An Ultra-Wideband Indoor Localization Algorithm with Improved Cubature Kalman Filtering Based on Sigmoid Function

Abstract

In this paper, an improved cubature Kalman filtering (CKF) is proposed using the Sigmoid function to address the problems of positioning accuracy degradation and large deviations in ultra-wideband (UWB) indoor positioning in non-line-of-sight environments. The improved CKF is based on the squared range difference (SRD) model of the time difference of arrival (TDOA) algorithm. The inaccurate impact of model estimation under non-Gaussian noise is reduced by updating the measurement noise matrix in real time. The covariance matrix is estimated using singular value decomposition (SVD) to solve the problem of degraded state estimation performance. The filtering effect of the improved CKF algorithm is evaluated by referring to the checkpoints in the dynamic trajectory. The experimental results show that the proposed algorithm effectively mitigates the impact of UWB ranging outliers in the occluded experimental environment, which makes the dynamic positioning trajectory smoother, better fitted, and more stable. The algorithm improves the positioning accuracy by up to 39.29% compared with the SRD model used alone.

1. Introduction

With the development of smart cities and Internet of Things (IoT) technology, there is a high application demand for indoor location services. High-precision indoor positioning systems are the basis for providing high-quality location services [1] and are different from traditional positioning techniques, which rely on the strength of signals received by the target object to meet the indoor high-precision demand. Ultra-wideband (UWB) positioning is a new type of short-range and high-rate wireless communication technology [2] that adopts the carrier-free spread spectrum technology, which uses an impact pulse with a very low duty cycle as the information carrier. It has a strong anti-jamming ability and can reach a centimeter-level positioning accuracy [3]. However, UWB signals are easily reflected, diffracted, or blocked during transmission, resulting in signal weakness. In addition, multipath interference and the non-light-of-sight (NLOS) phenomenon also cause large errors in the positioning results. There are three main positioning technologies for UWB: RSSI technology, AOA technology, and TOA/TDOA technology [4].

UWB position estimation methods are divided into two kinds [5]: estimating the position of the point to be measured based on the existing measurement information, or optimally estimating the position based on the existing redundant measurement information. For example, He Shijun et al. [6] proposed an algorithm based on time-of-arrival (TOA) ranging information to estimate the entire region of target existence and select the center of mass of the region as the estimation result. Svecova M et al. [7] used Kalman filtering to process UWB ranging data, which increased the localization accuracy by 21.96%. However, the Kalman filter algorithm was used under the premise that the model is a linear Gaussian model, which can optimally estimate the linear Gaussian system state and reduce its robustness in the case of complex environmental noise. To reduce the nonlinear error problem, Canadian scholar Arasaratnam proposed a nonlinear filtering tracking cubature Kalman filtering (CKF) algorithm based on the spherical–radial cubature rule [8], achieving significantly superior accuracy and efficiency to those of other filters. Li Z et al. [9] used the three-dimensional spherical–radial rule for the CKF algorithm to approximate the Gaussian system with Taylor series third-order accuracy under non-Gaussian noise conditions. It obtained better numerical accuracy. Zhang M. et al. [10] proposed a new positioning algorithm based on the current statistics (CS) model and CKF algorithm for the nonlinear tracking problem in traditional UWB indoor positioning. The algorithm improves the positioning accuracy of UWB indoor positioning. Hongqiang Z. et al. [11] designed an adaptive square root cubature Kalman filter (ASRCKF) positioning algorithm to solve the problem of target tracking in the research field, and the modified algorithm achieved a positioning accuracy of 15 cm in the dynamic tracking environment. However, the CKF algorithm often reduces the numerical accuracy when calculating the estimated mean square deviation because it does not satisfy the positive definiteness [12]. In addition, it is not adaptive, which will significantly impact or even cause divergence in the filtering performance when the model does not match the target.

This study uses the CKF algorithm to process UWB localization data based on the squared range difference (SRD) model of distance, which is fundamental to the time difference of arrival (TDOA) algorithm, to address the problem of accuracy degradation in UWB dynamic moving target localization. Moreover, this paper proposes an improved adaptive algorithm by combining the Sigmoid function and the singular value decomposition (SVD) method to address the poor positioning accuracy under non-line-of-sight conditions. The Sigmoid adaptive function is used to update the measurement noise matrix, and singular value decomposition is used to decompose the covariance estimation. They are employed to improve the accuracy and stability of the UWB positioning system. Finally, the improved CKF algorithm demonstrated a stronger dynamic stability performance and higher positioning accuracy via dynamic trajectory and static reference checkpoint experiments.

The rest of this paper is organized as follows: The SRD model based on the TDOA algorithm is presented in Section 2. The improved adaptive CKF algorithm based on the Sigmoid function and SVD decomposition method is given in Section 3. The experimental testing and evaluation of the proposed algorithm are detailed in Section 4. Finally, conclusions and future research arrangements are drawn in Section 5.

2. SRD Model Based on TDOA Algorithm

The TDOA algorithm is calculated by locating the difference in timestamps between the signal sent by the tag and the signal received at each base station [13]. Based on the knowledge of mathematical geometry, it is known that at least three distance differences are required to determine the position of a point in three-dimensional space. The trajectory of a dynamic fixed point at a distance difference from two base stations is a hyperbola [14,15]. The schematic diagram of the TDOA localization algorithm is shown in Figure 1.

UWB positioning tags send a UWB signal to the outside, and all base stations within the wireless coverage range of the tags will receive the wireless signal. The distance between the tags and different UWB positioning base stations is different, and the time nodes at which different base stations receive the signal of the same tag are different, thus obtaining an “arrival time difference”. There is a target M with coordinates $(x,y,z)$ and four base stations, which include one primary base station and three secondary base stations with coordinates $(x_i,y_i,z_i),i=1,2,3,4$. The distance $d_i$ from the target M to each base station is written as follows (1):

$$d_i=\sqrt{{(x-x_i)}^2+{(y-y_i)}^2+{(z-z_i)}^2}$$

(1)

In this way, three or more localization base stations are required to determine the location of the target to be measured [16]. $d_{i,1}$ represents the difference in distance from the target M to the primary base station, and each of the secondary base stations is defined as follows (2):

$${d_{i,1}}^2={(d_i-d_1)}^2$$

(2)

where $d_1$ represents the distance from target M to the main base station as $d_1=\sqrt{{(x-x_1)}^2+{(y-y_1)}^2+{(z-z_1)}^2}$.

Because it is too complex, the calculation process of the localization algorithm is based on the squared-range-based least squares method. The SRD is simple to calculate and has strong stability. In conjunction with the SRD model, Equation (2) is rewritten in vector form as follows (3):

$${\widehat{d}}_{i,1}^2={(d_1^2-||x-d_i|{|}^2)}^2$$

(3)

where $\left|\right|•\left|\right|$ denotes the $•$ Euclidean distance to be obtained.

In this study, only planar coordinates are considered for UWB indoor localization in conjunction with the characteristics of the selected indoor environment. The least squares method could be used to solve the hyperbolic nonlinear system of equations using the TDOA method [17]. Equation (3) can be equated to the following constraints (4):

$$\min {\displaystyle \sum _{i=2}^4(2d_{i}x-{\left|\right|x\left|\right|}^2+}{d}_1^2-{\left|\right|d_i\left|\right|}^2{)}^2$$

(4)

where $x=\left[\begin{array}{c}x\\ y\end{array}\right]$. Letting $A=\left[\begin{array}{l}2d_2-1\\ 2d_3-1\\ 2d_4-1\end{array}\right],$ $b=\left[\begin{array}{c}\left|\right|d_2|{|}^2-d_1^2\\ \left|\right|d_3|{|}^2-d_1^2\\ \left|\right|d_4|{|}^2-d_1^2\end{array}\right]$, the above problem can be written as follows (5):

$$\min \left|\right|Ax-b|{|}^2$$

(5)

The residuals of the function are $\epsilon =Ax-b$. Then, the formula for the sum of squares of the residuals of the function is defined as follows (6):

$$f(x)={(Ax-b)}^2=(Ax-b){(Ax-b)}^{\mathrm{T}}$$

(6)

Deriving both sides of the above Equation (6) with respect to $x$, the position coordinates of the target to be measured can be obtained, as shown in Equation (7):

$$x={(A^{\mathrm{T}}A)}^{-1}{A}^{\mathrm{T}}b$$

(7)

In practice, the measurement error is constantly changing. The disadvantage of the least squares algorithm is that each term in the residual sum-of-squares function has the same weight [18], which makes the whole localization unsatisfactory. To solve this problem, the weighted least squares method is used to control the accuracy of the whole measurement process [19], where the larger the residuals, the smaller the weight of the term, and vice versa. The weighted least squares expression is shown in Equation (8), where $W$ represents the weighting matrix:

$$x={(A^{\mathrm{T}}WA)}^{-1}{A}^{\mathrm{T}}Wb$$

(8)

3. Improved CKF Localization Algorithm Based on Volume Criterion

In UWB indoor localization, the position of the phase centroid of the localization tag and the velocity information are used as state vectors for filtering, as follows [20]:

$$x_k=F_{k,k-1}{x}_{k-1}+{\omega }_k$$

(9)

where $x=[p_x,p_y,v_x,v_y]$ represents the state vector. $(P_x,P_y)$ represents the planar position coordinates of the phase center point of the positioning tag. $v_x,v_y$ represents the planar velocity of the phase center point of the positioning tag. ${\omega }_k$ represents the noise vector at time $k$, and its corresponding covariance matrix is $Q_k$.

$F_{k,k-1}$ represents the state transfer matrix at time $k$, as follows: $F_{k,k-1}=\left[\begin{array}{cccc}1& 0& \Delta T& 0\\ 0& 1& 0& \Delta T\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$.

The measurement model of the system is as follows (10):

$$Z_k=H_k{x}_k+v_k$$

(10)

where $Z_k$ represents the measured distance from the UWB positioning tag to the ith base station at time $k$, where $i=1,2,\dots ,\mathrm{N}$. $\mathrm{N}$ represents the number of base stations. $d_{i,k}$ represents the true distance from the positioning tag to the ith base station at time $k$. $v_k$ represents the observation noise vector of the positioning tag at time $k$, and its corresponding covariance matrix is $R_k$.

$H_k$ represents the observation matrix at time $k$, where $H_k=\left[\begin{array}{cccc}\frac{\partial d_{i,1}}{\partial p_{x,1}}& \frac{\partial d_{i,1}}{\partial p_{y,1}}& 0& 0\\ \frac{\partial d_{i,2}}{\partial p_{x,2}}& \frac{\partial d_{i,2}}{\partial p_{y,2}}& 0& 0\\ ⋮& ⋮& ⋮& ⋮\\ \frac{\partial d_{i,k}}{\partial p_{x,k}}& \frac{\partial d_{i,k}}{\partial p_{y,k}}& 0& 0\end{array}\right]$.

UWB ranging still has some outliers under non-line-of-sight error conditions, which seriously affects the accuracy and stability of the filtering results [21]. To solve this problem, this study combines the SRD model based on the TDOA algorithm with the CKF algorithm, utilizes the SVD decomposition method to solve the problem of the estimated covariance matrix being unable to meet the positive qualitative requirements in real time, and uses the Sigmoid adaptive function to update the measurement noise matrix $R$ in real time. They reduce the problem of inaccuracy of the model estimation affecting the filtered results under non-Gaussian noise, which allows for higher accuracy.

3.1. SVD Method Decomposes Estimated Covariance Array

The volume criterion is used to pass a series of point sets with a nonlinear function and then approximate the posterior mean and variance of the nonlinear function based on the weighted summation [22]. When the standard CKF algorithm is used to calculate the volume points, Cholesky matrix decomposition is usually used to solve the estimated covariance matrix $P_{k-1|k-1},P_{k|k-1}$. After several filters, the estimated covariance matrix in the algorithm easily loses the positive characterization. When the system covariance matrix is affected by the complex environment to form a pathological matrix, the CKF algorithm cannot decompose the covariance matrix via Cholesky decomposition, which leads to the failure of the whole filtering algorithm. In this study, SVD decomposition is used to decompose and estimate the covariance matrix based on the CKF algorithm. Its estimation accuracy is improved by up to 37.4% over three degrees of freedom compared with Cholesky decomposition [23]. In numerical matrix decomposition, the SVD method reduces the dimensionality by retaining only the most important singular values and vectors and estimates the pure signal by removing the noisy signal vector components that fall in the noise space. Therefore, the SVD decomposition method can improve the robustness of numerical computation and enhance the stability of the CKF algorithm.

3.1.1. State Prediction Section

The estimated covariance is decomposed using the SVD method as follows (11):

$$P_{k-1|k-1}=U_{k-1}\left[\begin{array}{ll}{S}_{k-1}& 0\\ 0& 0\end{array}\right]V_{K-1}^{\mathrm{T}}$$

(11)

where $S_{k-1}$ generally denotes the diagonal matrix. $S_i$ represents the eigenvalue of $P_{k-1|k-1}$. The volume point is calculated as follows (12):

$$X_{i,k-1|k-1}=U_{k-1}\sqrt{S_{k-1}}{\epsilon }_i+{\widehat{x}}_{k-1|k-1}$$

(12)

where ${\epsilon }_i=\sqrt{n}{\left[1\right]}_i$. For a two-dimensional sphere $(n=2)$, the sphere has intersections with both axes, as follows: ${\left[1\right]}_i=\left[{(1,0)}^{\mathrm{T}},{(0,1)}^{\mathrm{T}},{(-1,0)}^{\mathrm{T}},{(0,-1)}^{\mathrm{T}}\right]$. ${\left[1\right]}_i$ denotes the ith column of the set of $\left[1\right]$.

$X_{i,k-1|k-1}^{*}$ represents the new sampling point obtained by propagating the volumetric point through the state equation.

$$X_{i,k-1|k-1}^{*}=F(X_{i,k-1|k-1})$$

(13)

The further state prediction value ${\widehat{x}}_{k|k-1}$ and prediction covariance $P_{k|k-1}$ at moment $k$ are as follows (14):

$$\{\begin{array}{l}{\widehat{x}}_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{X}_{i,k-1|k-1}^{*}}\\ P_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{X}_{i,k-1|k-1}^{*}{X}_{i,k-1|k-1}^{*T}}-{\widehat{x}}_{k|k-1}{\widehat{x}}_{k|k-1}^T+Q_{k-1}\end{array}$$

(14)

3.1.2. Measurement Component

The SVD decomposition of the further prediction covariance $P_{k|k-1}$ is performed, and the time $k$ volume points are passed through the quantile equation to obtain new sampling points $Z_{i,k|k-1}$ as follows (15):

$$Z_{i,k|k-1}=H(X_{i,k|k-1})$$

(15)

${\widehat{Z}}_{k|k-1}$ represents the predicted value of the measurement at time $k$.

$${\widehat{Z}}_{k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{Z}_{i,k|k-1}}$$

(16)

The autocorrelation covariance matrix $P_{zz,k|k-1}$ and the inter-correlation covariance matrix $P_{xz,k|k-1}$ for ${\widehat{Z}}_{k|k-1}$ are calculated as follows (17):

$$\{\begin{array}{l}{P}_{zz,k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{Z}_{i,k|k-1}}{Z}_{i,k|k-1}^{\mathrm{T}}-{\widehat{Z}}_{k|k-1}{\widehat{Z}}_{k|k-1}^{\mathrm{T}}+R_k\\ P_{xz,k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{X}_{i,k|k-1}}{Z}_{i,k|k-1}^{\mathrm{T}}-{\widehat{x}}_{k|k-1}{\widehat{Z}}_{k|k-1}^{\mathrm{T}}\end{array}$$

(17)

3.2. CKF Parameter Adaptive Algorithm Based on Sigmoid-like Functions

The Sigmoid function is a common S-shaped growth curve in biology, which is characterized by strict monotonicity and good continuity. It is an excellent threshold function [24], with the following expression (18):

$$f(x)=\frac{1}{1+e^{-x}}$$

(18)

In the CKF algorithm, the calculation of the measurement noise matrix $R$ and error covariance matrix $P$ is closely related to the accuracy of individual measurement errors. Based on the CKF algorithm, this study defines a kind of Sigmoid function, combines the advantages of the function to freely adjust the boundaries of the function, and updates the measurement noise matrix $R$ in real time. Thus, the proposed algorithm can obtain higher estimation accuracy and stronger stability.

$$R[e(k)]=\lambda \frac{{\mathrm{e}}^{\mu e(k)}-{\mathrm{e}}^{-\mu e(k)}}{{\mathrm{e}}^{\mu e(k)}+{\mathrm{e}}^{-\mu e(k)}}$$

(19)

where $e(k)=|x(k)-\widehat{x}(k)|$ represents the absolute value function of the difference in the position of the label at a given moment. $\mathrm{e}$ is the base of the natural logarithm, and $\lambda ,\mu$ represents a real number.

In the standard CKF algorithm solving process, the ranging anomalies have a significant impact on the results of the measurement update. The proposed algorithm is based on a kind of Sigmoid function of the CKF parameter adaptive method and the real-time update of the measurement noise matrix $R$. Thus, the correction of $P_{zz,k|k-1}$ is as follows (20):

$${\overline{P}}_{zz,k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{Z}_{i,k|k-1}}{Z}_{i,k|k-1}^T-{\widehat{Z}}_{k|k-1}{\widehat{Z}}_{k|k-1}^{\mathrm{T}}+R[e(k)]$$

(20)

${\overline{K}}_k$ represents the system Kalman filter gain matrix.

$${\overline{K}}_k=P_{xz,k|k-1}{\overline{P}}_{zz,k|k-1}^{-1}$$

(21)

The state estimate $x_{k|k}$ and the estimated covariance $P_{k|k}$ at moment $k$ are as follows (22):

$$\{\begin{array}{l}{x}_{k|k}={\widehat{x}}_{k|k-1}+\overline{K}(z_k-{\widehat{z}}_{k|k-1})\\ P_{k|k}=P_{k|k-1}+{\overline{K}}_k{\overline{P}}_{zz,k|k-1}{\overline{K}}_k^{\mathrm{T}}\end{array}$$

(22)

3.3. Precision Evaluation Methods

In this study, some fixed reference checkpoints are selected to evaluate the performance of the positioning algorithm of the UWB system during the dynamic trajectory measurement of the positioning tags. The differences in the distance between the best valuation obtained by the localization algorithm and the base station location, as well as the distance between the reference checkpoint and the base station location are compared using the distance residual value.

$${\delta }_l=\sqrt{\frac{v_i-{\widehat{x}}_i}{n}}$$

(23)

where ${\delta }_l$ represents the residual of the distance between the best valuation of the localization algorithm and the base station location, as well as that of the reference checkpoint and the base station location. $v_i=\sqrt{l_i-x_i}$ represents the distance between the reference checkpoint and the base station location. ${\widehat{x}}_i=\sqrt{{\widehat{l}}_i-x_i}$ represents the distance between the best valuation of the localization algorithm of the reference checkpoint and the base station location. $l_i$ represents the true coordinates of the reference checkpoint. $x_i$ represents the base station location coordinates. ${\widehat{l}}_i$ represents the best valuation of the location of the localization algorithm, and $n$ represents the number of times the reference checkpoint has been localized, where $i=1,2,3\dots$.

In this study, the Sigmoid-like adaptive function is used to update the measurement noise matrix $R$ in real time. Based on the CKF algorithm, the estimation accuracy of the localization results is improved when the position error is large, and the stability of the algorithm is maintained when the position error is small to satisfy the parameter change requirements of the system. It improves the performance of the whole localization algorithm. The specific process is shown in Figure 2.

4. Experimental Validation and Analysis

4.1. Experimental Validation of the Improved Algorithm Model

The effectiveness of the proposed improved CKF algorithm compared with the traditional SRD algorithm model for solving UWB indoor localization position was experimentally verified and evaluated in an indoor environment in this study.

In the experiment, the indoor environment was selected as an experimental space with a size of 9 m × 6 m. The laboratory was equipped with tables, computers, and other obstacles that could be verified for non-line-of-sight distance. The experimental platform was the UWB indoor localization system developed by Zhengzhou Lianrui Electronic Science and Technology Co. (Zhengzhou, China). The system utilized the TDOA algorithm to obtain the UWB location tag result, enabling subsequent data processing. In the experimental process, the relevant personnel held UWB positioning tags along the table edge to carry out dynamic trajectory measurements. The coordinates of the four positioning base stations were represented as (0 cm, 0 cm, 189 cm), (0 cm, 720 cm, 189 cm), (400 cm, 0 cm, 189 cm), and (400 cm, 720 cm, 189 cm). When reading data from a UWB ranging sensor, the refresh frequency of programmable logic controller (PLC), set at 50 Hz, is followed according to communication protocol requirements. The ranging accuracy of UWB is 50~100 cm. The data processing software was matlab2016b. The experimental scenario is shown in Figure 3.

This study only considered the localization results on the two-dimensional plane since this experiment was carried out on a smooth indoor floor and the UWB positioning tag changes extremely little under elevation. In the case of no obstacle occlusion, the UWB positioning signal was not occluded by the personnel when they held the UWB positioning tag for trajectory measurement along the table edge. During the measurement process, some feature points (such as the table corner, etc.) were selected as the reference checkpoints of the whole localization process to evaluate the algorithmic accuracy of the UWB localization system. In the case of obstacle occlusion, such as the relevant personnel holding tightly onto the UWB positioning tags, non-line-of-sight conditions were realized. The rest of the experimental operations were the same as above. The UWB positioning trajectories are shown in Figure 4 and Figure 5, and the residual comparison curve plots of the reference checkpoints are shown in Figure 6, Figure 7 and Figure 8.

Figure 4 represents the dynamic trajectory before filtering is carried out by the SRD algorithm and after filtering is carried out by the SRD + CKF algorithm, and the real motion trajectory. The dynamic trajectory after the combined SRD + CKF algorithm is smoother and more concentrated than the trajectory before filtering. The whole algorithm has fewer points with larger deviations away from the real trajectory, which indicates that the CKF algorithm can improve the stability of the whole localization system. Figure 5 shows the dynamic trajectories of the SRD algorithm before filtering, the SRD + CKF algorithm, and the improved SRD + CKF algorithm trajectory. The dynamic trajectory points of the improved SRD + CKF algorithm are more intensive compared with the unimproved algorithm, and have fewer dispersion points that deviate from the trajectory, which is closer to the real trajectory. In the case of obstacles, the discrete points of the improved filtering algorithm can remain concentrated for some time. There are fewer dispersion points than in the unimproved algorithm, which indicates that the Sigmoid function can strengthen the system’s overall stability based on the CKF algorithm. The advantages of the Sigmoid function are also better reflected. The parameters of the Sigmoid function in this study were obtained as $\lambda =1,\mu =0.1$ by conducting several experiments to better verify the specific effect of the proposed algorithm. Figure 6, Figure 7 and Figure 8 show the X-axis error, Y-axis error, and total residual distribution curve of the reference checkpoints of each algorithm and the improved algorithm. The distance residuals of the reference checkpoints of the SRD + CKF algorithm and the improved SRD + CKF algorithm are smaller than those of the SRD algorithm, and the improved algorithm demonstrates a much better accuracy than the SRD algorithm and the unimproved SRD + CKF algorithm. Comparing the SRD algorithm and the unimproved SRD + CKF algorithm, the accuracy was improved by a maximum of 39.29% and 30.08%. The specific values are shown in

Table 1. Comparison of average distance residual values for each algorithm.

Distance Parameter	Reference Checkpoint	SRD	SRD + CKF	Improved SRD + CKF
${\delta }_l/dm$	Point 1	4.6407	4.3768	3.3083
	Point 2	3.3149	2.9288	2.1716
	Point 3	3.7049	3.2166	2.2489
	Point 4	4.2658	3.9851	3.1267

.

4.2. Experimental Analysis of Inaccurate Algorithmic Model Estimation

This study compares the experiments of the SRD + CKF algorithm and the improved SRD + CKF algorithm when the estimated covariance matrix is decomposed without using the SVD decomposition method to better verify the performance of the SVD decomposition method, which can solve the matrix non-positive characterization problem, and the Sigmoid function, which can update the measurement noise matrix in real time. Moreover, this study also compares the experiments wherein the measurement noise covariance matrix of the improved algorithm is not updated using the Sigmoid function but continues to use the noise $R(299th)$ from the previous moment at the reference checkpoint in the interval of (300, 400th) times. At this point in the interval, the estimated covariance matrix ${\overline{P}}_{zz,k|k-1}$ is expressed as follows (24):

$${\overline{P}}_{zz,k|k-1}=\frac{1}{2n}{\displaystyle \sum _{i=1}^{2n}{Z}_{i,k|k-1}}{Z}_{i,k|k-1}^T-{\widehat{Z}}_{k|k-1}{\widehat{Z}}_{k|k-1}^{\mathrm{T}}+R(299th)$$

(24)

The X-axis and Y-axis residual values as well as the total distance residual values for the SRD algorithm, the SRD + CKF algorithm, and the improved SRD + CKF algorithm without using SVD decomposition are given in

Table 2. Average distance residuals for each algorithm without SVD decomposition.

Prediction Algorithms	State Variable	Distance Residual Value/dm
SRD	X-direction error	9.5757
	Y-direction error	10.1263
	Total residuals	10.5804
SRD + CKF	X-direction error	12.5269
	Y-direction error	9.9273
	Total residuals	15.9823
Improved SRD + CKF	X-direction error	10.7342
	Y-direction error	8.8997
	Total residuals	13.9473

. Based on Table 2, when SVD decomposition is not used, Cholesky decomposition cannot satisfy the estimation of the covariance matrix to maintain positive definiteness. Moreover, when the system covariance matrix forms a pathological matrix under the influence of the complex environment, the CKF algorithm is ineffective. The two-dimensional residual values and the total residual values of the CKF algorithm and the improved CKF algorithm in the table are larger than those of the pre-filtering, and the improved algorithms cannot achieve the expected results.

Figure 9 presents the distribution curve of the total residual values when the measurement noise matrix is updated in real time without using the Sigmoid function at approximately the (300, 400th) localization of the reference checkpoint. The measurement noise matrix of the 299th time is constantly used in this interval. The total residual value of the improved CKF algorithm without using the Sigmoid function update is larger than that of the SRD algorithm and the SRD + CKF algorithm, and the stability of the CKF algorithm cannot be better reflected when the environment changes. The maximum residual value of the improved CKF algorithm can reach 23 dm, which is much higher than that of the remaining two algorithms. After the 400th localization, the distance parameter value of the improved algorithm is lower than that of the remaining two algorithms while continuing to use the Sigmoid function to update the measurement noise matrix, which, in turn, verifies that the improved CKF algorithm based on the Sigmoid function improves the accuracy and robustness of indoor UWB localization.

5. Conclusions

Despite its high positioning accuracy, the positioning accuracy of the UWB positioning system when used for indoor positioning is still affected by the non-line-of-sight environment during ranging. Based on the SRD model of the TDOA algorithm, this paper proposed an improved CKF parameter adaptive indoor positioning algorithm, using the Sigmoid function to update the measurement noise matrix in real time. This algorithm was used to correct the dynamic trajectory of the UWB positioning system in the non-line-of-sight environment. The stability and positioning accuracy of the whole positioning system were greatly improved. In the experimental process, in this study, we designed dynamic trajectory static reference checkpoints for UWB positioning in the non-line-of-sight environment and compared the trajectory fitting and accuracy of the proposed algorithm. This study also compared the residual comparison experiments without using SVD decomposition to estimate the covariance matrix, and without using the Sigmoid function to update the measurement noise covariance matrix.

The experimental results show that the improved UWB indoor positioning algorithm based on the Sigmoid function proposed in this paper can solve the problem of maintaining the positive characterization of the estimated covariance. Furthermore, it can reduce the dispersion points away from the real trajectory and improve the outliers in the non-visual range environment during the ranging process of UWB dynamic positioning. Moreover, it also enhances the stability and accuracy of the UWB positioning system. The elevation was not considered in this experiment, which could be enhanced in future research. The next step is to use this method for multi-sensor fusion to improve the accuracy of multi-source fusion indoor positioning.