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

Chankyo Kim, Minghan Zhu, Tzu-Yuan Lin, Avantika Rattan, Maani Ghaffari · N/A · 2026

Summary

Neural inertial odometry increasingly uses networks as learned measurements inside filtering pipelines.

Abstract Summary

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.

Key Points

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.

Problem Setup and Methodology

Let $\bar{\mathbf{z}}^{s}_{i}=[(\bar{\mathbf{f}}^{s}_{i})^{\top},(\bar{\boldsymbol{\omega}}^{s}_{i})^{\top}]^{\top}\in\mathbb{R}^{6}$ denote a bias-corrected IMU sample with specific force and angular velocity expressed in the sensor frame.

A coordinate rotation $\mathbf{R}\in\mathrm{SO}(3)$ acts on the two vector channels by $\rho(\mathbf{R})=I_{2}\otimes\mathbf{R}$, applied componentwise to the window $\bar{\mathcal{Z}}^{s}=\{\bar{\mathbf{z}}^{s}_{i}\}_{i=1}^{n}$.

Problem Setup and Methodology — GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry — Figure 2: Filter-connected GINIO pipeline. The estimator propagates raw IMU measurements and maintains nuisance states such as IMU bias. The learned predictor receives the corresponding calibrated, bias-corrected window in a Direct/Body, World, or Last-Frame-Aligned representation and returns a vector-valued motion measurement together with its covariance for the measurement update. The exact $\mathrm{SO}(3)$ transformation law is enforced by the learned motion-and-covariance interface, not by the choice of filter backend.

Experiments

We evaluate GINIO along three axes: filter-connected learned inertial odometry, aerial inertial prediction, and physical IMU remounting.

TLIO/RIDI test whether the interface improves standard filter-based NIO; NanoBench tests whether the same $\mathrm{SO}(3)$ principle transfers to aggressive nano-quadrotor dynamics; and Fetch remounting tests whether the learned predictor remains robust when the IMU is physically reinstalled.

Experiments — GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry — Figure 3: Fetch robot with the IMU mounted inside the mobile base.

Ablations and Stress Tests

We isolate the mechanisms behind the main results: architectural equivariance rather than augmentation, tensor-consistent covariance rather than axis-aligned uncertainty, and the boundary between coordinate-frame rotations and frame-estimation mismatch. In our controlled TLIO ablation, architectural equivariance is more reliable than $\mathrm{SO}(3)$ augmentation for arbitrary 3D rotations.

A non-equivariant ResNet fails under ID/$\mathrm{SO}(3)$ (1.817 m $\rightarrow$ 108.235 m ATE).

Ablations and Stress Tests — GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry — Figure 6: Filter-connected TLIO controls with the SCEKF backend held fixed. (a) Attitude/frame-estimation error sweep comparing a TLIO-style ResNet+SCEKF baseline with GINIO. (b) Covariance-only sweep with an identical frozen mean predictor, comparing a coordinate-diagonal uncertainty head with the equivariant spectral-derived parameterization. Holding the backend fixed—and in (b) the learned mean as well—isolates the learned interface and covariance parameterization from filter changes.

Sources and Demonstrations

Figures are reproduced from Kim et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.