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Extending Ground-Constraint LiDAR-IMU Calibration to Tilted Surfaces in a Continuous-Time Framework

Vassili Korotkine, Pierre Chamoun, Mohammed Ayman Shalaby, James Richard Forbes

arXiv:2608.25135v1cs.RO

TL;DR

Ground-vehicle LiDAR-IMU calibration loses observability under planar motion, while existing ground constraints assume flat terrain. The paper adds distance and inclination residuals to a continuous-time calibration framework, retaining validity on tilted surfaces; experiments report improved repeatability on tilted and flat ground, with stronger improvement on tilted ground.

  • Problem

    Ground vehicles commonly lack full motion excitation, and existing ground constraints rely on gravity and surface-normal colinearity, restricting calibration to flat ground.

  • Method

    The method adds distance and inclination ground-plane residuals to the continuous-time OA-Calib framework for targetless LiDAR-IMU calibration.

  • Results

    Improved repeatability is demonstrated on tilted and flat-ground datasets, with stronger improvement reported for tilted ground.

  • Takeaways & Limitations

    The method provides a ground-constraint calibration approach for planar vehicle motion on non-flat terrain and includes an open-source implementation.

Abstract

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This paper presents a novel method that extends targetless LiDAR-IMU calibration for ground vehicles to non- flat environments. Calibration typically necessitates full exci- tation of the sensor rig, a requirement that is not fulfilled by ground vehicles in normal operation. To address the degenerate planar motion, state-of-the-art methods propose residuals that assume the colinearity of the gravity and physical surface normal vectors, restricting usage to cases where the ground is assumed flat. This paper proposes ground-plane residuals that do not require this assumption, and are applicable for planar motion on a tilted surface. Results are demonstrated on a dataset collected from a Husky ground vehicle, on the M2DGR dataset, as well as on an offroad vehicle dataset. Repeatability is shown to be improved both in tilted and flat-ground scenarios, with strong improvement demonstrated for the tilted case. The implementation and experiments are open-sourced at https://github.com/vkorotkine/licalib_tilted_ground.

I. INTRODUCTION

Targetless LiDAR-IMU calibration avoids deployment targets but must address observability loss during ground-vehicle planar motion. This paper introduces ground constraints valid on tilted surfaces and evaluates them against flat-ground calibration approaches.

  • Motivation: LiDAR and IMU data fusion requires accurate spatial-temporal calibration of the sensors’ extrinsic pose and time offset.The calibration aligns measurements to a common reference frame and clock; inaccuracies can affect downstream robotics pipelines.
  • Existing calibration approaches: Target-based calibration can be accurate but requires custom targets, whereas targetless methods need no prior setup and may be less accurate.Targetless methods lack the geometric information supplied by known targets.
  • Planar-motion degeneracy: Full motion excitation is generally required for observability, but ground vehicles’ single-axis rotation leaves position calibration unobservable along the rotation axis.This degeneracy motivates adding ground constraints to the calibration problem.
  • Limitations of prior methods: Existing ground constraints assume gravity and the physical surface normal are colinear, limiting their use to flat ground; pitch excitation does not address strictly planar motion.On tilted surfaces, the gravity and surface normal vectors are not colinear.
  • Paper contribution: The proposed distance and inclination residuals remain valid without the flat-ground assumption, addressing planar-motion observability loss on tilted terrain.The distance residual constrains position-extrinsic projection along the surface normal, while the inclination residual uses the dot product of surface normal and gravity.
  • Paper contribution: The paper claims the first method addressing planar-motion observability loss on tilted terrain and provides an open-source continuous-time ground-constraint implementation.It evaluates the approach on flat- and inclined-ground vehicle datasets and the M2DGR dataset, comparing against GRIL-Calib.

II. CONTINUOUS-TIME LIDAR-IMU CALIBRATION

The proposed system inherits OA-Calib’s continuous-time trajectory representation and optimization framework. It models the IMU trajectory with a B-Spline and jointly estimates calibration-related states and biases.

  • Base framework: OA-Calib is used as the unchanged continuous-time base system because it naturally uses all measurements without downsampling or finite-differencing.The proposed contribution is limited to adding ground-constraint residuals within this framework.
  • Trajectory representation: OA-Calib parametrizes the IMU trajectory with a B-Spline containing N control points.The spline time knots remain fixed throughout optimization.
  • Trajectory representation: Each control point is represented by a pose transformation, with the world frame initialized from the IMU frame at the first IMU measurement.The world reference frame is set to the first control-point frame and the world reference position to its physical position.
  • Optimization state: The full optimization state includes spline parameters, intrinsic sensor parameters, constant IMU biases, gravity, and LiDAR-IMU spatial-temporal extrinsics.The constant-bias and state assumptions are inherited from OA-Calib.
  • Residual framework: IMU angular-velocity and acceleration residuals compare measurements with spline-derived quantities, while LiDAR uses point-to-plane map residuals.These residual constructions belong to the inherited continuous-time calibration pipeline.

III. PROPOSED METHOD

The method extracts ground-plane information from LiDAR data and adds residuals to the OA-Calib pipeline. Two novel residuals handle tilted ground, while a simplified flat-ground orientation residual is also retained.

  • Residual construction: Ground-plane residuals are computed at a given time using pose quantities obtained from the B-Spline state.The method extracts ground-plane information from the LiDAR point cloud before forming calibration residuals.
  • Residuals: The flat-ground orientation residual assumes colinearity between gravity and the surface normal, matching the flat-ground constraint used by prior systems.It is included alongside the tilted-ground residuals.
  • Residuals: The distance residual constrains extrinsic translation along the vehicle’s rotation axis and independently resolves position-extrinsic unobservability caused by planar motion.The distance residual compares the relevant calibration geometry with the ground-plane information.
  • Residuals: The inclination residual provides an additional orientation constraint by using the ground inclination angle rather than assuming flat ground.It extends the orientation ground constraint to tilted surfaces.
  • Residuals: Both the distance and inclination residuals are novel tilted-ground residuals, while the flat-ground residual avoids introducing extraneous ground reference frames.Ablation studies assess the relative contribution of the residuals.

A. Required Quantities and Assumptions

The method requires measured IMU height, LiDAR-extracted ground geometry, and ground inclination information for tilted surfaces. Its flat-ground orientation residual uses gravity resolved through the IMU trajectory and assumes surface normal–gravity colinearity.

  • Required quantities: The IMU height db must be measured beforehand because it cannot be refined on strictly flat ground.Its accuracy directly affects the position extrinsic along the vehicle’s rotation axis.
  • Required quantities: Patchwork++ extracts the LiDAR-frame surface normal nl and LiDAR height dl at each time instant.Ground extraction requires the driven surface to be reasonably planar; robust losses reject outliers during optimization.
  • Required quantities: The proposed inclination residual requires ground inclination cos(ϕ), obtained from a separate sensor or estimator providing pitch and roll.The paper computes it using gravity information resolved in the IMU frame.
  • Flat-ground orientation residual: The flat-ground orientation residual simplifies the prior formulation by avoiding extraneous ground reference frames.It carries the same information as the prior orientation residual.
  • Flat-ground orientation residual: On flat surfaces, the surface normal is assumed colinear with the gravity vector.The IMU gravity vector depends on the B-Spline-computed IMU rotation and world-frame gravity.

C. Distance Residual

The distance residual compares the IMU and LiDAR heights above the same ground plane, constraining the position extrinsic along the surface normal. This remains valid on tilted surfaces and addresses the single-axis-motion unobservability.

  • Residual construction: The distance residual equates the IMU–LiDAR height difference with the position extrinsic projected onto the surface normal.It uses the measured IMU height db and LiDAR height dl.
  • Residual construction: Resolving the ground-plane relation in the LiDAR frame yields the distance-residual formulation.The LiDAR frame provides the surface normal directly from ground segmentation.
  • Constraint effect: The residual constrains the position extrinsic along the direction of the surface normal.This constraint is independent of the flat-ground assumption because the normal is measured in the LiDAR frame regardless of inclination.
  • Constraint effect: Under typical operating conditions, the surface normal aligns with the vehicle’s rotation axis, directly constraining the otherwise unobservable position component.This resolves the position-extrinsic unobservability caused by single-axis ground-vehicle motion.

D. Inclination Residual

The inclination residual extends the orientation constraint to tilted surfaces, where gravity and the physical surface normal are not colinear. It remains valid on tilted ground, unlike the flat-ground residual.

  • Validity by surface type: On tilted ground, the inclination residual remains valid while the flat-ground residual does not.The two residuals disagree on tilted surfaces, whereas both are valid for flat ground.
  • Residual construction: The inclination residual replaces the flat-ground colinearity condition with a known surface inclination cos(ϕ).The inclination is incorporated by extending the flat-surface orientation residual.
  • Required information: The required inclination can be computed from a known flat-ground IMU gravity vector and reused for tilted sequences.This assumes the IMU remains rigidly mounted and does not shift between calibration sequences.
  • Required information: In the experiments, the reference gravity vector is obtained by calibrating on a known flat-ground sequence and extracting the optimized world-frame gravity.The resulting vector is then used to compute inclination for subsequent tilted-ground calibration.

IV. RESULTS

The proposed continuous-time method is evaluated against GRIL-Calib using multiple datasets, measured-extrinsic initialization, and repeatability as the primary quality metric. Results report mean extrinsics for sanity checking and spread across repeated sequences for performance assessment.

  • Experimental setup: The proposed algorithm is compared with GRIL-Calib across multiple datasets using physically measured IMU height and LiDAR ground geometry.The LiDAR ground distance and normal are extracted with Patchwork++ for both methods.
  • Experimental setup: Measured extrinsics initialize both calibration procedures, providing a more realistic starting condition than heavily perturbed initialization.The paper motivates this choice by typical engineering knowledge of rough sensor placement.
  • Metrics: Repeatability is assessed from the spread of calibration results across multiple sequences because physical measurements are insufficiently accurate as ground truth.Lower standard deviation indicates improved repeatability.
  • Metrics: The mean rotation is represented by a 1−2−3 Euler-angle decomposition, while standard deviations quantify result spread in degrees.The standard deviations are the performance metric used for comparing calibration methods.
  • Metrics: The reported evaluation consolidates mean extrinsics across all datasets in Table III.Mean calibration values serve as a sanity check alongside spread-based performance measurements.

B. Husky Clearpath Ground Vehicle

The Husky UGV evaluation covers flat and inclined driving sequences. The proposed method improves repeatability on flat ground and remains close to measured extrinsics on inclined ground, unlike GRIL-Calib.

  • Evaluation setup: Three indoor flat-ground and seven outdoor inclined-ground sequences evaluate nominal performance and performance on a tilted surface.The Husky carries a Velodyne VLP-16 LiDAR and 3DM-GX5-AHRS IMU.
  • Flat-ground results: On flat-ground sequences, both methods converge reasonably close to measured extrinsics, while the proposed method achieves increased repeatability across all extrinsic parameters.The improvement is especially pronounced for yaw orientation and x-y position extrinsics.
  • Inclined-ground results: Inclined-ground results show GRIL-Calib yields an incorrect answer, whereas the proposed method remains reasonably close to measured extrinsics and mean flat-ground calibration results.These results are identified as the primary evidence for improvement on inclined ground.

C. M2DGR

The M2DGR evaluation tests the continuous-time method on flat-ground sequences corresponding to the GRIL-Calib analysis. Position repeatability improves substantially, while yaw repeatability is comparable but slightly degraded.

  • Dataset: The M2DGR dataset uses a Velodyne VLP-32C and Handsfree A9 IMU, with sequences and time segments selected to correspond directly to GRIL-Calib's analysis.The dataset was collected on flat ground, and its calibration provides the measured extrinsics.
  • Calibration results: The mean extrinsics of both methods are comparable to the measured extrinsics, while the proposed method significantly improves x-y position extrinsic repeatability.The per-sequence results and extrinsic spread are reported in Figures 5 and 6.
  • Calibration results: Both methods yield about one degree of standard deviation in yaw calibration extrinsic, with the proposed method slightly degrading relative to the GRIL-Calib baseline.This yaw repeatability differs from the Husky flat-ground sequences, where the proposed method reached a tenth of a degree.

D. Offroad Vehicle Dataset

The offroad evaluation finds broadly improved extrinsic spread with the proposed method, except for one orientation component. Ablation and misinitialization studies expose limits of repeatability and sensitivity to initialization and robust-loss settings.

  • Offroad results: Across indoor and outdoor offroad environments, the proposed method improves extrinsic spread across the board except for a slightly worsened y-component of orientation error.The two methods have very similar mean extrinsics.
  • Ablation study: The ablation compares combinations of flat-ground orientation, inclined orientation, and distance residuals to assess their relative contributions.Flat-ground residuals produce significantly higher position-extrinsic spread, while OA-Calib has the lowest spread among proposed methods.
  • Ablation study: Repeatability can favor a system without additional constraints because initialization suppresses drift, even though the axis-of-rotation position extrinsic remains unobservable without ground constraints.This demonstrates that repeatability alone can misrepresent calibration quality.
  • Misinitialized extrinsics: With misinitialized extrinsics, omitting the distance residual leaves the z-position error around 25 centimeters, while using it can produce larger spread.The insideGarage sequence shows a large error even with the distance residual because the robust Cauchy loss may reject its update.

V. CONCLUSION

The paper introduces continuous-time targetless LiDAR-IMU calibration for planar motion on non-flat ground and reports improved robustness and flat-ground performance. It also identifies practical limitations and evaluation challenges that motivate future work.

  • The proposed method calibrates LiDAR-IMU systems during planar motion on non-flat ground using a continuous-time formulation.
  • Robustness to tilted ground is improved relative to the state-of-the-art flat-ground calibration system.
  • Continuous-time modeling also improves performance in flat-ground scenarios.
  • The distance residual requires known IMU height, while flat-ground inclination computation requires a known IMU-resolved gravity vector.
  • Evaluation remains nontrivial because precise ground truth is lacking, calibration effects on estimation accuracy are difficult to measure, and initialization sensitivity affects results.
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