## 4 Experimental Validation and Analysis
### 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 \(\mathbf{R}(299 th)\) from the previous moment at the reference checkpoint in the interval of (300, 400 th) times. At this point in the interval, the estimated covariance matrix \(\overline{\mathbf{P}}_{zz,k|k-1}\) is expressed as follows (24):  
\[\overline{\mathbf{P}}_{zz,k|k-1}=\frac{1}{2n}\sum_{i=1}^{2n}\mathbf{Z}_{i,k|k-1}\mathbf{Z}_ {i,k|k-1}^{T}-\mathbf{\hat{Z}}_{k|k-1}\mathbf{\hat{Z}}_{k|k-1}^{T}+\mathbf{R}(299 th) \tag{24}\]  
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. 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, 400 th) localization of the reference checkpoint. The measurement noise matrix of the 299 th 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 400 th 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.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Prediction Algorithms** & **State Variable** & **Distance Residual Value/dm** \\ \hline \multirow{4}{*}{SRD} & X-direction error & 9.5757 \\  & Y-direction error & 10.1263 \\  & Total residuals & 10.5804 \\  & X-direction error & 12.5269 \\  & Y-direction error & 9.9273 \\  & Total residuals & 15.9823 \\  & X-direction error & 10.7342 \\ Improved SRD + CKF & Y-direction error & 8.8997 \\  & Total residuals & 13.9473 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Average distance residuals for each algorithm without SVD decomposition.