Article Figures & Data
Figures
- FIGURE 1
Flowchart on the proposed 3DMA GNSS
- FIGURE 2
(a)–(d) Location of experiment 1 to 5, respectively; and (e)-(i) Sky-pointing fisheye camera image of experiment 1 to 5, respectively
- FIGURE 3
L1- and L5-band pseudorange difference between simulated pseudoranges [e.g., (3), on Satellite G01, G03, G08, and G30] and the weighted average (Weighted AVG) of all available satellites at ground truth for Experiment 3. For each data set, L1- and L5-band pseudorange differences are separated into the upper row in the red and the lower row in the blue frame. The Y-axis label in the satellite’s ID, the elevation angle inside the bracket, and the labeled satellite visibility, LOS and NLOS, are also presented. It should be noted that the color axis is limited within the range of 20 m
- FIGURE 4
Ray-tracing results for satellite (a) G03, a multipath satellite in which the LOS path is not blocked, and the reflection path is found; and (b) G08, an NLOS diffracted satellite in which the LOS path is blocked, and the diffraction path is found (the red point illustrates the ground truth, the yellow line shows the LOS path, and the blue line represents the reflection/diffraction path)
- FIGURE 5
Multipath labeled satellite G03’s (a) pseudorange error in the entire experiment, and (b) probability density function plot
- FIGURE 6
Cumulative distribution function (CDF) of L1- and L5-band pseudorange error by using the data collected in all experiments
- FIGURE 7
(a)–(c) are heatmaps of SKY, LBR, and SKY+LBR using Method 1, respectively; (d)–(f) are heatmaps of SKY, LBR, and SKY+LBR using Method 4, respectively; (g) is the skymask with satellite, in which color points represent the 𝐶∕𝑁0 value for the corresponding satellites
- FIGURE 8
CDF plots on 2D positioning error of different receiver clock error estimation methods contribute to (a) Skymask 3DMA (SKY), (b) likelihood-based ranging GNSS (LBR), (c) integration of SKY and LBR (SKY+LBR), and (d) a combination of all ranging-based 3DMA (SKY, LBR, and SKY+LBR)
- FIGURE 9
Positioning results for Experiments 1 through 5 utilizing both L1- and L5-band measurements
- FIGURE 10
Heatmap of Experiment 5 in (a) SKY, (b) LBR, (c) SDM, (d) SKY+LBR (RNG), and (e) SKY+LBR+SDM. The green star refers to the truth location, and the purple point represents the positioning solution of the corresponding algorithm. The value inside the bracket denotes the 2D positioning error
- FIGURE 11
Heatmap of Experiment 1 in (a) SKY, (b) LBR, (c) SDM, (d) SKY+LBR (RNG), and (e) SKY+LBR+SDM. The green star represents the truth location, and the purple point refers to the positioning solution of the corresponding algorithm. The value inside the bracket denotes the 2D positioning error
- FIGURE B1
LBR heatmaps based on different receiver clock error estimation methods: (a) Method 1, (b) Method 2, and (c) Method 4
Tables
Criteria Skymask 3DMA Likelihood-based ranging GNSS Input parameters Satellite position
Positioning candidate position and its enhanced skymaskPseudorange difference Correction-based Geometry Statistical Available correction Reflection delay Reflection delay and some noise Uncertainty Building model accuracy The healthiness of the selected reference satellite Experiment name (Exp) Environment NLOS reception Multipath effect Mean elevation angle (degree) STD elevation angle (degree) Building height to street width ratio Average no. of satellite Average no. of L5 signal No. of epoch (1 Hz) 1 One side building Light Severe 38.07 40.01 0.93 13 2 600 2 One side building Light Severe 38.41 34.17 2.3 16 1 600 3 Sub-urban Moderate Moderate 42.60 17.52 0.68 16 6 1080 4 Deep urban Severe Severe 61.71 24.59 2.77 34 6 600 5 Deep urban Severe Severe 48.76 19.75 2.61 32 8 600 3DMA GNSS algorithm Abbreviation Symbol of the score at j-th candidate, Sj,* Related section or equation 1. Shadow matching (Groves et al., 2020) SDM Sj,SDM 2.2 2. Skymask 3DMA (Ng et al., 2020a) SKY Sj,SKY 2.3, 2.3.1, 2.3.2 3. Likelihood-based ranging GNSS (Groves et al., 2020) LBR Sj,LBR 2.3, 2.3.1, 2.3.3 4. Ranging-based 3DMA (RNG), integrate SKY and LBR SKY+LBR Sj,RNG 2.4(15) 5. Integrate SKY and SDM SKY+SDM Sj,SKY+SDM 
6. Integrate LBR and SDM LBR+SDM Sj,LBR+SDM 7. Integrate all 3DMA GNSS algorithm SKY+LBR+SDM Sj,SKY+LBR+SDM - TABLE 4
3DMA GNSS positioning RMS error of Experiment 3 based on different receiver clock error estimation methods
Positioning RMSE results (meter) Method Inter-constellation clock offset Satellite selection Candidate based SKY LBR SKY+LBR Average RMSE (meter) 1. SD-IC ✓ All 12.55 12.86 10.50 11.97 2. P-SD-IC ✓ LOS only ✓ 14.07 16.04 13.83 14.65 3. WLS-IC ✓ All 11.70 12.23 11.35 11.76 4. P-WLS-IC ✓ LOS only ✓ 12.16 11.82 10.64 11.54 5. P-WLS LOS only ✓ 14.41 15.73 11.13 13.76 - TABLE 5
Statistics of 3DMA GNSS positioning results using Methods 1 and 4 for Experiment 3, in meters
Method 1: SD-IC Method 4: P-WLS-IC SKY LBR SKY+LBR SKY LBR SKY+LBR RMS 12.55 12.86 10.50 12.16 11.82 10.64 Mean 11.38 11.45 9.79 11.52 11.38 10.14 STD 5.30 5.86 3.80 3.90 3.21 3.24 Max 56.80 53.03 52.90 42.39 36.74 32.72 Min 4.04 4.72 0.80 1.28 6.06 0.93 Experiment SKY LBR SDM SKY+LBR SKY+SDM LBR+SDM SKY+LBR+SDM 1 RMS 19.73 3.85 12.84 3.85 4.89 3.72 3.73 Mean 15.74 3.37 12.83 3.37 3.30 3.08 3.08 STD 11.91 1.86 0.46 1.86 3.61 2.09 2.10 Max 33.44 17.87 16.35 17.87 33.25 9.92 9.92 Min 0.23 0.23 11.76 0.23 0.25 0.33 0.33 2 RMS 39.90 8.56 9.26 9.76 12.13 5.47 5.83 Mean 34.52 7.80 7.94 8.44 8.42 4.61 4.71 STD 20.02 3.54 4.77 4.91 8.74 2.94 3.44 Max 64.00 21.75 45.50 26.93 57.17 15.99 22.26 Min 0.50 1.52 0.52 1.91 0.51 0.45 0.45 3 RMS 12.16 11.82 16.09 10.64 10.76 9.69 9.17 Mean 11.52 11.38 13.07 10.14 10.19 9.46 8.62 STD 3.90 3.21 9.38 3.24 3.45 2.07 3.11 Max 42.39 36.74 38.74 32.72 39.50 26.10 28.52 Min 1.28 6.06 0.79 0.93 1.86 3.75 0.99 4 RMS 7.65 5.72 10.21 6.37 5.12 4.19 4.28 Mean 5.86 5.12 7.56 5.31 4.40 4.08 3.87 STD 4.92 2.54 6.86 3.53 2.61 0.94 1.83 Max 59.64 15.74 53.34 25.25 48.23 10.34 15.34 Min 0.39 2.00 0.12 0.44 0.20 3.01 0.23 5 RMS 7.74 9.09 22.89 5.54 7.39 11.59 5.53 Mean 6.04 8.10 21.64 5.15 6.07 10.55 5.14 STD 4.84 4.12 7.45 2.05 4.21 4.80 2.03 Max 34.88 28.87 38.33 15.00 27.30 27.16 21.66 Min 0.80 0.18 4.80 0.20 1.20 3.01 1.57 - TABLE B1
3DMA GNSS positioning RMS error of Experiment 4 based on different receiver clock error estimation methods, in meters
Positioning RMSE results Method Inter-constellation clock offset Satellite selection Candidate based SKY LBR SKY+LBR 1. SD-IC ✓ All 15.77 12.84 16.41 2. P-SD-IC ✓ LOS only ✓ 11.13 6.93 11.40 3. WLS-IC ✓ All 20.77 9.31 10.26 4. P-WLS-IC ✓ LOS only ✓ 7.65 5.72 6.37 5. P-WLS LOS only ✓ 10.69 8.09 9.99 Separate constellation With inter-constellation correction Exp SKY LBR SDM SKY + LBR SKY + SDM LBR + SDM SKY + LBR + SDM SKY LBR SDM SKY + LBR SKY + SDM LBR + SDM SKY + LBR + SDM 1 RMS 13.76 4.86 13.26 4.87 4.01 4.07 4.08 26.23 6.74 13.26 6.74 18.13 7.26 7.26 Mean 9.06 4.19 13.20 4.19 3.48 3.56 3.57 24.81 5.69 13.20 5.69 13.92 6.48 6.48 STD 10.36 2.46 1.36 2.47 2.00 1.97 1.98 8.51 3.62 1.36 3.62 11.62 3.27 3.27 Max 39.18 12.25 30.01 12.25 10.80 10.80 10.80 33.03 19.85 30.01 19.85 33.78 16.13 16.13 Min 0.32 0.15 11.87 0.15 0.02 0.29 0.29 1.05 0.32 11.87 0.32 0.28 0.32 0.32 2 RMS 35.61 9.22 11.47 9.18 15.23 6.08 6.19 42.30 10.58 11.47 12.49 17.99 7.20 8.16 Mean 29.57 8.15 9.33 8.07 11.24 5.14 5.17 38.52 9.61 9.33 10.92 13.45 6.24 6.78 STD 19.87 4.30 6.68 4.37 10.27 3.25 3.41 17.49 4.44 6.68 6.06 11.96 3.60 4.54 Max 60.07 34.41 45.50 34.41 58.78 18.59 17.85 64.00 22.14 45.50 34.41 64.66 19.13 22.41 Min 0.47 0.95 0.52 0.95 0.49 0.37 0.39 1.20 0.80 0.52 0.50 0.53 0.46 0.35 3 RMS 21.07 18.86 17.11 18.15 15.96 14.08 14.60 18.64 13.35 17.11 13.15 11.93 9.97 9.52 Mean 18.99 16.79 13.41 15.42 13.81 12.24 11.82 16.55 11.43 13.41 10.91 11.08 9.23 8.47 STD 9.14 8.58 10.62 9.58 8.00 6.95 8.57 8.57 6.89 10.62 7.34 4.44 3.77 4.34 Max 43.51 40.22 38.74 37.97 40.69 33.75 35.27 41.23 37.41 38.74 36.74 34.67 33.61 32.75 Min 0.47 3.87 0.73 0.10 0.54 0.36 0.14 0.59 0.68 0.73 0.09 0.52 0.89 0.17 4 RMS 11.81 8.83 10.44 12.59 6.96 5.60 7.11 10.69 8.09 10.44 9.99 6.32 4.95 5.14 Mean 9.13 7.30 7.96 9.02 5.46 4.95 5.15 8.09 6.53 7.96 7.28 5.00 4.45 4.13 STD 7.51 4.98 6.76 8.79 4.31 2.62 4.90 7.00 4.78 6.76 6.84 3.87 2.16 3.06 Max 44.41 28.63 53.84 40.56 29.34 16.30 35.29 59.64 30.20 53.84 38.51 48.15 22.81 15.66 Min 0.53 0.72 3.26 0.30 0.27 1.48 0.17 0.59 2.04 3.26 0.31 0.21 1.48 0.14 5 RMS 8.13 7.87 23.03 5.48 5.38 8.88 5.16 7.01 11.23 23.03 5.96 5.91 11.68 5.71 Mean 6.06 6.96 21.83 5.24 5.15 8.07 4.99 5.66 10.01 21.83 5.48 5.39 10.71 5.28 STD 5.42 3.68 7.33 1.63 1.55 3.70 1.31 4.13 5.08 7.33 2.35 2.43 4.68 2.19 Max 42.89 23.02 37.53 14.58 17.62 24.96 13.15 42.98 46.12 37.53 42.90 35.14 46.59 43.03 Min 0.74 0.58 4.91 1.44 1.89 2.44 1.72 1.73 1.69 4.91 1.57 2.31 2.98 1.80 Separate constellation With inter-constellation correction Exp SKY LBR SDM SKY + LBR SKY + SDM LBR + SDM SKY + LBR + SDM SKY LBR SDM SKY + LBR SKY + SDM LBR + SDM SKY + LBR + SDM 1 RMS 11.75 4.73 12.84 4.75 4.14 4.09 4.10 19.73 3.85 12.84 3.85 4.89 3.72 3.73 Mean 7.71 4.13 12.83 4.13 3.60 3.58 3.58 15.74 3.37 12.83 3.37 3.30 3.08 3.08 STD 8.87 2.32 0.46 2.33 2.05 1.99 2.00 11.91 1.86 0.46 1.86 3.61 2.09 2.10 Max 39.15 11.77 16.35 11.77 11.56 11.56 11.56 33.44 17.87 16.35 17.87 33.25 9.92 9.92 Min 0.41 0.23 11.76 0.23 0.20 0.20 0.20 0.23 0.23 11.76 0.23 0.25 0.33 0.33 2 RMS 29.56 7.74 9.26 7.59 12.70 5.15 5.25 39.90 8.56 9.26 9.76 12.13 5.47 5.83 Mean 22.85 6.89 7.94 6.83 8.69 4.42 4.47 34.52 7.80 7.94 8.44 8.42 4.61 4.71 STD 18.77 3.52 4.77 3.31 9.26 2.65 2.76 20.02 3.54 4.77 4.91 8.74 2.94 3.44 Max 59.89 34.41 45.50 23.14 58.72 19.01 16.75 64.00 21.75 45.50 26.93 57.17 15.99 22.26 Min 0.49 1.15 0.52 0.47 0.46 0.47 0.42 0.50 1.52 0.52 1.91 0.51 0.45 0.45 3 RMS 14.41 15.73 16.09 11.13 11.76 10.17 8.89 12.16 11.82 16.09 10.64 10.76 9.69 9.17 Mean 12.95 13.61 13.07 10.01 10.68 9.55 8.19 11.52 11.38 13.07 10.14 10.19 9.46 8.62 STD 6.33 7.89 9.38 4.86 4.92 3.51 3.45 3.90 3.21 9.38 3.24 3.45 2.07 3.11 Max 35.43 42.26 38.74 33.92 38.93 36.10 29.34 42.39 36.74 38.74 32.72 39.50 26.10 28.52 Min 1.33 5.75 0.79 0.83 1.12 2.61 0.74 1.28 6.06 0.79 0.93 1.86 3.75 0.99 4 RMS 7.53 6.03 10.21 7.25 4.47 4.41 4.40 7.65 5.72 10.21 6.37 5.12 4.19 4.28 Mean 5.62 5.42 7.56 5.16 3.95 4.31 3.73 5.86 5.12 7.56 5.31 4.40 4.08 3.87 STD 5.02 2.65 6.86 5.09 2.10 0.93 2.33 4.92 2.54 6.86 3.53 2.61 0.94 1.83 Max 47.30 17.32 53.34 48.36 13.35 9.25 15.47 59.64 15.74 53.34 25.25 48.23 10.34 15.34 Min 0.33 1.95 0.12 0.32 0.20 1.52 0.21 0.39 2.00 0.12 0.44 0.20 3.01 0.23 5 RMS 5.33 5.35 22.89 4.71 5.17 7.07 4.68 7.74 9.09 22.89 5.54 7.39 11.59 5.53 Mean 5.02 4.81 21.64 4.37 4.93 6.40 4.56 6.04 8.10 21.64 5.15 6.07 10.55 5.14 STD 1.78 2.35 7.45 1.74 1.54 3.01 1.06 4.84 4.12 7.45 2.05 4.21 4.80 2.03 Max 24.15 15.92 38.33 11.49 26.27 25.45 8.72 34.88 28.87 38.33 15.00 27.30 27.16 21.66 Min 0.09 0.39 4.80 0.44 0.52 0.72 0.44 0.80 0.18 4.80 0.20 1.20 3.01 1.57
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