[Paper Review] A Mathematical Framework for IMU Error Propagation with Applications to Preintegration

Axel Barrau, Silvère Bonnabel|arXiv (Cornell University)|Mar 8, 2020

Inertial Sensor and Navigation39 references56 citations

TL;DR

This paper develops a Lie-group based framework using SE2(3) to model and propagate IMU errors for extended poses (position, velocity, orientation) and derives exact preintegration formulas including rotating Earth effects, with uncertainty and bias handling.

ABSTRACT

To fuse information from inertial measurement units (IMU) with other sensors one needs an accurate model for IMU error propagation in terms of position, velocity and orientation, a triplet we call extended pose. In this paper we leverage a nontrivial result, namely log-linearity of inertial navigation equations based on the recently introduced Lie group $SE_2(3)$, to transpose the recent methodology of Barfoot and Furgale for associating uncertainty with poses (position, orientation) of $SE(3)$ when using noisy wheel speeds, to the case of extended poses (position, velocity, orientation) of $SE_2(3)$ when using noisy IMUs. Besides, our approach to extended poses combined with log-linearity property allows revisiting the theory of preintegration on manifolds and reaching a further theoretic level in this field. We show exact preintegration formulas that account for rotating Earth, that is, centrifugal force and Coriolis effect, may be derived as a byproduct.

Motivation & Objective

• Extend the estimation of IMU error propagation from pose (SE(3)) to extended pose (SE2(3)) encompassing position, velocity, and orientation.
• Demonstrate log-linearity of IMU navigation equations within SE2(3) to enable robust preintegration on manifolds.
• Provide exact preintegration formulas accounting for rotating Earth (Coriolis and centrifugal effects) and relate to on-manifold filtering.
• Develop a framework to associate uncertainty with extended poses and derive propagation equations for noise and bias in IMU data.
• Address biases in IMU preintegration and propose first-order bias correction within the exponential coordinates on SE2(3).

Proposed method

• Model extended poses as elements of SE2(3) and use the exponential map to represent perturbations in nine-dimensional Lie algebra.
• Show group affine property for IMU dynamics and derive exact solution form T_t = Γ_t Φ_t(T_0) Υ_t.
• Derive preintegration formulas that incorporate rotating Earth by introducing velocity augmentation V′ = V + Ω×X and embedding into SE2(3).
• Derive exact discrete-time error propagation for concentrated Gaussians on SE2(3) and obtain linearized error propagation via Ad_Υ^{-1} and F operators.
• Define Gaussian distributions on SE2(3) as T = T̄ exp(ξ) with ξ ∼ N(0, Σ) and propagate through IMU models.
• Provide an exact error accumulation formula exp(ξ_k) = exp(F_0^{k-1} ξ_0) · ∏_{i=0}^{k-1} exp(F_{i+1}^{k-1} η_i) and discuss implications for uncertainty.
• Discuss first-order bias correction in preintegration using exponential coordinates and derive update rules.

Experimental results

Research questions

• RQ1 How can IMU error propagation be formulated for extended poses (position, velocity, orientation) within SE2(3)?
• RQ2 Does the log-linear property hold for IMU dynamics on SE2(3) when incorporating Coriolis and centrifugal effects due to rotating Earth?
• RQ3 Can exact or closed-form preintegration formulas be derived on SE2(3) that account for Earth rotation and bias/noise?
• RQ4 How can uncertainty be attached to extended poses, and how does it propagate through IMU-based preintegration?
• RQ5 What is the impact and correction mechanism for IMU biases within the SE2(3) preintegration framework?

Key findings

• IMU navigation equations on SE2(3) exhibit log-linearity, enabling preintegration on manifolds with rotating Earth considerations.
• Exact preintegration formulas accounting for Coriolis and centrifugal effects are derived, including a velocity augmentation trick V′ = V + Ω×X.
• A Gaussian on SE2(3) defined as T = T̄ exp(ξ) with ξ ∼ N(0, Σ) can be propagated exactly through noise-free IMU models, yielding a linear-like error propagation in exponential coordinates.
• An explicit error accumulation formula exp(ξ_k) = exp(F_0^{k-1} ξ_0) · ∏_{i=0}^{k-1} exp(F_{i+1}^{k-1} η_i) is established for noisy IMU data, enabling exact or closed-form uncertainty tracking (to first order via BCH).
• First-order bias correction for preintegration is facilitated by the exponential coordinates on SE2(3), providing refined Jacobians for bias updates.
• The framework connects SE2(3) based uncertainty handling with preintegration theory and supports practical factor-graph approaches for high-rate IMU fusion.