076] With this configuration, it becomes possible to measure with high accuracy the three-dimensional position of the aircraft 40. Furthermore, in the factor graph, even if the number of RDFs increases, only the number of nodes increases, and the amount of calculation is only proportional to the number of RDFs, so the effect of reducing the amount of calculation is high compared to methods such as LSE, in which the amount of calculation is proportional to the cube of the number of RDFs.

[0077] In the above description, the three-dimensional positioning is performed with one aircraft as the target, but the present invention can also be extended to the three-dimensional positioning of multiple targets. In this case, a problem arises as to which FG to input the multiple DOAs observed by each sensor. As one solution to this problem, a method described in an unpublished patent application by the same applicant as the present application (Patent Application No. 2020-089292, filing date: May 22, 2020) can be used. Therefore, it is clear that the present invention can also be used in multi-target positioning.

[0078] In the above description, direction estimation is performed based on a radio signal, and target positioning is performed based on the result. However, in the present invention, direction estimation may be performed based on something other than a radio signal. For example, a target is photographed by a plurality of optical cameras, and the direction of the target is estimated using image processing techniques such as pattern matching and deep learning for each camera image, and the positioning method of the present invention is applied to the estimated direction to estimate the position. In addition, for example, the surroundings of the vehicle body are photographed by a plurality of cameras attached to the vehicle body, and the direction of surrounding vehicles and obstacles is estimated from the images of each camera, and the positioning method of the present invention is applied to the estimated direction to estimate the position. Furthermore, in addition to optical cameras, a radar that estimates the direction of a target by rotating a sharp directional antenna mechanically or electronically may be used. In this way, the problem can be solved by a method similar to that described above, even by direction estimation using pattern recognition.

[0079] Although the present invention has been described in detail above, it goes without saying that the present invention is not limited to the above-mentioned configuration, and may be realized with configurations other than those described above. Furthermore, the present invention can also be provided as, for example, a method including technical procedures related to the above-mentioned processing, a program for causing a processor to execute the above-mentioned processing, or a storage medium for storing such a program in a computer-readable manner.

[0080] The scope of the present invention is not limited to the exemplary embodiments shown and described, but includes all embodiments that achieve equivalent effects to those intended by the present invention. Moreover, the scope of the present invention may be defined by any desired combination of specific features among all the respective features disclosed. [Industrial Applicability]

[0081] The present invention can be used in a positioning system for measuring the position of a target. [Explanation of symbols]

[0082] 10-1 to 10-6, 110-1 to 110-6: RDF, 20, 120: Positioning unit, 30, 130: Radar console, 40, 140: Aircraft, 21-1 to 21-M: Sub-FG, 22: Target position node P, 1D, 2D: Observation node D, 1N, 2N: DOA variable node N, 1F, 2F, 3F: Trigonometric function node F, 1d, 2d, 3d: Relative position variable node Δ, 1R, 2R, 3R: Relative position function node R

Claims

1. A plurality of sensors each estimating a direction of arrival of a wireless signal transmitted from a target; a positioning unit that measures a position of the target based on the arrival direction estimated by each of the plurality of sensors and the respective positions of the plurality of sensors; the positioning unit has a sub-factor graph node of a factor graph structure provided for each of the sensors, the sub-factor graph node calculating a position of the target based on a direction of arrival estimated by the sensor and a position of the sensor, and a target position node calculating a position of the target based on a calculation result by each of the sub-factor graph nodes; Each sensor estimates the azimuth angle φ and elevation angle θ of the target as seen by the sensor as a direction of arrival; Each sub-factor graph node of the positioning unit is determined by calculating the average μ φ and variance σ φ 2 and the average μ of the elevation angle θ θ and variance σ θ 2 Based on this, a trigonometric function node F Ａ Node, F B Node, F C A message is propagated between the node and relative position variable nodes, that is, a Δx node, a Δy node, and a Δz node, to calculate a relative position of the target as seen from the sensor, and the relative position is converted into an absolute position, which is output to the target position node as a calculation result; F A The nodes are the average μ φ and variance σ φ 2 and the message μ propagated from node Δy Δy→FA and σ Δy→FA 2 and a message based on μ FA→Δx and σ FA→Δx 2 is propagated to the Δx node, and the average μ φ and variance σ φ 2 and the message μ propagated from the Δx node Δx→FA and σ Δx→FA 2 and a message based on μ FA→Δy and σ FA→Δy 2 to a Δy node; F B The nodes are the average μ φ and variance σ φ 2 and the average μ of the elevation angle θ θ and variance σ θ 2 and the message μ propagated from the Δz node Δz→FB and σ Δz→FB 2 and a message based on μ FB→Δx and σ FB→Δx 2 is propagated to the Δx node, and the average μ φ and variance σ φ 2 and the average μ of the elevation angle θ θ and variance σ θ 2 and the message μ propagated from the Δx node Δx→FB and σ Δx→FB 2 Message μFB→Δz and σ based on FB→Δz 2 to the Δz node, F C The nodes are the average μ φ and variance σ φ 2 and the average μ of the elevation angle θ θ and variance σ θ 2 and the message μ propagated from the Δz node Δz→FC and σ Δz→FC 2 and a message based on μ FC→Δy and σ FC→Δy 2 is propagated to the Δy node, and the average μ φ and variance σ φ 2 and the average μ of the elevation angle θ θ and variance σ θ 2 and the message μ propagated from node Δy Δy→FC and σ Δy→FC 2 and a message based on μ FC→Δz and σ FC→Δz 2 A positioning system comprising a three-variable trigonometric function node configured to propagate a Δz node to a Δz node.

2. 2. The positioning system according to claim 1, F A Node, F B Node, F C Each message propagated from a node to a Δx node, a Δy node, and a Δz node is expressed as follows: [0010] [0025] [0030] [0045] [0050] [006] A positioning system characterized by being represented by:

3. 3. The positioning system according to claim 1, A positioning system, characterized in that each sub-factor graph node of the positioning unit feeds back a positioning result by the target position node to a calculation of a relative position of the target.

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