Patent Publication Number: US-2023160994-A1

Title: Positioning system, positioning method, and computer readable medium

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a Continuation of PCT International Application No. PCT/JP2020/034582, filed on Sep. 11, 2020, which is hereby expressly incorporated by reference into the present application. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a positioning system, a positioning method, and a positioning program. 
     BACKGROUND ART 
     In areas such as offices, factories, and commercial facilities, position information of people and things is attracting attention. In order to acquire position information, various positioning techniques are being considered 
     Non-Patent Literature 1 proposes a positioning technique that combines angle information and radio wave intensity. 
     CITATION LIST 
     Non-Patent Literature 
     Non-Patent Literature 1: S. Tomic, M. Beko, R. Dinis and L. Bernardo, “On Target Localization Using Combined RSS and AoA Measurements,” Sensors, Vol. 18, Issue.4, pp.1-25, 2018 
     SUMMARY OF INVENTION 
     Technical Problem 
     In the positioning technique described in Non-Patent Literature 1, positioning accuracy is improved by a weighted least-squares method using weights based on radio wave intensity. Therefore, a problem is that positioning accuracy cannot be improved in an environment where radio wave intensity cannot be obtained or radio wave intensity fluctuates greatly. 
     An object of the present disclosure is to realize highly accurate positioning using only angle information. 
     Solution to Problem 
     A positioning system according to the present disclosure performs positioning of a communication device that wirelessly communicates with each base station of a plurality of base stations, and the positioning system includes
         a relative angle acquisition unit to acquire a relative angle between each base station of the plurality of base stations and the communication device from each base station of the plurality of base stations;   a provisional position calculation unit to calculate a position of the communication device as a provisional position, using the relative angle between each base station of the plurality of base stations and the communication device and a position of each base station of the plurality of base stations;   a weight calculation unit to calculate a distance between each base station of the plurality of base stations and the communication device, using the position of each base station of the plurality of base stations and the provisional position, and calculate a weighting coefficient for correcting the provisional position for each base station of the plurality of base stations, based on the distance between each base station of the plurality of base stations and the communication device; and   a device position calculation unit to calculate the position of the communication device as a device position, using the relative angle between each base station of the plurality of base stations and the communication device, the position of each base station of the plurality of base stations, and the weighting coefficient for each base station of the plurality of base stations.       

     Advantageous Effects of Invention 
     A positioning system according to the present disclosure calculates the position of a communication device as a device position, using a relative angle between each base station and the communication device and a weighting coefficient obtained based on the distance between each base station and the communication device. Therefore, the positioning system according to the present disclosure can realize highly accurate positioning using only angle information. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG.  1    is a diagram illustrating a communication device that is a positioning target and base stations according to Embodiment 1; 
         FIG.  2    is an example of a configuration of a positioning system according to 
       Embodiment 1; 
         FIG.  3    is a flowchart illustrating operation of the positioning system according to Embodiment 1; 
         FIG.  4    is a diagram illustrating an example of an angular relationship between a base station and the communication device according to Embodiment 1; 
         FIG.  5    is a flowchart illustrating a positioning process using three-dimensional positioning by the positioning system according to Embodiment 1; 
         FIG.  6    is a diagram illustrating an example of base station information according to Embodiment 1; 
         FIG.  7    is a diagram depicting a relative angle containing an error according to Embodiment 1; 
         FIG.  8    is a flowchart illustrating a positioning process using two-dimensional positioning by the positioning system according to Variation 1 of Embodiment 1; 
         FIG.  9    is a diagram illustrating a communication device that is a positioning target and base stations according to Variation 2 of Embodiment 1; and 
         FIG.  10    is an example of a configuration of the positioning system according to Variation 3 of Embodiment 1. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     An embodiment of the present disclosure will be described hereinafter with reference to the drawings. Throughout the drawings, the same or corresponding parts are denoted by the same reference signs. In the description of the embodiment, description of the same or corresponding parts will be suitably omitted or simplified. In the drawings hereinafter, the relative sizes of components may be different from actual ones. In the description of the embodiment, directions or positions such as “up”, “down”, “left”, “right”, “front”, “rear”, “top side”, and “back side” may be indicated. 
     These terms are used only for convenience of description, and are not intended to limit the placement and orientation of components such as devices, equipment, or parts. 
     Embodiment 1 
     *** Description of Configurations *** 
       FIG.  1    is a diagram illustrating a communication device  10  that is a positioning target and base stations  20  according to this embodiment. In this embodiment, a positioning system  100  that performs positioning of the communication device  10  that wirelessly communicates with each base station  20  of a plurality of base stations will be described. The communication device  10  is the positioning target and is referred to also as a positioning target terminal. It is assumed that there are M communication devices  10 . It is assumed that there are N base stations. M and N are natural numbers.  FIG.  1    illustrates a case where there are one communication device  10  and three base stations  20 . 
     The communication device  10  wirelessly communicates with each base station  20  of the plurality of base stations. Specifically, the communication device  10  is a terminal that communicates with the base station  20  by wireless communication, such as a smartphone terminal, a tablet terminal, or a smartwatch. 
     The base station  20  can calculate a relative angle  31  with the communication device  10  or another base station. In  FIG.  1   , a base station  1  calculates a relative angle  1  with the communication device  10 , a base station  2  calculates a relative angle  2  with the communication device  10 , and a base station  3  calculates a relative angle  3  with the communication device  10 . 
     Referring to  FIG.  2   , an example of a configuration of the positioning system  100  according to this embodiment will be described. 
     The positioning system  100  is a computer. The positioning system  100  includes a processor  910 , and also includes other hardware such as a memory  921 , an auxiliary storage device  922 , an input interface  930 , an output interface  940 , and communication equipment  950 . The processor  910  is connected with other hardware components via signal lines and controls these other hardware components. 
     The positioning system  100  may be installed in the communication device  10  or may be installed in the base station  20 . Alternatively, the communication device  10  may be installed in a computer other than the base station  20 . 
     The positioning system  100  includes, as functional elements, a relative angle acquisition unit  110 , a provisional position calculation unit  120 , a weight calculation unit  130 , a device position calculation unit  140 , and a storage unit  150 . In the storage unit  150 , base station information  151  is stored. 
     The functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  are realized by software. The storage unit  150  is provided in the memory  921 . The storage unit  150  may be provided in the auxiliary storage device  922 , or may be divided and provided in the memory  921  and the auxiliary storage device  922 . 
     The processor  910  is a device that executes a positioning program. The positioning program is a program that realizes the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140 . 
     The processor  910  is an integrated circuit (IC) that performs operational processing. Specific examples of the processor  910  is a central processing unit (CPU), a digital signal processor (DSP), and a graphics processing unit (GPU). 
     The memory  921  is a storage device to temporarily store data. Specific examples of the memory  921  are a static random access memory (SRAM) and a dynamic random access memory (DRAM). 
     The auxiliary storage device  922  is a storage device to store data. A specific example of the auxiliary storage device  922  is an HDD. Alternatively, the auxiliary storage device  922  may be a portable storage medium, such as an SD (registered trademark) memory card, CF, a NAND flash, a flexible disk, an optical disc, a compact disc, a Blu-ray (registered trademark), or a DVD. HDD is an abbreviation for Hard Disk Drive. SD (registered trademark) is an abbreviation for Secure Digital. CF is an abbreviation for CompactFlash (registered trademark). DVD is an abbreviation for Digital Versatile Disk. 
     The input interface  930  is a port to be connected with an input device such as a mouse, a keyboard, or a touch panel. Specifically, the input interface  930  is a Universal Serial Bus (USB) terminal. The input interface  930  may be a port to be connected with a local area network (LAN). 
     The output interface  940  is a port to which a cable of an output device such a display is to be connected. Specifically, the output interface  940  is a USB terminal or a High Definition Multimedia Interface (HDMI) (registered trademark) terminal. Specifically, the display is a liquid crystal display (LCD). The output interface  940  is referred to also as a display interface. 
     The communication equipment  950  has a receiver and a transmitter. The communication equipment  950  is connected to a communication network such as a LAN, the Internet, or a telephone line. Specifically, the communication equipment  950  is a communication chip or a network interface card (NIC). 
     The positioning program is executed in the positioning system  100 . The positioning program is read into the processor  910  and executed by the processor  910 . The memory  921  stores not only the positioning program but also an operating system (OS). The processor  910  executes the positioning program while executing the OS. The positioning program and the OS may be stored in the auxiliary storage device  922 . The positioning program and the OS that are stored in the auxiliary storage device  922  are loaded into the memory  921  and executed by the processor  910 . Part or the entirety of the positioning program may be embedded in the OS. 
     The positioning system  100  may include a plurality of processors as an alternative to the processor  910 . These processors share execution of the positioning program. Each of these processors is, like the processor  910 , a device that executes the positioning program. 
     Data, information, signal values, and variable values that are used, processed, or output by the positioning program are stored in the memory  921  or the auxiliary storage device  922 , or stored in a register or a cache memory in the processor  910 . 
     “Unit” of each of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  may be interpreted as “process”, “procedure”, or “step”. The positioning program causes a computer to execute a relative angle acquisition process, a provisional position calculation process, a weight calculation process, and a device position calculation process. “Process” of each of the relative angle acquisition process, the provisional position calculation process, the weight calculation process, and the device position calculation process may be interpreted as “program”, “program product”, “computer readable storage medium storing a program”, or “computer readable recording medium recording a program”. A positioning method is a method performed by execution of the positioning program by the positioning system  100 . The positioning program may be stored and provided in a computer readable recording medium. Alternatively, the positioning program may be provided as a program product. 
     *** Description of Operation *** 
     Referring to  FIG.  3   , operation of the positioning system  100  according to this embodiment will be described. 
     A procedure for the operation of the positioning system  100  is equivalent to the positioning method. A program that realizes the operation of the positioning system  100  is equivalent to the positioning program. 
     &lt;Relative Angle Acquisition Process: Step S 101 &gt; 
     The relative angle acquisition unit  110  acquires the relative angle  31  between each base station  20  of the plurality of base stations and the communication device  10  from each base station  20  of the plurality of base stations. As described in  FIG.  1   , the base station  20  can calculate the relative angle  31  with the communication device  10 . 
       FIG.  4    is a diagram illustrating an example of an angular relationship between the base station  20  and the communication device  10  according to this embodiment. 
       FIG.  5    is a flowchart illustrating a positioning process using three-dimensional positioning by the positioning system  100  according to this embodiment. In  FIG.  4   , it is assumed that there are N base stations  20  and one communication device  10  in a three-dimensional space represented by a vector R 3  with three elements. The position of the base station  20  is a i =[x i , y i , z i ] T ∈R 3 , where i=1, . . . , N. N is a natural number of 2 or more. The position of the communication device  10  is x=[x, y, z] T ∈R 3 . A relative position vector r i  from the communication device  10  to the position of the base station  20  is r i =x−a i . 
     Specifically, in step S 11  of  FIG.  5   , the relative angle acquisition unit  110  acquires an azimuth angle φ i  from the communication device  10  to the base station  20  and an elevation angle Ψ i  from the communication device  10  to the base station  20  as the relative angles  31 . 
     Measurement noise occurs in angle measurement values. Therefore, the azimuth angle φ i , the elevation angle Ψ i , and expected values of measurement noise for the angle measurement values are expressed by Formula 1 below, where E(·) represents an expected value of ·. 
       ϕ i =ϕ i   0 +{tilde over (ϕ)} i  
 
       Ψ i =Ψ i   0 +{tilde over (Ψ)} i  
 
       {tilde over (ϕ)} i , {tilde over (Ψ)} i : Measurement noise
 
         E ({tilde over (ϕ)} i )=0
 
         E ({tilde over (Ψ)} i )=0
 
         E ({tilde over (ϕ)} i   2 )=σ ϕ     i     2  
 
         E ({tilde over (Ψ)} i   2 )=σ Ψ     i     2    (1)
 
     &lt;Provisional Position Calculation Process: Step S 102 &gt;The provisional position calculation unit  120  calculates the position of the communication device  10  as a provisional position  32 , using the relative angle  31  between each base station  20  of the plurality of base stations and the communication device  10  and the position of each base station  20  of the plurality of base stations. The provisional position calculation unit  120  estimates the position of the communication device  10  by a least-squares method or the like, using the relative angle  31  acquired from the base station  20  and the position of the base station  20  and using geometric relationships as constraints. 
     Specifically, the provisional position calculation unit  120  calculates the provisional position  32  by performing the least-squares method using errors in the measurement noise of the position of the communication device  10 . The errors in the measurement noise of the position of the communication device  10  are obtained based on the relative angles  31  between each base station  20  of the plurality of base stations and the communication device  10  and the position of each base station  20  of the plurality of base stations. 
       FIG.  6    is a diagram illustrating an example of the base station information  151  according to this embodiment. 
     As indicated in  FIG.  6   , identifiers that identify base stations and positions of the base stations are set in the base station information  151 . The positions of the base stations are represented using a format such as coordinates of a three-dimensional space, coordinates of a two-dimensional space, CAD data, or BIM data. The positions of the base stations may be represented using other formats. CAD is an abbreviation for computer-aided design. BIM is an abbreviation for building information modeling. In this embodiment, it is assumed that the positions of the base stations are represented by coordinates of the three-dimensional space. 
     Specifically, in step S 12  of  FIG.  5   , the provisional position calculation unit  120  performs three-dimensional positioning of the communication device  10 , using the least-squares method. 
     A relative position vector r i  from the communication device  10  to the base station  20  is r i =x−a i . Based on the relative angles between the base station  20  and the communication device  10 , orthogonal vectors c 1i  and c 2i  with respect to the relative position vector r i  are expressed by Formula 2 below. 
         c   1i =[−sin(ϕ i ), cos(ϕ i ), 0] T  
 
         c   2i =[cos(ϕ i )cos(Ψ i ), sin(ϕ i )cos(Ψ i ), −sin(Ψ i )] T    (2)
 
     By transforming the relative position vector r i =x−a i  using the orthogonal vectors c 1i  and c 2i  of Formula 2, the relational expressions of Formula 3 below are obtained, where ϵ 1i  and ϵ 2i  are errors associated with measurement noise. 
         c   1i   T ( x−a   i )=ϵ 1i  
 
         c   2i   T ( x−a   i )=ϵ 2i    (3)
 
     As described above, the errors ϵ 1i  and ϵ 2i  in the measurement noise of the position of the communication device  10  are obtained based on the relative angles  31  between each base station  20  and the communication device  10  and the position of each base station  20 . The provisional position  32 , which is an estimated position of the communication device  10  obtained by the least-squares method, is represented by Formula 4 below. The provisional position  32  represented by Formula 4 is given as a solution to the minimum value problem of Formula 5 below. 
         {circumflex over (x)}   LS   =[{circumflex over (x)}   LS   , ŷ   LS   , {circumflex over (z)}   LS ] T    (4)
 
     {circumflex over (x)} LS : Provisional position  32  of the communication device  10   
     
       
         
           
             
               
                 
                   
                     
                       x 
                       ˆ 
                     
                     
                       L 
                       ⁢ 
                       S 
                     
                   
                   = 
                   
                     
                       
                         argmin 
                         x 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         
                           
                             ( 
                             
                               
                                 c 
                                 
                                   1 
                                   ⁢ 
                                   i 
                                 
                                 T 
                               
                               ( 
                               
                                 x 
                                 - 
                                 
                                   a 
                                   i 
                                 
                               
                               ) 
                             
                             ) 
                           
                           2 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       
                         
                           ( 
                           
                             
                               c 
                               
                                 2 
                                 ⁢ 
                                 i 
                               
                               T 
                             
                             ( 
                             
                               x 
                               - 
                               
                                 a 
                                 i 
                               
                             
                             ) 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     5 
                   
                   ] 
                 
               
             
           
         
       
     
     {circumflex over (x)} LS : Provisional position  32  of the communication device  10   
     When Formula 5 is represented in a matrix format, the expression of Formula 6 below is obtained. 
     R M×N  is a matrix with M rows and N columns. Therefore, R 2N×3  is a matrix with 2N rows and three columns. R 2N×1  is a matrix with 2N rows and one column. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           ˆ 
                         
                         
                           L 
                           ⁢ 
                           S 
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               
                                 A 
                                 T 
                               
                               ⁢ 
                               A 
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           A 
                           T 
                         
                         ⁢ 
                         b 
                       
                     
                     ; 
                     
                       
                         where 
                         ⁢ 
                         
                             
                              
                         
                         ⁢ 
                         A 
                       
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   c 
                                   11 
                                   T 
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     1 
                                     ⁢ 
                                     N 
                                   
                                   T 
                                 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     2 
                                     ⁢ 
                                     1 
                                   
                                   T 
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     2 
                                     ⁢ 
                                     N 
                                   
                                   T 
                                 
                               
                             
                           
                           ] 
                         
                         ∈ 
                         
                           ℝ 
                           
                             2 
                             ⁢ 
                             N 
                             × 
                             3 
                           
                         
                       
                     
                   
                   , 
                   
                     b 
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   c 
                                   11 
                                   T 
                                 
                                 ⁢ 
                                 
                                   a 
                                   1 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   c 
                                   11 
                                   T 
                                 
                                 ⁢ 
                                 
                                   a 
                                   N 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   c 
                                   
                                     2 
                                     ⁢ 
                                     1 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   a 
                                   1 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   c 
                                   21 
                                   T 
                                 
                                 ⁢ 
                                 
                                   a 
                                   N 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ∈ 
                       
                         ℝ 
                         
                           2 
                           ⁢ 
                           N 
                           × 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     6 
                   
                   ] 
                 
               
             
           
         
       
     
     {circumflex over (x)} LS : Provisional position  32  of the communication device  10   
     &lt;Weight Calculation Process: Step S 103 &gt; 
     The weight calculation unit  130  calculates the distance between each base station  20  of the plurality of base stations and the communication device  10 , using the position of each base station  20  of the plurality of base stations and the provisional position  32 . Based on the distance between each base station  20  of the plurality of base stations and the communication device  10 , the weight calculation unit  130  calculates a weighting coefficient  33  for correcting the provisional position  32  for each base station  20  of the plurality of base stations. The weight calculation unit  130  calculates the weighting coefficient  33  so that the greater the distance, the smaller the influence of the error in the least-squares method. 
       FIG.  7    is a diagram depicting a relative angle containing an error according to this embodiment. 
     There is an error in a relative angle calculated by the base station  20 . As indicated in  FIG.  7   , it can be seen that the greater the distance between the base station  20  and the communication device  10 , the greater the deviation of the estimated position of the communication device  10  from the real position may be. Therefore, the weighting coefficient  33  is assumed to be any function that monotonically decreases with distance. Specifically, the reciprocal of the distance is used as the weighting coefficient  33 . 
     Specifically, in step S 13  of  FIG.  5   , the weight calculation unit  130  calculates the weighting coefficients  33  of each base station  20  as W 1i  and W 2i . 
     It is assumed that the measurement noise in each of the azimuth angle and the elevation angle from the communication device  10  to the base station  20  is sufficiently smaller than 1, as indicated in Formula 7 below. Using geometric relationships and the addition theorem of trigonometric functions, Formula 3 can be transformed as indicated in Formula 8 below. 
       |{tilde over (ϕ)} i |, |{tilde over (Ψ)} i |«1   (7)
 
       ϵ 1i   ≅−R   2i {tilde over (ϕ)} i  
 
       ϵ 2i   ≅−R   3i {tilde over (Ψ)} i    (8)
 
     R 2  is the Euclidean distance between the base station and the communication device in an x-y plane, and is represented by Formula 9 below. 
         R   2i √{square root over (( {circumflex over (x)}−x   i ) 2 +( ŷ−y   i ) 2 )}  (9)
 
     R 3i  is the Euclidean distance between the base station and the communication device in a three-dimensional space and is represented by Formula 10 below. 
         R   3i =√{square root over (( {circumflex over (x)}−x   i ) 2 +( ŷ−y   i ) 2 +( {circumflex over (z)}−z   i ) 2 )}  (10)
 
     It is assumed that the measurement noise in each of the azimuth angle and the elevation angle between the communication device  10  and the base station  20  is sufficiently smaller than 1, as indicated in Formula 11 below. 
       {tilde over (ϕ)} i «1, {tilde over (Ψ)} i «1   (11)
 
     Variances σ 1i   2  and σ 2i   2  of ϵ 1i  and ϵ 2i  of Formula 8 can be calculated as indicated in Formula 12 below, where E(·) represents an expected value of ·. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                           
                             1 
                             ⁢ 
                             i 
                           
                           2 
                         
                         = 
                           
                         
                           E 
                           ⁡ 
                           ( 
                           
                             
                               ( 
                               
                                 
                                   ε 
                                   
                                     1 
                                     ⁢ 
                                     i 
                                   
                                 
                                 - 
                                 
                                   E 
                                   ⁡ 
                                   ( 
                                   
                                     ε 
                                     
                                       1 
                                       ⁢ 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         ≅ 
                           
                         
                           
                             R 
                             
                               2 
                               ⁢ 
                               i 
                             
                             2 
                           
                           ⁢ 
                           
                             σ 
                             
                               
                                 ϕ 
                                   
                               
                               i 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           σ 
                           
                             2 
                             ⁢ 
                             i 
                           
                           2 
                         
                         = 
                           
                         
                           E 
                           ⁡ 
                           ( 
                           
                             
                               ( 
                               
                                 
                                   ε 
                                   
                                     2 
                                     ⁢ 
                                     i 
                                   
                                 
                                 - 
                                 
                                   E 
                                   ⁡ 
                                   ( 
                                   
                                     ε 
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         ≅ 
                           
                         
                           
                             R 
                             
                               3 
                               ⁢ 
                               i 
                             
                             2 
                           
                           ⁢ 
                           
                             σ 
                             
                               ψ 
                               i 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     12 
                   
                   ] 
                 
               
             
           
         
       
     
     Assuming that the variance of the measurement noise of the azimuth angle and the variance of the measurement noise of the elevation angle are roughly equal, the variances σ 1i   2  and σ 2i   2  can be represented using a constant C, as indicated in Formula 13 below. 
       σ 1i   2   ≅R   2i   2   C  
 
       σ 2i   2   ≅R   3i   2   C    (13)
 
     In this embodiment, the weighting coefficients  33  are calculated using Formula 6 representing the provisional position  32  in a matrix format and a position a i  of the base station  20 , as indicated in Formula 14 below. 
     
       
         
           
             
               
                 
                   
                     
                       w 
                       
                         1 
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       
                         
                           C 
                         
                         
                           σ 
                           
                             1 
                             ⁢ 
                             i 
                           
                         
                       
                       ≅ 
                       
                         1 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       ˆ 
                                     
                                     
                                       L 
                                       ⁢ 
                                       S 
                                     
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       y 
                                       ˆ 
                                     
                                     
                                       L 
                                       ⁢ 
                                       S 
                                     
                                   
                                   - 
                                   
                                     y 
                                     i 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                     
                   
                     
                       w 
                       
                         2 
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       
                         
                           C 
                         
                         
                           σ 
                           
                             2 
                             ⁢ 
                             i 
                           
                         
                       
                       ≅ 
                       
                         1 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       ˆ 
                                     
                                     
                                       L 
                                       ⁢ 
                                       S 
                                     
                                   
                                   - 
                                   
                                     x 
                                     i 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       y 
                                       ˆ 
                                     
                                     
                                       L 
                                       ⁢ 
                                       S 
                                     
                                   
                                   - 
                                   
                                     y 
                                     i 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       z 
                                       ˆ 
                                     
                                     
                                       L 
                                       ⁢ 
                                       S 
                                     
                                   
                                   - 
                                   
                                     z 
                                     i 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     14 
                   
                   ] 
                 
               
             
           
         
       
     
     Here, the weighting coefficients  33  are W 1i  and W 2i . W 1i  is a weight corresponding to the orthogonal vector c 1i  and W 2i  is a weight corresponding to the orthogonal vector c 2i . When the position of the communication device  10  is calculated, this allows weighting to be performed taking into consideration the influence of noise depending on the distance between the base station  20  and the communication device  10 . 
     &lt;Device Position Calculation Process: Step S 104 &gt; 
     The device position calculation unit  140  calculates the position of the communication device  10  as a device position  34 , using the relative angle  31  between each base station  20  of the plurality of base stations and the communication device  10 , the position of each base station  20  of the plurality of base stations, and the weighting coefficients  33  for each base station  20  of the plurality of base stations. The device position calculation unit  140  calculates the device position  34  by performing the least-squares method by multiplying each error by the corresponding weighting coefficient  33 . 
     Specifically, in step S 14  of  FIG.  5   , the device position calculation unit  140  calculates the position of the communication device  10  as the device position  34  by the weighted least-squares method, using the weighting coefficients W 1i  and W 2i  of each base station  20 . The weighted least-squares method is referred to also as the weighted least-squares method. 
     In the weighted least-squares method, the device position  34 , which is the estimated position of the communication device  10 , is given as a solution to the minimum value problem of Formula 15 below. This allows the influence of the base station with a great noise influence to be reduced for the device position  34 , which is the estimated position of the communication device  10 . 
     
       
         
           
             
               
                 
                   
                     x 
                     ˆ 
                   
                   = 
                   
                     
                       
                         argmin 
                         x 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                         
                           
                             ( 
                             
                               
                                 w 
                                 
                                   1 
                                   ⁢ 
                                   i 
                                 
                               
                               ⁢ 
                               
                                 
                                   c 
                                   
                                     1 
                                     ⁢ 
                                     i 
                                   
                                   T 
                                 
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     a 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       
                         
                           ( 
                           
                             
                               w 
                               
                                 2 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               
                                 c 
                                 
                                   2 
                                   ⁢ 
                                   i 
                                 
                                 T 
                               
                               ( 
                               
                                 x 
                                 - 
                                 
                                   a 
                                   i 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     15 
                   
                   ] 
                 
               
             
           
         
       
     
     {circumflex over (x)}: Device position  34  of the communication device  10   
     When Formula 15 is represented in a matrix format, Formula  16  is obtained. 
         {circumflex over (x)} =( A   T   WA ) −1   A   T   Wb;    (16)
 
     where W=diag(w 11   2 , . . . , w 1N   2 , w 21   2 , . . . , w 2N   2 )∈   2N×2N  
         {circumflex over (x)}: Device position  34  of the communication device  10         

     *** Other Configurations *** 
     &lt;Variation 1&gt; 
     In this embodiment, the positions of the communication device  10  and the base station  20  are represented by three-dimensional coordinates, and the position of the communication device  10  is calculated by three-dimensional positioning. 
     In Variation 1 according to this embodiment, the positions of the communication device  10  and the base station  20  may be represented by two-dimensional coordinates, and the position of the communication device  10  may be calculated by two-dimensional positioning. A specific example of two-dimensional positioning will be described below. 
       FIG.  8    is a flowchart of a positioning process using two-dimensional positioning by the positioning system  100  according to a variation of this embodiment. It is assumed here that there are N base stations  20  and one communication device  10  in a two-dimensional space represented by a vector R 2  with two elements. The position of the base station  20  is a i =[x i , y i ] T ∈R 2 , where i=1, . . . , N. N is a natural number of 2 or more. The position of the communication device  10  is x=[x, y] T ∈R 2 . A relative position vector r i  from the communication device  10  to the position of the base station  20  is r i =x−a i . 
     In step S 21 , the relative angle acquisition unit  110  acquires an azimuth angle φ i  from the communication device  10  to the base station  20  from each base station  20 . 
     In step S 22 , the provisional position calculation unit  120  performs two-dimensional positioning of the communication device  10 , using the least-squares method. 
     An orthogonal vector c 1i  with respect to the relative position vector r i  is represented by Formula 17 below, based on the azimuth angle φ i . 
         c   1i =[−sin(ϕ i ), cos(ϕ i )] T    (17)
 
     By transforming the relative position vector r i =x−a i  using the orthogonal vector c 1i , the relational expression of Formula 18 below is obtained, where ϵ 1i  is an error associated with measurement noise. 
         c   1i   T ( x−a   i )=ϵ 1i    (18)
 
     The provisional position  32 , which is the estimated position of the communication device  10 , obtained by the least-squares method is represented by Formula 19 below. The provisional position  32  represented by Formula 19 is given as a solution to the minimum value problem of Formula 20 below. 
         {circumflex over (x)}   LS   =[{circumflex over (x)}   LS   , ŷ   LS ] T    (19)
 
     {circumflex over (x)} LS : Provisional position  32  of the communication device  10   
     
       
         
           
             
               
                 
                   
                     
                       x 
                       ˆ 
                     
                     
                       L 
                       ⁢ 
                       S 
                     
                   
                   = 
                   
                     
                       argmin 
                       x 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       
                         
                           ( 
                           
                             
                               c 
                               
                                 1 
                                 ⁢ 
                                 i 
                               
                               T 
                             
                             ( 
                             
                               x 
                               - 
                               
                                 a 
                                 i 
                               
                             
                             ) 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     20 
                   
                   ] 
                 
               
             
           
         
       
     
     {circumflex over (x)} LS : Provisional position  32  of the communication device  10   
     When Formula 20 is expressed in a matrix format, the expression of Formula 21 below is obtained. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           ˆ 
                         
                         LS 
                       
                       = 
                       
                         
                           
                             ( 
                             
                               
                                 A 
                                 T 
                               
                               ⁢ 
                               A 
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           A 
                           T 
                         
                         ⁢ 
                         b 
                       
                     
                     ; 
                     
                       
                         where 
                         ⁢ 
                             
                         A 
                       
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   c 
                                   11 
                                   T 
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     1 
                                     ⁢ 
                                     N 
                                   
                                   T 
                                 
                               
                             
                           
                           ] 
                         
                         ∈ 
                         
                           ℝ 
                           
                             N 
                             × 
                             3 
                           
                         
                       
                     
                   
                   , 
                   
                     b 
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   c 
                                   11 
                                   T 
                                 
                                 ⁢ 
                                 
                                   a 
                                   1 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   c 
                                   11 
                                   T 
                                 
                                 ⁢ 
                                 
                                   a 
                                   N 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ∈ 
                       
                         ℝ 
                         
                           N 
                           × 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     21 
                   
                   ] 
                 
               
             
           
         
       
     
     {circumflex over (x)} LS : Provisional position  32  of the communication device  10   
     In step S 23 , the weight calculation unit  130  calculates the weighting coefficient  33  of each of the base stations  20  as W 1i , using the provisional position  32 . 
     It is assumed that the measurement noise of the azimuth angle from the communication device  10  to the base station  20  is sufficiently smaller than 1, as indicated in Formula 7 above. Using geometric relationships and the addition theorem of trigonometric functions, Formula 18 can be transformed as indicated in Formula 22 below. 
       ϵ 1i   ≅−R   2i {tilde over (ϕ)} i    (22)
 
     R 2i  is the Euclidean distance between the base station and the communication device in the x-y plane, and is represented as in Formula 9 above. 
     A variance σ 1i   2  of ϵ 1i  of Formula 22 can be calculated as indicated in Formula 23 below, where E(·) represents an expected value of ·. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                           
                             1 
                             ⁢ 
                             i 
                           
                           2 
                         
                         = 
                           
                         
                           E 
                           ⁡ 
                           ( 
                           
                             
                               ( 
                               
                                 
                                   ε 
                                   
                                     1 
                                     ⁢ 
                                     i 
                                   
                                 
                                 - 
                                 
                                   E 
                                   ⁡ 
                                   ( 
                                   
                                     ε 
                                     
                                       1 
                                       ⁢ 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         ≅ 
                           
                         
                           
                             R 
                             
                               2 
                               ⁢ 
                               i 
                             
                             2 
                           
                           ⁢ 
                           
                             σ 
                             
                               ϕ 
                               i 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     23 
                   
                   ] 
                 
               
             
           
         
       
     
     From Formula 23, the relationship in Formula 24 below is derived. 
       σ 1i   2 ∝R 2i   2    (24)
 
     In this embodiment, the weighting coefficient  33  is calculated using Formula 21 representing the provisional position  32  and the position a i  of the base station  20 , as indicated in Formula 25 below. 
     
       
         
           
             
               
                 
                   
                     w 
                     
                       1 
                       ⁢ 
                       i 
                     
                   
                   = 
                   
                     1 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   x 
                                   ˆ 
                                 
                                 
                                   L 
                                   ⁢ 
                                   S 
                                 
                               
                               - 
                               
                                 x 
                                 i 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 
                                   y 
                                   ˆ 
                                 
                                 LS 
                               
                               - 
                               
                                 y 
                                 i 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     25 
                   
                   ] 
                 
               
             
           
         
       
     
     Here, the weighting coefficient  33  is W 1i . 
     In step S 24 , the device position calculation unit  140  calculates the device position  34  of the communication device  10  by the weighted least-squares method, using the weighting coefficient W 1i  of each base station  20 . In the weighted least-squares method, the device position  34  of the communication device  10  is treated as an estimated position, and the device position  34  of the communication device  10  is given as a solution to the minimum value problem of Formula 26 below. 
     
       
         
           
             
               
                 
                   
                     x 
                     ˆ 
                   
                   = 
                   
                     
                       argmin 
                       x 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       
                         
                           ( 
                           
                             
                               w 
                               
                                 1 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               
                                 c 
                                 
                                   1 
                                   ⁢ 
                                   i 
                                 
                                 T 
                               
                               ( 
                               
                                 x 
                                 - 
                                 
                                   a 
                                   i 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                         
                     26 
                   
                   ] 
                 
               
             
           
         
       
     
     {circumflex over (x)}: Device position  34  of the communication device  10   
     When Formula 26 is expressed in a matrix format, Formula 27 is obtained. 
         {circumflex over (x)} =( A   T   WA ) −1   Z   T   Wb;    (27)
 
     where W=diag(w 11   2 , . . . , w 1N   2 )∈   N×N    
     {circumflex over (x)}: Device position  34  of the communication device  10   
     &lt;Variation 2&gt; 
     In this embodiment, a terminal that communicates with the base station  20  by wireless communication, such as a smartphone terminal, a tablet terminal, or a smartwatch, is assumed as a specific example of the communication device  10 . However, the communication device  10  may be one base station included in a plurality of base stations. 
       FIG.  9    is a diagram illustrating the communication device  10  that is a positioning target and the base stations  20  according to Variation 2 of this embodiment. A base station  20   a  of the plurality of base stations is an example of the communication device  10  that is the positioning target. The base station  20   a  wirelessly communicates with other base stations of the plurality of base stations. 
     As described above, a base station may be regarded as the positioning target. As indicated in  FIG.  9   , the position of the base station  20   a  whose position is unknown may be calculated based on the position of each base station  20  of the plurality of base stations and the relative angle between each base station  20  and the base station  20   a.    
     &lt;Variation 3&gt; 
     In this embodiment, the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  are realized by software. As a variation, the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  may be realized by hardware. 
     Specifically, the positioning system  100  includes an electronic circuit  909  in place of the processor  910 . 
       FIG.  10    is a diagram illustrating a configuration of the positioning system  100  according to Variation 3 of this embodiment. 
     The electronic circuit  909  is a dedicated electronic circuit that realizes the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140 . Specifically, the electronic circuit  909  is a single circuit, a composite circuit, a programmed processor, a parallel-programmed processor, a logic IC, a GA, an ASIC, or an FPGA. GA is an abbreviation for Gate Array. ASIC is an abbreviation for Application Specific Integrated Circuit. FPGA is an abbreviation for Field-Programmable Gate Array. 
     The functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  may be realized by one electronic circuit, or may be distributed among and realized by a plurality of electronic circuits. 
     As another variation, some of the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  may be realized by the electronic circuit, and the rest of the functions may be realized by software. Alternatively, some or all of the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  may be realized by firmware. 
     Each of the processor and the electronic circuit is also called processing circuitry. That is, the functions of the relative angle acquisition unit  110 , the provisional position calculation unit  120 , the weight calculation unit  130 , and the device position calculation unit  140  are realized by the processing circuitry. 
     *** Effects of This Embodiment *** 
     As described above, the positioning system  100  according to this embodiment first calculates the provisional position of the communication device by the least-squares method or the like, using the relative angle acquired from each base station and the position of each base station and using geometric relationships as constraints. Then, the positioning system  100  calculates the distance between the communication device and each base station, based on the provisional position of the communication device and the position of each base station. Then, the positioning system  100  estimates the final position of the communication device as the device position by the weighted least-squares method or the like, using geometric relationships as constraints and using a weight based on the distance. 
     As described above, the positioning system  100  according to this embodiment can realize highly accurate positioning using only angle information by performing weighting such that the influence of a distant base station is reduced. 
     In Embodiment 1 above, each unit of the positioning system has been described as an independent functional block. However, the configuration of the positioning system may be different from the configuration in the above embodiment. The functional blocks of the positioning system may be configured in any way, provided that the functions described in the above embodiment can be realized. The positioning system may be a system composed of a plurality of devices instead of one device. 
     Portions of Embodiment 1 may be implemented in combination. Alternatively, one portion of this embodiment may be implemented. This embodiment may be implemented as a whole or partially in any other combination. 
     That is, in Embodiment 1, portions of the embodiment may be freely combined, any constituent element of the embodiment may be modified, or any constituent element may be omitted in the embodiment. 
     The embodiment described above is an essentially preferable example and is not intended to limit the scope of the present disclosure, the scope of applications of the present disclosure, and the scope of uses of the present disclosure. The embodiment described above can be modified in various ways as necessary. 
     REFERENCE SIGNS LIST 
       10 : communication device;  20 ,  20   a : base station;  31 : relative angle;  32 : provisional position;  33 : weighting coefficient;  34 : device position;  100 : positioning system;  110 : relative angle acquisition unit;  120 : provisional position calculation unit;  130 : weight calculation unit;  140 : device position calculation unit;  150 : storage unit;  151 : base station information;  909 : electronic circuit;  910 : processor;  921 : memory;  922 : auxiliary storage device;  930 : input interface;  940 : output interface;  950 : communication equipment.