## 4 Experiments and Results
### Experiment Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Navigation Trips** & **Epochs (s)** & **Algorithm** & **RMSE (m)** & **STD (m)** \\ \hline  & & 1. NMEA & 31.09 & 14.47 \\  & & 2. WLS & 38.30 & 20.00 \\
1 & 952 & 2. 3 DMA GNSS & 19.70 & 15.51 \\  & & 3. LC-FGO & 24.66 & 14.95 \\  & & 4. LC-FGO-PP & 15.54 & 12.25 \\ \hline  & & 1. NMEA & 74.81 & 31.57 \\
2 & 979 & 2. WLS & 59.15 & 26.94 \\  & & 2. 3 DMA GNSS & 29.14 & 16.75 \\  & & 3. LC-FGO & 33.56 & 17.13 \\  & & 4. LC-FGO-PP & 24.66 & 13.08 \\ \hline  & & 1. NMEA & 19.87 & 7.35 \\
3 & 574 & 2. 3 DMA GNSS & 27.62 & 16.40 \\  & & 3. LC-FGO & 22.98 & 11.04 \\  & & 4. LC-FGO-PP & 21.38 & 9.51 \\ \hline  & & 1. NMEA & 17.20 & 11.08 \\
4 & 607 & 2. 3 DMA GNSS & 21.08 & 12.26 \\  & & 3. LC-FGO & 13.01 & 6.48 \\  & & 4. LC-FGO-PP & 14.09 & 6.85 \\ \hline  & & 1. NMEA & 29.01 & 7.43 \\
5 & 599 & 2. WLS & 30.34 & 10.46 \\  & & 2. 3 DMA GNSS & 22.64 & 13.21 \\  & & 3. LC-FGO & 20.38 & 10.25 \\  & & 4. LC-FGO-PP & 18.90 & 11.18 \\ \hline  & & 1. NMEA & 36.89 & 18.01 \\
6 & 934 & 2. 3 DMA GNSS & 18.27 & 11.28 \\  & & 3. LC-FGO & 15.32 & 10.44 \\  & & 4. LC-FGO-PP & 14.56 & 8.73 \\ \hline  & & 1. NMEA & 33.36 & 15.61 \\  & & 2. WLS & 44.25 & 25.89 \\
7 & 885 & 2. 3 DMA GNSS & 18.64 & 11.27 \\  & & 3. LC-FGO & 25.17 & 11.03 \\  & & 4. LC-FGO-PP & 12.17 & 6.08 \\ \hline  & & 1. NMEA & 39.09 & 11.05 \\  & & 2. WLS & 36.43 & 15.94 \\
8 & 513 & 2. 3 DMA GNSS & 16.55 & 9.46 \\  & & 3. LC-FGO & 21.30 & 8.36 \\  & & 4. LC-FGO-PP & 14.22 & 7.47 \\ \hline  & & 1. NMEA & 24.17 & 10.38 \\
9 & 878 & 2. 3 DMA GNSS & 41.50 & 26.91 \\  & & 3. LC-FGO & 44.62 & 29.99 \\  & & 4. LC-FGO-PP & 37.67 & 24.91 \\ \hline  & & 1. NMEA & 36.01 & 16.88 \\
10 & 742 & 2. WLS & 49.43 & 31.11 \\  & & 2. 3 DMA GNSS & 26.72 & 15.29 \\  & & 3. LC-FGO & 25.49 & 13.08 \\  & & 4. LC-FGO-PP & 20.46 & 11.15 \\ \hline  & & 1. NMEA & 46.78 & 18.12 \\  & & 2. WLS & 62.33 & 37.96 \\
11 & 733 & 2. 3 DMA GNSS & 36.85 & 28.13 \\  & & 3. LC-FGO & 37.82 & 27.85 \\  & & 4. LC-FGO-PP & 32.13 & 26.08 \\ \hline  & & 1. NMEA & 46.78 & 18.12 \\  & & 2. WLS & 62.33 & 37.96 \\
11 & 733 & 2. 3 DMA GNSS & 36.85 & 28.13 \\  & & 3. LC-FGO & 37.82 & 27.85 \\  & & 4. LC-FGO-PP & 32.13 & 26.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics on positioning results of all experiments. Figure 6: (a) map plot, (b) positioning errors, (c) magnified map plot of last 300 epochs, and (d) number of received satellites and average skymask elevation angle of good positioning trip (Trip 6). Figure 7: (a) map plot, (b) positioning errors, and (c) number of received satellites and average skymask elevation angle of bad positioning trip (Trip 2). Similar to Trip 6, both 3 DMA GNSS and two LC-FGO algorithms outperform WLS. The RMSE of WLS, 3 DMA GNSS, LC-FGO, and LC-FGO-PP are 59.15 m, 29.14 m, 33.56 m, and 24.66 m, respectively. However, the LC-FGO is not outperforming the 3 DMA GNSS. The overall positioning error is larger than that in Trip 6 because the environment is more complex. The average skymask elevation angle is higher in Trip 2, resulting in a more severe NLOS reception that mostly occupies a large portion of the total received satellites. The main error comes from the last 200 epochs. The performance of 3 DMA GNSS keeps fluctuating during this period. The natural difference between 3 DMA GNSS and FGO results in the average performance of this experiment. 3 DMA GNSS is a snapshot estimation, and each epoch performance is independent of the others. However, the FGO is different, especially for the forward FGO. The fluctuation of the forward FGO will keep accumulating errors in batch optimization. Therefore, the future estimation is affected. However, the combined FGO, LC-FGO-PP, has a much stronger constraint that tries to optimize the solution in both directions. As a result, the positioning error can be suppressed. Therefore, if the performance of LC-FGO has to be improved, marginalization analysis must be done to find the acceptable error of this graphical problem.