As shown in Figure 1, in the offline process, the terminal device will collect fingerprint data at the preset sampling point and preprocess it, and then use AAPC algorithm to divide the obtained fingerprint into different fingerprint members and corresponding fingerprint database. The online phase will also be divided into two processes—“positioning” and “tracking”. The positioning stage is divided into two stages: coarse positioning and fine positioning. In the coarse positioning stage, when the mobile terminal sends a positioning query request, the positioning engine will determine which feature set its data belongs to by calculating

the similarity between the location data corresponding to the request and the related signals in the reference database. In the process of fine positioning, “CS + positioning algorithm” is used to obtain the position estimation. Finally, in the tracking phase, a method combining Kalman filter and physical constraints is used to track the trajectory.

In order to ensure the accuracy and reliability of the experimental data, when the mobile terminal reaches the reference point, it needs to collect several data of an AP in four directions, East, West, North and South (40 in the article), and process the data obtained in each direction to obtain the average value of the signal strength of an AP in each direction corresponding to the mobile terminal at this point, using r s s i , j d . Where d represents the direction of the mobile terminal, i represents the AP number corresponding to the data, and j represents the location of the reference point where the mobile terminal is located. Then the mathematical expression of the average signal strength of all APS corresponding to this point is shown in formula (1), and l represents the number of APS that can be detected by the mobile terminal at this reference point.

r s s j d = [ r s s 1 , j d , r s s 2 , j d , ⋯ , r s s i , j d , ⋯ , r s s L , j d ] (1)

So the data of multiple APS detected at the r-th test point can be expressed as:

r s s r = [ r s s 1 , r , r s s 2 , r , ⋯ , r s s i , r , ⋯ , r s s L , r ] (2)

Then in the coarse positioning process, the similarity between the calculated signal strength and the class representation can be expressed as:

s i m r , c = | r s s r − c h c | 1 (3)

where the number of clusters is C, Ch_c represents the class representation.

In the process of fine positioning, we calculate rss_r and the final positioning result obtained by matching the similarity between each member in the class.

Using compressive sensing algorithm to process signals first requires clustering processing of the signals [17] . Here, we use AAPC clustering algorithm [18] , after the location fingerprint is collected, AAPC clustering algorithm is used to generate different fingerprint categories.

p e = p n , C n > C t p s = p n , C n < C t p n + 1 = 0.5 ⋅ ( p e + p s ) } (4)

Formula (4) shows that AAPC uses dichotomy to dynamically change the bias parameter p to get the clustering result, and the clustering number is [2, N ]. p_e and p_s represents the starting and ending values of the dichotomy respectively. N is the number of iterations, C_t is the number of clusters generated. When C_i = C_t, the clustering result Ch_c is obtained, N is the total number of reference points.

Because the location of the mobile terminal at a certain time in the positioning area is unique with the corresponding best reference point, the current user’s location can be expressed as a 1 sparse vector, so the problem of indoor positioning can be transformed into how to recover the sparse vector. Sparse vector 𝒂 is shown in formula (5):

a = [ 0, ⋅ ⋅ ⋅ ,0,1,0, ⋅ ⋅ ⋅ ,0 ] ⊤ (5)

According to the principle of compressed sensing rss_r can be expressed as:

r s s r = c c n ⋅ a + ε (6)

In formula (6), c_cn epresents the best matching class generated by AAPC clustering algorithm using formula (3), and ε represents the environmental noise in practical applications. Multiply both sides of the equation by the perception matrix P, where P is the AP selection matrix:

Y = P ⋅ r s s r = P ⋅ c c n ⋅ a + ε (7)

Due to the certain spatial correlation between P and c_cn epresents the best matching class generated by AAPC clustering algorithm using for the non correlation condition of the sensing matrix is not satisfied. Therefore, the perception matrix P and c_cn are orthogonalized. Let Z = MY, and matrix A be an orthogonalized matrix. Therefore, the problem is solved by minimizing the l₁ norm in formula (8).

a ^ = arg min ‖ a ‖ 1 ,   s .   t .     Z = A ⋅ a + M ⋅ ε (8)

‖   ‖ is the norm of vector l₁ ,formula (9) represents the set of values in a that are greater than the threshold value, denoted by D.

D = { n | a ( n ) > λ } (9)

The final position estimate is:

L O C ( x , y ) = ∑ n ∈ D a ( n ) ( x n , y n ) ∑ n ∈ D a ( n ) (10)

The WKNN positioning method ensures the positioning accuracy of the algorithm to a certain extent when the external conditions of the experiment meet the requirements, and due to its simplicity, it has become a relatively classic and commonly used algorithm in indoor positioning. The principle is as follows: first, select the K best matched candidate reference point positions, and then use these K positions to weighted sum up to obtain the final position estimation as shown in formula (11):

L O C ( x , y ) = ∑ l = 1 K L O C ( x l , y l ) ∑ l = 1 K ω l (11)

Let 𝒂 solve for non-zero positions as a ˙ , In an ideal environment, when there is little or no environmental noise, a ˙ can be obtained. However, in practical application environments, the impact of environmental noise is inevitable, resulting in a value of a ˙ , not being 1, and the degree of deviation is positively correlated with the size of environmental noise. Based on this conclusion, the accuracy of the localization results obtained by the CS algorithm can be determined. Take threshold δ = 0.1, determine the size relationship between a ˙ and threshold. When a ˙ is less than the threshold, it can be considered that the deviation from 1 is small, that is, the CS algorithm is less affected by environmental noise; on the contrary, when a ˙ is greater than the threshold, the environmental impact is greater. Therefore, the joint localization method proposed in the article adopts the CS algorithm results when the environmental noise impact is small, because the signal reconstruction effect of CS is better at this time; When subjected to significant environmental noise, the results of online positioning algorithms are more accurate. So the final position estimation is shown in formula (12):

L O C ( x , y ) = { ∑ l = 1 K L O C ( x l , y l ) ∑ l = 1 K ω l | a ˙ − 1 | > δ ∑ n ∈ D a ( n ) ( x n , y n ) ∑ n ∈ D a ( n ) | a ˙ − 1 | ≤ δ (12)

The traditional Kalman filter [19] includes two parts: the prediction process and the correction process. Based on the minimum mean square error, the optimal state of the current time state of the system is estimated through the observation data and the previous time state of the system. But in the process of indoor positioning, the noise in the environment is uncontrollable, and the noise matrix q_k and r_k is usually not zero, which will affect the accuracy of traditional Kalman filter prediction results. Aiming at the disadvantage of traditional Kalman filter, this paper uses sage Husa adaptive Kalman filter algorithm to realize the user’s trajectory tracking. Sage Husa adaptive Kalman filter is a filtering method with the function of restraining filter divergence. It adds the idea of dynamic statistical estimation in the process of Kalman filter prediction and correction. On the one hand, it uses the observation value to constantly correct the prediction value, and on the other hand, it estimates and corrects the unknown system model parameters and noise statistical parameters. The formula is shown in (13)-(22). P is the variance of state estimation error and K is the gain matrix.

x ^ k − = F x ^ k − 1 − + q k + 1 (13)

P k − = F P k − 1 F ⊤ + Q k − 1 (14)

K k = P k − H ⊤ ( H P k − H ⊤ + R k ) − 1 (15)

e k = Z k − H x ^ k − − r k − 1 (16)

x ^ k = x ^ k − + K k e k (17)

P k = ( I − K k H ) P k − (18)

Add a forgetting factor to adaptively adjust the noise matrix:

q k = ( 1 − d k ) q k − 1 + d k ( x ^ k − F x ^ k − 1 − ) (19)

Q k = ( 1 − d k ) Q k − 1 + d k ( K k e k e k T K k T + P k − F P k − 1 F T ) (20)

r k = ( 1 − d k ) r k − 1 + d k ( Z k − H x ^ k − ) (21)

R k = ( 1 − d k ) R k − 1 + d k ( e k e k T − H P k − H T ) (22)

In the formula d k = ( 1 − b ) ( 1 − b k + 1 ) , b is the forgetting factor, with a value range of (0, 1).

The experimental environment used in this paper is shown in Figure 2. There are 538 reference points (RPS) marked by red dots. The experimental environment covers an area of 50 * 18 square meters, and the number of available APS is 339. During the experiment, when an AP signal cannot be detected, the default value is −100 dbm. The parameters of the experimental equipment are shown in Table 1.

In this paper, the box chart is used to display the error distribution of positioning results. Each box chart contains six values: the upper edge (1.5 times), the upper quartile (75%), the mean value (represented by a small square in the figure), the median value, the lower quartile (25%), and the lower edge (1.5 times). In the box graph, we can clearly see the average and edge distribution of the positioning error of the algorithm. As shown in Figure 3, this paper compares the positioning accuracy of WKNN, SAWKNN and RWKNN algorithms before and after CS processing, and it can be seen that the positioning accuracy of the positioning algorithm processed by CS algorithm has been improved to a certain extent. The upper quartile of the three algorithms has decreased from 1.45 M,

1.39 M and 1.45 M to 1.17 M, 1.14 M and 1.18 M respectively, while the lower quartile is 0, which means that the overall larger error points of the algorithm are decreasing and the smaller error points are increasing, The positioning accuracy is improved by 23%, 10% and 22% respectively.

Combined with the static positioning results of the user, the current position information of the user is used as the prediction value of the adaptive Kalman filter, and the observation value is the result of the algorithm proposed in this paper, that is, L O C ′ ( x r , y r ) is used as the input of the filtering system, in which the forgetting factor B is 0.5, and x ˙ k , y ˙ k , x k , y k is used to represent the displacement and velocity on the X and Y axes respectively. In the ideal state, the motion state is uniform linear motion, so there is the state equation (23), which shows the state transition matrix F, The observation vector is shown in formula (24), where Z_k1 and Z_k2 represent the observation values in the x-axis and y-axis respectively. Equation (24) gives the expression of the observation equation, which shows that the observation matrix H, the positioning interval T in the experiment is 1s, q_k and r_k initial value of is 0. Formulas (25) and (26) are the initial values of Q k , R k respectively.

[ x ˙ k y ˙ k x k y k ] = [ 1 0 T 0 0 1 0 T 0 0 1 0 0 0 0 1 ] [ x ˙ k − 1 y ˙ k − 1 x k − 1 y k − 1 ] + W k ,   W k ~ ( 0 , Q ) (23)

[ Z k 1 Z k 2 ] = [ 1 0 0 0 0 0 1 0 ] [ x ˙ k y ˙ k ] + V k ,   V k ~ ( 0 , R ) (24)

Q k = 50 ⋅ [ T 3 3 0 T 2 2 0 0 T 3 3 0 T 2 2 T 2 2 0 T 0 0 T 2 2 0 T ] (25)

R k = [ 2 0 0 0 2 0 ] (26)

Figures 4-6 show the trajectory comparison of three algorithms, WKKN, SAWKNN, and RWKNN, processed by Sage-Husa adaptive Kalman filtering. The red line represents the actual trajectory, and the black line represents the trajectory tracking trajectory. The error in trajectory optimization can be divided into two categories: normal error and abnormal error. For normal errors, adaptive Kalman filtering can be directly applied for optimization, as it follows the laws of physical motion. For abnormal errors, physical constraints need to be added first to ensure that the positioning point does not collide with obstacles, while limiting the maximum speed to avoid excessive displacement. After

experiments, the ideal trajectory tracking effect was ultimately achieved.

Figure 7 shows the distribution of positioning errors of WKNN, SAWKNN, and RWKNN algorithms after Sage Husa adaptive Kalman filtering. The average positioning errors are 0.65 m, 0.49 m, and 0.61 m, respectively, further improving the positioning accuracy.