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NavFormer: IGRF Forecasting in Moving Coordinate Frames

Yoontae Hwang, Dongwoo Lee, Minseok Choi, Heechan Park, Yong Sup Ihn, Daham Kim, Deok-Young Lee

TL;DR

NavFormer addresses the challenge of forecasting IGRF total intensity from rotating sensor frames by introducing a geometry-aware front end. It combines rotation-invariant scalar features with a Canonical SPD module that stabilizes the spectrum of windowed triad second moments, implemented through a Gram-based canonical frame and state-conditioned SPD scaling. This front end feeds a Patch-Channel Grid Transformer with FiLM conditioning, enabling robust long-horizon forecasting and strong generalization in few-shot and zero-shot settings. Empirical results across five flights show consistent accuracy gains over strong baselines, demonstrating improved data efficiency and cross-frame transfer in magnetometer-based navigation. The approach offers a practical, geometry-respecting pathway for robust sensor fusion in moving platforms.

Abstract

Triad magnetometer components change with sensor attitude even when the IGRF total intensity target stays invariant. NavFormer forecasts this invariant target with rotation invariant scalar features and a Canonical SPD module that stabilizes the spectrum of window level second moments of the triads without sign discontinuities. The module builds a canonical frame from a Gram matrix per window and applies state dependent spectral scaling in the original coordinates. Experiments across five flights show lower error than strong baselines in standard training, few shot training, and zero shot transfer. The code is available at: https://anonymous.4open.science/r/NavFormer-Robust-IGRF-Forecasting-for-Autonomous-Navigators-0765

NavFormer: IGRF Forecasting in Moving Coordinate Frames

TL;DR

NavFormer addresses the challenge of forecasting IGRF total intensity from rotating sensor frames by introducing a geometry-aware front end. It combines rotation-invariant scalar features with a Canonical SPD module that stabilizes the spectrum of windowed triad second moments, implemented through a Gram-based canonical frame and state-conditioned SPD scaling. This front end feeds a Patch-Channel Grid Transformer with FiLM conditioning, enabling robust long-horizon forecasting and strong generalization in few-shot and zero-shot settings. Empirical results across five flights show consistent accuracy gains over strong baselines, demonstrating improved data efficiency and cross-frame transfer in magnetometer-based navigation. The approach offers a practical, geometry-respecting pathway for robust sensor fusion in moving platforms.

Abstract

Triad magnetometer components change with sensor attitude even when the IGRF total intensity target stays invariant. NavFormer forecasts this invariant target with rotation invariant scalar features and a Canonical SPD module that stabilizes the spectrum of window level second moments of the triads without sign discontinuities. The module builds a canonical frame from a Gram matrix per window and applies state dependent spectral scaling in the original coordinates. Experiments across five flights show lower error than strong baselines in standard training, few shot training, and zero shot transfer. The code is available at: https://anonymous.4open.science/r/NavFormer-Robust-IGRF-Forecasting-for-Autonomous-Navigators-0765
Paper Structure (39 sections, 2 theorems, 27 equations, 4 figures, 8 tables)

This paper contains 39 sections, 2 theorems, 27 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Geometric Constraints in Time Series Forecasting
  4. Stable Canonicalization via SPD Manifold
  5. NavFormer
  6. Problem Setting
  7. Scalar Geometric Features
  8. Canonical SPD module
  9. Canonical Frame from Aggregated Gram Matrix
  10. State-Dependent SPD Scaling
  11. Patch-Channel Grid Transformer
  12. Temporal Patching per Channel
  13. Patch Embedding
  14. FiLM Conditioning of Token Embeddings
  15. Grid Self-Attention Encoder
  16. ...and 24 more sections

Key Result

Proposition 3.1

Let $G = U \Lambda U^\top$ be the aggregated Gram matrix and $M(\mathbf{s}) =U \Sigma(\mathbf{s}) U^\top$ the canonical SPD transform. Define the Gram matrix of the modulated triads as Then In particular, $\tilde{G}(\mathbf{s})$ has eigenvectors $U$ and eigenvalues$\sigma_i(\mathbf{s})^2 \lambda_i$ for $i=1,2,3$.

Figures (4)

  • Figure 1: Overview of the NavFormer architecture.
  • Figure 2: Spectrum and stability statistics of the aggregated Gram matrix $G$ over test windows.
  • Figure 3: Forecasting comparison on NV2 with lookback $L=60$ and prediction horizon $L_{\text{pred}}=120$.
  • Figure 4: Flight path. Adapted from the MagNav repository (https://github.com/Naatyu/MagNav).

Theorems & Definitions (4)

  • Proposition 3.1: Spectral Alignment of Canonical SPD
  • proof : Proof
  • Proposition 2.1: Spectral Alignment of Canonical SPD
  • proof