Indoor visible light high-precision positioning method based on user rapid self-correction fingerprint database

Abstract

The invention discloses an indoor visible light high-precision positioning method based on a user rapid self-correction fingerprint database, and the method comprises the specific steps: 1, building a reference point direct-view receiving power model based on a link loss index under vertical receiving, constructing fingerprint points in the fingerprint database, and obtaining the actual receiving power of each fingerprint point; 2, correcting the obtained actual receiving power of each fingerprint point to obtain the direct-view receiving power of each fingerprint point in the real-time receiving direction of the user; 3, constructing a feature fingerprint database containing the position and the receiving direction of the user; and 4, estimating the two-dimensional position of the user by using the feature fingerprint database obtained in the step 3 and a weighted KNN algorithm. The method provided by the invention is more suitable for high-precision and low-complexity positioning and navigation service of an actual scene.

Description

Technical Field

[0001] The invention belongs to the technical field of indoor visible light positioning, and in particular relates to an indoor visible light high-precision positioning method based on a user's fast self-correction fingerprint library. Background Art

[0002] The Global Positioning System (GPS) has been widely used in outdoor scenes for navigation, positioning, tracking, and timing of targets. Since GPS signals cannot penetrate walls and have multipath effects, it is difficult to provide accurate positioning services for targets such as people or objects in indoor scenes.

[0003] Visible Light Positioning (VLP) uses a light-emitting diode (LED) as a transmitter, a photodetector as a receiver, and visible light with a wavelength range of 380nm to 780nm as a positioning signal to perform centimeter-level estimation of the position of indoor targets. It can be widely used in advanced application scenarios such as position-assisted channel estimation, drone swarms and fast-moving robots, and positioning frequencies of hundreds of hertz.

[0004] The diffraction and reflection of visible light signals in indoor scenes are usually ignored, and it is generally believed that the line-of-sight (LOS) component is dominant. The trilateral positioning method based on LOS received signal strength (RSS) estimates the distance between the LED and the PD according to the measured received light intensity. This positioning method is low-cost, easy to implement, and can generally achieve centimeter-level accuracy. However, most existing RSS-based VLPs assume that the normal vector of the receiver is vertically upward. Under the ideal assumption of vertical reception, the distance model between the LED and the PD can be expressed as a simple function of RSS. The distance between the LEDs and the PD is estimated based on the received RSS, but this functional relationship cannot correctly characterize the analytical relationship between the distance between the LEDs and the PD and the RSS when the receiver is tilted. The distance between the LEDs and the PD estimated based on the distance model of vertical reception and the RSS under tilted reception must be different from the actual distance, resulting in a decrease in positioning accuracy. To address this problem, the researchers derived the analytical relationship between the distance between the transmitter and the receiver under tilted reception and the received RSS. Based on this relationship, the three-sided positioning method is used to estimate the two-dimensional position of the user under tilted reception. The limitation of this method is that when the receiving pitch angle is greater than 40 degrees, the light signal from one or two LEDs exceeds the field of view of the PD, resulting in zero RSS for direct-view reception, inability to estimate the distance between the LED and the PD, and failure of the positioning system. Therefore, most RSS-based indoor VLPs are limited by the ideal assumption of vertical reception and are difficult to use for high-precision positioning of users with tilted receiving directions in actual scenarios.

[0005] The existing fingerprint positioning combined with artificial neural networks can complete the complex mapping of the measured received RSS to the two-dimensional or three-dimensional coordinates of the receiving terminal, but the positioning accuracy depends on the size of the fingerprint library established in the offline stage. The smaller the grid of the fingerprint library, the higher the positioning accuracy. The fingerprint libraries currently established based on vertical reception are all multi-parameter sets measured under the condition that the normal vectors of the transmitter and receiver are parallel. In order to reduce the time complexity of fingerprint library establishment, researchers proposed an artificial fingerprint library creation method based on the link loss index. This method only needs to measure a few samples to create a dense fingerprint library with high positioning accuracy. However, the link index loss model used to create the fingerprint library is only applicable to vertical reception, and is not suitable for positioning in dynamic scene tilted reception scenarios. Summary of the invention

[0006] The purpose of the present invention is to provide an indoor visible light high-precision positioning method based on a user's fast self-correction fingerprint library, which takes into account the randomness of the receiving azimuth and tilt angle caused by random behavior and can be used for accurate positioning of dynamically receiving users.

[0007] The technical solution adopted by the present invention is a method for indoor visible light high-precision positioning based on a user's fast self-correction fingerprint library, which is specifically implemented in the following steps:

[0008] Step 1: Establish a reference point direct-view receiving power model based on the link loss index under vertical reception, construct fingerprint points in the fingerprint library, and obtain the actual receiving power of each fingerprint point;

[0009] Step 2, correcting the actual receiving power of each fingerprint point obtained in step 1 to obtain the direct-view receiving power of each fingerprint point in the user's real-time receiving direction;

[0010] Step 3, construct a feature fingerprint library containing the location and the user's own receiving direction;

[0011] Step 4: Use the feature fingerprint library obtained in step 3 and the weighted KNN algorithm to estimate the two-dimensional position of the user.

[0012] The present invention is also characterized in that:

[0013] The specific process of step 1 is:

[0014] Step 1.1, assuming that the number of indoor LED light sources is N, the projection of each LED light source on the indoor receiving plane is used as the receiving reference point corresponding to the LED light source, that is, the number of receiving reference points is N;

[0015] Step 1.2, establish a reference point direct-view receiving power model based on the link loss index under vertical reception, and calculate the direct-view receiving power of each receiving reference point through the established reference point direct-view receiving power model based on the link loss index.

[0016] The expression of the reference point direct-view received power model based on the link loss index is:

[0017]

[0018] In formula (1), P t is the transmission power of the LED light source, A PD is the effective detection area of ​​the photodetector, h is the vertical height between the LED light source and the receiving plane, d r,i is the distance between the LED light source and the receiving reference point, ψ r,i is the incident angle of the optical signal at the receiving end at the receiving reference point, and m is the radiation index of the LED light source;

[0019] Step 1.3: Divide the indoor receiving plane into M×M grids. The center of each grid is the actual receiving position of each fingerprint point in the fingerprint library. Use formula (2) to obtain the actual receiving power of each fingerprint point.

[0020] The expression is:

[0021]

[0022] In formula (2), d r,i is the distance between the ith LED light source and its corresponding receiving reference point, d a,ji is the distance between the i-th LED light source and the j-th fingerprint point, i = 1, 2, 3, ... N, j = 1, 2, ..., M × M, and m is the radiation index of the light source.

[0023] The specific process of step 2 is:

[0024] Step 2.1, assuming that the yaw angle, pitch angle and roll angle of the receiving terminal held by the user are α, β and γ, the receiving tilt angle θ and azimuth angle ω generated by the rotation of the receiving terminal are calculated by formula (3) and formula (4);

[0025] The expression is:

[0026] θ=cos -1 (cosβ·cosγ) (3)

[0027]

[0028] Step 2.2: Calculate the factors for correcting multiple RSS parameters at each fingerprint point in the user receiving direction using formula (5):

[0029] The expression is:

[0030]

[0031] In formula (5), x i and i Represents the horizontal and vertical coordinates of the LED light source, x j and j Indicates the horizontal and vertical coordinates of each fingerprint point in the fingerprint library;

[0032] Step 2.3, the factors obtained in step 2.2 Correct the actual receiving power of each fingerprint point obtained in step 1 to obtain the direct-view receiving power of each fingerprint point in the user's real-time receiving direction.

[0033] The expression is:

[0034]

[0035] The specific process of step 3 is:

[0036] Step 3.1, use a gyroscope to measure the yaw angle α, pitch angle β and roll angle γ of the receiving terminal held by the user, and use a photodetector to measure the received electrical power from the i-th LED light source

[0037] Step 3.2: Measure the received electrical power of the i-th LED light source according to step 3.1 The received optical power from the i-th LED is calculated by formula (7):

[0038] The expression is:

[0039]

[0040] In formula (7), η is the photoelectric conversion efficiency of the photodetector;

[0041] Step 3.3, according to the yaw angle α, pitch angle β and roll angle γ measured in step 3.1, calculate the current receiving tilt angle θ and the current azimuth angle ω by formula (3) and formula (4);

[0042] Step 3.4, calculate the RSS parameter set using formula (6): The RSS parameter set is used as a feature fingerprint library to estimate the user's current location;

[0043] Among them, the RSS parameter set includes the fingerprint point location information (x j ,y j ,z j ), and also represents the user’s current receiving direction information (θ, ω).

[0044] The specific process of step 4 is:

[0045] Step 4.1, calculate the received optical power from the i-th LED With the fingerprint library The Euclidean distance of each fingerprint point in

[0046] The expression is:

[0047]

[0048] Step 4.2, Euclidean distance Arrange from small to large to get d j , record the horizontal coordinate x of the corresponding fingerprint point k and the ordinate y k ;

[0049] Step 4.3, calculate the weighting coefficient using formula (9);

[0050] The expression is:

[0051]

[0052] Step 4.4: Estimate the user's position with the optimal number of nearest neighbors using formula (10):

[0053] The expression is:

[0054]

[0055] In formula (10), K is the optimal number of nearest neighbors.

[0056] The beneficial effects of the present invention are as follows: the indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library of the present invention corrects the reference point direct-view receiving power model based on the link loss index, and performs fast self-correction of the fingerprint library, and quickly constructs a feature fingerprint library that simultaneously contains the fingerprint position and the user's current receiving pitch angle and receiving tilt angle, and uses a weighted KNN algorithm to accurately estimate the user's two-dimensional position; the method of the present invention avoids the laborious construction, high calculation complexity and generalization problems of establishing a fingerprint library containing position, pitch angle and azimuth in the offline stage, and is more suitable for high-precision, low-complexity positioning and navigation services in actual scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 It is a flow chart of the indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library of the present invention;

[0058] Figure 2 This is a hardware structure diagram used in the indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library of the present invention;

[0059] Figure 3 The graph is a variation curve of the average positioning error based on different distance standards with the number of nearest neighbors in Example 3 of the present invention when the user receiving direction is random and the size of the fingerprint grid is 20cm×20cm;

[0060] Figure 4 The cumulative distribution curve of positioning error based on different distance standards under the conditions of 20cm×20cm fingerprint grid size and optimal number of neighbors when the user receiving direction is random in Example 3 of the present invention;

[0061] Figure 5 is a curve diagram showing the average positioning error as a function of the receiving pitch angle and roll angle in a fingerprint grid size of 20 cm×20 cm in Example 3 of the present invention;

[0062] Figure 6It is the cumulative distribution function of the positioning error obtained by simulation based on the corrected fingerprint library and the uncorrected fingerprint library under the ED, SED and SCD distance standards in Example 3 of the present invention with a fingerprint grid size of 20 cm×20 cm.

[0063] In the figure, 1. LED light source, 2. gyroscope, 3. photodetector, 4. receiving terminal. DETAILED DESCRIPTION

[0064] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0065] Example 1

[0066] The present invention is based on the indoor visible light high-precision positioning method of the user's fast self-correction fingerprint library, which is specifically implemented according to the following steps:

[0067] Step 1: Establish a reference point direct-view receiving power model based on the link loss index under vertical reception, construct fingerprint points in the fingerprint library, and obtain the actual receiving power of each fingerprint point;

[0068] The specific process is:

[0069] Step 1.1, assuming that the number of indoor LED light sources is N, the projection of each LED light source on the indoor receiving plane is used as the receiving reference point corresponding to the LED light source, that is, the number of receiving reference points is N;

[0070] Step 1.2, establish a reference point direct-view receiving power model based on the link loss index under vertical reception, and calculate the direct-view receiving power of each receiving reference point through the established reference point direct-view receiving power model based on the link loss index.

[0071] The expression of the reference point direct-view received power model based on the link loss index is:

[0072]

[0073] In formula (1), P t is the transmission power of the LED light source, A PD is the effective detection area of ​​the photodetector 3, h is the vertical height between the LED light source and the receiving plane, d r,i is the distance between the LED light source and the receiving reference point, ψ r,i is the incident angle of the optical signal at the receiving end at the receiving reference point, m is the radiation index of the LED light source, and the half-power angle Φ of the LED light source 1 / 2 related:

[0074]

[0075] Step 1.3: Divide the indoor receiving plane into M×M grids. The center of each grid is the actual receiving position of each fingerprint point in the fingerprint library. Use formula (2) to obtain the actual receiving power of each fingerprint point.

[0076] The expression is:

[0077]

[0078] In formula (2), d r,i is the distance between the ith LED light source and its corresponding receiving reference point, d a,ji is the distance between the i-th LED light source and the j-th fingerprint point, i=1,2,3,…N, j=1,2,…,M×M, m is the radiation index of the light source, which is related to the half-power angle of the LED;

[0079] Step 2, correcting the actual receiving power of each fingerprint point obtained in step 1 to obtain the direct-view receiving power of each fingerprint point in the user's real-time receiving direction;

[0080] The specific process is:

[0081] Step 2.1, assuming that the yaw angle of the receiving terminal 4 held by the user is α, the pitch angle is β, and the roll angle is γ, calculate the receiving tilt angle θ and the azimuth angle ω generated by the rotation of the receiving terminal 4 by using formula (3) and formula (4);

[0082] The expression is:

[0083] θ=cos -1 (cosβ·cosγ) (3)

[0084]

[0085] Step 2.2: Calculate the factors for correcting multiple RSS parameters at each fingerprint point in the user receiving direction using formula (5):

[0086] The expression is:

[0087]

[0088] In formula (5), x i and i Represents the horizontal and vertical coordinates of the LED light source, x j and j Indicates the horizontal and vertical coordinates of each fingerprint point in the fingerprint library;

[0089] Step 2.3, the factors obtained in step 2.2 Correct the actual receiving power of each fingerprint point obtained in step 1 to obtain the direct-view receiving power of each fingerprint point in the user's real-time receiving direction.

[0090] The expression is:

[0091]

[0092] Step 3, construct a feature fingerprint library containing the location and the user's own receiving direction;

[0093] The specific process is:

[0094] Step 3.1, using the gyroscope 2 to measure the yaw angle α, pitch angle β and roll angle γ of the receiving terminal held by the user, and using the photodetector 3 to measure the received electrical power from the i-th LED light source

[0095] Step 3.2: Measure the received electrical power of the i-th LED light source according to step 3.1 The received optical power from the i-th LED is calculated by formula (7):

[0096] The expression is:

[0097]

[0098] In formula (7), η is the photoelectric conversion efficiency of the photodetector 3;

[0099] Step 3.3, according to the yaw angle α, pitch angle β and roll angle γ measured in step 3.1, calculate the current receiving tilt angle θ and the current azimuth angle ω by using formula (3) and formula (4);

[0100] Step 3.4, calculate the RSS parameter set using formula (6): The RSS parameter set is used as a feature fingerprint library to estimate the user's current location;

[0101] Among them, the RSS parameter set includes the fingerprint point location information (x j ,y j ,z j ), and represents the user's current receiving direction information (θ, ω);

[0102] Step 4, using the feature fingerprint library obtained in step 3 and the weighted KNN algorithm to estimate the two-dimensional position of the user;

[0103] The specific process is:

[0104] Step 4.1, calculate the received optical power from the i-th LED With the fingerprint library The Euclidean distance of each fingerprint point in

[0105] The expression is:

[0106]

[0107] Step 4.2, Euclidean distance Arrange from small to large to get d j , record the horizontal coordinate x of the corresponding fingerprint point k and the ordinate y k ;

[0108] Step 4.3, calculate the weighting coefficient using formula (9);

[0109] The expression is:

[0110]

[0111] Step 4.4: Estimate the user's position with the optimal number of nearest neighbors using formula (10):

[0112] The expression is:

[0113]

[0114] In formula (10), K is the optimal number of nearest neighbors.

[0115] Example 2

[0116] The indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library described in Example 1, the main hardware used and its functions are as follows:

[0117] Light source: white light LED, half-power angle is 60 degrees;

[0118] Photodetector 3: converts light signals into current signals with a sensitivity of 0.4A / W;

[0119] Gyroscope 2: measures the yaw angle, pitch angle and roll angle of the receiving terminal 4;

[0120] Receiving terminal 4: including but not limited to a smart phone, tablet computer, etc. equipped with a photoelectric detector 3.

[0121] Example 3

[0122] The indoor space used in this embodiment is as follows Figure 1 As shown, the size is 5m×5m×3m.

[0123] Step 1: Use the projections of the four LED light sources 1 (LED light source 1, LED light source 2, LED light source 3 and LED light source 4) on the receiving plane as receiving reference points. The height of the receiving reference points is 0.85m. The coordinates of the receiving reference points are expressed as Indicates that i=1, 2, …, N, where N is the number of receiving reference points, which is the same as the number of LED light sources. In this embodiment, N=4, and the coordinates of the receiving reference points are (-1.25, 1.25, 0.85), (1.25, 1.25, 0.85), (1.25, -1.25, 0.85), and (-1.25, -1.25, 0.85).

[0124] The receiving plane is divided into M×M grids, and the center of each grid is the coordinate of the fingerprint point. Indicates that all fingerprint points and the receiving reference point are located in the same receiving plane, that is, M is the number of fingerprint points. In this embodiment, M=25;

[0125] The direct-view receiving power of each receiving reference point is obtained by formula (1):

[0126] The actual received power of each fingerprint point is obtained by formula (2): i=1,2,3,4;

[0127] Step 2: Use the gyroscope 2 to measure the yaw angle α, pitch angle β and roll angle γ of the receiving terminal held by the user, and use the photodetector 3 to measure the received electrical power from the i-th LED light source.

[0128] The received light power from the i-th LED is obtained by formula (7): The photoelectric conversion efficiency is η=0.4W / A in the embodiment;

[0129] The current receiving tilt angle θ and the current azimuth angle ω are calculated by formula (3) and formula (4);

[0130] The formula (6) contains both the fingerprint point position information The RSS parameter set that simultaneously represents the user's current receiving direction information (θ, ω) The RSS parameter set is used as a feature fingerprint library to estimate the user's current location;

[0131] Step 3: Calculate the received optical power from the i-th LED using formula (8): With the fingerprint library The distance between each fingerprint point

[0132] In this embodiment, three distance standards are selected, namely Euclidean distance (ED), squared Euclidean distance (SED) and squared chord distance (SCD);

[0133] Distance Arrange from small to large to get d j , record the horizontal and vertical coordinates of the corresponding fingerprint points;

[0134] The weighting coefficient w is calculated by formula (9): j , j = 1, 2, ..., 625;

[0135] Through formula (10) and weighting coefficient w j Estimate the user's position with the optimal number of nearest neighbors

[0136] Unless otherwise specified, the Matlab simulation parameters are set according to the steps in the embodiment. Figure 3 The fingerprint grid size is 20cm×20cm. The average positioning error of the user varies with the number of nearest neighbors under the ED, SED and SCD distance standards. It can be seen that the optimal number of nearest neighbors under the three distance standards is 4;

[0137] When the optimal number of nearest neighbors K = 4, Figure 4 It is the cumulative distribution function of the positioning error under the ED, SED and SCD distance standards when the fingerprint grid size is 20cm×20cm. It can be seen that under the same scene and the same fingerprint library, the positioning accuracy based on the SCD distance standard is better than that of the SED and ED standards;

[0138] When the optimal number of nearest neighbors K = 4, Figure 5 The fingerprint grid size is 20cm×20cm. The average positioning error of the user changes with the azimuth and inclination angle of the receiving terminal 4 under the SCD distance standard. It can be seen that when the azimuth and inclination angle of the receiving terminal 4 are both 40°, a positioning error of about 5.1cm can be obtained based on the proposed user fast self-correction fingerprint library method.

[0139] When the optimal number of nearest neighbors K = 4, Figure 6 The cumulative distribution function of the positioning error obtained by simulating the corrected fingerprint library and the uncorrected fingerprint library under the ED, SED and SCD distance standards is shown in Figure 2. The positioning accuracy obtained based on the corrected fingerprint library is much better than that obtained based on the uncorrected fingerprint library.

Claims

1. Indoor visible light high-precision positioning method based on user fast self-correction fingerprint library, It is characterized in that Follow the steps below to implement it: Step 1: Establish a reference point direct-view receiving power model based on the link loss index under vertical reception, construct fingerprint points in the fingerprint library, and obtain the actual receiving power of each fingerprint point; Step 2, correcting the actual receiving power of each fingerprint point obtained in step 1 to obtain the direct-view receiving power of each fingerprint point in the user's real-time receiving direction; Step 3, construct a feature fingerprint library containing the location and the user's own receiving direction; Step 4: Use the feature fingerprint library obtained in step 3 and the weighted KNN algorithm to estimate the two-dimensional position of the user.

2. According to claim 1, the indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library, It is characterized in that The specific process of step 1 is: Step 1.1, assuming that the number of indoor LED light sources is N, the projection of each LED light source on the indoor receiving plane is used as the receiving reference point corresponding to the LED light source, that is, the number of receiving reference points is N; Step 1.2, establish a reference point direct-view receiving power model based on the link loss index under vertical reception, and calculate the direct-view receiving power of each receiving reference point through the established reference point direct-view receiving power model based on the link loss index. The expression of the reference point direct-view received power model based on the link loss index is: In formula (1), P t is the transmission power of the LED light source, A PD is the effective detection area of ​​the photodetector, h is the vertical height between the LED light source and the receiving plane, d r,i is the distance between the LED light source and the receiving reference point, ψ r,i is the incident angle of the optical signal at the receiving end at the receiving reference point, and m is the radiation index of the LED light source; Step 1.3: Divide the indoor receiving plane into M×M grids. The center of each grid is the actual receiving position of each fingerprint point in the fingerprint library. Use formula (2) to obtain the actual receiving power of each fingerprint point. The expression is: In formula (2), d r,i is the distance between the ith LED light source and its corresponding receiving reference point, d a,ji is the distance between the i-th LED light source and the j-th fingerprint point, i=1,2,3,…N, j=1,2,…,M×M, and m is the radiation index of the light source.

3. The indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library according to claim 1, It is characterized in that The specific process of step 2 is: Step 2.1, assuming that the yaw angle, pitch angle and roll angle of the receiving terminal held by the user are α, β and γ, the receiving tilt angle θ and azimuth angle ω generated by the rotation of the receiving terminal are calculated by formula (3) and formula (4); The expression is: θ=cos -1 (cosβ·cosγ) (3) Step 2.2: Calculate the factors for correcting multiple RSS parameters at each fingerprint point in the user receiving direction using formula (5): The expression is: In formula (5), x i and i Represents the horizontal and vertical coordinates of the LED light source, x j and j Indicates the horizontal and vertical coordinates of each fingerprint point in the fingerprint library; Step 2.3, the factors obtained in step 2.2 Correct the actual receiving power of each fingerprint point obtained in step 1 to obtain the direct-view receiving power of each fingerprint point in the user's real-time receiving direction. The expression is:

4. The indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library according to claim 3, It is characterized in that The specific process of step 3 is: Step 3.1, use a gyroscope to measure the yaw angle α, pitch angle β and roll angle γ of the receiving terminal held by the user, and use a photodetector to measure the received electrical power from the i-th LED light source Step 3.2: Measure the received electrical power of the i-th LED light source according to step 3.1 The received optical power from the i-th LED is calculated by formula (7): The expression is: In formula (7), η is the photoelectric conversion efficiency of the photodetector; Step 3.3, according to the yaw angle α, pitch angle β and roll angle γ measured in step 3.1, calculate the current receiving tilt angle θ and the current azimuth angle ω by formula (3) and formula (4); Step 3.4, calculate the RSS parameter set using formula (6): The RSS parameter set is used as a feature fingerprint library to estimate the user's current location; Among them, the RSS parameter set includes the fingerprint point location information (x j ,y j ,z j ), and also represents the user’s current receiving direction information (θ, ω).

5. The indoor visible light high-precision positioning method based on the user's fast self-correction fingerprint library according to claim 4, It is characterized in that The specific process of step 4 is: Step 4.1, calculate the received optical power from the i-th LED With the fingerprint library The Euclidean distance of each fingerprint point in The expression is: Step 4.2, Euclidean distance Arrange from small to large to get d j , record the horizontal coordinate x of the corresponding fingerprint point k and the ordinate y k ; Step 4.3, calculate the weighting coefficient using formula (9); The expression is: Step 4.4: Estimate the user's position with the optimal number of nearest neighbors using formula (10): The expression is: In formula (10), K is the optimal number of nearest neighbors.

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