Neural Inertial Odometry from Lie Events

TL;DR

This work tackles drift and rate-generalization challenges in neural inertial odometry by replacing raw IMU inputs with Lie Events that live on the $SE(3)$ manifold. By extending level-crossing concepts and event polarities to Lie groups, the authors derive a canonical, rate-robust sampling strategy that yields fixed-format inputs for neural displacement priors, improving generalization across motion profiles and sampling rates. Applied to TLIO and RoNIN and validated on TLIO, Aria, RIDI, and OxIOD datasets, the method achieves up to about a 21% reduction in trajectory error with minimal preprocessing, and demonstrates strong robustness in rate-sensitivity tests. This event-based, manifold-aware canonicalization opens the door to broader sensor applicability beyond IMUs and cameras, enabling more robust, real-time inertial odometry in diverse settings.

Abstract

Neural displacement priors (NDP) can reduce the drift in inertial odometry and provide uncertainty estimates that can be readily fused with off-the-shelf filters. However, they fail to generalize to different IMU sampling rates and trajectory profiles, which limits their robustness in diverse settings. To address this challenge, we replace the traditional NDP inputs comprising raw IMU data with Lie events that are robust to input rate changes and have favorable invariances when observed under different trajectory profiles. Unlike raw IMU data sampled at fixed rates, Lie events are sampled whenever the norm of the IMU pre-integration change, mapped to the Lie algebra of the SE(3) group, exceeds a threshold. Inspired by event-based vision, we generalize the notion of level-crossing on 1D signals to level-crossings on the Lie algebra and generalize binary polarities to normalized Lie polarities within this algebra. We show that training NDPs on Lie events incorporating these polarities reduces the trajectory error of off-the-shelf downstream inertial odometry methods by up to 21% with only minimal preprocessing. We conjecture that many more sensors than IMUs or cameras can benefit from an event-based sampling paradigm and that this work makes an important first step in this direction.

Figures (8)

• Figure 1: Neural Inerial Odometry from Lie Events. We train Neural Displacement Priors (NDPs), which enable low-drift inertial odometry, with Lie Events derived from acceleration $\boldsymbol{a}(t_i)$ and angular rate $\boldsymbol{\omega}(t_i)$ measurements from an Inertial Measurement Unit (IMU). These events enhance the robustness of NDPs due to their favorable properties under varying sampling rates and trajectory profiles. To generate these events, we produce pre-integrations $\mathbf{x}(t)$ which reside in the special Euclidean group $SE(3)$ and then perform level-crossing on this signal which prompts the generalization of level-crossing and event polarities (red and blue arrows) to higher dimensional manifolds.
• Figure 2: Event timestamps $\tau_j$ generated from reference signal $\mathbf{x}(t)=\mathbf{x}^*(\phi(t))$ can be computed from the event timestamps $\sigma^*_j$ of the path $\mathbf{x}^*(s)$, by applying the inverse mapping $\phi^{-1}$. Moreover, references $\mathbf{x}_{\text{ref},j}=\mathbf{x}(\tau_j)$ and $\mathbf{x}^*_{\text{ref},j}=\mathbf{x}^*(\sigma^*_j)$ are equal and independent of a specific $\phi$.
• Figure 3: Illustration of on-manifold event generation. (A) For a reference signal $\mathbf{x}(t)$ on $SE(3)$ we find the tangent space at the last reference pose $\mathbf{x}_\text{ref,i}$. In this space, we find the moment it exits the $\theta$-ball and record the polarity$\mathbf{p}(\tau_j)$ with unit norm perpendicular to that ball, and update the reference pose to $\mathbf{x}_{\text{ref},j}=\mathbf{x}_{\text{ref},j-1}\text{Exp}(\mathbf{p}(\tau_j)\theta)$. (B) Two trajectories $\mathbf{x}_1(t)=\mathbf{x}^*(\phi_1(t))$ and $\mathbf{x}_2(t)=\mathbf{x}^*(\phi_2(t))$ on the simplified manifold $\mathbb{R}^2$ with time parametrizations $\phi_1,\phi_2$. (C) After projecting both trajectories onto the $xy$-plane, the reference signals $\mathbf{x}_{\text{ref},j},\mathbf{x}_{\text{ref},j-1}$ and polarities $\mathbf{p}(\tau_j),\mathbf{p}(\tau_{j-1})$ are equal.
• Figure 4: IMU rate sensitivity analysis. Each method is trained on the TLIO training set. Methods + interp. and + splat. were trained with IMU rate augmentation and TLIO and TLIO + events were trained without data rate augmentation.
• Figure 5: Hyperparameter Sensitivity: (left) the contrast threshold determines the distance to the reference when events are fired, (middle) initial velocity $\mathbf{v}_0$ and (right) polarity $\mathbf{p}(\tau_j)$ noise determine the range of uniform noise perturbing these quantities during training. In particular, $\mathbf{v}_0$ noise affects pre-integration.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2