Article Figures & Data
Figures
- FIGURE 1
Our proposed framework, which consists of two modules: a GNN and a learned Kalman filter
In the learned Kalman filter, we first perform prediction to propagate the state and uncertainty forward. During the measurement update, we receive position corrections, which are computed by first aggregating the embeddings of all nodes in the GNN graph, averaging the embeddings via mean pooling, and then passing them through fully connected layers. These corrections are used as measurements during the state and covariance update step of the BKF. The output from the learned Kalman filter is the 3D position. Using the predicted and true positions, we compute the MSE loss and backpropagate the gradients of the loss with respect to parameters of the GNN and the Kalman filter to jointly update their parameters. In the next time step, we use the updated GNN parameters to perform prediction.
- FIGURE 2
Integration of the GNN with the learned Kalman filter
GNSS measurements are processed, passed through the GraphSAGE layers, normalized via batch normalization, and activated through a ReLU function. The node embeddings are then aggregated by a mean aggregation method, followed by a fully connected layer that outputs the measurement vector, which is then utilized in the learned Kalman filter state update to learn the absolute position.
- FIGURE 3
A sample feature vector and feature matrix for a small-scale GNN graph.
Here, GLONASS refers to the Russian global navigation satellite system and GALILEO refers to the navigation system created by the European Union.
We construct a comprehensive feature vector for each satellite, also referred to as a node, in the network. This vector is formulated by concatenating several critical metrics: the LOS vector, range residual, carrier-to-noise density ratio, and pseudorange uncertainty. The feature matrix is a result of combining the feature vectors for all nodes in the graph.
- FIGURE 4
Trajectory tracking from our algorithm and the loosely coupled GNN/Kalman filter baseline for one test data set collected in San Jose
The left plot shows an aerial view of the entire trajectory. The right plots show magnified versions of selected portions of the entire trajectory. Our trajectory closely matches ground truth, outperforming the baseline, and achieves a 2.2-m mean error in the north and 0.6-m error in the east direction.
- FIGURE 5
Trajectory tracking from our algorithm and the loosely coupled GNN/Kalman filter baseline for one test data set collected in Sunnyvale
The left plot shows an aerial view of the entire trajectory. The right plots show magnified versions of selected portions of the entire trajectory. Our trajectory closely matches ground truth, outperforming the baseline, and achieves a 0.9-m mean error in the north and 1.1-m error in the east direction.
Tables
Data Set Devices 2021-08-04-US-SJC-1 GooglePixel4, GooglePixel5, SamsungGalaxyS20Ultra 2021-04-02-US-SJC-1 GooglePixel4, GooglePixel5, SamsungGalaxyS20Ultra, XiaomiMi8 2021-08-24-US-SVL-1 GooglePixel4, GooglePixel5, SamsungGalaxyS20Ultra, XiaomiMi8 2021-04-26-US-SVL-2 SamsungGalaxyS20Ultra, XiaomiMi8 Data Set Devices 2021-12-07-US-LAX-1 GooglePixel5, GooglePixel6Pro, SamsungGalaxyS20, XiaomiMi8 2021-12-07-US-LAX-2 GooglePixel5, GooglePixel6Pro, SamsungGalaxyS20, XiaomiMi8 2021-12-08-US-LAX-1 GooglePixel5, GooglePixel6Pro, SamsungGalaxyS20, XiaomiMi8 2021-12-08-US-LAX-3 GooglePixel5, GooglePixel6Pro, SamsungGalaxyS20, XiaomiMi8 2021-12-08-US-LAX-5 GooglePixel5, GooglePixel6Pro, SamsungGalaxyS20, XiaomiMi8 2021-12-09-US-LAX-2 GooglePixel5, GooglePixel6Pro, SamsungGalaxyS20, XiaomiMi8 - TABLE 3
Experimental Parameters for the BKF Module
The process and measurement noise parameters are iteratively updated in our algorithm through backpropagation.
Parameter Value State_dim 3 Input_dim 6 Initial process_noise I3 × 1e – 3 Initial measurement_noise I3 × 1e – 2 - TABLE 4
Key Hyperparameters for the GNN Module
The edge threshold similarity parameter indicates the minimum similarity score required to establish an edge between nodes in the graph. Note that the final hyperparameter values were determined after multiple Monte Carlo empirical validations had been conducted on a subset of the training data sets.
Parameter Value Hidden_dimension 128 No. of nodes 20 No. of epochs 11 No. of layers 8 Optimizer Adam Learning rate 0.001 Weight decay 5e – 4 Loss function MSE Minimum value for edge threshold similarity 0.5 - TABLE 5
Key Parameters in the GNN Architecture
The weights of different layers in the GNN are jointly updated with the BKF parameters during training.
Layer Name Number of Parameters Linear 1 × 3 P (covariance matrix) 3 × 3 Linear (input layer) 6 × 128 SAGEConv 128 × 128 BatchNorm1d 128 × 1 Linear (output layer) 1 × 128 × 3 - TABLE 6
Summary of Positioning Error in the East Direction on Test Data Sets
Our algorithm outperforms all of the baselines on the test data sets. Note that the median, minimum, and maximum errors in the east direction were unavailable for the attention-based neural network baseline.
Error Metric (m) Kalman Filter Attention (Kanhere et al., 2022) Loose GNN + Kalman Filter (Mohanty & Gao, 2022) Our Approach Mean 4.6 5.9 3.5 1.1 Median 4.5 - 3.4 1.1 Minimum 1.2 - 2.1 0.6 Maximum 14.1 - 4.8 1.8 - TABLE 7
Summary of Positioning Error in the North Direction on Test Data Sets
Our algorithm outperforms all of the baselines on the test data sets with respect to mean, median and minimum metric and provides a comparable maximum positioning error. Note that the median, minimum, and maximum errors in the north direction were unavailable for the attention-based neural network baseline.
Error Metric (m) Kalman Filter Attention (Kanhere et al., 2022) Loose GNN + Kalman Filter (Mohanty & Gao, 2022) Our Approach Mean 3.0 6.4 2.0 1.9 Median 2.1 - 1.8 1.8 Minimum 1.3 - 1.0 0.7 Maximum 7.7 - 5.7 5.8 - TABLE 8
Comparison of Horizontal Localization Errors (m) on Test Data Sets
Our approach consistently outperforms the baseline, demonstrating significant improvements in localization accuracy. These results underscore the efficacy of our method in enhancing positioning accuracy across a range of conditions and devices.
No. Test Data Set Name Device Loose GNN + Kalman Filter (Mohanty & Gao, 2022) Our Approach 1 2021-04-02-US-SJC-1 GooglePixel4 3.7 2.7 2 2021-04-02-US-SJC-1 GooglePixel5 5.0 2.2 3 2021-04-02-US-SJC-1 SamsungGalaxyS20Ultra 4.1 2.5 4 2021-04-02-US-SJC-1 XiaomiMi8 5.1 2.6 5 2021-04-26-US-SVL-2 SamsungGalaxyS20Ultra 3.7 2.1 6 2021-04-26-US-SVL-2 XiaomiMi8 5.1 2.3 7 2021-08-04-US-SJC-1 GooglePixel4 2.5 1.4 8 2021-08-04-US-SJC-1 GooglePixel5 6.6 5.9 9 2021-08-04-US-SJC-1 SamsungGalaxyS20Ultra 3.0 1.4 10 2021-08-24-US-SVL-1 GooglePixel4 4.2 1.3 11 2021-08-24-US-SVL-1 GooglePixel5 4.8 1.4 12 2021-08-24-US-SVL-1 SamsungGalaxyS20Ultra 2.9 1.9 13 2021-08-24-US-SVL-1 XiaomiMi8 2.8 2.2 - TABLE 9
Generalization Error in the East Direction on Previously Unseen Data Sets in the City of Los Angeles
Note that both our approach and the baseline were trained on an identical number of data sets (49).
Error Metric (m) Loose GNN + Kalman Filter (Mohanty & Gao, 2022) Our Approach Mean 12.4 1.7 Median 12.4 1.8 Minimum 10.8 0.8 Maximum 13.9 3.0 - TABLE 10
Generalization Error in the North Direction on Previously Unseen Data Sets in the City of Los Angeles
Note that both our approach and the baseline were trained on an identical number of data sets (49).
Error Metric (m) Loose GNN + Kalman Filter (Mohanty & Gao, 2022) Our Approach Mean 5.5 1.9 Median 5.4 1.7 Minimum 2.7 1.0 Maximum 8.2 4.1 - TABLE 11
Positioning Error in the East Direction as a Function of Different Similarity Threshold Values for Edge Creation
The threshold value of 0.5 m is found to be optimal.
Error Metric (m) Threshold = 0.3 m Threshold = 0.5 m Threshold = 0.9 m Mean 1.4 1.1 1.2 Median 1.3 1.1 1.2 Minimum 0.8 0.6 0.7 Maximum 2.2 1.8 2.0 - TABLE 12
Positioning Error in the North Direction as a Function of Different Similarity Threshold Values for Edge Creation
Even in the north direction, the threshold value of 0.5 m is found to be optimal.
Error Metric (m) Threshold = 0.3 m Threshold = 0.5 m Threshold = 0.9 m Mean 2.0 1.9 2.0 Median 1.8 1.9 1.8 Minimum 0.7 0.7 0.7 Maximum 5.8 5.8 5.8
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