# Comprehensive Evaluations of NLOS and Linearization Errors on UWB Positioning
## 3 Tests and Evaluation
### Evaluations Based on the Simulation Test
In general, the running time for the linearization estimation methods such as LS and EKF was much shorter than these of the nonlinearization filters (i.e., UKF, CKF, PF, and RPF). Even for the same type of estimation methods, the running times were also different. For example, the running times of LS and EKF were different. This was because a time-update phase was needed in EKF. Meanwhile, the CKF algorithm presented higher execution efficiency and less running time than the other nonlinearization filters, which was due to the fewer iterations. In general, RPF costs more running time, with about 16.4 times, 20.5 times, 6.5 times, 2.3 times, 3.6 times, and 1.3 times than that of Bancroft, LS, EKF, UKF, CKF, and PF, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Bancroft** & **LS** & **EKF** & **UKF** & **CKF** & **PF** & **RPF** \\ \hline Time (s) & 1.5 & 1.2 & 3.8 & 10.7 & 6.8 & 19.3 & 24.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Algorithm running times in static simulation experiment. Figure 7: CDF of the 3D localization error. horizontal was much more reasonable than that in the vertical direction, which can be found in Figure 10. As is shown, the HDOP and VDOP were 0.79 and 77.35 on average. All of these reasons resulted in lower positioning accuracies in practical UWB tests. According to the CDF of the localization error in Figure 11, 95% of 3D position errors of the Bancroft, LS, EKF, UKF, CKF, PF, and RPF were less than 8.90 m, 4.95 m, 4.15 m, 4.05 m, 3.31 m, 3.89 m, and 2.91 m, respectively. Figure 12 shows the statistical results of UWB position errors in horizontal and vertical components of the seven algorithms in terms of RMSE and average errors, respectively. Similar to the simulation results, the horizontal positioning accuracy was much higher than the vertical positioning accuracy. Meanwhile, solutions from UKF and CKF were more accurate than that of EKF. This was because UKF and CKF acted as definite sampling filtering algorithms. While dealing with nonlinear system, sigma point sets were generated according to the sampling strategy based on the mean and covariance of the system state prior probability density. Therefore, it did not need to linearize UWB observation equations leading to no linearization errors. Meanwhile, the Jacobi matrices will not be needed during filtering algorithm iteration. However, the principle of avoiding linearization for UKF and CKF is different. For UKF, it is realized by computing the approximation term of the probability statistical eigenvalues of the nonlinear equation instead of ignoring the higher order term of the nonlinear equation after Taylor expansion [47]. The CKF is based on the theory of numerical integration and approximates the Gaussian integral by using the third-order spherical radial volume criterion [48]. Therefore, CKF is more rigorous and stable than UKF in theory. The statistical results in Figure 12 prove it visibly. For the least square method, it avoids the explicit inversion of the matrix, and guarantees the symmetry and positivity of the covariance matrix. It can be seen from Figure 12a that the 3D positioning RMSE values of Bancroft, LS, EKF, UKF, CKF, PF, and RPF were 2.86 m, 3.38 m, 2.51 m, 2.45 m, 1.89 m, 1.85 m, and 1.41 m, respectively. Compared with the solutions of Bancroft, LS, EKF, UKF, CKF, and PF, RPF upgraded the position accuracy by 50.7%, 58.3%, 43.8%, 42.4%, 25.4%, and 23.8%, respectively. This indicates that the truncation errors would degrade UWB position accuracy significantly, with the percentages of 2.5%, 3.9%, and 26.1% in the north, east, and vertical components on average. Such accuracy-decrease rates caused by NLOS error were 6.1%, 5.2%, and 25.1% in the three directions. Therefore, when UWB ranging distance contained NLOS errors, the robust algorithm in RPF can remove the impact of NLOS on UWB position accuracy. Table 6 lists the running times required for the seven algorithms.