*5.2. Case One: The Narrow Viaduct*

In case one, the GNSS signal was slightly affected by multipath and NLOS signals when the vehicle went through the narrow viaduct. The trajectory of the vehicle is shown in Figure 6a. In our analysis, the positioning error is higher in the period from 149th s to 164th s. It happens to be similar that according to the onboard RTK data, the solution type of GNSS positioning in the period from 149th s to 164th s was not "NARROW INT" (the "NARROW INT" indicates that the multi-frequency RTK is a fixed solution in the NovAtel receiver). Therefore, we needed to perform fault detection in this period. The positioning errors in the east and north are shown in Figure 6b. The maximum errors in the east and north were 1.1156 m and −0.84 m, respectively.

**Figure 6.** Trajectory and positioning errors in the east and north. (**a**) The trajectory of the vehicle in case 1. (**b**) The positioning errors in the east and north.

The top view of the 3D LiDAR global point cloud map built by the LOAM algorithm is shown in Figure 7. Figure 8 shows 31 tree trunks and lampposts marked in the 3D LiDAR global point cloud map as detection targets. These targets are marked with colored points in Figure 8 and numbered 0–30 from left to right in the east view.

**Figure 7.** Three-Dimensional LiDAR global point cloud map in case one.

**Figure 8.** Target detection in map numbered 0–30, including tree trunks and lampposts, derived from the east view.

Two frame point clouds are randomly selected to present the point cloud target detection results, as shown in Figure 9. Figure 9a shows the 645th frame, and a total of 10 targets were detected, including targets No. 8–10 marked in Figure 8. Figure 9b shows the 1023rd frame, and a total of four targets were detected, including targets No. 17–18 marked in Figure 8.

**Figure 9.** Single frame point cloud target detection. (**a**) The 645th frame and No. 8–10 targets in Figure 8. (**b**) The 1023rd frame and No. 17–18 targets in Figure 8.

Figure 10a shows the matched targets between the single frame target detection outputs and the global map-based target detection results. Figure 10b shows the number of detected targets and matched targets in each frame. It can be seen that at least one matched target is present in each frame. This meets the time continuity requirement of matched targets and can support subsequent fault detection processes. Table 1 summarizes the number of detected targets and matched targets obtained with single frame detection.

**Figure 10.** Matched targets between the single frame and map-based target detection results. (**a**) The No. 0–30 of matched targets for target detection in global map. (**b**) The number of detected and matched targets in each frame.

**Table 1.** The number of detected and matched targets for single frame target detection.


The position deviations of all matched targets in each frame in the east and north directions are shown in Figure 11a,b, respectively. Figure 12 shows the mean position deviation of the matched targets in each frame, which is used as the test statistic for the proposed fault detection algorithm.

**Figure 11.** The position deviations of the matched targets in the east and north. (**a**) The position deviation of the matched targets in the east. (**b**) The position deviation of the matched targets in the north.

**Figure 12.** The mean position deviations of the matched targets in the east and north in case one.

Sensitivity is an important fault detection performance evaluation index, and the missed detection rate and false alarm rate are important integrity assessment indices. Figure 13a shows the fault detection result produced by the residual chi-square test, and Figure 13b is the result of the proposed LiDAR aided real-time fault detection algorithm. The fault detection performance of the two algorithms is compared in Table 2. The fault could not be detected by the residual chi-square test, while the proposed algorithm could detect the fault from 160.02th to 164.75th s. Therefore, it can be proven that the proposed algorithm has higher sensitivity. The percentages of false alarms and missed detections were reduced by 42.67% and 31.2%, respectively, relative to those of the residual chi-square test.

**Figure 13.** Fault detection results of the residual chi-square test and the proposed algorithm. (**a**) The fault detection result of the residual chi-square test. (**b**) The LiDAR aided real-time fault detection result. (**c**) Local magnification for fault detection of proposed algorithm.


**Table 2.** The fault detection performance of the chi-square test and the proposed algorithm.

The proposed algorithm can mainly be used to solve the existing problems faced by the EKF algorithm during the fault period. The positioning errors of the EKF, OFFAF and the proposed algorithm in the east and north are shown in Figure 14. The results are analyzed in Table 3. Compared with the EKF, the mean, maximum and root mean square error (RMSE) positioning errors of the proposed algorithm were reduced by 67.66%, 51.9% and 71.58% in the east and 12.93%, 27.02% and 33.6% in the north, respectively. Compared with the OFFAF, the mean, maximum and root mean square error (RMSE) positioning errors of the proposed algorithm were reduced by 51%, 39.1% and 60.3% in the east and −5.2%, 21.1% and 19.2% in the north, respectively. The proposed algorithm is more effective than the OFFAF.

**Figure 14.** Positioning errors in the east and north for the EKF, OFFAF and the proposed algorithm.


**Table 3.** The GNSS/INS integrated positioning performance of the EKF and the proposed algorithm.

The error bound, which is the upper limit of positioning error, is an important integrity assessment index. When the positioning error exceeds the error bounds, an alarm is triggered. The existing research on integrated navigation error bounds were mainly performed by simulation experiments in [38]. In this study, the error bound of integrated navigation is innovatively verified with real test data. The error bounds and horizontal positioning error for the EKF and the proposed algorithm are shown in Figure 15a,b, respectively. In the two algorithms, the error bounds could overbound the horizontal error. From Table 4, the mean value and maximum value of the error bounds of the proposed algorithm were reduced by 53.03% and 57.88%, respectively, relative to the EKF during the fault period.

**Figure 15.** The error bounds and horizontal error of the EKF and the proposed algorithm in case one. (**a**) The error bounds and horizontal error of the EKF. (**b**) The error bounds and horizontal error of the proposed algorithm.

**Table 4.** The error bounds of the EKF and proposed algorithm during the fault period.

