EqNIO: Subequivariant Neural Inertial Odometry

TL;DR

A neural network is designed that respects the physical roto-reflective symmetries inherent in IMU data and converts equivariant covariances and displacements into equivariant covariances and displacements, achieving an invariant canonicalization that can be directly used with off-the-shelf inertial odometry networks.

Abstract

Neural networks are seeing rapid adoption in purely inertial odometry, where accelerometer and gyroscope measurements from commodity inertial measurement units (IMU) are used to regress displacements and associated uncertainties. They can learn informative displacement priors, which can be directly fused with the raw data with off-the-shelf non-linear filters. Nevertheless, these networks do not consider the physical roto-reflective symmetries inherent in IMU data, leading to the need to memorize the same priors for every possible motion direction, which hinders generalization. In this work, we characterize these symmetries and show that the IMU data and the resulting displacement and covariance transform equivariantly, when rotated around the gravity vector and reflected with respect to arbitrary planes parallel to gravity. We design a neural network that respects these symmetries by design through equivariant processing in three steps: First, it estimates an equivariant gravity-aligned frame from equivariant vectors and invariant scalars derived from IMU data, leveraging expressive linear and non-linear layers tailored to commute with the underlying symmetry transformation. We then map the IMU data into this frame, thereby achieving an invariant canonicalization that can be directly used with off-the-shelf inertial odometry networks. Finally, we map these network outputs back into the original frame, thereby obtaining equivariant covariances and displacements. We demonstrate the generality of our framework by applying it to the filter-based approach based on TLIO, and the end-to-end RONIN architecture, and show better performance on the TLIO, Aria, RIDI and OxIOD datasets than existing methods.

Figures (14)

• Figure 1: Predicted trajectories and covariance ellipsoids from TLIO (left) and subequivariant EqNIO (right) for 5 identical trajectories with different IMU frames. The de-rotated trajectories and ellipsoids of TLIO demonstrate significant inconsistency, while the ones by EqNIO are perfectly aligned.
• Figure 2: EqNIO (a) processes gravity-aligned IMU measurements, $\{(a_i,\omega_i)\}_{i=1}^n$. An equivariant network (blue) predicts a canonical equivariant frame $F$ into which IMU measurements are mapped, i.e. canonicalized, yielding invariant inputs $\{(a'_i, \omega'_i)\}_{i=1}^n$. A conventional neural network then predicts invariant displacement ($d'$) and covariance ($\Sigma'$) which are mapped back yielding equivariant displacement ($d$) and covariance ($\Sigma$). The equivariant network (b) takes as input $n\times C_0^s$ scalars, and $n\times C_0^v$ vectors: Vectors are processed by equivariant layers (Eq-L, Eq-Conv, Eq-LN), while scalars are separately processed with conventional layers. Eq-L (green) uses two weights $W_1,W_2$ for SO(2) equivariance, and only $W_1$ 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.
• Figure 3: Symmetries in neural inertial odometry. (a) An IMU undergoes three trajectories in $xy$-plane, each related to a 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-aligned frame (red-green) differ only by an unknown yaw roto-reflection $R_{3\times3}$. 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 $a'=R_{3\times3}a$ and $\omega'=\text{det}(R_{3\times3})R_{3\times3}\omega$. 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 $\omega=v_1\times v_2$ and process $v_1,v_2$ instead, which transform as $v_{1/2}'=R_{3\times3}v_{1/2}$
• Figure 4: Trajectory errors for EqNIO applied to TLIO compared to vanilla TLIO trained with and without yaw augmentations on TLIO and Aria Datasets visualized with a box plot.
• Figure 5: Groundtruth (black), and predicted trajectories on the TLIO Dataset by baseline TLIO (Blue), our best method applied to TLIO (Red). Left and right are difficult trajectories, while the middle trajectory has medium difficulty.