---
kind: concept
slug: lidar-inertial-odometry
title: LiDAR-inertial odometry
summary: >-
  LiDAR-inertial odometry estimates a moving platform's pose by fusing laser scans with IMU readings. The IMU predicts motion between scans, and each scan corrects the prediction. Three papers show two designs: iterated error-state Kalman filters (R2LIVE, R3LIVE) and an iSAM2 factor graph with scan matching (LVI-SAM). Accuracy figures are author-reported.
tags: [state-estimation, sensor-fusion, lidar, slam]
papers: ["2102.12400", "2104.10831", "2109.07982"]
related: [error-state-kalman-filter, unit-quaternion-rotation, visual-inertial-odometry, simultaneous-localization-and-mapping]
status: draft
updated_at: 2026-10-31
---

## Overview

LiDAR-only odometry degenerates where there are few geometric features, which
matters most for small-field-of-view LiDARs [arxiv:2102.12400 §I]. Long
corridors and open fields are the usual examples [arxiv:2104.10831 §1]. An IMU
alone drifts when trusted by itself, so the two sensors are combined.

The shared idea: the IMU propagates the state between scans, and each LiDAR
scan is matched against a local or global map to correct that prediction. The
scan enters either as a filter update, as in [R2LIVE](/papers/2102.12400)
[arxiv:2102.12400 §IV-D] [arxiv:2102.12400 §IV-I], and [R3LIVE](/papers/2109.07982)
[arxiv:2109.07982 §IV], or as a constraint in a factor graph, as in the
LiDAR-inertial part of [LVI-SAM](/papers/2104.10831) [arxiv:2104.10831 §2.3].
All three cards describe systems that also use a camera. The variants below are
therefore compared on their LiDAR-inertial parts where the paper gives them.
For R3LIVE, the reported numbers belong to the full LiDAR-inertial-visual system.

## Variants

### Iterated error-state Kalman filter (R2LIVE)

The state holds the IMU pose, velocity, the camera-IMU extrinsic and the gyro
and accelerometer biases. Between sensor events it is propagated through the
IMU readings [arxiv:2102.12400 §IV-C]. At each LiDAR frame the point-to-plane
residuals against nearby map planes are linearized and fused in an iterated
MAP update [arxiv:2102.12400 §IV-G] [arxiv:2102.12400 §IV-I]. The error state
lives on the manifold's tangent space [arxiv:2102.12400 §IV-A]. The authors
report a median relative pose error of 0.27° and 0.21% on a 300 m D-GPS
sequence, against 0.41° and 0.34% for FAST-LIO [arxiv:2102.12400 fig. 11].
Card: [R2LIVE](/papers/2102.12400).

### Iterated error-state Kalman filter with a global colour map (R3LIVE)

R3LIVE splits the work into a LiDAR-inertial odometry (LIO) and a
visual-inertial odometry (VIO). The LIO minimises point-to-plane residuals in
an error-state iterated Kalman filter and appends each scan to a global map of
voxels [arxiv:2109.07982 §II] [arxiv:2109.07982 §III-B1]. The paper refers to the
authors' earlier work for the LIO equations, so the card does not specify them
[arxiv:2109.07982 §IV]. The VIO then colours the map. It runs a reprojection
update on tracked points, then a photometric update against the stored map
colours, and fuses the observed colours into each map point
[arxiv:2109.07982 §V-A] [arxiv:2109.07982 §V-B] [arxiv:2109.07982 §V-C].
The VIO projects points from the map that the LIO builds, so it needs the LIO
to run first [arxiv:2109.07982 §II].

The authors report translation drift of 0.093 to 0.164 m and rotation drift of
0.285 to 3.925° over four campus trajectories of 1191 to 1524 m, with no
loop-closure processing [arxiv:2109.07982 tab. II]. In a seaport test on a 300 m
sequence, R3LIVE's high-resolution setting is reported at 0.21° and 0.17%
[arxiv:2109.07982 tab. III]. The LIO takes 18.40 ms per frame on a desktop and
39.56 ms on the onboard UAV computer [arxiv:2109.07982 tab. IV]. Card:
[R3LIVE](/papers/2109.07982).

### Factor graph with scan matching (LVI-SAM's LiDAR-inertial system)

LVI-SAM adapts LIO-SAM. Its factor graph holds IMU preintegration factors,
lidar odometry from scan matching, visual odometry and loop closure
[arxiv:2104.10831 §2.3]. A new keyframe is added when the pose has changed
enough, and the feature map is a sliding window of keyframes, which bounds the
cost [arxiv:2104.10831 §2.3]. The graph is solved with iSAM2 [arxiv:2104.10831 §2.1].
Scan matching gives a degeneracy flag from the smallest eigenvalue of AᵀA
[arxiv:2104.10831 §2.3.2]. Loop candidates come from DBoW2 and are verified by
ICP [arxiv:2104.10831 §2.2.4] [arxiv:2104.10831 §2.3]. On its Jackal dataset
with loop closure, LVI-SAM reports 0.67 m RMSE against GPS, the lowest of the
systems compared there [arxiv:2104.10831 tab. 2]. Card:
[LVI-SAM](/papers/2104.10831).

## When to use which

| Situation | Use | Why |
|---|---|---|
| You need a coloured, dense map as the output | Iterated ESKF with a colour map (R3LIVE) | Its VIO colours each map point by photometric fusion, and maps export as pcd, ply and obj [arxiv:2109.07982 §I] [arxiv:2109.07982 §VII-A]. |
| You need one state estimate at IMU rate on an embedded CPU, with the camera tightly fused | Iterated ESKF (R2LIVE) | R2LIVE's LiDAR-inertial odometry runs in 15.98 to 30.64 ms per experiment onboard [arxiv:2102.12400 tab. I] [arxiv:2102.12400 §VI-F]. R3LIVE's VIO is 15.00 ms onboard at 320 × 256 and 0.10 m point spacing, but 166.08 ms at 1280 × 1024 and 0.01 m [arxiv:2109.07982 tab. IV], so its setting decides the cost. |
| You need loop closure and a global correction of drift | Factor graph | LVI-SAM verifies loop candidates with ICP inside the graph [arxiv:2104.10831 §2.3]. R2LIVE's own 876 m loop closed without a loop-closure step [arxiv:2102.12400 §VI-D]. The R3LIVE card records loop-closed trajectories without loop-closure processing [arxiv:2109.07982 §VI-C]. |
| You may reuse code and the license matters | Factor graph | The LVI-SAM repository is BSD-3-Clause [arxiv:2104.10831 §3]. The R2LIVE and R3LIVE code is GPL-2.0, a copyleft license, as recorded on [R2LIVE](/papers/2102.12400) and [R3LIVE](/papers/2109.07982). |
| The camera is unavailable or unreliable | None, as tested | R3LIVE's VIO needs the camera, and the card records no LIO-only result of its own. LVI-SAM's LIS-only ablation has the highest end-to-end translation error of its Urban settings, 290.43 m [arxiv:2104.10831 tab. 1]. Test any LiDAR-inertial-only setup on your own data. |
| The scene has bare walls with few visual features | Iterated ESKF with a colour map (R3LIVE) | R3LIVE colours the map from raw pixel values and does not depend on extracted visual features [arxiv:2109.07982 §I]. Its texture-less passage test ends with 1.62° and 4.57 cm drift [arxiv:2109.07982 §VI-B] [arxiv:2109.07982 fig. 8]. That is the authors' only such test. |
| Your LiDAR differs from the papers' sensors | Re-check the degeneracy handling and the thresholds | R2LIVE and R3LIVE both used a Livox AVIA with a 70.4° × 77.2° field of view [arxiv:2102.12400 §VI-A] [arxiv:2109.07982 §VI-A]. LVI-SAM used a Velodyne VLP-16 [arxiv:2104.10831 §3]. Each card reports one LiDAR model and no test with another. |

## Implementation sketch

Pseudocode in our own words. The common core is one motion propagation and
one scan correction. The variants differ in how the correction is applied and
in how the map is kept.

```pseudocode
# Common core of one LiDAR-inertial step  [arxiv:2102.12400 §IV-D]
# Propagate the state through the IMU readings since the last scan
state, cov = propagate_imu(state, cov, imu_since_last_scan)

# Motion-compensate the scan with the propagated state  [arxiv:2102.12400 §IV-B]
points = deskew(scan, imu_since_last_scan)

# Filter variant: iterate linearize-and-update at this scan time
#   [arxiv:2102.12400 §IV-G] [arxiv:2102.12400 §IV-I]
for _ in range(max_iterations):    # cap not stated in the paper
    residuals, H = point_to_plane(points, state, local_map)
    state, cov = kalman_update(state, cov, residuals, H)
    if update_is_small(state):     # threshold not stated in the paper
        break

# Map variant: append the scan to a global voxel map  [arxiv:2109.07982 §III-B1]
# global_map = append_points(global_map, points, state)

# Graph variant: add factors instead of updating a filter
#   [arxiv:2104.10831 §2.3]
# add_imu_preintegration_factor(graph, keyframe_prev, keyframe_new)
# add_lidar_odometry_factor(graph, scan_match_pose)  # see degeneracy below
# optimize_incrementally(graph)                      # iSAM2 [arxiv:2104.10831 §2.1]
```

Points to decide when implementing either variant:

- **Degeneracy.** LVI-SAM's text drops the lidar-odometry factor when a
  degenerate direction is detected [arxiv:2104.10831 §2.3.2]. Its released
  code keeps the factor and projects out the degenerate directions instead.
  Pick one behaviour on purpose.
- **Iteration and stop rules.** The paper does not state a cap or stop
  threshold for the filter update [arxiv:2102.12400 §IV-I]. The R2LIVE code
  uses 4 LiDAR iterations, which is a code value and not a paper value. The
  R3LIVE paper also gives no convergence threshold for its iterated updates
  [arxiv:2109.07982 §V-A2].
- **Map lifetime.** R3LIVE keeps a global map and marks voxels active for a
  recent window, which its paper gives as 1 s [arxiv:2109.07982 §III-B1]. The
  card records that the shipped code defaults to 0.0 and does not set it, so
  an implementation that follows the paper must keep that window itself.
- **Noise densities.** R2LIVE does not state its IMU noise values, and its
  covariance input `Q` is not defined in the paper [arxiv:2102.12400 eq. 7].
  Take them from your own sensor calibration.

## Contradictions

No single result is contradicted, but two comparisons point in different
directions. Each side is cited, and the disagreement is not resolved here.

- **R2LIVE against FAST-LIO on a 300 m sequence.** The R2LIVE paper reports R2LIVE
  at 0.27° and 0.21%, against 0.41° and 0.34% for FAST-LIO [arxiv:2102.12400 fig. 11].
  The R3LIVE paper, on its own seaport sequence (a), reports R2LIVE at 0.34° and
  0.21%, and FAST-LIO2 at 0.21° and 0.20% [arxiv:2109.07982 tab. III]. The
  datasets, the baseline version (FAST-LIO against FAST-LIO2) and the error
  definitions differ, so the two tables do not test the same thing.
- **LVI-SAM on its own data against LVI-SAM as a baseline.** The LVI-SAM paper
  reports 0.67 m RMSE against GPS on Jackal [arxiv:2104.10831 tab. 2]. The
  R3LIVE paper reports LVI-SAM at 0.43° and 2.40% on its seaport sequence (a)
  [arxiv:2109.07982 tab. III], and it uses a modified LiDAR front end for that
  baseline [arxiv:2109.07982 §VI-D]. The metrics differ, so the numbers are not
  comparable.
- **Ablation text against table in LVI-SAM.** The text claims that depth reduces
  error in ablation A3 [arxiv:2104.10831 §3.1.3]. The rotation error in that row
  rises from 6.41° to 7.66° [arxiv:2104.10831 tab. 1].

## Papers

- [R2LIVE](/papers/2102.12400) (Jiarong Lin et al., arXiv:2102.12400): an iterated error-state Kalman filter with a sliding-window back end, tested in a tunnel and a campus building.
- [R3LIVE](/papers/2109.07982) (Jiarong Lin and Fu Zhang, arXiv:2109.07982): a LiDAR-inertial odometry plus a visual-inertial colouring stage on a global voxel map, tested on campus, seaport and UAV runs.
- [LVI-SAM](/papers/2104.10831) (Tixiao Shan et al., arXiv:2104.10831): a factor graph (iSAM2) that combines a visual-inertial and a LiDAR-inertial system, with DBoW2 loop closure verified by ICP.