GNSS Positioning using Cost Function Regulated Multilateration and Graph Neural Networks

TL;DR

The paper tackles GNSS first-fix localization in urban environments by replacing heuristic error weighting with a Graph Neural Network that estimates per-measurement errors, paired with a Cost Function Regulator that ensures the multilateration objective is minimized at the true location. The architecture represents an epoch as a graph of measurements, uses GraphSAGE for message passing to predict residuals, and employs an adaptive measurement selector to maintain robust inputs for WLS. The key contributions are the Cost Function Regulator, the error-estimating GNN, and the adaptive measurement selection, all validated on a large real-world dataset where the method achieves 40–80% improvements in horizontal localization error over strong baselines, including LSTM-based error estimators. This approach enhances urban GNSS reliability and demonstrates practical gains for first-fix positioning in challenging environments by integrating learned error models with principled optimization regulation.

Abstract

In urban environments, where line-of-sight signals from GNSS satellites are frequently blocked by high-rise objects, GNSS receivers are subject to large errors in measuring satellite ranges. Heuristic methods are commonly used to estimate these errors and reduce the impact of noisy measurements on localization accuracy. In our work, we replace these error estimation heuristics with a deep learning model based on Graph Neural Networks. Additionally, by analyzing the cost function of the multilateration process, we derive an optimal method to utilize the estimated errors. Our approach guarantees that the multilateration converges to the receiver's location as the error estimation accuracy increases. We evaluate our solution on a real-world dataset containing more than 100k GNSS epochs, collected from multiple cities with diverse characteristics. The empirical results show improvements from 40% to 80% in the horizontal localization error against recent deep learning baselines as well as classical localization approaches.

Figures (7)

• Figure 1: Pipeline of the proposed solution. $\mathbf{F}$: input features, $\mathbf{r}$: residuals at a guess location, $\mathbf{\hat{e}}$: error estimates, $\mathbf{\hat{e}_f}$: selected error estimates, $\mathbf{H}$: geometry matrix, $\mathbf{\hat{w}}$: adjusted weights, $\mathbf{\hat{r}}$: adjusted residuals, $\mathbf{\Delta x}$: predicted displacement from guess location.
• Figure 2: On the left, we show with an example how we define a graph over 5 measurements in an epoch. The thickness of the edges represents the value of the angular proximity between each pair of satellites. On the right, we draw a simplified diagram for the Error Estimator neural network. First, measurement features $\mathbf{f}$ are independently encoded into latent vectors $\mathbf{x}$. Then, the Graph Neural Network propagates information across all measurements in the epoch, and an MLP outputs the estimated residuals at the ground truth location $\mathbf{\hat{e}}$. The $L_2$ loss is used for the supervised training given the ground truth residual labels $\mathbf{e}$.
• Figure 3: Proposed measurement selection algorithm
• Figure 4: Cumulative distribution of horizontal localization error from our proposed solution compared against several baselines.
• Figure 5: Predicted location from our solution (red) compared to least squares (blue).