MID360 IMU-LiDAR Calibration¶

This page reports Calibrex's native calibration of the Livox MID360 built-in IMU against LiDAR odometry.

The rotation run estimates the rotation of T_lidar_imu, the clock offset (t_imu = t_lidar + dt), and the gyro bias.
A second run estimates the translation (the IMU lever arm) from the accelerometer; see Translation (IMU lever arm).

The reference is the MID360 design value: IMU axes aligned with the LiDAR, with the IMU at (11.0, 23.29, -44.12) mm. It is used only for comparison, and it is not a measurement.

Two recordings from two dataset families are used:

Recording	Motion	Artifact
RTK-SLAM `stadtgarten_seq2`	hand-held, 15 min	`mid360_imu_lidar_rotation_rtk_slam_seq2.yaml`
Zenodo 14841855 "Driving SLAM Test with Livox MID360" (K. Koide, CC BY 4.0)	vehicle, 277 s	`mid360_imu_lidar_rotation_driving_slam.yaml`

Results¶

RTK-SLAM seq2 (hand-held): pass. Every rotation axis is estimated, and held-out windows detect every known-bad shift.

Quantity	Estimate (relative to design)	Reported std	Held-out control
roll	-0.080 deg	0.025 deg	1 deg detected (Δχ² 870)
pitch	+0.139 deg	0.024 deg	1 deg detected (Δχ² 1024)
yaw	-0.155 deg	0.031 deg	1 deg detected (Δχ² 643)
clock offset	+9.82 ms	0.03 ms	10 ms detected
gyro bias	(0.0156, -0.0025, 0.0017) rad/s	≤ 0.0002 rad/s	—

The held-out median rate residual is 0.012 rad/s, the same as the training residual.
The gyro-deskew iteration converged: in the last pass, every axis moved by at most 0.006 deg.

Driving (vehicle): inconclusive.

Roll is estimated at -0.31 ± 0.09 deg.
Pitch (± 0.31 deg) and yaw are not.
A vehicle rotates almost only about the vertical axis, so a 1 deg yaw shift is invisible to held-out windows (Δχ² 3.0). The yaw value wanders between deskew passes, and the artifact reports it as unobservable.
The clock offset is +2.3 ± 0.2 ms.

The two recordings come from different MID360 units, so their rotations need not agree.

What made the hand-held result possible¶

The first version of this page (PR #70) reported the hand-held recording as inconclusive: yaw sat 1.4 deg from the design value, and the held-out residual was 2.5 times the training residual. It attributed this to coning error in a mean-rate approximation. That diagnosis was wrong. Replacing mean rates with on-manifold gyro preintegration fixes coning in synthetic data, but it left the real-data result unchanged (yaw -1.39 deg).

The residual grew with the rotation rate: 0.007 rad/s below 0.3 rad/s, 0.18 rad/s above. The cause was deskewing. Sweeps were motion-compensated with the LiDAR's own previous motion at constant velocity, which cannot follow fast hand-held changes of rotation. Deskewing with gyro rotations instead cut the residual during rotation by about ten times, and the rotation moved to within 0.16 deg of the design value.

Deskewing with the gyro feeds the extrinsic under test into the LiDAR odometry, so the evaluation guards against the estimate merely confirming its own deskew model:

Iteration. Pass 0 deskews with LiDAR-only motion. Each later pass deskews with the previous estimate. Passes continue until every axis moves less than 0.25 of its reported std, up to five passes.
Convergence gate. A pass is refused, and downgraded to warn, while the last pass still moves any axis by more than its reported std. Before this gate existed, an unconverged second pass (yaw still moving 0.22 deg) had been labelled pass.
Feedback check. A final pass deskews with the estimate rotated by 2 deg on purpose. The artifact records the fraction of that error that survives in the result (deskew_feedback_ratio): 0.20 for the hand-held recording and 0.04 for the vehicle. Because the fraction is well below 1, the fixed point is not an artefact of its starting value; the iteration converges geometrically at that rate.

Comparison with LI-Init¶

The first audited SOTA claim of Calibrex compares this calibration with LI-Init (commit 66b157a, GPL-2.0). LI-Init runs only in a container (tools/external/li_init/), on the same recordings converted to ROS 1 bags, and its result is imported as a slac.external_calibration_run/v0.1 artifact (seq2, driving).

Recording	LI-Init result
RTK-SLAM seq2	refined rotation (1.25, -4.03, -0.45) deg from design, clock offset -55.8 ms
Driving	no result: its excitation meter stopped at 81/73/72 %

Pre-registration. The claim, metric, requirements, and thresholds were committed in mid360_imu_lidar_preregistration.yaml before any held-out score of LI-Init was computed. The audit protocol pins that file's SHA-256, and no threshold has changed since.

Scoring. Each method's rotation and clock offset are held fixed, the LiDAR odometry is recomputed with sweeps deskewed by that method's own extrinsic, and only the gyro bias is refit on the train windows. Every method is scored on the same held-out time spans: every third 10-second window of the design-reference odometry. Each span is one paired benchmark split.

Method (RTK-SLAM seq2, 29 held-out spans)	Mean of span medians	Median	p95
Calibrex	0.0123 rad/s	0.0117 rad/s	0.018 rad/s
MID360 design value (sanity row)	0.0299 rad/s	0.0285 rad/s	0.052 rad/s
LI-Init	0.148 rad/s	0.106 rad/s	0.347 rad/s

The paired improvement of Calibrex over LI-Init is 0.136 rad/s (95 % bootstrap CI 0.090-0.185). On the driving recording, Calibrex's held-out span median is 0.0052 rad/s over 9 spans.

Audit: supported (4/4 gates, protocol, result).

Calibrex returns an estimate on every held-out span of both recordings.
Its paired improvement over LI-Init has a 95 % CI lower bound ≥ 0.
Its mean held-out residual on RTK-SLAM is ≤ 0.02 rad/s.

The first build of this audit was refuted, by a pairing bug.

Each method's own deskewing changes its odometry segmentation. LI-Init's run had 89 windows instead of 88, so its every-third-window held-out choice landed on different stretches of the recording, about 11 s away.
The builder took the union of all held-out windows and counted a window missing from Calibrex's list as a Calibrex failure, which gave a failure rate of 0.5.
The fix scores every method on the same time spans (holdout_spans_s), and the builder now refuses scores taken on different spans.
The thresholds and requirements were not touched. On its own segmentation, LI-Init's held-out median is 0.089 rad/s; that is a different set of spans, and it is also far above Calibrex's.

What this claim does not say:

It is a single LI-Init run with its default MID360 parameters, on one hand-held recording where it returned a result.
Held-out fit to gyro-deskewed LiDAR odometry is the metric, not agreement with a surveyed ground truth. No ground truth exists for these units.
LI-Init also estimates translation; this claim covers only rotation and clock offset.

Translation (IMU lever arm)¶

This section describes the first version of the method (accelerometer_lever_arm/v0.1: one gravity vector per window, 1 s and 4 s sensitivity refits). The current version, v0.2, fits gravity per segment; see Revising the lever arm.

calibrex imu-lidar livox-translation estimates the translation of T_lidar_imu (the IMU origin in the LiDAR frame) from the accelerometer. It builds on a rotation artifact, whose rotation, clock offset, and gyro bias it takes as given. The digest of that artifact is recorded.

calibrex imu-lidar livox-translation BAG_DIR --profile rtk-slam \
  --rotation seq2.yaml \
  --dataset-family rtk_slam --dataset-license "see the RTK-SLAM dataset page" \
  --output seq2-translation.yaml

Model. The IMU follows the LiDAR odometry as p_imu = p_lidar + R_odom_lidar t. Over 2-second segments, the double integral of the gyro-rotated accelerometer must match that motion. The fit has the following unknowns:

the lever arm t, shared by every window;
gravity and the accelerometer bias, one per 10-second window;
a start velocity per segment.

The system is linear. The window nuisances are projected out, and a Huber IRLS fits the 3x3 system for t. Two modelling choices come from the real-data diagnosis:

Averaged start orientation. The double-integrated gravity is metres long over a segment, so a 0.05 deg error in a single scan's orientation moves t by centimetres. The start orientation is therefore the chordal mean of the orientation implied by every scan of the segment.
Per-window accelerometer bias. With one bias for the whole 15 minutes, the fit depended more strongly on the segment duration, and the residual grew. Both point to bias drift.

Uncertainty. The reported std of each axis is the largest of three:

the analytic std;
a window jackknife;
the segment sensitivity: the largest change when the fit is repeated with 1 s and 4 s segments.

The segment sensitivity is included because the estimate still moves with the segment duration by more than the jackknife spread. That is modelling error, possibly odometry errors correlated with motion, which resampling windows cannot reveal. An axis is estimated only if its reported std is ≤ 10 mm. Held-out windows must detect a 20 mm shift of each axis.

RTK-SLAM seq2 (hand-held): `pass`	x	y	z
estimate	21.3 mm	24.3 mm	-38.5 mm
reported std (analytic / jackknife / segment)	6.1 (0.7 / 4.0 / 6.1) mm	5.2 (0.6 / 1.4 / 5.2) mm	5.3 (1.1 / 5.3 / 4.2) mm
MID360 design value	11.0 mm	23.29 mm	-44.12 mm
difference to design	+10.3 mm	+1.0 mm	+5.6 mm
20 mm held-out control	detected (Δχ² 668)	detected (Δχ² 677)	detected (Δχ² 181)

Every axis is within 1.7 reported std of the design value.
The median held-out position residual is 5.6 mm (training 4.9 mm).
The median fitted gravity norm is 9.79 m/s².
The segment refits are (27.3, 22.1, -38.4) mm at 1 s and (15.9, 29.4, -42.7) mm at 4 s.

Artifact.

Driving (vehicle): inconclusive. Every axis is unobservable, with reported std of 97-155 mm. A vehicle barely rolls or pitches, so the accelerometer bias along the vertical and gravity cannot be told apart. The lever arm sees rotation almost only about the vertical axis. The fitted gravity norm of 9.11 m/s² is that degeneracy, not a unit error. Artifact.

Other lever arms on the same held-out windows (descriptive, not an audited claim). With the Calibrex rotation and nuisances refit, the held-out chi-square rises as follows:

Lever arm	Held-out Δχ²
MID360 design value	66
LI-Init refined `T_lidar_imu` translation (-2.5, -24.6, -103.5) mm	4668

The LI-Init value was seen while the method was being developed, so no SOTA claim is made for translation. Such a claim needs a pre-registered protocol on a recording whose scores have not been looked at.

Extrinsic audit against LI-Init: refuted¶

The second claim adds the lever arm to the comparison. The claim, procedure, metric, recordings, and thresholds were committed in mid360_imu_lidar_translation_preregistration.yaml (commit 5ca9989) before either method had been run on the evaluation recordings. stadtgarten_seq2 was development data and gates nothing.

Evaluation recordings. Two hand-held RTK-SLAM recordings of the same rig that had not been used before: stadtgarten_seq1 (27 min, park) and construction_seq1 (12 min, construction site).

Scoring. Each candidate's complete extrinsic (rotation, clock offset, and lever arm) is held fixed, and scoring works as follows:

One common odometry per recording, deskewed with the design reference, is shared by every candidate. With each method deskewing its own odometry, a handful of spans in which one run's registration failed dominated the development comparison.
Only per-span gravity, accelerometer bias, and segment velocities are refit.
The metric is the RMS of the position residual on every third 10-second span.

Verdict: refuted (protocol, result).

Calibrex's lever-arm run was inconclusive on both recordings, because x is unobservable. The pre-registration counts that as no estimate, so calibrex-estimate is contradicted on both recordings.
Every comparison gate therefore has no finite value.

Recording	Rotation (roll, pitch, yaw)	Lever arm x	y	z
stadtgarten_seq2 (development)	`pass` -0.08, +0.14, -0.16 deg	21.3 ± 6.1 mm	24.3 ± 5.2 mm	-38.5 ± 5.3 mm
stadtgarten_seq1	`pass` -0.06, +0.16, -0.17 deg	23.6 ± 11.0 mm (unobservable)	18.6 ± 1.7 mm	-37.4 ± 4.0 mm
construction_seq1	`pass` -0.10, +0.13, -0.19 deg	20.4 ± 13.3 mm (unobservable)	17.4 ± 3.9 mm	-35.0 ± 4.1 mm
MID360 design value	0, 0, 0	11.0 mm	23.29 mm	-44.12 mm

The rotation replicates across all three recordings to within 0.05 deg.
The lever-arm values replicate too. x is 20-24 mm every time, about 10 mm from the design value.
x's segment-duration sensitivity (11.0 and 13.3 mm) exceeds the 10 mm observability bound, however. The x estimate moves between about 7 and 35 mm when the fit uses 1 s or 4 s segments. Calibrex's own policy therefore declines to call x calibrated, and the audit takes that verdict at face value.

Descriptive only (gates nothing). The Calibrex candidate was scored anyway, and its scores are kept as descriptive_calibrex_native_inconclusive.json in mid360_imu_lidar_extrinsic_scores. Its paired improvement, with 95 % bootstrap CI, is:

Recording	over LI-Init	over LI-Init's lever arm	over design value
stadtgarten_seq1	2.56 mm (1.97-3.16)	0.78 mm (0.51-1.07)	-0.03 mm (-0.10-0.04)
construction_seq1	6.72 mm (4.70-9.13)	6.26 mm (4.20-8.58)	-0.03 mm (-0.14-0.08)

The metric cannot tell Calibrex's lever arm from the design value.
LI-Init's refined lever arm varies strongly between recordings: (-39, -40, -67) mm on seq1, (63, -171, -111) mm on construction, and (-2, -25, -103) mm on seq2.

What would change the verdict is a method change evaluated on new data, not a new threshold. That is what the next two sections do.

Revising the lever arm (v0.2)¶

The method was revised on the three spent recordings only: seq2, seq1, and construction_seq1. Each hypothesis was tested by refitting with 1, 2, and 4 s segments and checking whether x stopped moving:

Hypothesis	Result
accelerometer scale per axis	no: scale estimates were unphysical (3-5 % at 1 s), and x still moved
accelerometer-only delay (±20 ms)	no
LiDAR positions lagging orientations (fitted lag -8 to -11 ms)	no; the lag is consistent but does not stabilise x
position error proportional to acceleration (constant-velocity deskew)	no
gyro scale vs LiDAR rotation	LiDAR rates were 2-6 % below the gyro on the design-deskewed odometry, but 1.000 on the Calibrex-deskewed one; scaling the gyro made the fit worse
odometry tilt drift within a 10 s window	yes: with gravity fitted per 2 s segment, the 2 s and 4 s refits of x agree within 0.5-3 mm on all three recordings

Version accelerometer_lever_arm/v0.2 therefore fits gravity per segment and needs at least 10 scans per segment. Its segment sensitivity uses 3 s and 4 s refits. 1 s segments are excluded: with gravity and velocity per segment they are poorly constrained, and on every development recording they moved x by +7 to +16 mm for a reason that is still unidentified. With v0.2, all three development recordings pass, with x 28-36 mm and reported std 4-7.5 mm.

Second extrinsic audit: supported¶

The revised method was pre-registered in mid360_imu_lidar_translation_v2_preregistration.yaml (commit 5f7e694), before anything ran on construction_seq2, the last unused RTK-SLAM recording. The scoring procedure, metric, thresholds, and requirements are copied unchanged from the first round, and the scoring nuisances are pinned in tools/score_mid360_imu_lidar_extrinsics.py.

Verdict: supported, 4/4 gates (protocol, result).

construction_seq2 (10 min, 20 held-out spans)	Mean span RMS	Paired improvement of Calibrex (95 % CI low)	Spans where Calibrex is better
Calibrex	4.34 mm	—	—
MID360 design value (sanity row)	4.38 mm	—	12/20
LI-Init's lever arm with Calibrex's rotation	8.45 mm	4.11 mm (2.87)	18/20
LI-Init	8.47 mm	4.13 mm (2.79)	17/20

Calibrex on construction_seq2	x	y	z
lever arm ± reported std	27.0 ± 5.6 mm	17.9 ± 4.6 mm	-33.0 ± 7.9 mm
difference to design	+16.0 mm	-5.4 mm	+11.1 mm
20 mm held-out control	Δχ² 317	Δχ² 396	Δχ² 271

The rotation passes at (-0.10, +0.11, -0.23) deg.
LI-Init's lever arm is (-27, -172, +70) mm.

Rotation, lever arm, LI-Init.

What this claim does not say.

It is one recording. The first round's two recordings failed, and the method was changed after seeing them. Only construction_seq2 is untouched evidence for v0.2.
x disagrees with the design value. x is 16 mm from the design value here and 17-25 mm on the development recordings, 3-4.5 reported std, on every recording. Either the design value does not describe this unit's IMU position, or the method still has a bias.
The metric cannot tell the lever arm apart from the design value. It separates Calibrex from LI-Init, whose lever arm is 8-23 cm from the design value.
The per-segment gravity absorbs more than gravity. Its median norm ranges 9.68-10.05 m/s² across recordings.

Method¶

Odometry. Scan-to-local-map point-to-plane LiDAR odometry runs on deskewed sweeps: first with LiDAR-only constant velocity, then with gyro rotations mapped by the current extrinsic estimate.
Rotation increments. For consecutive scans, the LiDAR rotation increment is compared with the preintegrated gyro increment R dR_imu(b, dt) R^T. The raw gyro is integrated once; interval increments and first-order bias Jacobians come from cumulative sums, so every evaluation is vectorized.
Fit. A Procrustes initialization handles any mounting. It is followed by Huber IRLS over the rotation, gyro bias, and clock offset.
Held-out evidence.
Every third 10-second window is held out.
An 8-group window jackknife sets the reported std when it is larger than the analytic std.
Each rotation axis is shifted by 1 deg and the clock offset by 10 ms, and held-out windows must detect each shift (Δχ² ≥ 9).

Recording conventions are explicit per stream profile:

Profile	Point topic	Point time	Acceleration
`rtk-slam`	`/livox/points`	`offset_time`, seconds after the header	m/s²
`livox-ros-driver2`	`/livox/lidar`	`timestamp`, absolute nanoseconds	g

calibrex imu-lidar livox BAG_DIR --profile rtk-slam \
  --dataset-family rtk_slam --dataset-license "see the RTK-SLAM dataset page" \
  --output seq2.yaml

Limitations¶

The design reference is not a measurement.
The v0.1 lever arm failed the first pre-registered audit. The v0.2 lever arm passes on one unseen recording, but x is 16-25 mm from the design value on every recording.
1 s segments still bias x by +7 to +16 mm for an unidentified reason.
Accelerometer scale and axis misalignment are not modelled.
Gyro deskewing makes the LiDAR odometry depend on the IMU. The feedback ratio quantifies this dependence but does not remove it.
The external baseline is a single tool (LI-Init). The SOTA leaderboard records the claim with the scope above.