EqNIO: Subequivariant Neural Inertial Odometry
Authors: Royina Karegoudra Jayanth
Yinshuang Xu
Ziyun Wang
Evangelos Chatzipantazis
Kostas Daniilidis
Daniel Gehrig
Venue: ICLR 2025 Poster (2025)
Keywords: equivariance
inertial odometry
subequivariance
Primary area: applications to robotics, autonomy, planning
TL;DR: Our framework robustly regresses an equivariant frame which  improves the state-of-the-art in neural IO, when coupled with off-the-shelf filter-based or end-to-end models.

Abstract: Neural network-based odometry using accelerometer and gyroscope readings from a single IMU can achieve robust, and low-drift localization capabilities, through the use of _neural displacement priors (NDPs)_. These priors learn to produce denoised displacement measurements but need to ignore data variations due to specific IMU mount orientation and motion directions, hindering generalization.
This work introduces EqNIO, which addresses this challenge with _canonical displacement priors_, i.e., priors that are invariant to the orientation of the gravity-aligned frame in which the IMU data is expressed. We train such priors on IMU measurements, that are mapped into a learnable canonical frame, which is uniquely defined via three axes: the first is gravity, making the frame gravity aligned, while the second and third are predicted from IMU data.  The outputs (displacement and covariance) are mapped back to the original gravity-aligned frame. To maximize generalization, we find that these learnable frames must transform equivariantly with global gravity-preserving roto-reflections from the subgroup $O_g(3)\subset O(3)$, acting on the trajectory, rendering the NDP $O(3)$-_subequivariant_. We tailor specific linear, convolutional, and non-linear layers that commute with the actions of the group. Moreover, we introduce a bijective decomposition of angular rates into vectors that transform similarly to accelerations, allowing us to leverage both measurement types. Natively, angular rates would need to be inverted upon reflection, unlike acceleration, which hinders their joint processing. We highlight EqNIO's flexibility and generalization capabilities by applying it to both filter-based (TLIO), and end-to-end (RONIN) architectures, and outperforming existing methods that use _soft equivariance from auxiliary losses or data augmentation on various datasets. We believe this work paves the way for low-drift and generalizable neural inertial odometry on edge devices. The project details and code can be found at [https://github.com/RoyinaJayanth/EqNIO](https://github.com/RoyinaJayanth/EqNIO).

Published as a conference paper at ICLR 2025
EQNIO: SUBEQUIVARIANT NEURAL INERTIAL ODOMETRY
Royina Karegoudra Jayanth; Yinshuang Xu, Ziyun Wang. Evangelos Chatzipantazis, Kostas Daniilidis', Daniel Gehrig!
University of Pennsylvania (royinakj, xuyin, 2 yinuw, vaghat, dgehrig @seas upenn edu kostas@cis upenn. edu
ABSTRACT
Neural network-based odometry using accelerometer and gyroscope readings from single IMU can achieve robust, and low-drift localization capabilities, through the use neural displacement priors (NDPs). These priors learn to produce denoised displacement measurements but need to ignore data variations due to specific IMU mount orientation and motion directions, hindering generalization. This work introduces EqNIO, which addresses this challenge with canonical displacement priors, i... priors that are invariant to the orientation of the gravity-aligned frame in which the IMU data is expressed. We train such priors IMU measurements, that are mapped into a learnable canonical frame, which is uniquely defined via three axes: the first gravity, making the frame gravity aligned, while the second and third are predicted from IMU data. The outputs (displacement and covariance) are mapped back the original gravity-aligned frame. To maximize generaliza- tion, we find that these learnable frames must transform equivariantly with global prevetypr roto-reflections from the subgroup (3) C O(3), acting on the trajectory, rendering the NDP ((3))ssuequivariin convolutional, and non-linear layers that commute with the actions of the group. Moreover, introduce bijective decomposition of angular rates into vectors that transform similarly to accelerations, allowing us to leverage both measurement types. Natively, angular rates would need to be inverted upon reflection, unlike acceleration, which hinders their joint processing. We highlight EqNIO's ' flexibility and generalization capabilities by applying it to both filter-based (TLIO), and end-to-end (RONIN) architectures, and outperforming existing methods that use sofi equivariance from auxiliary losses data augmentation We believe this work paves the way for low-drift and generalizable neural in- ertial odometry on edge devices. The project details and code can be found https: / thub..b... com Roy nwwayanthh RgNIO
We tailor specific linear,
various datasets.
TLIO
TLIO running rotated data
EqNIO running on rotated data
EqNIO (this work)
s012) equivariant frame
SO(2) alignment
S0(2) alignment
trojectory covariance
trajectory
covariance
inconsistent trajectories and covariance
consistent trajectories and covariance
Figure 1: Neural Displacement Priors (NDPs) that rely on data augmentation (TLIO, left) produce different trajectory for different reference frames while strictly equivariant approaches (EqNIO, right) yield one trajectory independent of the reference frame. EqNiO achieves this by learning an equivariant canonical frame aligned, per definition, with the predicted covariance (ellipsoidss In test time, the gravity-aligned IMU orientation estimate the Kalman filter is the reference frame.
denotes equal contribution, denotes equal advising

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1 INTRODUCTION
Inertial Measurement Units (IMUs) are commodity sensors that measure the accelerations and angular velocities of body. Due to their low cost, they are widely used in diverse applications such as robot navigation and Mixed Reality for precise and rapid tracking of body frames. However, since they are differential sensors, relying only on IMUs invariably results in drift. Traditional Visual Inertial Odometry (VIO) approaches can effectively mitigate this drift by combining IMU measurements with features extracted from camera images. Still, these images are of limited use in high-speed scenarios with challenging lighting conditions, since they can suffer from saturation, and blurring artifacts.
Recently, a novel class of methods has emerged that instead mitigates this drift with neural displace- ment priors (NDPs), that are directly learned from IMU data alone (Liu et al. 2020 Herath et al.. 2020). These methods perform competitively with VIO methods despite using only a single IMU sensor. However, learning generalizable priors proves challenging: while identifying specific motion patterns, they must learn to ignore data differences due to particular IMU mount orientations, and the direction of the motion patterns. However, they often fail do so in practice (see Fig. D left state-of-the-arr method TLIO (Liu et al. (020 yielding large trajectory variations when observing simply rotated input data. These variations persist, in spite of data augmentation strategies (TLIO Liu et al.. (2020), RONIN Herath et al. (2020) that include random data rotations or the enforcement of equivariance constraints via auxiliary consistency losses during training (RIOCao et al. (2022)).
In this work, we simplify this task by introducing EQNIO, which leverages canonical displacement priors (CDPs). CDPs are invariant to IMU orientation and, as a result, easier to learn, more robust, and more generalizable than regular NDPs. EqNIO first maps the IMU data into a learnable, equivariant, and gravity-aligned frame F, before passing them to the CDP, and mapping the outputs back to the original frame. Our design can flexibly integrate arbitrary off- the-shelf methods such TLIO (Liu ctt al. 2020) and RONIN (Herath et al. 2020). provided that suitable change of basis maps are defined for the network outputs (displacement and covariance of TLIO, and only linear velocity for RONIN). aligns the z-axis with the gravity direction, estimated from an off-the-shelf filter (Liu et al. 2020), and is traditionally used in inertial odometry. The second is a learnable, pravity-preserving roto-reflection.
The frame F is composed of two transformations: The first, Sravity--lignmeen
To be maximally generalizable, we show that this roto-reflection must generalize to arbitrary gravity- preserving rotorreflections of the input data. We design specific model that achieves this by processing IMU data equivariantly, i.e., in way that commutes with the action of roto-reflections on the input data. We identify these roto-reflections as elements of the group (3). a subgroup of the orthogonal group (3). which is isomorphic to O(2). Equivariance ensured by a preliminary, unique preprocessing step that maps accelerometer and gyroscope measurements into a space that transforms consistently under the group action, and subsequent processing with equivariant MLPs, convolutions, and non-linear layers. Due to the isomorphism with ((2) these layers are O(2) equivariant, and, due to the subgroup property, also called ((3) subequivariant.
Contributions: (i) We apply a canonicalization scheme that maps IMU measurement, and NDP outputs to and from a gravity-aligned, subequivariant, canonical frame. This procedure can be flexibly applied to arbitrary off-the-shelf network architectures. NDPs trained on such canonical data produce inherently more robust, and generalizable results than previous work. (ii) We formalize the group actions of gravity-preserve unique preprocessing steps that map both accelerometer and gyroscope measurements into a space in which these actions are consistent. (iii) We tailor an (2) equivariant network that regresses canonical roto-reflcctions from 2D vector features, and ID scalar derived from IMU data. It leverages specialized (2) equivariant MLPs and convolution, to process vector features, conventional layers to process scalar features, and equivariant non- non-linearssies to mix vector and scalar features.
roto-reflection from (3) on IMU measurements and derive
We demonstrate the generality of our framework by applying it to two neural inertial odometry methods, TLIO (Liu et al.; 2020), and RONIN (Herath et al. 2020). Extensive qualitative and quantitative results comparing EqNIO against previous works across diverse benchmarks establish a new state-of-the-art in inertial-only odometry. EqNIO significantly enhances the accuracy, reliability, and generalization of existing methods.
RELATED WORK
Neural inertial odometry is highly related to inertial navigation systems (INS) which can be broad classified into estimation-based multi-sensor data fusion-based (MSDF) techniques, and learning-
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based techniques. Recent works in inertial attitude estimation (Asgharpoor Golroudbari & Sabour 2023: Ge et al. (024). VIO (van Goor & Mahony| 2023), LIDAR odometry (Zheng et al. 2024 and SLAM (Kim& & Sukkarich 2003) are highly related and are hence discussed in App. A.I
Purely Inertial Odometry can be broadly classified into two categories: kinematics- and learning- baaedd approaches. Kinematics-basee approaches (Leish-
Model-based Inertial Odometry:
man et al.) 2014 Titterton et al. 2004 [Bortz] 1971) leverage analytical solutions based on double integration that sufters from drift accumulation over time when applied consumer-grade IMUs.
To mitigate this drift, loop closures (Solin et al. 2018), and other pseudo measurements derived
from IMU data that are drift-free have been explored (Groves) 2015 Hartley et al. 2020 Brajdic &
Harle 2013). In the context of Pedestrian Dead Reckoning (Jimenez et al. (2009), these include step
counting (Ho et al. 2016, Brajdic& & Harle 2013), detection of the system being static (Foxlin 2005
Rajagopal 2008) and gait estimation (Beautils et al. 2019).
Learning- Inertial Odometry: Recently, RIDI Yan al. 2018). PDRNet (Asraf et al. 2022) and RONIN (Herath | (020) proposed CNN, RNN, and TCN-based velocity regression. RIDI uses these velocities to correct the IMU measurements, while RONIN directly integrates them while as- suming given orientation information. Denoising networks either regress IMU biases (Brossard et al.
2020b; Buchanan et al. 2023; Brossard et al.
2020a) denoised IMU measurements directly (Stein 2022). While Buchanan et al.
brener et al.
(2023) uses constant covariance, AI-IMU (Brossard
et al. 2020a) estimates the covariance for automotive applications. Displacement-basse methods like
1ONet (Chen et al. 2018aa], TLIO (Liu et al.) 2020). RNIN-VIO (Chen et al. 2021a), and IDOL (Sun)
ct al. 2021) directly estimate 2D/3D displacement. Unlike TLIŌ which regresses a diagonal covari- ance matrix, Russell & Reale (2021) estimates the full covariance matrix parameterized via Pearson correlations. RNIN-VIO extends TLIO to continuous human motion adding a loss function for long-term accuracy. Unlike these methods, EqNIO learns canonical displacement priors and thus generalizes better to arbitrary IMU orientation and motion directions.
The previous learning-basse based approaches (Liu al., 2020 Chen et al. 2021a) use S0( augmentation strategies to achieve approximate SO(2) equivariance. Mo- tion'l ransformer (Chen al. 2019) used GAN-based RNN encoder to transfer IMU data into domain-invariant space by separating the domain-related constant. Recently, RIO (Cao et al. 2022) demonstrated the benefits of approximate SO(2) equivariance with an auxiliary loss, introduced Adaptive Test Time Training (TTT), and uncertainty estimation via ensemble of models (See Ap- pendix A.4 more details). We propose integrating strict equivariance by design directly into the framework. Additionally, no prior work has addressed reflection equivariance, which requires specific preprocessing of gyroscope data for it to adhere to the right-hand rule. Our novel ((2) equivariant framework can be seamlessly integrated with existing saaming-based inertial navigation systems.
Equivariant Inertial Odometry:
Group equivariant networks (Cohen & Welling 2016) commute by design with group actions on the input, and have been tailored to variety of inputs and architecture designs. These include point clouds (Thomas et al.
Equivariant Networks:
2018 Chen et al. 20216 Deng et al., 202T). 2D (Worrall
al., 2017, Weiler Cesa 2019). 3D (Weiler et al. 2018
Esteves et al.) 2019), and spherical
images (Cohen et 2018 Esteves et al.
2018: 2020; 2023). ) graphs (Satorras et al. (021) and
general manifolds (Cohen et al. [20 90 a
Weiler et al. 2021 Xu et al.) (024). Yet, equivariant
networks tailored to IMU data and their symmetries have not been studied, and are introduced in
the current work. Our method draws on general theories and methods developed for equivariance
to E(n) its subgroups. Cesa et al. 2202 ); Xu et al. (2022) use Fourier analysis to design steerable CNN kernels on homogeneous space, while While/Finzi et al. (202T) proposed an algorithm for finding a kernel by solving a linear equivariant map constraint. Villar et al. (202)) demonstrated that any O(n) equivariant function can be represented using a set of scalars and vectors. However, applying these to the neural integration of IMUs is not straightforward as gravity's presence introduces subequivariance, angular velocity in the input data follows the right-hand rule, and the input is a sequence with a time dimension. Related approaches (Han al. 2002]] Chen et al., 2023) tackle subequivariance using equivariant graph networks and calculating gram matrices achieving simple O(2) equivariance. However, dealing with data that obey the right-hand rule (e.g. angular rates), has been underexplored, and is addressed in the current work. Canonicalization is a prevalent approach to equivariance. (aba jet al. (2023) first introduced learned canonicalization functions for equivariant networks, which we adopt in our design. We discuss additional works on equivariant canonicalization in Appendix A.
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EqNIO
May
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Eq. Freme model
Bay d.1
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Figure 2: EqNIO (a) processes gravity- IMU measurements, (a,,,,))) w,)) An equivariant network (blue) predicts canonical equivariant frame F into which IMU measurements mapped, ie. canonicalized, yielding invariant inputs ((a), ) conventional neural network then predicts invariant displacement (d") and covariance (2") which are mapped back yielding equivariant displacement (d) and covariance (E), The equivariant network (b) takes as input n X C. scalars, and n x Ci vectors: Vectors are processed by equivariant layers (Eq-L. Eq Eq-Conv. Eq-LN), while scalars are separately processed with conventional layers. Eq-L (green) uses two weights W1, W2 for SO(2) equivariance, and only W, for O(2) equivariance. Eq-Conv (pink) uses Eq-L. to perform 1-D convolutions over time. The equivariant non-linear layer (orange) mixes vector and scalar features.
PROBLEM SETUP
This paper targets neural inertial odometry using data from single IMU, comprised of accelerom- eter (giving linear accelerations E 3 and gyroscope (giving angular velocity E R IMU's measure sequences of data {(a, w;) } T each expressed in the local IMU body frame, b, time f, These are related the true IMU acceleration a, and angular rates W; via
W R. + b% o
à = ", " +
(1)
where g is gravity pointing downward world frame w. RR is the transformation between b and w at time f4, and bg, be and v/ n/ are IMU biases and noises respectively. Naively integrating angular rates and accelerations to get positions p; and orientations R, leads to significant drift due to sensor noise and unknown biases. We thus turn attention neural displacement priors & which regress accurate 2D linear velocities (Herath et al. 2020) or 3D displacement E IR' and covariances E 3 (Liu et al. 2020) from sequences of bias and gravity aligned IMU measurements
Wi - R, 69) a, = - R, b²)
(2)
where RR aligns the z-axis with gravity, and defined as S. R2 RR with some unobservable yaw rotation R... The neural displacement prior has the form
d, Σ ({(a,, H..)
(3)
where d 3 denotes displacement on the time interval [t tx]. and 3 denotes associated covariance prediction. For instance, [Luu et al. (2020), uses these network predictions as measurements and fuses them an EKF estimating the IMU state w, i.e. orientation, position, velocity, and IMU biases. Preliminaries terms used in inertial odometry are included in App. A.2.2 and details of EKF and IMU measurement model are included App. A.6
We simplify the learning of informative priors by suitably canonicalizing the IMU measurements in two steps: First, we gravity-alige IMU measurements by rotating them into the frame R. such that the z-axis of the IMU frame and world frame coincide. We the EKF orientation state A.6)). Later we empirically show
from (Liu et al. 2020) to find the gravity direction (see App. App. the robustness of method noise originating from this estimation. In what follows we thus
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reflected trojectory
K
Raxadu w
of
(R.1))
reference trajectory
Rascs
rotated trajectory
rot. sense
rot. sense
RascPP
P2
gravity-aligned frame z
tononicalttame
incorrect rot. correct rot.
proposed decomposition
Figure 3: Symmetries in neural inertial odometry. (a) An IMU undergoes three trajectories in zy- plane, each related to reference (blue) via rotation (purple) and/or reflection (orange) around gravity (parallel to the z-axis). At a fixed time, IMU measurements on different trajectories, expressed in the corresponding local gravity- frame (red-green) differ only by an unknown yaw roto-reflection Rax3. Mapping these measurements to a canonical frame (yellow-red) that transforms equivariantly under roto-reflections of the trajectory eliminates this ambiguity enhancing the sample efficiency of downstream neural networks. (b) Expressed in alternative roto-reflected frames, acceleration, and angular rates transform as a' Rax3a and w' = det R3xa) Rax3w. Angular rate must follow the right-hand-rule, and thus be also inverted when reflected. To ensure a similar transformation rule as a, we decompose w = V1 x 2/2 and process U1 02 instead, which transform as 0//22 = Rax381/2
assume accelerations and angular rates to be expressed in the gravity-aligned frame and illustrate these frames in Fig. 3 (a) for three rotated trajectories. Gravity alignment reduces data variability by two degrees of freedom. However, this frame is not unique, since simply rotating it around z or reflecting it across planes parallel
(applications of rotations or roto-reflections from the R E Rg g) and (3) = {R ( g}) result in new valid gravity-aligned frames. Hence, secondly, we predict a canonical frame and map the IMU data into this frame, which we later show to be subequivariant to roto-reflections. In what follows we will restrict our discussion to the 0g(3) case but note that, where not explicitly stated, this discussion carries over to SO₂ well. Next, we will introduce our canonicalization procedure to ensure better network generalization.
groups SO,
4 METHODOLOGY
[))(() E (3) from data, which generalizes across arbitrary yaw orientations. We use this frame to map IMU data into canonical frame before giving as input to the NN, and mapping the outputs back, (see Fig. 2 (a))..e
Our goal is to predict a canonical yaw frame F = (r(((s,
(4) where a', w/ are expressed in the canonical frame. Finally, we map d', E' back to the original frame via d Pa(F)d and vec(E) pr veccE"e)"" Here vec(.) stacks the columns of Σ into a single vector, and p is homomorphism that maps group elements F to corresponding matrices, called matrix representations. These capture the transformation of a, w, and Σ under the action of F.
d' (b (())
with d: Pa (F )a: and w/ Pw Jwi,
While Pa (F) = PF(F)) F3x3, with Fax3 E 3x33 being the rotation matrix corresponding to element F, covariances transform as (F( = Fax3 @ Fax3, where @ is the Kronecker product. Unfortunately, reflections (det(Faxa)
-1) induce reflection and inversion of angular rates to preserve the right-hand-rule (see Fig. b), i.e. (F) = det( F3xx3)Fxxx and for reflections. As discussed later P= (F)= # Pa (F) hinders joint processing of accelerations and angular rates. Next, we will discuss the design of V which ensures generalization across arbitrary yaw rotations.
4.1 EQUIVARIANT FRAME
Here we derive a property of the frame network I such that it can generalize to arbitrary roto- reflections of the IMU body frame. To generalize, canonical IMU measurement inputs a/ w to the network must look identical under arbitrary roto-reflections R E Og(3 (3). Let a,, 4;, and at w denote quantities before and after application of R. Then ot = Pa (R)a, and w = (W Enforcing identical inputs under both rotations, i.e. a = a' we have
a' = Pa (F* )o; wh = p Jwi
(5)
We that choosing F+ R.F, i.e. that F transforms equivariantly leads to
a² = Pa PF PF R Pa(F-1R RR)aa oñ
(6)
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where we have used the fact that Pa is homomorphism. This shows the invariance of de and similar proof can be done for w/. This equality puts a constraint on the NN that estimates F, namely
RF = w (H(ea (f))a))))
(7)
ie. must be a function that is equivariant with respect to group actions by elements from 0g (3). Since this is subgroup of 0(3) we also say that v must be subequivariant with respect to (3).
In addition, this equivariance property of V induces end-to-end equivariance to predicted displace-
ments d Pa( Pa(F)d (F)d' and covariances vec(I) = p=(F))ve(E'). This is because
d' pa(F")dd' pa (F())) pa Pa()
(8)
vee(E") = pr:(F") pm::
(vee)"""))") pv: (R.()vvee((E") ps)s)) (Fee)))sss)))c)ce (F)vec(E'>= pv: p>: (R)vee(E)
(9)
using, again, the homomorphism of p and the fact that d', vec(E") are, by construction of equation 6
invariant to rotations by R.
We show empirically in Sec. 6 the diagonal parameterization of E' aids in stabilization and convergence of the network. Therefore, we assume the displacement uncertainties Ed,az = Ed.y= = O and without loss in generality choose Σ' = diag(e?"z, e2",, e22x), ) where Ug My. U2 are learnable, in TLIO (Liu et al. [2020). Since our network predicts vec(E) (ecc22e))"") where both >' and F are learned, the resulting covariance is > = FE'F FT (in matrix format). Via the transformation F' we can learn arbitrarily rotated Σ in the zy-plane. We posit that this forces the frame network T/ to learn F that aligns with the principle axes of the statistical uncertainty in displacement Σg. See App. [A.2.3 for details on covariance parameterizations. Writing the singular value decomposition (SVD) we see that Σn- = Udiag( (En,, Eny Egg) By inspection, this uncertainty is matched when F aligns with principle directions U and E' aligns with the true uncertainties in those directions.
Diagonal Covariance:
In the next section, let us now discuss the specific issue that arises when designing an equivariant frame network to process both a; and wi, and the specific preprocessing step to remedy
4.2 DECOMPOSITION OF ANGULAR RATES
As previously discussed and w transform under different representations Pa # Pw- This hinders joint feature learning since this would entail forming linear combinations of a and w. and these linear combinations will not transform under Pa or Pw. We propose a preprocessing step that decomposes wi into perpendicular vectors U1 22 via bijection F
R.
R x R
F(w) = (vi, v2) =
|
|wl|
F (mi, v2) = w = 0 x v2
(10)
|m I
We define wi = [- w2 0 and w2 = w x wT. uf Wy = O, we use Wt a x w and if both
W/r wy = 0 and a x w = 0, we use wh = w x [100]".
= Poz (F) F3x3. Let variables with * denote transformed vectors according rotation R. Their cross product has the desirable property w v7 x v (R3x3vi) x (Rax312)) det( [ax3 33 R x v2) = det( [a ( Raxaw a (R),,, using the standard pross-product property (Ax) x (Ay) = det( (A) A(x x y), and recalling that Rax3 is the matrix representation of R. The group action on exactly coincides with what was derived in Sec. 3 We only use this decomposition when with 0g(3), where we process a, U1 D2 in a unified way. For (3), we process a, which transform similarly since det(R = 1.
Fig. 3 (b) shows that U1 and D2 transform with representation Pe,, (F)
4.3 TRANSITION TO O(2) EQUIVARIANCE AND BASIC NETWORK LAYERS
Expressed in the gravity- aligned frame, representations Rax3 of R E Og leave the z-axis unchanged, and can thus be decomposed into Rax3 = Rax2 @ 1, where the direct sum @ constructs a block- diagonal matrix of its arguments. This decomposition motivates the decomposition a; = di.ry @ at,z and 0122,; U1/2.i.ay + 01/2.i.2. where @ concatenates the zy-coordinates of each vector which transform with representation R2x2 and its 2 component which transforms with representation 1, i.e is invariant. This means that the cry-components transform according to representations of group ((2) and implies that ((2) _ (3). Inspired by Villar et al. (2021). we design our frame network to learn universally 0(2) equivariant outputs from invariant teatures alongside 2D vector features.
We convert the sequence of n IMU measurements into n x Ce rotation invariant scalar features and n x 2 x Cj vector features. As vector features we select the zry-components of each input

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Published as a conference paper at ICLR 2025
vector (Cj- 2 for SO(2) corresponding with di,ays Wa, and Cj 3 for ( corresponding with as 5,2.. V22,,zy). Instead, as scalar features we select (i) the z-components of each vector, (ii) the norm of the zy-compponets of each vector, and (iti) the pairwise dot-product of the ry-components of each vector. For S0(2) we have Cj 2+ 2+2+1 1 = 5. while for ((2) we have Cě = 3 + 3 + = 9. We process scalar features with MLPs and standard 1-D convolutions, vector features with specific linear and convolution layers, and combine both with specialized non-linear layers.
Equivariant Linear Layer Following Villar et al. (2021), we design 2D version of vector neuron (Deng et al., (022)) to process the vector features, enhancing efficiency. Following Finzi et al.) (2021), we consider learnable linear mappings your = We'", with input and output vector features e" 2 E IR² and seek a basis of weights W E 22x2 which satisfy Rex = W W2xx20, i... cquivariantly transform vector features V E IR2. This relation yields the constraint
(R2x2 @ Rexe) ( = vec(W),
(11)
Solving the above equation amounts to finding the eigenspace of the left-most matrix with eigenvalue
1. Such analysis for R2x2 E SO(2) yields W Wso(2) = w1y 12x2 + Wy where Roo denotes a 90 degree counter-olockwise rotation in 2D, and w1, W2 E IR are learnable weights. Similarly, for ((2) we find Wo(2)= = W1 2222x2. Vectorizing this linear mapping to multiple input and output vector features we have the following SO(2) and ((2) equivariant linear layers:
SO(2):
: n ===="" W1 Reov W2
O(2)
W1
(12)
with n'Th R22x y 22 and W1, W2 E RC, Note that the SO(2) layer has twice as many parameters as the ((2) layer. We stack the above linear components into a kernel to design equivariant I-D convolution layers. Since the IMU data forms time sequence, we implement convolutions across time. We visualize our Linear and Convolutional Layers in Fig.
Nonlinear Layer Previous works (Weiler et al. 2018, WWiler & Cesa 2019) propose various nonlinearities such as norm-nonlineartty tensor- product nonlinearity, and gated nonlinearity for SO(3) and (2)) equivariance in an equivariant convolutional way: while Deng et al. (2021) applies per-point nonlinearity for vector features only. Since we already apply convolutions over time we simply apply a non-linearity pointwise. Unlike Deng et al. (2021), we need to mix scalar and vector features and thus adapt the gated nonlinearity (Weiler et al. 20188) to pointwise nonlinearity. Specifically, for vector and scalar features yd E | .t E Raxe we concatenate the norm features || || E RxxxC with si" We run a single MIP with an output of size n x 2C, and split it into new norm features 7 E RR" and new activations E =xxx'' which we modulate with a non-linearity s = (3). Finally, we rescale the original vector features according to the new norm:
tensor-product
7.8 = mlp(I || Oe )
out
(3)
(13)
where Be concatenates along the feature dimension. See Fig. 2p for more details.
5 EXPERIMENTS
We apply framework to two types of neural inertial navigation systems: (i) end-to-end deep learning approach (RONIN), and (ūi) a filter-based approach with a learned prior (TLIO). Both networks process IMU samples in gravity-aligned frame without gravity compensation, i.c., removing the gravity vector from the accelerometer reading. While RONIN regresses only a 2D velocity, TLIO estimates the orientation, position, velocity, and IMU biases using an EKF in 3D, which propagates states using raw IMU measurements and applies measurement updates with predicted displacement and uncertainty from NN. Sec. 6 presents extensive ablations.
Datasets: Our TLIO variant is trained on the TLIO Dataset (Liu et al. 2020) and tested on TLIO and Aria Everyday Activities (Aria) Datasets (Lv et al. 2024). Our RONIN variant is trained on RONIN Dataset (Herath et al. 2020). We train on the 50% open-sourced data, We test our RONIN variant on three popular pedestrian datasets RONIN (Herath et al. 2020), RIDI (Yan et al. 2018 and OxIOD (Chen et al. (0188)), which specifically target 2D trajectory tracking. RONIN-U (Unseen) contains IMU measurements from people who did not participate in the training and validation data collection. The people used record RONIN-S (Seen) overlap with those from the training and validation set, but their data is disjoint from these sets. See App[A.3 A.5 and Fig. for more details and visualizations of the equivariant network.
AA.3 for more dataset details, and
Baselines: We compare EqNIO with TLIO with yaw augmentation (Liu et al. 2020), on the 3D benchmarks, and RONIN and RIO (Cao et al. 2022) on the 2D benchmarks. We also report the naive
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TLIO Datasct
Aria Datasct
Model
MSE"
ATE
ATE*
RTE
RTE"
AYE
MSE
ATE
ATE"
RTE
KTE*
"m²")
(m)
(m))
(m)
(n)
(dog)
(10"Fm²)
(n)
(t1)
(01)
(deg)
TLIO
3.333
1.722
3.079
0.521
0.542
2.366
15.248
1.969
4,560
0.834
0.977
2.309
aug.
2,073
SO12) Eq. Frame
3.194
1.480
2.401
0.490
0.501
2.428
2.457
1.178
1.864
0.449
0.484
2.084
(2) Eq. Frame
2.389
Table I: Trajectory errors, lower being better. + rot. aug. is trained with yaw augmentations. Lowest, and second lowest values are marked in red and orange. * no EKF.
TUD rot. aug.
TUO
TUD 5012) Eq. Trame
TUO O1ZI Eq. Frame
3.0
2.5-
2.0-
Att/mm
De
1.5
10
TUD
TUO
ArM
Aria
TLID
Ada
Figure 4: Trajectory emrors for NQ applied TLIO compared vanilla TLIO trained with and without yaw augmentations on TLIO and Aria Datasets visualized with box plot.
Ground Truth
TUO
TLJO
Starting Point
2.
Y(ma
Y0m)
vimi
-8
-30
-20
-10
0
-10
D
10
Xlm)
Figure 5: Groundtruth (black), and predicted trajectories the TLIO Dataset by TLIO (Blue), and EqNIO (Red). Left and right have high, and the middle has medium difficulty.
IMU double integration (NDI) from Herath et al. (2020), and RONIN results trained on 100% the RONIN training dataset. All other methods use only the public 50% of this dataset. RIO extends RONIN with two features: Joint optimization of an MSE loss on velocity predictions and cosine similarities with equivariance constraint modeled using auxiliary loss (+-J) and Adaptive Test-Time- Lime-Training strategy (+T7T). Finally, +-J + TTT combines both.
Metrics: The NN performance (indicated with +) is reported in terms of Mean Squared Error (MSE) in 10-2m² Absolute Translation Error (ATE) in m, and Relative Translation Error (RTE) in m, on the cumulative sum of the displacements predicted by the network done in prior work. For TLIO, we also report the ATE in m, RTE in m, and Absolute Yaw Error (AYE) in degrees when rumning the EKF, See App. A.7 for more metric details. In what follows, we abbreviate EqNIO with +Eq. Frame.
5.1
RESULTS USING THE TLIO ARCHITECTURE
Tab. compares baseline TLIO, trained with yaw augmentations in (Liu et al. 2020) (+ rot.agg.),
TLIO without augmentation (termed TLIO), and our two methods applied to TLIO without yaw augmentations (termed +SO(2) and +0(2) Eq. Frame). As seen in Tab. [ +0(2) Eq. Frame outperforms TLIO on metrics that ignore the EKF (with *) by a large margin of 57%, 12%, and 11% MSE* ATE*, and RTE* respectively. The +SO(2) Eq. Frame model follows closely with 54%, 11% and 7% respectively on the Aria Dataset. The performance of our methods is consistent TLIO and Aria Datasets illustrating our generalization ability. Tab. and Fig. 4 show our method surpasses the baseline on most metrics while remaining comparable in AYE. The superior performance of our model as compared to baseline TLIO when the NN is combined with EKF (i.e., performance on ATE, RTE, and AYE metrics) is attributed to its generalization ability when the orientation estimate is not very accurate as well as the equivariant covariance predicted by the network. See Fig. and App. A.8 for trajectory plots from the TLIO test dataset.
5.2 RESULTS USING THE RONIN ARCHITECTURE
On the RONIN dataset, we compare against RONIN and RIO (indicated with +J, +TTT, and +J+TTT). RIO does not provide results for the RONIN Seen Dataset (RONIN-S) RIDI Cross
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RONIN-U U
RONIN'S S
RIDI-T
RIDI C
OxIOD
Model (RONIN)
ATE
RTB
ATE
RTE
ATE
RTE
ATE
RTE
ATE
RTE
(m)
(m)
(m)
(m)
(m)
(m)
(m)
(m)
(m)
(m)
100% data
5.14
4.37
3.54
2.67
1.63
1.67
1.62
3.46
4.39
50% data
5.57
4.38
1.19
1.75
3.52
4.42
50% data
5.02
4.23
1.13
1.65
3,59
4.43
50% data TTT
5,05
4.14
1.04
1.53
2.92
3.67
50% data + +TTT+
5.07
4.17
1.03
1,51
2.96
3.74
550% data S0(2) Eq. Frame
5.18
4.35
3.67
2.72
0.86
1.59
0.63
1.39
1.22
2.39
50%% data (12) Eq. Frame
4,42
3,95
3.32
2.66
0.82
1,52
0.70
1.41
1.28
2.10
Naive Double Integration (NDI)
458.06
117.06
675.21
1,6948
31.06
37,53
32.01
38.04
1941.41
848.55
Table 2: Trajectory errors with the RONIN architecture (lower is better), on the RONIN Unseen (-U),
Seen (-S) (people used to record the test dataset are the same as train and validation), RIDI Test
(-T), Cross Subject (-C), and the Oxford Inertial Odometry Datasets (OxIOD). Red (lowest), orange
(second lowest). i results from Cao et al. (022). indicates no-EKF.
TLJO Dataset
Aria Dataset
Model
MSE (10 (10-Fm²²
ATE
ATE
RTE (m)
RTE*
AYE
MSE (10-Pm²)
ATE
ATE
RTE
RTE
AYE
(m)
(m)
(dog)
(m)
(m)
(m)
(m)
(dag)
TLO
3,333
1,722
3.079
0.521
0.542
2.365
15.248
1.969
4.560
0.834
0.977
2.309
aug.
3.242
1.812
3.722
0.500
0.551
2.376
5.322
1.285
2.102
0.464
0.521
2.0733
ang more layers
3.047
1,613
2,766
0.524
0.519
2.397
2,403
1.189
2.541
0.472
0.540
2,081
+OL aug. Non Eq. Frame
3,008
1,429
2.443
0.495
0.496
2.411
2.437
1.213
2.071
0.458
0,508
2.096
aug. PCA Frume
3,473
1,506
2.709
0.523
0.535
2.459
6.558
1.717
4.635
0.771
0.976
2.232
+ +80(2) Eq. Frame +S
3.331
1,626
2.796
0.524
0.536
2,400
2.591
1,146
2,067
0.466
0,517
2.089
+80(2) Eq. Frame P
3,298
1,842
2.652
0.588
0.523
2.537
2.635
1.592
2,303
0.585
0,539
2.232
+SO(2) Frame
3.194
1,480
2,401
01490
0.501
2,457
1.178
1.864
0,449
0,484
2.084
+ .0(2) Eq. Frame S
3.061
1,484
2.474
0.462
0.481
2.390
2,421
L.175
1.804
0.421
0,458
2.043
+ (0(2) Eq. Frame P
2.990
1.827
2,316
0.578
0.478
2,534
2.373
1.755
1,859
0,564
0,468
2:223
Eq. Frame
2.982
1,433
2.382
0.458
0.479
2.301
L118
0.416
0.465
Eg CNN
3.194
1,580
3.385
0.564
0.610
2.394
8.946
3.223
6.916
1.091
1,251
2.299
Table 3: Ablations with the TLIO architecture, lower is better. We test non-equivariant (+Non Eq. Frame), PCA-based (+PCA Frame), S0(2) equivariant (+SO(2) Eq. Frame), and 0(2) equivariant (+0(2) Eq. Frame) frames, and yaw augmentation (+ rot. aug.). We also test ry-isotropic (+S) and Pearson-based (+P) covariance parameterizations. Eq CNN is fully equivariant CNN. Red (lowest), orange (second lowest), and yellow (third lowest). indicates no-EKF.
Subject Dataset (RIDI-C). As the RONIN base model does not use an EKF, we only report metrics with As Seen in Tab. 2 our methods significantly outperform the original RONIN by large margin of 14% and 9% on ATE* and RTE* respectively even on the RONIN-U dataset. Our methods have better generalization of 56% and 43% on ATE and RTE* respectively on OxIOD Dataset. The +2))) Eq. Frame model converges at 38 epochs compared to over 100 in RONIN implying faster network convergence with our framework as compared to data augmentation. This demonstrates superior generalization of our strictly equivariant architecture. RIO's approach, involving multiple data rotations, test optimization, and deep ensemble at test time, would result in higher computational and memory costs as compared to our method. Finally, the comparison with NDI highlights the need for a neural displacement prior.
seen RIDI-T and OxIOD, outperforming even +J+TTT; by a margin
ABLATION STUDY
Here, we show the necessity of incorporating equivariance in inertial odometry, the choice of equivariant architecture, and covariance parameterization. We present all the ablations using the TLIO base model in Tab. 3 both with and without integrating the EKF. AprrtA.9furthher the performance of all models above on test dataset which is augmented with rotations and/or reflections. App. A. A.13 A. A. 14 present sensitivity studies on the input sequence length, estimated gravity direction, IMU biases and IMU sampling rate.
further contains
Baseline Ablation: Is yaw augmentation needed when the input is in a local gravity-aligned frame? We trained TLIO both with and without yaw augmentation using identical hyperparameters and the results in Tab. 3 (rows I and 2) reveal that augmentation enhances the network's 's generalization. improving all metrics for the Aria dataset with the lowest margin of 10% on AYE and highest margin of 65% for MSE* This underscores the importance of equivariance for network generalization. Does a Deeper TLIO with a comparable number of parameters match the performance of equivariant methods? We enhanced the residual depth of the original TLIO architecture from

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Published as a conference paper at ICLR 2025
-6
Ground Tush
Ground
Ground
frome
10
-12
-14
12
12 10
12
10
20m1
2010
Xino
Figure 6: Trajectories and frame basis vectors (red/blue) on the Aria Dataset.
residual blocks of depth 2 each to depth each (row 3) match the number of parameters with our +S0(2) Eq. Frame model (row 8). Despite having fewer parameters due to the removal of the orthogonal basis in SO(2) vector neuron-basen architecture, +0(2) Eq. Frame model (row 11) still outperformed the deeper TLIO. The data from Tab. 3 show that merely increasing the network's size, without integrating true equivariance, insufficient for achieving precise inertial odometry.
Frame Ablation:
Can a non-equivariant MLP predict meaningful frames? We trained TLIO with yaw augmentation and identical hyperparameters alongside an additional MLP mirroring the architecture of our method to predict a frame and term this baseline +Non Eq. Frame (row 4). We observed that +Non Eq. Frame tends to overfit to the TLIO dataset, and thus produce worse results on the Aria dataset. The predicted frames also poorly correlate with the underlying trajectory, as illustrated in Fig. [ Can frames predicted using PCA (handcrafted equivariant frame) achieve the same performance? Using PCA to generate frames leads to underperformance on the Aria dataset, and worse results than the original TLIO which is likely due to PCA's noise sensitivity as Additionally, PCA cannot distinguish between SO( and (22) transformations. Fig. 6 also shows that ((2) does not have frames as smooth as SO(2) as the reflected bends have reflected frames. See App. 16
shown in Fig. 6
additional ablation using tramaanve for canonicalization.
Architecture Ablation:
Are fully equivariant architectures better ? We trained a fully equivariant I-D CNN using the layers in Sec. 4.3 Tab. 3 (row 12) shows our frame-based methods outperforming the equivariant CNNs, likely by leveraging the power of scalars and conventional backbones. We believe the fully equivariant architecture is overly restrictive, while also requiring full network redesign. By contrast, our method can flexibly integrate existing ofte--of-the-ar displacement priors.
Covariance Ablation: Do we need equivariant covariance? We investigated the importance of equivariant covariance for both SO(2) and 0(2) groups, as described in Sec. 4. App. A.2.3 App.[A.2.3 for covariance parameterizations). In Tab. 3, the models +S (rows and 9) are trained with invariant covariance parameterized as Σ = diag( Zux , e²2x)), )) that is unaffected by application of F. The results show that the equivariant covariance yields better performance, especially when combined with EKF, as it provides a more accurate estimate of the prediction covariance. Can a full covariance matrix with Pearson parameterization improve performance? In Tab. 3 our model outperforms +P (rows 7 and 10) in most cases indicating that by aligning the principal covariance axis into the basis of the equivariant frame, we intrinsically force covariance in the equivariant frame to be diagonal, which reduces ambiguity while training. Diagonal covariances improve convergence stability during optimization as stated in Liu et al. (2020). App. AI visualizes the covariance consistency of EqNIO and we conduct an analysit sis on the covariance parameterization in App. AA.17
7 CONCLUSION
We introduce a robust and generalizable neural displacement prior that combats drift in IMU-only neural inertial odometry via equivariant canonicalization. Our canonicalization scheme generally applicable and eliminates the underlying yaw ambiguity in gravity-alignes frames which arise from roto-reflections in the plane around gravity. We fully characterize actions from this group on all relevant inputs and outputs of the prior and leverage this insight to design a network that produces learned frames that are O (3) equivariant to these actions. By reducing the data variability seen by neural networks these frames boost the generalization of existing networks and enforce exact equivariance, unlike existing strategies that use data augmentation or equivariant consistency losses to enforce approximate equivariance. We demonstrate the generality of our framework through extensive validation on various datasets and applications to two base architectures (TLIO and RONIN). We believe this work paves the way for robust, and low-drift odometry running edge devices.
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ACKNOWLEDGEMENTS
We gratefully acknowledge support by the following grants: NSF FRR 2220868, NSF IIS-RI 2212433,
ONR X00014-22-1-2677 and SNF 225354.
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A APPENDIX
A.1 MORE RELATED WORKS
Inertial Navigation Systems (INS): The research in INS can be put into three broad categories: estimation-basse techniques, multi-sensor data fusion (MSDF) techniques, and learning- -basdd al- gorithms. The Kalman Filter (Kalman & Others (960) and it's variants, EKF, UKF and MEKF,
have been implemented for inertial attitude estimation (Jing et al.; 2017: Chiella et al. 2019 Hall
et al. 2008), purely inertial odomety (Leishman al.)
2014 Titterton et al.) 2004 Bortz 1971).
Visua Inertial odometry (VIO) (Mourikis & Roumeliotis) 2007). LIDAR Odometry /Zhang & Singh
2014
Zheng al., 2024) and SLAM (Newman 1999 Williams al.] 2000; Kim & Sukkarich
2003
but suffer from drift when using IMU data alone. Learning-based algorithms can reduce this
drift, and have been designed for inertial attitude estimation (Weber et al.
2021: Brotchie al. Asgharpoor Golroudbari & Sabour 2023), inertial odometry (Chen et al. 201 Liu et al.
2022:
2020
Buchanan et al.j [023] [
Tang al.) 2024; Qiu et al. (2224) LIDAR odometry (Dong et al.
2024
lvanmard-GGh. et al.] 202 Okawara et al. 2004), VIO (Chen et al.) 202 Pan et al.. 2024)
and SL.AM (Wang et al. 2024 Peng et al. |l.| 20024. The consistency of VINS and EKF-SL.AM has
been studied in Hesch et al. (2014): 1): Huang et al. (2008) 2009). Recent works have also explored equivariant observer design (Ng et al. 2023 Mahony et al. 2022]] Bouazza et al. 2024), equivariant
IMU Preintegration (Delama et al. (024). the design of equivariant filters (van Goor al. 2023 van
Goor & Mahony [203] Fornasier et al. (004) and geometric data fusion (Ge et al.
2024) for attitude
estimation that do include any learning-based components. They work on equivariant dynamics and use the lift function to map the state and the extended velocity to the Lie algebra associated with the system's symmetry group, enabling the construction of lifted system the Lie group. The error dynamics are linearized around fixed origin, which makes it independent of the current state. To the best of our knowledge, we are the first to bake the equivariance into the NN for the inertial odometry problem. Integrating our equivariant network into these models would be potential future work.
Canonicalization: Both hand-crafted and learned canonicalization (owe 2004 Yüceer Oilazer 1993] Jaderberg et al. (015) has been widely studied and practiced. Recently, canonicalization has gained popularity in equivariant models, improving training efficiency and enabling seamless integration with standard model architectures. [Kaba et al. (2023) first generally and theoretically proposed using Learned Canonicalization Functions to achieve equivariance. (Pnny et al. (2021) established standardizing the input by calculating the smallest set of frame set. Further, Ma et al. shows the connection between the averaging method and canonical forms. In practical applications, [i et al. (021): Baker al. (2024); [uo al. (2022), Du al. (2022) canonicalize 3D point clouds by constructing global and local frames. Deng et al. (2021) introduces an equivariant canonicalization method in PointNet and et al. (2024) integrates canonicalization into the Perceiver 10 architecture. Additionally, with the rise of large pre-trained models, recent work (Mondal et al. 2023 applies canonicalization as a preprocessing step before inputting such models.
A.2 PRELIMINARY
A.2.1
EQUIVARIANCE
In this section, we introduce more preliminaries of group and representation theory which form the mathematical tools for equivariance.
Group The group G is set equipped with associative binary operation which maps two arbitrary two elements in G to an element in G. It includes an identity element, and every element in the set has an inverse element.
In this paper, we focus on the group SO(2) and 0(2). (2( is the set of all 2D planar rotations, represented by orthogonal matrices Rxx2 R2x2 with det( R2x2) 1. This group operation is matrix multiplication, and each rotation matrix has an inverse, which its transpose. The identity element is the matrix representing rotation.

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(2) consists of all distance- preseevvng transformations in Euclidean 2D space, including both rotations and reflections. Elements of ((2) are orthogonal matrices R2x2 E m with det( (Rxx2) = =1, with the group operation being matrix multiplication. Each transformation matrix has inverse, and the identity element is the matrix representing no transformation.
Group Representation and Irreducible Representation The group representation a homomor- phism from the group G to the general linear map of a vector space V of a field K, denoted GL(V). In this work, we use R and K R
An irreducible representation (irrep) of a group G is representation in which the only invariant subspaces under the action of G the trivial subspace {0} and the entire space In other words, an irreducible representation cannot be broken down into smaller, nontrivial representations,ie. it cannot be the direct sum of several nontrivial representations.
For SO(2), we can use A E (0, 2x] to represent SO(2), for any 8, the irreducible representation of the frequency n E N is:
(0) = cos no
sin no cos n&
sin no
In this work, we use 1. For O(2), we can use r (-1,1) to denote reflection and (0, 2n] to denote rotation. The trivial representation po(r, 0) 1. For the nontrivial representation of frequency n e N+
(cos(n0) sin(n0) sin(no)) cos(n1)) (1) .)
Pro Pn( ()
There is another one-dimensional irrep for O(2), p(r, 0) r which corresponds to the trivial representation of rotation.
The introduction to group representations has been covered extensively in previous work on equiv- ariance (Cohen & Welling 2016 Weiler al. 2018: Xu al. 2024). Specifically, for SO(2) and 0(2), Weiler & Cesa (2019) provide a detailed introduction.
Invariance and Equivariance Given a network : + 3, for any x E X
( ((x),
implies the group representation D of the output space is trivial, i.e. identity, and the input does not transform (i.e. the input is invariant) under the action of the group. Note that in our paper X' and ) can be seen the direct sum (concatenation) of all input and output vectors respectively. In our paper, the coordinates/ projections 3D vector to the gravity axis z-axis are invariant, therefore we call them invariant scalars.
network कb : X' + y is equivariant if satisfies the constraint
2) = p²)((())
In this paper, the input is the sequence of accelerations and angular velocities, and the output is composed of displacement and covariance. For displacement, the z-component is invariant while zyycoomponents are acted under the representation of defined in the above section. Hence, for displacement, x² P1 + 1 and for covariance 3D covariance, p² (pI + (p1 $
1)
Subequivariance As mentioned in prior works (Chen et al. 2023 Han et al. 2022), the existence of
gravity breaks the symmetry in the vertical direction, reducing 0(3) to subgroup O(2). We formally characterize this phenomenon of equivariance relaxation subequivariance. We have mathematically defined the subequivariance in Section of the paper. In simpler terms, the gravity axis decoupled separate 2D vector. Upon
and treated an invariant scalar while the other two axes are handled rotation, the invariant scalar remains constant while the other two axes are transformed under rotation. So are limited now to SO(2) rotations and roto-reflections, In the general case of equivariance, the 3D vector would be considered dineensiinnii and an SO(3) rotation would act on it. The transformation would be along all three axes.
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A.2.2 INERTIAL ODOMETRY
In this section, we introduce more preliminaries on the terms used in inertial odometry.
Inertial Measurement Unit Inertial Measurement Unit (IMU) is an electronic device that measures and reports linear acceleration, angular velocity, orientation, and other gravitational forces. An IMU typically consists of 3-axis accelerometer, a 3-axis gyroscope, and depending on the heading requirement 3-axis magnetometer.
An accelerometer measures instantaneous linear acceleration (a:). It can be thought of mass spring, however in micro-electro mechhticae systems (MEMS) it is beams that flex instead of spring.
gyroscope measures instantaneous angular velocity (w;) It measures the angular velocity of its frame, not any external forces. Traditionally, this can be measured by the fictitious forces that act on moving object brought about by the Coriolis effect, when the frame of reference is rotating. In MEMs, however, we use figh-freueeccy oscillations of mass to capture angular velocity readings by the capacitance sense cones that pick up the torque that gets generated.
World Frame world frame, also known cartesian coordinate frame, is fixed frame with known location and does not change over time. In this paper, we denote the fixed frame with z-axis perfectly aligned with the gravity vector as the world frame, denoted as w.
Local-gravity-alliner Frame A gocal-gravity-alignee frame has of its axes aligned with the gravity vector at all times but it is not fixed to known location.
Body Frame A body frame comprises the origin and orientation of the object described by the navigation solution. In this paper, the body frame is the IMU's frame. This is denoted as b for the IMU data.
Gravity-compensatiion Gravity compensation refers to the removal of the gravity vector from the accelerometer reading.
of IMU data refers to expressing the data in the gravity- aligned frame. This is done by aligning the z-axis of the IMU inertial frame with the gravity vector pointing downwards and is usually achieved by fixing the roll and pitch (rotations around the and axes) or by applying a transformation estimated by the relative orientation between the gravity vector and fixed z-axis pointing downwards. This is usually achieved with a simple rotation.
Gravity-alignment
A.2.3 UNCERTAINTY QUANTIFICATION IN INERTIAL ODOMETRY
In this section, we provide more context uncertainty quantification in odometry and detail the different parameterizations used for regressing the covariance matrix in the paper.
Homoscedastic Uncertainty Homoscedastic uncertainty refers to uncertainty that does not vary for different samples, i.e., it is constant.
Heteroscedastic Uncertainty Heteroscedastic uncertainty is uncertainty that is dependent on the sample, i.c., varies from sample to sample.
Epistemic Uncertainty Epistemic uncertainty is uncertainty in model parameters. This can be
reduced by training the model for longer and/or increasing the training dataset to include more diverse samples.
Aleatoric Uncertainty Aleatoric uncertainty is the inherent noise of the samples. This cannot be reduced by tuning the network or increasing the diversity of the data.
Why do we need to estimate uncertainty in inertial odometry? Inertial odometry Kalman filters need measurement and plant covariances that are traditionally manually tuned and considered
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constant cross- measurement (homoscedastic). Neural networks offer us way to provide uncer- tainty estimate for each measurement. In the case of TLIO, the measurement of the Kalman Filter is displacement and the neural network predicts both the measurement and its covariance.
What is the uncertainty we are estimating in inertial odometry? We are regressing the aleatoric uncertainty using the neural network.
How the uncertainty estimated in this paper? We regress alcatoric uncertainty as a covariance matrix jointly while regressing 3D displacement following the architecture of TLIO (Liu et al.) 2020). Since there no ground truth for the covariance, we use the negative log-likell log-likelihood loss of the prediction using the regressed Gaussian distribution. As this loss captures the Mahalanobis distance, the network gets jointly trained to tune the covariance prediction.
2020) regresses only the three diagonal elements of Diagonal covariance matrix TLIO (Liu al.
the covariance matrix as log v Eazz log v Eny and log v Ees and the off-diagonal elements are zero, This formulation constrains the uncertainty ellipsoid to along the local gravity-aligned frame.
Full covariance matrix using Pearson correlation Russell & Reale (2021) define a parameteriza- tion to regress the full covariance matrix. They regress six values of which three are the diagonal elements log VErr, log v /Egy and log /E.z and the remaining three are Pearson correlation coef- ficients Pay. Pys. and Pas. The diagonal elements are obtained by exponential activation while the off-diagonal elements are computed as follows
Eg PV En
where Pij passes through tanh activation.
Diagonal covariance matrix in canonical frame In our approach, we regress the three diagonal elements as log VLezz log /Eay and log VLya in the invariant canonical frame. Since the z-axis decoupled from the æy-axis, only Eax and Egy are mapped back using the equivariant frame to obtain a full 2D covariance matrix from the diagonal entries. The resulting matrix is as follows
LΦx E
Egy 0
Egy
0
0
Σ
A.3 DATASET DETAILS
In this section, provide detailed description of the 4 datasets used in this work TLIO and Aria for TLIO architecture, and RONIN, RIDI and OxIOD for RONIN architecture.
TLIO Dataset- The TLIO Dataset (iu al. 2020 a headset dataset that consists of IMU raw data at IkHz and ground truth obtained from MSCKF at 200 Hz for 400 sequences totaling 60 hours. The ground truth consists of position, orientation, velocity, IMU biases and noises in IR3 The dataset was collected using a custom rig where an IMU (Bosch BM1055) is mounted on headset rigidly attached the cameras. This dataset captures a variety of activities including walking, organizing the kitchen, going up and down stairs, on multiple different physical devices and more than 5 people for wide range of individual motion patterns, and IMU systematic errors. We use their data splits for training (80%), validation (10%), and testing( 10%).
Aria Everyday Dataset- Aria Everyday Dataset (Lv al. 2024 an open-sourced egocentric dataset that is collected using Project Aria Glasses. This dataset consists of 143 recordings accumu- lating to 7.3 hrs capturing diversity in wearers and everyday activities like reading, morning exercise, and relaxing. There are two IMUs on the left and right side of the headset of frequencies 800 and IkHz respectively. They have two sources of ground truth- open and closed loop trajectory at IkHz. Open loop trajectory is strictly causal while closed loop jointly processes multiple recordings to place them in a common coordinate system. The ground truth contains position and orientation R33 We test dataset. The raw right IMU data is used to compare closed- trajectory with EKF results. The data was downsampled to 200Hz and preprocessed using the closed-looy trajectory to test the NN trained on TL.O.
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RONIN Dataset- RONIN Dataset (Herath et al. 2020) consists of pedestrian data with IMU frequency and ground truth at 200Hz. RONIN data features diverse sensor placements, like the device placed in bag, held in hand, and placed deep inside the pocket, and multiple Android devices from three vendors Asus Zenfone AR, Samsung Galaxy S9, and Google Pixel 2 XI.. Hence, this dataset has different IMUs depending on the vendor. We use RONIN data splits train and test their model with and without our framework. RONIN-U (Unseen) is test set that contains IMU measurements from people who did not participate in the training and validation data collection. The people used record the RONIN-S (Seen) overlap with those from the training and validation set, but their data is disjoint from the training and validation set. The RONIN-U dataset, thus, tests the generalization capabilities of method.
RIDI Dataset- RIDI Dataset (Yan et al. 2018) is another pedestrian dataset with IMU frequency and ground truth at 200 Hz. This dataset features specific human motion patterns like walking forward/backward, walking sidewards, and acceleration//declerres
They also record data with four different sensor placements. We report test results of RONIN models both the RIDI test and cross-subject datasets. RIDI results are presented after post-po post-procsssing the predicted trajectory with the Umeyama algorithm (Umeyama 1991) for fair comparison against other methods.
OxIOD Dataset- OxIOD Dataset (Chen et al. 2018b) stands for Oxford Inertial Odometry Dataset consists of various device placements/attachmensss motion modes, devices, and users capturing everyday usage of mobile devices. The dataset contains 158 sequences totaling 42.5 km and 14.72 hours captured in motion capture system. We use their unseen aulti-attachhents test dataset for evaluating our framework applied to RONIN architecture.
A.4 COMPARISON WITH PREVIOUS WORK
We start off by describing the different types of equivariance used in neural inertial odometry, followed by an in-detail description of each the related works our method is compared.
A.4.1 COMPARISON OF TYPES OF EQUIVARIANCE
In the current literature on learning- based purely inertial odometry, there are primarily three ways of achieving equivariance: (1) data augmentation, (2) the imposition of equivariant constraints in the form of auxiliary losses, and (3) the tailoring of NN to respect exact equivariance by design. The latter is the strategy used in this work. These different types of equivariance used in inertial
Transfomation
Trnining
Traasing Inpui
Training GT Augmentation Yes
Tss Inpui Augmentition
Strsz/Appron.ne mate
Group
Loss
Angumentation Yes
Equivariance
20201
S0:21
MLE
No
Apgroximate
RONIN
MSE
Appruximale
RIO
Cao et
2022
G
Equivarimce oss
Ausilary
Yes
No
Yes
Appenximate
Olils
012)
MLE
No
No
No
Stricss
Table 4: Comparison of our method with previous methods from the perspective of equivariance
odometry are illustrated in Figure 7 Importantly, while data-augmentation and auxiliary losses enforce approximate equivariance, leading residual equivariance errors even after training, the of equivariant NNs enforces strict equivariance, and thus equivariance error.
A.4.2 TLIO
TLIO, described (Liu et al. 2020), consists of two components: an Extended Kalman Filter (EKF), estimate the current orientation, position, velocity, and IMU biases, and NN to predict displacement prior, which is defined via its mean displacement and associated uncertainty.
We treat the displacement prior virtual measurement (with associated realization and measurement noise given by the mean displacement and covariance) and fuse it with the EKF state at 20 Hz. The EKF details be found in Appendix A.6 At the IMU sampling rate (1kHz for the TL.O and
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O() Equivariant Network
Ry o 4 = oo o Ry by
Hesign
a, @ d
4
Ry
Ry
4
d
SO(2)
Data Augmentation
Ry 4 ~ 4 Ry through training
a,
d
4
Ry
L(K,
d
SO(2) Equivariant Loss
Ry o 4 ~ 4 o Ry through training
R' Ry
CL(
a, w
d
Ry
4
R' Ry
CL CL(R,
d
R'
4
Figure 7: Overview of different equivariance types in neural inertial odometry. Works can be split into those using SO(2) Data augmentation, such as Liu et al. (2020); Herath al. (2020); Cao et al. (2022), those using SO(2) Equivariant losses Cao et al. (2022). and our work, which tackles this challenge with an (2) equivariant network. Data augmentation methods apply random yaw (R.) on acceleration angular velocity w, and displacements d for each loaded sample, and thus achieve approximate equivariance to yaw rotations (p( R2) o V Ri o p( RR,)). By contrast, methods that use SO(2) equivariant losses pass multiple randomly rotated samples through the network and enforce consistency between respective outputs using a self-supervised equivariant loss (C'L). Finally, our method employs equivariant canonicalization scheme which employs specialized equivariant layers that render our network inherently exactly equivariant (p(R.) R v = W o p (R,)) )) to reflectitnt elements R. E O(2) (a more general group than SO(2)). Our network thus does not require data augmentation, or expensive equivariant losses and still achieves better results.

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Aria Dataset), the EKF state and covariance are propagated using the IMU measurements. Since the measurement equation depends on current and past states, we initiate EKF state augmentation and cloning at each measurement. To avoid yaw observability issues, we use strategy similar to Coen et al. (202)) to linearize at the beginning of the 1 1-second window of past update/clone states. The EKF current states at the update frequency are stored as clone states before the EKF measurement update step. This is referred to as state augmentation, and the augmented past states are referred to as clone states.
The neural displacement prior d and the associated prediction covariance E are predicted from a NN. The NN takes as input IMU samples linearly interpolated from IkHz to 200Hz. This implies that within a time window of second, the network takes as input 200 IMU samples of 6 dimensions of which 3 dimensions correspond to accelerometer readings and 3 correspond to gyroscope readings. The input samples are first expressed in the orientation using the noisy gyroscope measurements. Finally, they are expressed in the local gravity- aligned frame using the yaw-estimate from the EKF at the clone state i corresponding to the beginning of the I-second window. The NN input is bias compensated using factory calibration values of IMU accelerometer and gyroscope biases and does not the EKF bias estimates, to avoid instabilities. The predicted covariance is parametrized as a diagonal covariance matrix and claimed in TLIO to lead to better convergence and stability during training. The NN is trained with VIO estimates of bias and orientation and uses displacement calculated from VIO position estimates as supervision data.
world frame by estimating the IMU
During deployment with the EKF, (i) bias estimates may not be accurate as they set to factory calibration values, and (ii) gravity orientation may be noisy due to noise in orientation estimation. These noise sources lead drift. To mitigate this drift, during training TLIO uses three data augmentation strategies: (i) IMU bias augmentation using ng ~ (0..2, 0.2mm/5* additive noise for the accelerometer bias and nw ~ [0000, 0.05|rad/s additive noise for the gyroscope bias. (ii) Gravity direction perturbation rotates the samples around random horizontal rotation axis by an angle a (1 5] deg. (iii) Yaw (2) augmentation rotates the network inputs and ground truth by random yaw angle y ~ U[-1,, m..
The network is trained with a batch size of 1024 samples for 50 epochs, with 1770 iterations per epoch. The NN architecture presented in Liu et al. (2020) consists of ID convolutional ResNet consisting of dilated causal convolutions followed by two disjointed 3-layer MLPs after flattening the ResNet output. One head corresponds to the 3D displacement prediction and the other head corresponds to 3 diagonal covariance values. The NN is trained for 10 epochs with a Mean Squared Error loss and the remaining 40 epochs with Mean Log Likelihood Error loss (MLE) as want the displacement prediction to converge before indirect supervision of the predicted covariance.
A.4.3 RONIN
RONIN is the predecessor of TLIO and is an end-to-end architecture regressing 2D velocity from IMU samples over I-second windows. The RONIN training dataset consists of pedestrian data which has 2D position as ground-trutt (GT). The GT displacement is divided by the time interval to obtain ground truth velocity for training. The regressed 2D velocities are directly integrated to obtain 2D position estimates.
During evaluation the predicted and ground truth trajectories are aligned using the Umeyama Ummeymaa| (1991) (scaled Procrustes) algorithm, since RONIN does not estimate ori- entation. The NN architecture RONIN is the same as TLIO where the network takes 200 IMU inputs of dimension 6 and consists of ID convolutional ResNet architecture followed by single 3-layer MLP head to regress 2 values corresponding to the x-axis and y-axis of the predicted velocity. As there is EKF, they do not estimate covariance. The only augmentation done is yaw augmenta- tion similar to TLIO. The random yaw angles sampled from (0, 2T]] are used to rotate a single batch of input samples and corresponding GT velocities. The batch size is 1024, and the network is trained for 100 epochs. The NN is trained only on the MSE loss.
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A.4.4 RIO
RIO proposes a framework that can be easily integrated with existing learning-based inertial odometry architectures. During training, RIO computes an auxiliary equivariant loss in the following way. It passes the network input and a randomly yaw-rotated copy (yaw rotation from SO(2) sampled from (0, 2T) through the neural network, and computes two losses. The first compares the output from the original sample to ground truth (via an MSE loss), and the second measures the negative cosine similarity between the original prediction and the prediction obtained from rotated input. This training procedure is termed the joint-training strategy. When the magnitude of ground truth velocity less than 0.5 m/s (termed as nearly stationary), the auxiliary loss is ignored. During inference, RIO uses a prediction model, and a model ensemble to employ an adaptive ses-time--traininn strategy. The model ensemble is used to estimate the uncertainty in prediction and trigger retraining of the prediction model. This ensemble consists of NNs that were trained from different random initializations. The variance of the predictions from the models is considered as the uncertainty of the predictions. Retraining occurs with a history of 128 samples, by minimizing the auxiliary loss and using the prediction model for initialization. Each of the samples is rotated by the four angles [72, 144, 216, 288] degrees. When the model predicts velocities below 0.5 m/s the prediction model is reset. So at any instance of time, there is one copy of the model that is continuously tuned during inference, one copy of the original trained network, and set of randomly initialized networks to get an ensemble-based uncertainty estimate.
While our work and RIO both aim to address the equivariance of the neural inertial odometry by modeling rigid transformations of trajectory and IMU data, there are several distinct differences. Most significantly, RIO addresses approximate equivariance, while our method enforces strict equivariance. Approximate equivariance enforces equivariance via equivariance loss which penalizes inconsistencies in predictions on data from four rotated trajectories. By contrast, strict equivariance enforces inherent consistency by tailoring specific NN components to be exactly equivariant. This work proposes several specialized linear and non-linear layers to guarantee the strict equivariance of the predicted canonical frame, while RIO uses traditional layers.
RIO adopts a test-time training strategy, which adds an additional computational overhead and thus increases the inference time as they required store multiple versions of the prediction model and augmented trajectories. Since our method is strictly equivariant, we do not require a fest-Time Training strategy. Moreover, despite not using such a strategy our method outperforms RIO.
Finally, RIO only handles 2D displacement outputs and equivariance to rotations in SO(2), i.e. 2D planar rotations. By contrast, our method can handle both 2D and 3D outputs and thus addresses equivariance in 2D and subequivariance in 3D. Moreover, we also extend the modeling to roto- reflections, i.e., the group (2), which comprises rotations and reflections the plane perpendicular to gravity and requires a novel bijection for angular rate preprocessing.
A.5 EQUIVARIANT NETWORK IMPLEMENTATION DETAILS
In this section, we first provide an overview of the canonical displacement prior and then describe in detail the equivariant network implementation and how it is combined with TLIO and RONIN.
A.5.1 OVERVIEW OF CANONICAL DISPLACEMENT PRIOR
(1) Following Figure we first gravity align the IMU data, i.e. rotate it such that the z-axis is aligned with gravity (estimated online by an EKF).
(2) We then process this IMU data by "cquivariant frame model" that outputs two vectors. This network takes in vector and scalar features derived from the IMU data. The equivariant network is strietly O(2) or SO(2) equivariant by design. The predicted vectors are then converted into an orthogonal set of unit vectors using Gram-Schmin orthogonalization.
(3) Then, we transform the input IMU data using the equivariant frame from (2) ("canon." " block in Figure 2 to produce invariant (consistent) inputs within the canonical frame, ensuring robustness to any roto-reflection group transformations applied to the original data.
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(4) The invariant imput is fed into standard neural network architecture ("off-the-shelf model"), such as TLIO's ResNet, to generate a displacement and covariance, termed canonical displacement prior expressed with respect to the learned canonical frame. These outputs remain consistent under transformations from the roto-reflection group applied to the original input.
(5) Lastly, project back the predicted displacement and covariance using the canonical frame to obtain an equivariant displacement and covariances using the equivariant frame of (2).
Finally, depending on the backbone architecture which our framework is applied (in case for TLIO), the predicted displacement and covariances are fed into filtering algorithm, like an EKF as a measurement update the estimate of the IMU state (orientation, position, velocity and biases).
A.5.2 IMPLEMENTATION DETAILS
The input to the framework is IMU samples from the accelerometer and gyroscope for a window of 1 second with IMU frequency 200Hz resulting in n = 200 samples. All IMU samples within a window are gravity-aligned with the first sample at the beginning the window, previously referred to as the clone state. The network design, seen in Figure b,, differs in architecture for SO(2) and ((2) and hence described separately below.
SO())- We decouple the z-axis from the other two axes and treat linear acceleration and angular velocity along the z-axis as scalars (2). We also take the norm of the 2D accelerometer and gyroscope measurements (2), their inner product (1) resulting in invariant scalars g=x5, The x and y components of IMU measurements are passed as vector inputs R=x2x2 The vectors and scalars are then separately passed to the linear layer deseribed in Section 4.3 The equivariant network predicting the equivariant frame consists of 1 linear layer, nonlinearity, 1 convolutional block with convolution applied over time, non-linearity, and layer norm. The hidden dimension is 128 and the convolutional kernel is 16 x 1. Finally, the fully connected block of hidden dimension 128 consisting of linear, nonlincarity. layer norm, and output linear layer follows a pooling over the time dimension. The output of the final linear layer is 2 vectors representing the two bases of the equivariant frame. The input vectors of dimension 2=x2x2 are projected into the invariant space via the equivariant frame resulting in invariant features in Rax4 These features are combined with the input scalars and passed as input (Rax6) to TLIO or RONIN base architecture. The output of TLIO is invariant 3D displacement and diagonal covariance along the principal axis. The output of RONIN is 2D velocity. The x and components are back-projected using the equivariant frame to obtain displacement vector in IR² and the covariance in the original frame. The covariance is parameterized and processed mentioned in Section
O())- The preprocessing is as described in Section 4.1 where w is decomposed to two vectors on and 12 that each have magnitude V|w]]. The preprocessed input therefore consists 3 vectors a, Vi, and U2. This is then passed to the equivariant network by decoupling the s-axis resulting in vector input x3x3 which represents 3 vectors in 2D. The scalars passed to the linear layer described in Section 4.3 consist of the accelerometer 2-axis measurement (1), the component of the two vectors V1 and 02 (2), the norm of the vectors (3) and the inner product of the vectors(3) resulting in gax9 The network architecture is the same as SO(2) with hidden dimension 64 and convolutional blocks in order to make it comparable in the number of parameters to S0(2) architecture. The invariant features obtained by projecting the three vectors using the equivariant frame are processed as mentioned in Section to obtain 2 vectors in 3D that are fed as input to TLIO and RONIN. The postprocessing the same as S0(
The framework is implemented in Pytorch and all hyperparameters of the base architectures are used train TLIO and RONIN respectively. The 50( architecture has 8,884,870 while (22) has 6,020,230 parameters and the base TLIO architecture has 5, 5,424,646. The baseline TLIO and our methods applied to TLIO were trained on NVIDIA a40 GPU occupying 7-8 GB memory per epoch. The training took 5 mins per epoch over the whole training dataset. We train for 10 epochs with MSE Loss and the remaining 40 epochs with MLE Loss similar to TLIO (Liu et al., 2020). RONIN was trained on NVIDIA 2080ti for 38 epochs taking 2 mins per epoch. The loss function used MSE as mentioned in Herath et al. (2020). The EKF described in TLIO was run on NVIDIA 2080ti with the same initialization and scaling of predicted measurement covariance in TLJO (Liu et al. 2020).

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We compare the resource requirements of the SO(2), and O(2) variant of our method coupled with TLIO, with base TLIO without an equivariant frame. We report the floating point operations (FLOPs), the inference time (in milliseconds), and Maximum GPU memory (in GB) during inference, an NVIDIA 2080 Ti GPU for the NN averaged over multiple runs to get accurate results. While base TLIO uses 35.5 MFLOPs, 2.79 ms, and 0.383 GB per inference, our SO(2) equivariant method instead uses 531.9 MFLOPs, 5.43 ms, and 0.383 GB per inference. Finally, our O(2) equivariant method uses 638.5 MFLOPs, 5.70 ms, and 0.385 GB per inference. We further evaluate the Maximum GPU memory for the equivariant networks separately and report 0.255 GB per inference for SO(2) equivariant frame prediction and 0.257 GB per inference for O(2) equivariant frame prediction. The Maximum GPU memory is unaffected because the equivariant frame computation utilizes less memory than TLIO.
Finally, we also evaluate our method with downstream EKF on an NVIDIA 2080 Ti GPU. The EKF incorporates raw IMU measurements for propagation, and displacement measurements from the NN as measurement updates. For every 20 imu samples, we send the last 200 IMU measurements to the NN to provide this measurement update. The original TLIO requires 0.492 seconds and 1.113 GB of memory. For the SO(2) variant of our method, to process second real-world data. For the O(2) variant, we use 0.554 seconds and 1.115 GB of memory, showing that our method is faster than real-time. The increase in memory for the O(2) variant is due to the additional preprocessing step.
require 0.554 seconds and 1.109 GB of memory
With comparable computing resources, our equivariant model outperforms TLIO since we leverage symmetry, which is an intrinsic property in inertial odometry.
A.6 EKF DETAILS
The EKF continuously estimates the orientation, linear velocity, position, acceleration, and gyroscope biases. In TLIO architecture, the EKF propagates the IMU samples at higher frequency of kHz while the NN takes clone states at 200Hz to predict the displacement over second time window and its associated uncertainty. Throughout the paper, we denote the clone state indices with and the EKF propagation indices with k. We further outline the EKF state definition, propagation model, state augmentation, measurement, and update model below for completeness of the manuscript. must be noted that we follow TLIO (Liu et al. 2020) and do not make any modifications to this part of the architecture.
A.6.1 EKF STATE DEFINITION
The state of the EKF defined as X (E......:En,, En, where Ç., i 1,... represents the clone states whose corresponding IMU measurements are passed to the NN as input and s has the current propagation state.
= (", pp,) R,, ",, "p, bg: b..)
where R, represents the orientation estimate of EKF from IMU body frame to gravity world frame, and u represent the position and linear velocity estimates, and by and ba denote the bias estimates of the gyroscope and accelerometer respectively. The error state of the EKF is propagated as linearized error and so the error state is defined as
E (0,, 8p,) 3 = (0, 88, 8p, 86,, 5b..)
where the tilde indicates errors in every state. The errors for all states, except orientation, are approximated to simple subtraction even though the SE(3) parameterization
in the EKF process model is T(3)= x 50(3). For orientation, we use the logarithm map of rotation to find the error as 0 = (ogso(3) (RR (RR-1) E s3)).. The EKF and error state is of dimension (6n + 15) and the corresponding state covariance matrix P has dimension (6n + 15, 6n 15).
A.6.2 PROCESS MODEL
IMU's measure sequences of data {ak,, T each expressed in the local IMU frame at time 4 These are related to the true IMU acceleration âs and angular rates Wg via
Wkk
We b% n
ag
ax
Rk
T g
Dg 7h
(14)
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where ğ is gravity vector pointing downward in world frame, and mig and vi are IMU noises respec- tively. The EKF propagation uses raw IMU samples in the local IMU frame b, following strap-down inertial kinematics equations:
Rx-11 = ",Rx expso bg
VK-- "Vv gAA + Bi(aa bak)
1 A0 "aa - bak))
Pk+1
"x + Pk +
2
bg(k+1) bgk
ba(k+1) = ågk
where at timestep k. At is the time interval, "9 is the constant gravity vector, 7ydk and Nadk are the IMU noises that are assumed to be normally distributed, and expso(3) is the SO(3) exponential map.
The linearized error propagation is written as:
Sk+1 A 12)
where n [nwkk Takk Nydk: Badk and the subscript brackets (...) indicate matrix dimensions. The corresponding linearized propagation of the state covariance P is as follows:
Pa+1 = A& + B&W B
Ton 0
0 B₁
A
A
B
where / stands for identity matrix amd W(12,12) the covariance matrix of sensor noise and bias random walk.
A.6.3 STATE AUGMENTATION
As the NN inputs used to correct the EKF estimates frequency of 20Hz while the propagation of the EKF is at IkHz as mentioned TLIO, at the measurement update frequency new state is augmented with copy operation incrementing the dimension seen below:
Pa+1 A&PEAT + B&WBY B
Ion
0
4g
0
B
B
B
where A and B are the partial propagation matrices for rotation and position only with dimensions (6, 15) and (6, 12) respectively.
A.6.4 MEASUREMENT MODEL
The measurement model the EKF uses the displacement estimates provided by the NN, aligning them in a local glivity-aligned frame to ensure the measurements are decoupled from global yaw information:
((x) RC, (P, pp)) dey + Mky
where R, is the yaw rotation matrix, p, and p, are positions of the first and last clone state for the 1-second displacement prediction window, di indicates the predicted displacement which is the output of the NN and My represents the measurement noise modeled by the network's 's uncertainty output as Mi = (O, ). The orientation estimate of the EKF at clone state is decomposed using extrinsic "XYZ" Fuler angle convention as R; R2 RgRa.
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A.6.5 UPDATE MODEL
The Kalman gain computed based on the measurement and covariance matrices, and the state and covariance are updated accordingly. The key update equations involve the computation of the Kalman gain (K), updating the state (X), and updating the covariance matrix (P):
K PH (HPH" + Eey))
X ((h()) diz)
P- KH) KK
where denotes addition operation except for rotation where the update operation writes R exp(O)R and the linearized measurement matrix H 13,6 which has zeros other than
ah(X)
Hg.
L. p. H
00;
(X( - R2
Hap:
06p
Oh(X)
Hop
RT
where
0
H.
0
0
[coS 7 tan | sin tan 3
1
and [x]x x is a skew-symmetric matrix built from a vector s.
A.7 EVALUATION METRICS DEFINITION
We follow most metrics in TLIO [Lu al. 2020) and RONIN (Herath al. 2020), besides the M SE loss we reported in the paper. Here we provide the mathematical details of these metrics.
MSE (m²): Translation error per sample between the predicted and ground truth displace- ment averaged over the trajectory. It is computed 1 > "d, "d However, it should be noted that MSE mentioned TLIO (Liu et al. 2020) is the same as MSE Loss calculated as the squared error averaged separately for each axis 1 Σ "d,, dg,- where is an axis.
ATE (m): Translation Error assesses the discrepancy between predicted and ground truth (GT) positions across the entire trajectory. It is computed as ΣР "p: "2
RTE (m): Relative Translation Error measures the local differences between pre- dicted and GT positions over a specified time window of duration (1 minute).
Σ "Pe48t - "p; - (" w (2
AYE Absolute Yaw Error calculated as
1 Σ 11 - 1....
A.8 VISUALIZATION TLIO RESULTS
Figure 8 and Figure show only the NN results compared ground truth displacements. The ATE and RTE are calculated on the cumulative trajectory obtained from the predicted displacements. Figure D with whisker extended to include the outlier which is commonly calculated as 1.5 IQR (inter-quartile range). Figure I0 shows the results of EKF without excluding the outliers. We provide more trajectory visualizations of TLIO test data in Figure 11
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Eq.
35
30
0.6
25
ATE(m)
20
RrE(m)
MSE(m²)
15-
0.3
10
0.2-
0.1
0.0
TUO
TLO
Figure 8: The superior performance of framework applied to TLIO architecture when compared
to baseline TLIO trained with and without augmentations on TLIO and Aria Datasets visualized with a box plot. Blue, Orange, Green, and Red indicate +rot. aug.. TLIO, +SO(2) Eq. Frame and (0 Eq. Frame. ATE, RTE, and MSE indicate ATE*, RTE*, and MSE corresponding to only the NN results.
TLIO rot aug.
TLIO
TUO + S012) Eq. Frame
TUO 0/2) Eq. Frame
35
0.7
30
25
0.5
ATE(m)
RTE(m)
MSE(m²)
0.4
15-
0.1
TLIO
Aria
TUO
Aria
TUO
Aria
Figure 9: The superior performance of our framework applied to TLIO architecture when compared
baseline TLIO trained with and without augmentations TLIO and Aria Datasets visualized with box plot. Blue, Orange, Green, and Red indicate +rot. aug.. TLIO, +S0(2) Eq. Frame and +0(2) Eq. Frame. ATE, RTE, and MSE indicate ATE*, RTE*, and MSE* corresponding to only the NN results. The whisker is extended to 1.5 IQR (inter-quartile range).
A.9 AUGMENTED TLIO TEST DATASET RESULTS AND ANALYSIS
We also perform ablation study on data augmentation for our model. For NN results, we apply four random yaw rotations trajectory and random rotations plus reflection per trajectory. The results are detailed in Table Except for our equivariant model, all other methods show decreased performance compared to their results on non-augmented test data, whereas model maintains consistent performance and outperforms the other methods.
For the Extended Kalman Filter (EKF) results, we augment the test data using random SO(3) rotations. Notably, we do not include reflections due to the structural constraints of the Kalman filter. As shown in Table despite the +Non Eq. Frame model outperforming on ATE metrics. Our model exceeds +Non Eq. Frame on the augmented dataset. Our approach not only sets benchmark but also maintains consistent performance across random rotations.
in non-augmented tests
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TUQ + 10l wug.
TUO
TUD + S012) Eq. Framm
TUD + G12) Bo. Framm
17.5
3.0
15.0
15.0
2.5
15.0
12.5
12.5
2.0
100-
10.0
4.5
5.0
5.0
5.0
25
2.5
0.0
0.0
Figure 10: The superior performance our framework applied to TLIO architecture when compared to baseline TLIO trained with and without augmentations TLIO and Aria Datasets visualized with a box plot. Blue, Orange, Green, and Red indicate +rot. aug., TLIO, +SO(2) Eq. Frame and +0(2) Eq. Frame. The whisker is extended to 1.5 * IQR (inter-quartile range).
Ground Truth
TLIO
Eq. Starting Point
15
30m1
12
20
300g
Figure 11: More Visualizations of final estimated trajectories TLIO Dataset by baseline TLIO (Blue), our best method applied to TLIO (+0(2) Eq. (Rad))(Red), and the Ground -ruth trajectory (Black).
A.10 VISUALIZATION RONIN
The visualization of trajectories in RONIN is displayed Figure
A.11 COVARIANCE CONSISTENCY
Similar to TLIO (Lu et al. 2020). we plot the prediction error against the standard deviation (o) predicted by the network in the invariant space. As seen in Figure 14 and Figure [3 the
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Rotations
Rotations + Reflections
Model
MSE* (10 mm33
ATE*
RTE*
MSE*
ATE*
RTE*
(m)
(m)
(10 (10-2m²3
(m)
(m)
TLIO
0.2828
27.7797
3.1390
0.2989
23,4839
3.1313
rot. aug.
0.0327
3.3180
0.5417
0.0347
2.9110
0.5654
+ rot. aug. + more layers
0.0306
3.0264
0.5300
0.0332
2.3028
0.5592
rot. aug. Non Eq. Frame
0.0302
2.6379
0.5025
0.0331
2.3212
0.5446
+ rot. aug. + PCA Frame
0.2286
21.3795
2.5288
0.2467
10.1660
2.2283
SO(2) 2) Eq. Frame
0.0319
2.3218
0.4957
0.0339
1.8664
0.5178
(2)) Eq. Frame
0.0298
2.3305
0.4719
0.0298
1.6418
0.4361
Table Ablation Study For Neural Network with Random Rotation and Reflection Transformation (4 per trajectory) TLIO test dataset. A lower error indicates better model. The lowest values annotated with Red. Our proposed methods are bold.
Model
ATE (m)
RTE
Drift
AYE
(m)
(deg)
TLIO
10.3005
3.6263
2.9501
3.3684
rot. aug.
1.6744
0.4944
1.5526
2.7290
rot. aug. + more layers
1.6447
0.5466
1.2767
2.7279
rot. aug. + Non Eq. Frame
1.4924
0.5119
1.2721
2.7109
rot. aug. + PCA Frame
8.5787
2.9962
2.0872
3.0183
SO(2) Eq. Frame
1.4850
0.4901
1.3029
2.7615
(2) Eq. Frame
1.4316
0.4592
1.3096
2.7250
Table Results of evaluation of EKF with Random Rotation Transformations (4 per trajectory) on TLIO test dataset (i.e., results on augmentated test dataset). A lower error indicates a better model. The lowest values are annotated with Red. Our proposed methods are in bold.
covariance prediction of our method is consistently within the 3-a depicted by the red lines. These results show that our diagonal covariance prediction in the invariant space is consistent.
EqNio
Egpio grow perturtation
A.12 ABLATION ON IMU SEQUENCE LENGTH
We aligned the sequence length with baseline models for fair com- parison. However, in this Section, we ablate on the sequence length shown in Table 7 and Table 8 Table 7 varies sequence lengths and displacement prediction windows (c.g., 0.5s displacement with 0.5s of 200Hz IMU data results in sequence length of 100). Ta- fixes the prediction window at second and varies the context window (e.g., 2s context window with 200Hz IMU data results in sequence length of 400). Our results confirm TLIO (Liu et al. 2020) Sec. VII A. increasing the context window reduces MSE but not ATE. A lower MSE loss over the same displacement window lower ATE. Thus, the addition of the equiv- ariant framework does not change the characteristics of the base (off-the-shelf) model used.
Sensitivity to gravity axis perturbation, and effectiveness of perturbation augmentation during training.
Figure 15:
does translate
A.13 SENSITIVITY ANALYSIS TO GRAVITY DIRECTION PERTURBATION
As Wang et al. 2023 study the robustness of SO(3) subequivariance to slight gravity axis We perturb the gravity axis in the test set by uniformly sampled angles
perturbations in Fig. 15
u(-0,00 0),), where 0 2", 12,40,0 6" 8 As Liu al. 2020) observe that ATE*(m) can be
15

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Published as a conference paper at ICLR 2025
Figure 12: Visualization of RONIN Unseen Test Dataset Trajectories for our best method applied to RONIN, +0(2) Eq. Frame.
stabilized by training with 0 = gravity perturbation augmentation. The error of EqNIO trained without this augmentation increases with increasing degree of perturbation.
Table presents the sensitivity analysis gravity direction perturbation, applied for different ranges, i.e., for 2", the gravity direction perturbation of (-2%, 2") is applied to the test dataset. We also present results for + O(2) Eq. Frame model trained without the gravity direction perturbation of (-50,5% during training. We observe the same trend of stability in MSE* as reported in TLIO (Liu ct al. 2020) when trained with gravity direction perturbation.
A.
SENSITIVITY TO IMU SAMPLING RATE
To study the IMU sampling rate sensitivity, we deploy our pre-trained model on IMU data that is resampled from rate of (50, 100, 200, 250, 500, 1000) to 200 Hz, since TLIO requires fixed input size of 200 IMU measurements for second. This means that for r < 200 IMU data is interpolated. We report the results without the EKF on the Aria Dataset in Table 10
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Published as a conference paper at ICLR 2025
0.2
0.2
0.20
0.20
0.20
Sigma
0.30
0.30
0.10
0.00
0.00
0.09
Figure 13: Consistency of Covariance Prediction in the Invariant Space for TLIO test dataset
0.73
0.2
025
0.15
0.15
0.1
Sigma
0.30
0.10
0.05
0.65
0.00
0.00
0.00
-1.0
-0.3
0.0
0.5
1.0
-1.0
-05
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
Enror
Enor
Eirol
Figure 14: Consistency Covariance Prediction in the Invariant Space for Aria dataset
The results show the relative stability of our method for rates equal to and above 200 Hz, with maximal ATE increase of 0.2%. By contrast, going below 200 Hz leads to higher increase 15.8% (for 50 Hz). However, such low sampling rates unlikely in real-world scenarios, since most commodity IMUs provide kilohertz-level sampling rates.
A.15 SENSITIVITY TO IMU BIAS ESTIMATE
We study the sensitivity of our network to inaccurate bias estimation. For this, we monitor the MSE after perturbation of the biases used to de-bias the input IMU data our network, as done Liu
et al. (2020). We sample uniform noise from the range U[-r, r] where is defined in Table Our results seen Table II and Table are in accordance with the sensitivity study conducted in
TLIO [Liu et al. (2020)).
A. CANONICALIZATION VIA FRAME-AVERAGING
In this section, conduct technique in Puny et al. (202T). We first perform PCA on the IMU data, which results in a set of four frames, corresponding to the four solutions of PCA. We then use the equivariant format in Eqn.5
ablation for our frame ablation section with the Trame-averagee
17

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Published as a conference paper at ICLR 2025
TLIO Dataset
Aria Dataset
Model
Context Window (8)
MSE$
ATE*
RTE
MSE*
ATE*
RTE#
TLIO
(10 (L0-?m²)
(m7)
(m1)
(10 110-?m²²)
(m)
(m7)
rot. aug.
0.5
1.132
2.029
0.340
1.038
1.489
0.332
rot. aug.
I
3.242
3.722
0,551
5.322
2.103
0,521
rot. aug.
2
9.862
5.102
0.944
6.717
3,452
0.970
(2) Eq. Frame
0.5
1.124
0.711
0.175
1.040
0.673
0.190
SO((2) Eq. Frame 1
3.194
2.401
2.457
1.864
0.484
+SO(2) Eq. Frame 2
10.019
3.862
0.797
6,569
2.745
0.774
(2) Eq. Frame
0.5
1.040
0.595
0.136
1.002
0.589
0,148
(2) Eq. Frame 1
2.982
2.382
0.479
2.304
1.849
0.465
(2) Eq. Frame
2
9.804
4.268
0.762
6.112
2.556
0.709
Table 7: Results for ablation on changing prediction displacement window on TLIO architecture.
TLJO Dataset
Aria Dataset
Model
Context
MSE*
ATE*
RTE*
MSE*
ATE* (m)
RTE*
TLJO
Window (8)
(10-2m) 3
(m)
(m)
(10 (10-3m²)
(m)
rot. aug.
1
3.242
3.722
0.551
5.322
2.103
0.521
rot. aug.
3.199
2,555
0.511
3.790
2.895
0.713
rot. aug.
3
3,284
4,463
0.617
3.511
3.014
0.738
+ (2) Eq. Frame
1
3.194
2,401
0.501
2,457
1.864
0.484
SO)) Eq. Frame
2
2.886
1.837
0.429
2.187
1.533
0.444
+SO(2) Eq. Frame
2.790
3.090
0.492
1.986
1,684
0.447
( Eq. Frame
2,982
2.382
0.479
2.304
1.849
0.465
( Eq. Frame
2
2.382
1.895
0.367
1.307
1.382
0.338
(2) Eq. Frame
3
2.161
2.083
0.366
0.974
1.672
0.366
Table 8: Results for ablation on changing context window with fixed displacement window of 1-
second on TLIO architecture.
of Puny al. (2021) which canonicalizes the input with respect the four frames, runs a separate prediction on each input, and then averages the outputs. Note that this scales the inference time by factor of We show the results in Table [3 We see that while TLIO frame averaging achieves a low еτгоr the TLIO dataset it fails generalize to the Aria Dataset. Furthermore,
equivariant method (TLIO 0(2) Eq. Frame) outperforms it on both datasets and across metrics. We believe that the subpar performance of TLIO + frame averaging stems from the noise sensitivity of PCA. Due to this sensitivity, it likely overfits to the specific noise level present in the TLIO dataset, which does not match the one in the Aria dataset.
A.17 ANALYSIS OF THE COVARIANCE PARAMETERIIZATIONS
In addition to supporting our design choice, of diagonal covariance parameterization in the canonical frame empirically by running NN (see Table 3 add additional statistical analyses below, using two more techniques: (i) we report the percentage of displacement predictions by our network that have an error outside of the 3-sigma bound, in x.y, and direction (denoted dz. 6y. 3 percent), where sigma is given by our network, and (ii) we compute the median negative log- (median NLL) of the ground truth samples given the mean and covariance provided by our method. All results were calculated in the canonical frame and pertain to the Aria Dataset. In addition, we highlight the importance of learning the covariance, by introducing two additional baselines: First, similar to Liu et al. 2020), we use pre-trained method to generate displacement measurements but use constant isotropic covariance (a = 0.01) instead the learned one (indicated with +constant cov), and secondly train model only MSE for 20 epochs (indicated with (mse) +constant cov) and deploy with the same constant covariance. We see in Table [4 that TLIO + O(2) Eq. Frame, TLIO O(2) Eq. Frame P and TLIO O(2) Eq. Frame (mse) constant cov, all show low outlier
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Published as a conference paper at ICLR 2025
TLIO Dataset
Model
g direction perturbation
MSE#
ATE*
RTE#
(TLIO)
(deg)
(10 110-2m= 3 (mr)
(mn)
+ (2) Eq. Frame
0
3.194
2.401
0.501
+ S0(2) Eq. Frame
2
3.201
2.409
0.500
+ (2) Eq. Frame
4
3.206
2.404
0.498
(2) Eq. Frame
3.241
2.442
0.501
+ +2)(2) Eq. Frame
3.298
2.502
0.506
+ +((2) Eq. Frame
0
2.982
2.406
0.478
+ +0(2) Eq. Frame
2
3.198
2.663
0,498
+(() Eq. Frame
4
3.742
3.292
0.559
+0(2) Eq. Frame
6
4.505
4.228
0.659
+ 0() Eq. Frame
8
5.433
5.218
0.768
+ (2) Eq. Frame
0
2.981
2.382
0.479
(()) Eq. Frame
2
2.988
2.390
0,480
((2) Eq. Frame
4
3.010
2.415
0.486
( Eq. Frame
3.060
2.471
0.486
+ (2) Eq. Frame
8
3.095
2.506
0.489
Table 9: Results for ablation changing prediction displacement window TLIO +0(2) Eq.
Frame. & implies the network was trained without gravity direction perturbation.
Model (TLIO)
Sampling rate
MSE* (10 3m
ATE*
RTE*
(Hz)
(m)
(m)
(()) Eq. Frame
1000
2.324
1.876
0.470
+ (2) Eq. Frame
500
2.316
1.878
0.469
+ (()) Eq. Frame
250
2.291
1.882
0.468
(2)) Eq. Frame
200
2.276
1.881
0.468
((2) Eq. Frame
100
2.191
1.932
0.470
(() Eq. Frame
2.256
2.174
0.501
Table 10: Results for ablation on changing IMU sampling rate on TLIO + O(2) Eq. Frame. on Aria Dataset
counts. However, low outlier counts can also be achieved by predicting high covariances. However, this strategy increases the median NI.I, as seen in TLIO O(2) Eq. Frame + P and TLIO O(2) Eq. Frame (mse) constant cov, showing that they overestimate covariances. Our method (TLIO + O(2) Eq. Frame) shows the highest median NLL overall methods, with low outlier counts. For completeness, we also report the tracking performance of the new methods in Table 13 We that, first, O(2) Eq. Frame + constant cov performs worse than our method, indicating that our learned covariance adapts to the specific learned displacements. Second, we see that constaatn cov performs even worse. This is likely due to the network being overconfident in its displacement prediction before MLE finetuning, which results in significant outlier prediction.
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Published as a conference paper at ICLR 2025
Model
Accel Bias Range (m/s*)
MSE*
(TLIO)
(10 3mmm
(2) Eq. Frame
0.0
2,981
(2) Eq. Frame
2,984
+ (2) Eq. Frame
0.2
3.001
(() Eq. Frame
0.3
3.031
(2) Eq. Frame
0.4
3,066
(2) Eq. Frame
0.5
3.110
Table 11: Results for ablation on accelerometer bias sensitivity on TLIO + O(2) Eq. Frame. on TLIO
test Dataset
Model
Gyro Bias Range
MSE (10 m
(TLIO)
(rad/s)
(() Eq. Frame
0.000
2.981
(() Eq. Frame
0.025
2,981
+ (() Eq. Frame
0.050
2,981
(2) Eq. Frame
0.075
2,981
O() Eq. Frame
0.100
2.981
Table 12: Results for ablation on gyroscope bias sensitivity on TLIO + 0(2) Eq. Frame. on TLIO
test Dataset
TLIO Dataset
Aria Dataset
Model
MSE*
ATE* (m)
RTE* (m)
MSE* (10 (10=²m²)
ATE* (m)
RTE* (m)
(10 mm
TLIO
+ rot.
3.242
3,722
0,551
5.322
2.103
0.521
(2)() Eq. Frame
3.194
2,401
0.501
2.457
1,864
0.484
(2) Eq. Frame
2.982
2.382
0.479
2.304
1.849
0.465
+ frame averaging
3.208
3.057
0.536
5.821
4.554
0.992
Table 13: Results for the frame-averaging technique evaluating only the neural network performance on the TLIO and Aria Datasets.
Model (TLIO)
82
6y
82
Median NLL
(%)
(%)
(%)
(()) Eq. Frame +S
3.49
3.35
7.75
-7,0551
(2) Eq. Frame
0.59
0.71
0.51
-7.3191
+ +2(2) Eq. Frame +P
0.00
0.96
-4.1156
+ (2) Eq. Frame + constant cov
2.18
2.20
0.46
-4.1111
+ +2)) Eq. Frame (mse) + constant cov
0.72
0.72
0.04
-3,9189
Table 14: Analysis of covariance parameterizations by measuring the number of outliers outside of
the 3-a bound, and the negative log-likelihood (NLL) of ground truth samples on the Aria Dataset.
TLIO Dataset
Aria Dataset
Model
ATE (m)
RTE (m)
AYE (deg)
ATE
RTE (m)
AYE (deg)
(TLIO)
(m)
(2) Eq. Frame +S
1.484
0.462
2.390
1.175
0.421
2.043
(()) Eq. Frame
1.433
0.458
2.389
0.416
2.059
(() Eq. Frame +P
1.827
0.578
2.534
1.755
0.564
2.223
+2) Eq. Frame + constant cov
1.669
0.506
2.481
1.680
0.534
2.197
(2) Eq. Frame (mse) + constant
2.883
0.799
2.477
2.032
0.652
2.224
Table 15: EKF Results for the different covariance parameterizations
20


Reviews:
[{"review_id": "Ff3WU4OYpV", "reviewer": "Reviewer_cuLt", "summary": "The authors propose a method to adapt existing inertial odometry (IO) architectures to be invariant to the IMU orientation. This is done by making use of an $O_g(3)$/$SO_g(3)$ equivariant network that transforms the gravity-aligned IMU measurements to a canonical frame as a pre-processing step for non-equivariant IO. The predicted displacement and covariance from IO for these canonicalized measurements are then transformed back to the source frame using the inverse canonical frame. The proposed method leads to improved accuracy while maintaining comparable runtime and can in principle be applied to any IO method.", "questions": "- Why is this canonical. equiv. scheme chosen over other equiv. choices? e.g. frame averaging (Puny et al. ICLR '22) also allows adapting existing non-equiv. architectures.\n- I'm confused about the choice of metrics, especially for the TLIO experiments. From the definitions in A.5 (I believe squared norm is missing), it seems that MSE is just sqrt(ATE)? But the numbers don't reflect this. And I also think ATE, RTE, AYE would be sufficient. Do you do SE3 alignment with the GT trajectories?\n\nMinor non-critical comments:\n- Could you elaborate on the yaw augmentation procedure used for TLIO / RoNIN?\n- It is surprising to me that despite requiring 10x more FLOPs than the non-equiv. architectures, there is barely any increase in runtime (<1 ms). Since there is no code release, can you comment more on the reasons for this efficiency?\n- Writing: In Fig. 3b it is not clear what 'rot. sense' means; explain how the frame is constructed from the network outputs with gs-orth. for sake of clarity; Typo in conclusion: \"respects eliminates\"; Would be helpful to indicate that * means no-EKF in the table 2,3 captions or simply remove the * since it is not applicable to RoNIN.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- This is an interesting and novel application of equivariant networks to an under-explored but useful problem. The canonicalization approach is a good choice for this problem as it adapts existing sota IO architectures and keeps the pipeline interpretable.\n- The symmetry of the problem in terms of $O_g(3)$ equivariance (which is $O(3)$ subequivariant) is well presented. Care has been taken to consistently process the IMU measurements and specialized $O(2)$/$SO(2)$ architectures based on vector neurons have been developed, while more architectures are possible.\n- The approach was tested on two IO architectures which showed accuracy improvements on many datasets, with comprehensive ablation studies, while keeping the inference times comparable.", "weaknesses": "- The general canonicalization scheme has been proposed before by Kaba, Sékou-Oumar, et al. \"Equivariance with learned canonicalization functions.\" ICML 2023, and cannot not be presented as a contribution. The original work must be cited.\n- While there is a reduction in drift compared to the baselines, the remaining drift is still significant (>2m) which suggests that the main problem in IO is not in exact equivariance but elsewhere (most likely sensor noise).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SIbxasY6iv", "reviewer": "Reviewer_wVti", "summary": "This work introduces EqNIO, a neural network-based odometry system that enhances localization accuracy using accelerometer and gyroscope data from a single IMU. Traditional neural odometry methods face challenges with generalization, as variations in IMU orientation and motion direction can disrupt displacement predictions. EqNIO overcomes this by training a model with canonical displacement priors, aligning IMU data to a gravity-aligned frame with learnable yaw. This approach ensures that the system’s outputs are invariant to rotations and reflections in the gravity direction, supporting robust generalization. Through carefully designed layers and an innovative angular rate decomposition, EqNIO can effectively integrate both acceleration and angular data. Tested on TLIO, Aria, RONIN, RIDI, and OxIOD datasets, EqNIO demonstrates superior performance and adaptability over existing methods, marking a step forward in low-drift neural inertial odometry suitable for edge devices.", "questions": "What is the roto-reflection group, and why is it important? A more detailed explanation of this concept and its relevance would be helpful.\n\nWhat is the PCA(handcrafted equivariant frame)? A more detailed explanation of this concept and its relevance would be helpful.\n\nClarity in distinguishing from RIO: It appears that the figure is intended to convey the core idea of this work. However, the differences between this approach and RIO are unclear—elaborating on this distinction would strengthen the presentation.\n\nData specification in figure captions: It would be beneficial if each figure caption specified which data is seen and which is unseen to enhance the reader's understanding. Note, the performance of different method highly depends on how much data is seen or trained. \n\nSupplementary material vs. main paper clarity: The supplementary material provides much better clarity than the main paper. Including some of this contextual information directly in the main paper would make it easier for reviewers to follow your method.", "limitations": "", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "I think the method itself looks novel and interesting.  It introduces a canonicalization scheme that leverages gravity and an estimated sub-equivariant frame to map IMU measurements into a canonical orientation. This procedure can be flexibly applied to arbitrary off-the-shelf network architectures by mapping the inputs into the canonical space and mapping the outputs back into the original space.", "weaknesses": "The primary weakness of this paper is the clarity of its writing. I’m unable to fully understand the major differences between this work and RIO: Rotation-equivariance Supervised Learning of Robust Inertial Odometry.\n\nWhile the key idea of this paper is clear, it’s difficult to discern how it specifically diverges from the previous work. I strongly recommend that the authors begin by clearly outlining the main concepts, followed by a detailed description of the methodology. This structure would greatly help readers in understanding the unique contributions of this work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fHuXJgGP15", "reviewer": "Reviewer_AwiV", "summary": "This paper presents a new method for inertial odometry, which predicts the poses given IMU measurements. The method is called EqNIO, which brings the idea of the so-called canonical displacement priors. How it works, is (1) that IMU measurements are mapped into a gravity aligned, canonical frame with neural networks, and (2) mapping the outputs back to the original frame. Several contributions are presented. A canonicalization scheme is presented, that maps IMU measurements into a canonical orientation. A processing step is devised, which map both accelerometer and gyro readings into a space where gravity direction is preserved. Finally, a neural network designed is presented to perform regression tasks. Several experiments are presented, demonstrating advancements to the state of the art and ablation studies that motivate the overall approach.", "questions": "I wonder for sensor fusion in the form of visual-inertial odometry, it could be helpful to have a good uncertainty estimates from the inertial odometry module. Uncertainty estimates could consider the modelling errors of neural networks, and propagated to the final module. Here, priors can also be defined using physical properties of the system. Would it be a consideration to use Bayesian modelling tools?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Pros:\n\n-\tThe paper is written well with clear figures, and many intuitive explanations of complex concepts.\n\n-\tThe paper presents several technical contributions, which overall leads to more generalizable framework for inertial odometry. Mixing the physical properties with learning based regression module makes sense, which can boost generalization performance.\n\n-\tExperimental results are presented to a large extent, demonstrating advancements to the state of the art. Ablation studies presented are useful to better comprehend the research done.", "weaknesses": "Cons:\n\n-\tIt is not clear if ICLR is the best venue for such research, since learning components here is rather limited to a regression module. \n\n-\tUncertainty modelling assumes diagonal covariance. Validity of these assumptions are tested by looking at the final performance that it helps. Perhaps an in-depth analysis on this step could help, despite not the core focus of the paper. For example, there has been many evaluation tools from uncertainty quantification literature, and can be presented here.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "x5bAp11mVW", "reviewer": "Reviewer_EZfo", "summary": "This paper  presents a method to enhance inertial odometry by applying group equivariance to canonicalize IMU data and targeting yaw ambiguity in gravity-aligned frames through a subequivariant framework. \n\nThe authors design a neural network architecture that maintains equivariance under roto-reflections around the gravity axis, allowing integration with existing systems like RONIN and TLIO. By predicting canonical yaw frames and equivariant covariance matrices, EqNIO improves generalization across diverse motion patterns and reduces drift caused by sensor noise and biases. Experiments on publicly available datasets demonstrate that this method achieves reductions in Mean Squared Error and Absolute Translation Error compared to baseline models, while also exhibiting faster convergence and maintaining computational efficiency suitable for deployment on edge devices.", "questions": "Could you provide a technical explanation of how the canonical displacement priors are implemented in practice?\nHow exactly do the gravity-aligned frames and equivariant yaw frames work in your model architecture?\nWhat specific mechanisms in your learnable yaw orientation approach help address the yaw drift problem?\nCould you provide experimental evidence demonstrating how your method improves yaw estimation compared to existing approaches?\nHow does your model perform under varying IMU sampling rates?\nWhat are the performance characteristics under extreme motion scenarios where IMU biases may be significant?\nHow does the system behave in situations with poor gravity alignment?\nHow does your work relate to recent developments in learning-based inertial attitude estimation?\nCould you elaborate on the connections between EqNIO and current research in equivariant neural networks specifically applied to inertial navigation?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper introduces an approach by applying strict subequivariance in neural inertial odometry, addressing yaw ambiguity and gravity alignment limitations directly within the network architecture, an aspect that prior methods often handled indirectly. Its originality lies in developing a canonicalization scheme using the roto-reflection group to simplify IMU data processing. The authors integrate and extend existing methods by adapting the framework to both end-to-end and filter-based neural inertial odometry systems, which demonstrates its flexibility and scalability. \n\nIn terms of quality, the paper provides detailed, reproducible implementation notes and thorough ablation studies in the appendix that clarify design decisions and evaluate parameterization choices. It emphasizes empirical rigor by testing the model across multiple datasets with varied sensor placements and motion patterns, supporting claims of robustness and broad applicability. Clarity is maintained through structured explanations of complex mathematical formulations, specifically around group theory and its relevance to sensor data processing, while the significant computational efficiency results underscore the practical utility for edge-device applications. The combination of technical insights and comprehensive empirical validation underlines the paper's contribution to advancing neural inertial odometry, particularly in settings with challenging orientations and device constraints.", "weaknesses": "The abstract describes EqNIO as leveraging \"canonical displacement priors\" to generalize across arbitrary IMU orientations, but it lacks a clear technical explanation of how these priors work in practice. Generalization is claimed to stem from \"canonical gravity-aligned frames\" and \"equivariant yaw frames,\" but the abstract could benefit from a more precise explanation of these transformations and their operationalization in the model.\n\nThe learnable yaw orientation in canonical frames is a promising feature but lacks clarity on how it resolves yaw drift or improves orientation estimation, given that yaw is typically the most challenging to estimate accurately in inertial odometry due to the absence of an absolute reference.\n\nThe introduction highlights EqNIO’s generalization and robustness but does not discuss potential limitations or scenarios where the approach may struggle (e.g., handling different sampling rates, extreme motions where IMU biases may not be fully mitigated, or contexts with poor gravity alignment).\n\nWhile EqNIO is compared to existing neural odometry methods like TLIO and RONIN, the introduction does not delve into specific weaknesses in these prior approaches and how EqNIO addresses these limitations.\n\nThe paper covers a broad range of related works but may omit some recent or seminal papers in the domain of learning-based inertial odometry and equivariant neural networks. Ensure a comprehensive literature review by including all relevant and recent works that contribute to the field. This includes verifying that seminal papers and the latest advancements are adequately cited to position EqNIO within the current research landscape. Due to the inherent relationship between odometry and inertial attitude estimation, as well as the similar methods applied to both, I highly encourage you to explore these areas further, including learning-based approaches to inertial attitude estimation.\n\nThe descriptions of related methods (e.g., TLIO, RONIN) are somewhat high-level and lack technical depth.  Providing only superficial descriptions may not adequately highlight the nuances that differentiate EqNIO from these methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)