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

Motivation

• IMU mounting changes can express the same physical motion in different sensor frames, so learned measurements used inside estimators must remain meaningful across those conventions.Gravity-aligned inputs can reduce mounting variability but introduce a privileged vertical axis and dependence on external attitude estimates.Source: §1 · pp. 1–2

Geometric interface

• GINIO specifies how IMU windows map to geometric quantities consumed by an estimator, rather than prescribing a backbone or filter.The motion output transforms as a vector and its covariance as a tensor under rotations of the IMU measurement frame.Source: §1 · p. 2

Contributions

• Last-Frame Alignment expresses each window in its final sensor frame and is provably equivalent to world-frame training for SO(3)-equivariant predictors.The same interface is instantiated with ResNet-style, AirIO-style recurrent, and EqNIO-style canonicalized models.Source: §1 · p. 2
• The contributions include spectral covariance for uncertainty that rotates with learned motion and validation across TLIO/RIDI, NanoBench, and physical Fetch remounting.The evaluation also includes ablations for augmentation, covariance, capacity, and frame-estimation mismatch.Source: §1 · p. 2

2.1 Learned Inertial Odometry Across Motion Regimes

• IONet, RIDI, and RoNIN established neural inertial odometry for pedestrian and mobile sensing, while TLIO and AI-IMU use learned predictions as estimator updates or corrections.AirIO emphasizes body-frame IMU representation and attitude information for UAV motion; NanoBench provides real nano-quadrotor flights and aggressive maneuvers.Source: §2.1 · p. 3

2.2 Rotation Consistency and Equivariant Models

• Gravity-aligned inputs and augmentation can improve robustness, but EqNIO enforces only gravity-stabilizer subequivariance, whereas GINIO imposes full SO(3)-equivariance on motion and covariance.GINIO specializes geometric learning tools to temporal IMU odometry and filter-compatible uncertainty.Source: §2.2 · p. 3

Problem setup

• The exact guarantee applies when the same calibrated, bias-corrected vector measurements are re-expressed under a coordinate-frame rotation; additional physical remount effects lie outside it.Bias shifts, scale or misalignment changes, lever-arm effects, vibration changes, and timing offsets require calibration and/or the connected estimator.Source: §3 · p. 3

Equivariant temporal processing

• Equivariant temporal layers share scalar coefficients across the three spatial coordinates, while invariant gates and vector interactions provide nonlinear processing.The design avoids arbitrary learned 3 × 3 mixing of spatial axes and is used in residual temporal blocks.Source: §3 · pp. 3–4

Spectral covariance

• Spectral covariance predicts an equivariant frame and invariant log-standard deviations so the covariance rotates by congruence; diagonal heads are retained as fixed-frame ablations.The frame is obtained by SVD-based polar projection, with an invariant regularizer encouraging independent predicted vectors.Source: §3 · p. 4

Last-Frame Alignment

• Last-Frame Alignment expresses each sample in the final sensor frame and is equivalent to world-frame training for an SO(3)-equivariant predictor.It avoids aligning to world vertical, but depends on relative rotations supplied by the connected attitude/filter estimate.Source: §3 · p. 4

Filter-connected fusion

• The filter propagates raw IMU readings and estimates local states such as bias; the predictor uses the corresponding bias-corrected window and supplies motion and covariance for correction.The backend choice is a protocol or deployment choice, not a requirement of the equivariant representation; World and Sensor+LFA yielded identical ATE in the stated no-filter controlled ablation.Source: §3 · p. 4, Figure 2 · p. 5

4 Experiments

• The evaluation spans filter-connected human-motion NIO, NN-only aerial prediction, and physical remounting, comparing baselines in their native input frames and backends.The ID/SO(3) protocol rotates IMU vector channels and corresponding targets by sampled rotations.Source: §4 · p. 5, §4.1 · p. 5

4.1 Experimental Setup

• TLIO/RIDI use standard train/test splits, NanoBench reports benchmark-aligned ATE, and perturbation diagnostics report velocity RMSE.Additional uncertainty diagnostics, training details, and per-sequence results are provided in the appendix.Source: §4.1 · p. 5

4.2 Human-Motion Benchmarks: TLIO and RIDI

• On TLIO, EqNIO’s ID/ID ATE is 1.554 m but its ID/SO(3) ATE degrades to 76.389 m; GINIO-W stays near nominal with 17.81M FLOPs.The passage reports roughly 12× fewer FLOPs than EqNIO for GINIO-W.Source: §4.2 · p. 6, Table 1 · p. 6
• On RIDI, GINIO-W has the best nominal ATE among world-frame methods, while GINIO-S is most stable in the rotated sensor-frame configuration; NN-only GINIO reaches 0.927 m ATE versus IMUNet’s 1.43 m.The NN-only comparison uses the matched RIDI test protocol and no downstream filter.Source: §4.2 · p. 6

4.3 Aerial Benchmark: NanoBench

• Without external attitude, GINIO-A reduces NanoBench ATE from 5.579 m to 1.430 m while using 7.4× fewer parameters than AirIO.With attitude, GINIO-A improves ATE from 1.205 m to 1.045 m while using 8.7× fewer parameters.Source: §4.3 · p. 6, Table 2 · p. 7
• GINIO-RD reaches 0.581 m ATE versus ResNet1D’s 0.645 m and EqNIO O(2)’s 0.828 m, while GINIO variants remain unchanged under SO(3) perturbation.The temporal-backbone comparison reports GINIO-RD as the accuracy-oriented diagonal-covariance variant.Source: §4.3 · pp. 6–7, Table 2 · p. 7

4.4 Physical IMU Remounting on Fetch

• On Fetch, unseen-remount ATE is 0.50 m for GINIO-R versus 8.15 m for non-equivariant ResNet, with comparable nominal accuracy.The remount evaluation averages two unseen orientations and is trained only on the nominal mounting.Source: §4.4 · p. 7, Table 3 · p. 8

5.1 Architectural Equivariance vs. Augmentation

• Under TLIO ID/SO(3), the non-equivariant ResNet reaches 108.235 m ATE, SO(3) augmentation reaches 3.189 m, and the equivariant model remains at 1.828 m.The same ablation reports that LFA restores world-frame accuracy in the sensor-frame setting; with the same InEKF backend, GINIO-S remains stable at 2.704 m and 2.759 m while ResNet-S degrades to 36.593 m.Source: §5.1 · p. 8
• GINIO-RD/RS use 0.57M/0.80M parameters versus 5.4M for non-equivariant ResNet-style baselines; GINIO-A uses 0.045M parameters and improves ATE in both NanoBench attitude settings.The recurrent model uses 7.4×–8.7× fewer parameters than AirIO, supporting the paper's structural-regularization interpretation for data-limited aerial learning.Source: §5.1 · p. 8

5.2 Spectral Covariance and Uncertainty Consistency

• The spectral covariance head reduces covariance-equivariance error by more than three orders of magnitude and reduces velocity RMSE, but does not uniformly improve calibration.Its NEES is closer to the ideal value of 3, while the diagonal head's 95% coverage is closer to 0.95.Source: §5.2 · p. 8, Figure 4 · p. 9

5.3 Test-Time Frame-Estimation Mismatch

• Frame-estimation mismatch corrupts the gravity frame or LFA relative frame, so exact coordinate-frame equivariance does not make preprocessing errors disappear.The stress test injects angular error θerr into the preprocessing frame and reports ATE sweeps across TLIO, NanoBench, and RIDI for 3°, 5°, 10°, 15°, and 20°.Source: §5.3 · p. 9
• For NanoBench gravity-frame mismatch, GINIO-R has the smallest 3°→20° ATE increase (+0.03 m), versus +0.25 m for EqNIO O(2) and +0.10 m for its augmented variant.Across TLIO and RIDI, GINIO-R has the flattest or near-flattest degradation under gravity-frame and LFA relative-frame errors; augmentation does not consistently match its stability.Source: §5.3 · p. 9, Figure 5 · p. 9
• At 20°, filter-connected GINIO obtains 3.07 m ATE versus 4.85 m for the TLIO-style baseline; spectral covariance is more robust than diagonal covariance (1.349 vs. 1.826 m).With the learned mean frozen, the diagonal head is marginally better nominally (1.072 vs. 1.081 m); paired robustness differences are significant for θ ≥15°.Source: §5.3 · p. 9, Figure 6 · p. 10
• The network-only rotation test, frame-estimation sweeps, and filter-connected controls probe different things: exact coordinate-frame law, imperfect preprocessing attitudes, and robustness after downstream fusion.The filter-connected controls complement rather than replace the network-only sweeps; the magnitude of frame-estimation errors depends on the filter estimate.Source: §5.3 · p. 10

• The exact equivariance guarantee applies to calibrated, bias-corrected vector measurements under coordinate-frame rotations.Source: §6 · p. 10
• Physical remount changes such as bias shifts, scale/misalignment, lever-arm effects, vibration, timing offsets, and inaccurate preprocessing attitudes lie outside the guarantee and require calibration and/or connected-estimator handling.Source: §6 · p. 10
• LFA depends on relative attitude estimates, and severely corrupted relative rotations can degrade the sensor-frame representation; the Fetch evidence is not a paired same-motion repeatability test.Source: §6 · p. 10

7 Conclusion

• Across filter-connected TLIO/RIDI, aerial NanoBench prediction, and Fetch remounting, GINIO supports mounting-rotation robustness across backbones and estimators.The authors present SO(3) equivariance as a filter-compatible design principle, rather than a dataset- or architecture-specific robustness trick.Source: §7 · p. 10

B.2 Filter-Connected Inference Loop

• The filter-connected estimator uses bias-corrected IMU windows to predict a motion measurement and covariance, then rotates sensor-frame outputs into the backend measurement frame before updating.The estimator retains local nuisance states such as IMU bias; the filter-side invariant error representation does not itself provide frontend equivariance.Source: §B.2 · p. 13

C.1 Equivariant Temporal Layers

• Equivariant temporal layers share scalar weights across the three spatial coordinates, and invariant gates and equivariant vector interactions preserve the SO(3) action.This structural sharing makes each channel-mixing block 9× smaller; GINIO-RD backbones use 0.57M parameters versus 5.4M for non-equivariant ResNet-style baselines.Source: §C.1 · p. 14, §C.2 · p. 14

C.5 Equivariant Reinstantiation of Heterogeneous Backbones

• GINIO reinstantiates ResNet-style, AirIO-style recurrent, and EqNIO-style canonicalized backbones so the full learned motion computation obeys the SO(3) transformation law.The authors identify the equivariant gated recurrent cell and full-SO(3) canonicalized predictor as, to their knowledge, first formulations of their respective backbone families.Source: §C.5 · p. 15

C.7 GINIO-A: AirIO-Style Recurrent Instantiation

• The AirIO-style recurrent cell derives reset and update gates from invariant scalars, then broadcasts them across vector-channel coordinates to preserve equivariance.The decoder maps the equivariant hidden state to a velocity vector; the bidirectional cell therefore yields a mean that transforms as a vector.Source: §C.7 · p. 16

C.8 GINIO-E: EqNIO-Style Full-SO(3) Canonicalized Instantiation

• GINIO-E replaces EqNIO’s gravity-stabilizer O(2) frame with a learned SO(3)-equivariant canonical frame, lifting the canonicalized design to full SO(3).The predicted displacement and covariance are mapped back to the original frame.Source: §C.8 · p. 17

C.9 AirIO-Style Capacity Ablation on NanoBench

• A 6.1× smaller recurrent equivariant model slightly improves ATE over the original GINIO-A.The reduced model is also smaller than both AirIO and the original equivariant recurrent variant.Source: §C.9 · p. 17, Table 5 · p. 17
• The channel-reduced GINIO-A control addresses whether the AirIO-style equivariant result is merely due to greater model capacity.The comparison supports the authors’ view that equivariance acts as a structural regularizer in the aerial setting, rather than added representational capacity.Source: §C.9 · p. 17
• Table 5 reports the reduced model’s parameter-count and ATE comparison with the original equivariant recurrent model.The caption identifies this as an AirIO-style GINIO-A capacity ablation on NanoBench.Source: Table 5 · p. 17

D Proof of Last-Frame Alignment Equivalence

• LFA rotates every sample into the final sensor frame, making the gravity direction approximately constant across the window and reducing rotational nonstationarity.The representation remains sensor-centric rather than aligning to world vertical.Source: D Proof of Last-Frame Alignment… · pp. 17–18, Figure 8 · p. 18
• For an SO(3)-equivariant predictor, LFA and world-frame training differ only by the final rotation; the physical target is unchanged.The same change-of-frame argument preserves the covariance congruence law when the covariance-valued head is equivariant.Source: D Proof of Last-Frame Alignment… · p. 18

E EqNIO Compatibility: Og(3) vs. Full SO(3)

• EqNIO’s gravity-stabilizer O(2) symmetry is suited to accurately gravity-aligned inputs, whereas arbitrary sensor-frame rotations require full SO(3) equivariance.The paper states that EqNIO degrades more under out-of-plane remounting and frame-estimation mismatch outside its assumed symmetry group.Source: E EqNIO Compatibility: Og(3) vs. Full… · p. 18

F.1 Trajectory Metric Definitions

• ATE is aligned-position RMSE over the full trajectory; the TLIO protocol’s RTE uses fixed 1 s windows and removes the starting yaw error.RTE emphasizes local translational consistency, while drift ratio diagnoses normalized long-horizon drift.Source: §F.1 · p. 19, §F.2 · p. 19

F.3 Controlled TLIO Network Ablation

• SO(3) augmentation improves the non-equivariant TLIO baseline from 108.2 to 3.2 under ID/SO(3), but does not impose a continuous transformation law.With an equivariant predictor, LFA restores the sensor-frame model to world-frame accuracy.Source: §F.3 · p. 19, Table 7 · p. 20

F.4 Matched Backend and Filter Ablations

• With the same InEKF backend, non-equivariant ResNet-S diverges under ID/SO(3), whereas GINIO-S remains stable.This comparison is reported as evidence that rotational robustness is not a filter artifact.Source: §F.4 · p. 19, Table 8 · p. 20

G.2 NanoBench ID/SO(3) Velocity Stress

• Under test-time frame rotations, non-equivariant AirIO and ResNet1D vary with the input frame, whereas GINIO-A and GINIO-RS remain stable.Figure 9 reports velocity RMSE for this NanoBench stress test.Source: §G.2 · p. 20, Figure 9 · p. 20

H.1 Equivariant SPD Parameterization

• The spectral covariance parameterization combines an equivariant eigenbasis with invariant eigenvalues; near-collinear predicted vectors are stabilized by numerical regularization.The appendix reports covariance-equivariance error as a diagnostic, but these passages provide no comparative error values.Source: §H.1 · p. 21

H.4 Full Covariance Ablation Table

• Table 9 reports the raw NanoBench covariance ablation, including MSE-trained variants; calibration metrics are not meaningful for heads trained with MSE.The table covers RMSE, NLL, and equivariance errors.Source: §H.4 · p. 21, Table 9 · p. 22

H.5 Uncertainty Calibration

• On NanoBench, NLL-trained heads are better calibrated than MSE-trained heads; aggregate cov95 is 0.915 for diagonal and 0.837 for spectral.The spectral head is closest to ideal in the coverage-curve tail, while the heads are comparable in the mid-coverage range; it also learns input-dependent uncertainty.Source: §H.5 · p. 22, Figure 10 · p. 22
• With the mean predictor frozen, a 20° frame error yields 1.349 m for the equivariant spectral-derived marginal versus 1.826 m for the coordinate-diagonal head.Nominal ATE is nearly identical at 1.081 m and 1.072 m, respectively; paired differences are significant for θ ≥15°.Source: §H.5 · p. 22

I Test-Time Frame-Estimation Mismatch: Full Sweeps

• GINIO-R is the most stable curve on TLIO and RIDI under frame-estimation mismatch and stays near the noise floor on NanoBench.Table 10 separates gravity-frame perturbations from LFA-relative-rotation perturbations; EqNIO O(2) appears only in the gravity-frame columns.Source: I Test-Time Frame-Estimation Mismatch:… · p. 23, Table 10 · p. 23

J Additional Nominal and Filter-Connected Diagnostics

• Nominal TLIO per-sequence curves overlap substantially despite sharper separation under rotated-frame evaluation; NanoBench GINIO-RD’s favorable mean ATE is not driven by one sequence.Representative filter-connected trajectories illustrate alignment-dependent updates deviating and GINIO remaining close to ground truth, but the three examples are qualitative diagnostics.Source: J Additional Nominal and… · pp. 23–24, Figure 11 · p. 23, Figure 12 · p. 24

K.1 Platform and Protocol

• Fetch models were trained on 23 nominal-mount sequences and tested on two unseen 90° physical remounts, with onboard wheel odometry as reference.Because mount recordings are unpaired, nominal-to-remount ATE differences indicate robustness trends rather than paired per-trajectory comparisons.Source: §K.1 · p. 24

K.2 Full Hardware Results

• GINIO-R changes by less than 10^-3 m in ATE under additional synthetic test-time rotations of Fetch IMU streams.These eval SO(3) rotations are applied at test time, not used as training-time augmentation.Source: §K.2 · p. 24

L Additional Visually Degraded Evaluation

• Full SO(3) augmentation improves the non-equivariant ResNet’s unseen mean from 8.15 to 0.58 m at a nominal-accuracy cost; GINIO-R reaches 0.495 m without rotation augmentation.GINIO-R’s unseen-mount ATE is 0.226 m for TC1 and 0.764 m for TC2.Source: Table 12 · p. 25
• On a representative fast-circle Fetch sequence under TC1, the non-equivariant ResNet diverges while GINIO-R tracks the onboard odometry reference.The full diagnostic plots show the same contrast.Source: Figure 13 · p. 26, Figure 14 · p. 26
• Under test-time rotations, the equivariant predictor is invariant while the non-equivariant baseline degrades on AquaticVision.S04 and S06 are excluded from training in the unseen setting; visual-SLAM baselines have no ID/SO(3) entry because they are not subject to IMU-frame rotation.Source: L Additional Visually Degraded… · pp. 24, 26, Table 13 · p. 25