## 4 Experiments and Results
### Experiment Results  
The evaluation was aimed at comparing the proposed algorithms in both a real-time and post-processing manner, also with several conventional solutions:  
1. **NMEA**: receiver output solution. 2. **WLS**: weighted least squares method [52]; uses pseudorange to estimate receiver location. 3. **3 DMA GNSS**: snapshot state-of-the-art 3 DMA GNSS with positioning hypothesis candidates [35]. 4. **LC-FGO (proposed)**: real-time forward (instantaneous) processed loosely-coupled FGO solution with integrated 3 DMA GNSS and velocity. 5. **LC-FGO-PP (proposed)**: combined (forward and backward) processed loosely-coupled FGO solution with integrated 3 DMA GNSS and velocity. The optimization frame was under the ECEF coordinate system. The comparison is divided into root-mean-squared error (RMSE) and standard deviation (STD) positioning error in meters. Note that both LC-FGO and LC-FGO-PP share the same graph structure. Only LC-FGO-PP uses historical and future factors and is optimized in a combined direction forward and backwards. A total of 11 experimental navigation trips were conducted in New York City. The positioning results of different trips are shown in Table 1. In summary, the candidate-based 3 DMA GNSS always outperformed the conventional WLS. After integrating the velocity and optimizing it only for forward direction, the positioning accuracy was improved. If constraints optimization is performed in a combined manner, the positioning accuracy becomes higher. Meanwhile, in most cases, FGO outperformed the receiver's output solution (NMEA). From the overall performance of different experiments, the RMSE and STD of 3 DMA GNSS are 25.34 m and 19.46 m, respectively. At the same time, LC-FGO is 21.05 m and 14.60 m for RMSE and STD, respectively. LC-FGO-PP have a stronger constraint between epochs and obtained RMSE of 15.97 m while STD is 12.48 m. Two FGO have a smaller RMSE, which means that the overall performance is better than that of 3 DMA GNSS, and a lower STD implies that they are more robust. We selected two trips out of eleven (one good and bad case, respectively) to further discuss in this section. We first present a trip with a good positioning performance (Trip 6). It starts from a relatively open-sky area and travels along straight to a deep urban canyon. The plots are shown in Figure 6.  
In this experiment, it can be observed that the 3 DMA GNSS outperforms WLS by twofold, and the positioning RMSE are 18.27 m and 39.17 m, respectively. Many solutions for WLS were located on the opposite side or on the wrong street, as shown in Figure 6c. With the aid of 3D models, 3 DMA can correct the solution back to the correct street. If further integrated with the Doppler measurements, the positioning error can be suppressed in most cases, especially around epochs 200 s to 600 s. Results in the RMSE of the forward LC-FGO and combined FGO (LC-FGO-PP) are 15.32 m and 14.56 m, respectively. Trip 6 is followed by a navigation trip with bad positioning results (Trip 2). Trip 2 begins in a deep urban canyon with a walk along the street to a relatively open area which is the opposite to Trip 6. The map and error plot of this experiment are shown in Figure 7.