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On the Derivation of Tightly-Coupled LiDAR-Inertial Odometry with VoxelMap

Zhihao Zhan

Abstract

This note presents a concise mathematical formulation of tightly-coupled LiDAR-Inertial Odometry within an iterated error-state Kalman filter framework using a VoxelMap representation. Rather than proposing a new algorithm, it provides a clear and self-contained derivation that unifies the geometric modeling and probabilistic state estimation through consistent notation and explicit formulations. The document is intended to serve both as a technical reference and as an accessible entry point for a foundational understanding of the system architecture and estimation principles.

On the Derivation of Tightly-Coupled LiDAR-Inertial Odometry with VoxelMap

Abstract

This note presents a concise mathematical formulation of tightly-coupled LiDAR-Inertial Odometry within an iterated error-state Kalman filter framework using a VoxelMap representation. Rather than proposing a new algorithm, it provides a clear and self-contained derivation that unifies the geometric modeling and probabilistic state estimation through consistent notation and explicit formulations. The document is intended to serve both as a technical reference and as an accessible entry point for a foundational understanding of the system architecture and estimation principles.
Paper Structure (19 sections, 81 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 81 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator $\boxplus$ / $\boxminus$
  4. Kinematic Models
  5. Continuous Model
  6. Discrete Model
  7. Probabilistic VoxelMap Representation
  8. Probabilistic Plane Representation
  9. Uncertainty of point ${}^G \mathbf{p}_i$
  10. Uncertainty of plane
  11. Map Construction and Update
  12. Error-State Kalman Filter for LiDAR-Inertial Odometry
  13. IMU Forward Propagation
  14. Method 1: Direct Discrete-time Linearization
  15. Method 2: Continuous-time Error Dynamics and Discretization
  16. ...and 4 more sections

Figures (4)

  • Figure 1: System overview of the VoxelMap-based LIO framework. LiDAR and IMU measurements are preprocessed and fused within an iterated error-state Kalman filter. The estimated pose registers points to the global frame and incrementally updates the VoxelMap.
  • Figure 2: The uncertainty model of the registered point.
  • Figure 3: The Uncertainty model of the plane normal.
  • Figure 4: The forward and backward propagation.