\textbf{PLATO}: \emph{Preintegration Learning from Accurate Trajectory Observations} for Neural Inertial Odometry

Summary

Neural inertial odometry has demonstrated strong potential for motion estimation in challenging environments, yet inertial-only preintegration remains sensitive to IMU bias and uncertainty.

Abstract Summary

Neural inertial odometry has demonstrated strong potential for motion estimation in challenging environments, yet inertial-only preintegration remains sensitive to IMU bias and uncertainty. To this end, this paper introduces \textbf{PLATO}:~\emph{Preintegration Learning from Accurate Trajectory Observations}, a likelihood-based framework that leverages accurate trajectory observations to jointly learn IMU bias dynamics modeled by a neural ordinary differential equation~(NODE) and gyroscope and accelerometer noise covariances. Optimization exploits the sparse structure of the negative log-likelihood, with IMU noise-parameter gradients computed by forward differentiation. A tailored double-adjoint scheme couples a discrete invariant-error adjoint with a continuous-time adjoint for the bias NODE, enabling memory-efficient likelihood optimization over the nested bias-dynamics and IMU-preintegration rollouts. Validation on EuRoC shows improved performance, and underwater robot experiments demonstrate applicability under intermittent lighting failures and visual degradation.

Key Points

Neural inertial odometry has demonstrated strong potential for motion estimation in challenging environments, yet inertial-only preintegration remains sensitive to IMU bias and uncertainty.
To this end, this paper introduces \textbf{PLATO}:~\emph{Preintegration Learning from Accurate Trajectory Observations}, a likelihood-based framework that leverages accurate trajectory observations to jointly learn IMU bias dynamics modeled by a neural ordinary differential equation~(NODE) and gyroscope and accelerometer noise covariances.
Optimization exploits the sparse structure of the negative log-likelihood, with IMU noise-parameter gradients computed by forward differentiation.
A tailored double-adjoint scheme couples a discrete invariant-error adjoint with a continuous-time adjoint for the bias NODE, enabling memory-efficient likelihood optimization over the nested bias-dynamics and IMU-preintegration rollouts.
Validation on EuRoC shows improved performance, and underwater robot experiments demonstrate applicability under intermittent lighting failures and visual degradation.

Preliminaries

The Lie group and Lie algebra preliminaries [ 20 ] used throughout this paper are briefly reviewed. Let $\mathbf{G}$ be a matrix Lie group with Lie algebra $\mathfrak{g}$. We abuse the notation $(\cdot)^{\wedge}$ to denote the mapping from the vector space to the corresponding Lie algebra with its inverse map $(\cdot)^{\vee}$.

The exponential map is defined as $\operatorname{exp}(\xi)\triangleq\operatorname{exp}_{m}(\xi^{\wedge}):\mathfrak{g}\rightarrow G$, where $\exp_{m}(\cdot)$ denotes the matrix exponential.

Preliminaries — \textbf{PLATO}: \emph{Preintegration Learning from Accurate Trajectory Observations} for Neural Inertial Odometry — Figure 2: Training strategy and structured gradient computation.

Problem Formulation

IMU measurements are commonly incorporated through the following rigid-body kinematics: in which the measurement model of IMU is given by where $\omega(t)$ and $a(t)$ denote the angular velocity and linear acceleration of the rigid body, $g$ denotes the gravity, $b_{g}(t)$ and $b_{a}(t)$ denote the gyroscope and accelerometer bias, $n_{g}(t)$ and $n_{a}(t)$ are the corresponding white Gaussian noises, and $\tilde{\omega}(t)$, $\tilde{a}(t$) are the IMU measurements.

Likelihood Evaluation from Pure IMU Rollout

The pure IMU preintegration is used to compute $p(\bm{X}^{s}\mid\bm{U},X_{0},b_{0},\Theta)$ required in ( 5 ). The bias is obtained by integrating ( 3 ), and the bias-corrected inertial inputs are then used to propagate a nominal motion trajectory according to ( 1 ), represented by $\bar{\bm{X}}=\{\bar{X}_{0},\bar{X}_{1},\ldots,\bar{X}_{N}\},\bar{X}_{k}\in SE_{2}(3)$.

Define the right-invariant error $\eta_{k}=X_{k}\bar{X}_{k}^{-1}$ and its logarithmic coordinate $\xi_{k}=\log(\eta_{k})$.

Optimization Procedure

Before optimizing $\mathcal{L}_{\mathrm{NLL}}$, the bias dynamics parameters $\theta$ are first trained with the MSE objective $\mathcal{L}_{\mathrm{MSE}}(\theta)=\frac{1}{2}\sum_{i=1}^{M}\/r_{i}(\theta)\/_{2}^{2}$.

This serves as the warm-up for the subsequent NLL optimization, and stabilizes the bias trajectory mean before covariance parameters are optimized, preventing the use of covariance inflation to absorb large residuals [ 18 , 6 ] .

Experiments

This section evaluates the practical effectiveness of the proposed method on public and self-collected datasets. The experiments examine two aspects: raw IMU preintegration after debiasing and downstream OpenVINS [ 10 ] performance.

The evaluation metrics are provided by [ 11 ] , including absolute orientation error (AOE) and absolute position error (APE), defined as $\mathrm{AOE}=(\sum_{k=1}^{N}\/\log(\hat{R}_{k}^{\top}R_{k})\/^{2}/N)^{1/2}$ and $\mathrm{APE}=(\sum_{k=1}^{N}\/p_{k}-\hat{p}_{k}\/^{2}/N)^{1/2}$.

Sources and Demonstrations

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