# GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry

arXiv: https://arxiv.org/abs/2609.25338 · HTML: https://mlsanity.com/paper/2609.25338/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 25 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [RoNIN: Robust Neural Inertial Navigation in the Wild: Benchmark, Evaluations, and New Methods](https://mlsanity.com/paper/1905.12853/) (2019)
- [IONet: Learning to Cure the Curse of Drift in Inertial Odometry](https://mlsanity.com/paper/1802.02209/) (2018)
- [TLIO: Tight Learned Inertial Odometry](https://mlsanity.com/paper/2007.01867/) (2020)
- [RIDI: Robust IMU Double Integration](https://mlsanity.com/paper/1712.09004/) (2017)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [AI-IMU Dead-Reckoning](https://mlsanity.com/paper/1904.06064/) (2019)
- [Denoising IMU Gyroscopes with Deep Learning for Open-Loop Attitude Estimation](https://mlsanity.com/paper/2002.10718/) (2020)
- [IDOL: Inertial Deep Orientation-Estimation and Localization](https://mlsanity.com/paper/2102.04024/) (2021)
- [Contact-Aided Invariant Extended Kalman Filtering for Robot State Estimation](https://mlsanity.com/paper/1904.09251/) (2019)
- [The invariant extended Kalman filter as a stable observer](https://mlsanity.com/paper/1410.1465/) (2014)
- [RIO: Rotation-equivariance supervised learning of robust inertial odometry](https://mlsanity.com/paper/2111.11676/) (2021)
- [Deep IMU Bias Inference for Robust Visual-Inertial Odometry with Factor Graphs](https://mlsanity.com/paper/2211.04517/) (2022)
- [Learned Inertial Odometry for Autonomous Drone Racing](https://mlsanity.com/paper/2210.15287/) (2022)
- [EqNIO: Subequivariant Neural Inertial Odometry](https://mlsanity.com/paper/2408.06321/) (2024)
- [AirIO: Learning Inertial Odometry with Enhanced IMU Feature Observability](https://mlsanity.com/paper/2501.15659/) (2025)
- [Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras](https://mlsanity.com/paper/2310.04521/) (2023)
- [AquaticVision: Benchmarking Visual SLAM in Underwater Environment with Events and Frames](https://mlsanity.com/paper/2505.03448/) (2025)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [Proprioceptive Invariant Robot State Estimation](https://mlsanity.com/paper/2311.04320/) (2023)
- [NanoBench: A Multi-Task Benchmark Dataset for Nano-Quadrotor System Identification, Control, and State Estimation](https://mlsanity.com/paper/2603.09908/) (2026)

## Abstract (arXiv)

Neural inertial odometry increasingly uses networks as learned measurements inside filtering pipelines. Such measurements should transform consistently under arbitrary IMU mounting conventions: their mean must transform as a vector, and their covariance must transform congruently as a second-order tensor. We present GINIO, a geometric SO(3)-equivariant interface for neural inertial odometry under arbitrary rotations of the IMU measurement frame. Given calibrated IMU windows, our framework predicts a motion measurement and uncertainty obeying these tensorial laws. To support efficient sensor-frame learning, we introduce Last-Frame Alignment (LFA), a deterministic preprocessing step that is provably equivalent to world-frame training for SO(3)-equivariant predictors. The connected estimator tracks sensor-local states such as IMU bias, separating nuisance estimation from the geometric law enforced by the learned measurement. We instantiate the same interface in filter-connected NIO, AirIO-style recurrent aerial prediction, EqNIO-style full-SO(3) canonicalization, and ResNet-style temporal backbones. On TLIO, GINIO achieves 2.018 m ID/SO(3) ATE while EqNIO degrades to 76.389 m, using 11.6x fewer FLOPs. On NanoBench, our AirIO-style instantiation improves ATE from 5.579 m to 1.430 m without external attitude input, and our ResNet-style instantiation reaches 0.581 m ATE versus 0.645 m for ResNet1D. On Fetch, GINIO empirically reduces unseen physical-remount ATE from 8.15 m to 0.50 m without retraining, demonstrating robustness beyond the exact coordinate-frame guarantee. For uncertainty, spectral covariance reduces covariance-equivariance error by over three orders of magnitude compared with a diagonal head.