# Comprehensive Evaluations of NLOS and Linearization Errors on UWB Positioning
## 3 Tests and Evaluation
### Evaluations Based on the Simulation Test  
Figures 4 and 5 show the position error in horizontal and vertical in terms of scatter diagrams and time series, which were calculated by the seven adjustment algorithms (Bancroft, LS, EKF, UKC, CKF, PF, and RPF) based on the simulated UWB data. It can be seen that the position errors of the seven algorithms in horizontal (centimeter level) were much lower than those in vertical direction (meter level). This was majorly caused by the geometry distribution between eight base stations and tag. As shown in Figure 6, the HDOP values (0.73 on average) were much smaller than the corresponding VDOP (184.05 on average). Additionally, since only Gaussian noises and no NLOS errors were added in these simulated data, the position differences presented in the results can reflect the real performance of different adjustment methods directly. Accordingly, the Bancroft algorithm had the worst solutions because it solved the results of nonlinear system in non-optimal in the statistical properties. The results from the other algorithms were higher accuracy. Because the criteria of minimizing the squared observation-residuals and minimizing the squared parameter-residuals [46] were used to find the optimal solutions. Meanwhile, the position errors caused by linearization errors in both LS and EKF can be eliminated while using the UKF, CKF, and PF. Additionally, since only Gaussian noises and no NLOS errors were added in these simulated data, the robust algorithm in Equation (16) could not be activated. Therefore, it can be seen that no more position improvements can be obtained while using the RPF compared to that of PF. Figure 7 shows the cumulative distribution function (CDF) of position errors of the seven positioning algorithms. It can be seen that the CDF of RPF, PF, and CKF have the highest probability in three-dimensional positioning error under the same threshold value. According to the CDF values of the localization error in Figure 7, at the percentage of 50%, 3D position errors for the Bancroft, LS, EKF, UKF, CKF, PF, and RPF were less than 0.43 m, 0.85 m, 0.42 m, 0.42 m, 0.37 m, 0.36 m, and 0.35 m, respectively. These values on the  
Figure 5: Time-series of vertical position errors in simulation test. Figure 6: HDOP (**a**) and VDOP (**b**) of the simulation experiment. percentage of 68% for the Bancroft, LS, EKF, UKF, CKF, PF, and RPF were 2.07 m, 1.32 m, 0.65 m, 0.62 m, 0.45 m, 0.44 m, and 0.43 m, respectively. While setting the percentage to 95%, the 3D position errors for the seven algorithms were 3.77 m, 2.43 m, 1.32 m, 1.25 m, 0.69 m, 0.71 m, and 0.67 m. Since only Gaussian noises were added in these simulated UWB distances, it indicates that the truncation errors would degrade UWB position accuracy significantly, with a percentage of 13.6%, 16.3%, and 40.3% in the north, east, and vertical directions on average. Figure 8 shows the statistical results of position errors in terms of RMSE and average error. Comparing the horizontal positioning error in Figure 8b with the three-dimensional positioning error in Figure 8a, it can be seen that the RMSE of UWB positioning accuracy in horizontal was about 0.03 m-0.08 m, which was much higher than that in the vertical direction (0.36 m-1.91 m). Here, the 3D positioning RMSE of Bancroft, LS, EKF, UKF, CKF, PF, and RPF were 1.91 m, 1.28 m, 0.68 m, 0.64 m, 0.42 m, 0.41 m, and 0.40 m, respectively. Statistically, the RPF produced position accuracy improvements with percentages of 79.1%, 68.8%, 41.2%, 37.5%, 4.7%, and 2.4% to the solutions of Bancroft, LS, EKF, UKF, CKF, and PF, respectively. Additionally, the nonlinearization filters including UKF, CKF, PF, and RPF performed much higher accuracy than EKF and LS. It was because the nonlinear UWB positioning system needs to be linearized by using Taylor expansion which lead to the second-order and higher-order items being ignored. Such truncation errors could reduce positioning accuracy. Table 5 lists the running time of the seven algorithms to complete the data processing.