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arxiv: 2605.09383 · v2 · pith:E4PYRHYRnew · submitted 2026-05-10 · 💻 cs.RO

Safety-Critical LiDAR-Inertial Odometry with On-Manifold Deterministic Protection Level

Yueqi Zhu , Yan Pan , Chufan Rui , Jiasheng Luo , Shihua Li , Bo Zhou This is my paper

Pith reviewed 2026-05-12 04:41 UTC · model grok-4.3

classification 💻 cs.RO
keywords LiDAR-inertial odometryset-membership filterdeterministic protection levelsafety-critical navigationunknown but bounded noiseon-manifold estimationICP uncertaintyfeasible sets
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The pith

LiDAR-inertial odometry supplies deterministic protection levels by turning bounded point-cloud noise into feasible pose sets through an on-manifold filter.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a safety-critical LIO system that provides online deterministic protection levels instead of relying on offline probabilistic evaluations that may not transfer to unknown environments. It adopts an unknown-but-bounded noise model on point clouds to derive a closed-form relationship linking that noise directly to the uncertainty produced by the iterated closest point algorithm. This relationship supports an on-manifold ellipsoidal set-membership filter whose output feasible sets serve as concrete safety references for downstream robot operations. A sympathetic reader would care because current autonomous systems often lack reliable, real-time safety bounds once they leave controlled test conditions.

Core claim

By adopting the unknown but bounded assumption, we derive a neat closed-form relationship between point cloud noise and the uncertainty of the estimation from the iterated closest point algorithm. Using this relationship, we design an on-manifold ellipsoidal set-membership filter and implement it within the LIO system. Leveraging the properties of the set-membership filter, our system offers the feasible sets of the estimated locations as the deterministic protection levels, serving as safety references for the robots' downstream autonomous operations.

What carries the argument

on-manifold ellipsoidal set-membership filter that maps bounded point-cloud noise to feasible pose sets via the closed-form ICP uncertainty relation

Load-bearing premise

The unknown-but-bounded noise model for point clouds holds in real-world environments and the derived closed-form relationship accurately propagates uncertainty through ICP without hidden fitting steps or post-hoc exclusions.

What would settle it

A controlled experiment in which artificially bounded perturbations are added to real LiDAR scans and the true robot trajectory exits the computed ellipsoidal feasible set would falsify the closed-form propagation step.

Figures

Figures reproduced from arXiv: 2605.09383 by Bo Zhou, Chufan Rui, Jiasheng Luo, Shihua Li, Yan Pan, Yueqi Zhu.

Figure 1
Figure 1. Figure 1: Overview of the proposed safety-critical LiDAR-inertial odometry with deterministic protection level. An uncertainty resolving method is designed to calculate the ellipsoidal set-membership uncertainty of the estimated pose from ICP. An on-manifold set-membership filter is designed to fuse the measurements from IMU and the pose estimated from ICP. Finally, the outputs are the estimated pose with the determ… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the point model with UBB noise. The unknown ground-truth point is encompassed by a bounded ellipsoidal set whose center is the measured point. Unknown Bearing Noise Direction of Laser Measured Range Unknown GT Range Tangent Space at Measured Point (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of the point model with UBB noises. (a) Range measurement with UBB noise. The unknown ground-truth range is restricted within a bounded set. (b) Bearing measurement with UBB noise. The length of the unknown bearing noise is less than the upper bound. To overcome this drawback, as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The relationship between the estimated on-manifold pose and its corresponding ellipsoidal set-membership uncertainty in the tangent space. where W I ˜t ∗ ∈ R 3 is the translation part of W I T˜ ∗ , ∆ξ = [∆ρ T ∆ϕ T] T, ∆ρ ∈ R 3 and ∆ϕ ∈ so (3). Consequently, it yields ∂ 2Ji [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Estimated locations and protection levels from our system. Unlike other methods that only provide estimated locations, our system can provide deterministic protection levels. (a) gate 02 in M2DGR. (b) sbs 03 in NTU VIRAL. When the robot moves a certain distance away from the origin of the current local map, a new local map will be created. We use the ellipsoidal sets of the last error state within the loca… view at source ↗
Figure 6
Figure 6. Figure 6: Estimated protection levels and trajectories from our system and PALoc (σ 2 = 0.001), and the ground truth using different sequences in M2DGR. The protection levels estimated by PALoc cannot cover the ground truth inevitably, therefore, cannot effectively reflect the effectiveness of the navigation system. However, the protection levels obtained by our system can more effectively cover the ground truth. 7.… view at source ↗
Figure 7
Figure 7. Figure 7: Our quadruped robot platform. The RoboSense Helios32 LiDAR and the YIS130 IMU were used as the sensors. The quadruped robot was remotely controlled and walked along multiple different trajectories in the campus environment [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimated protection levels and mapping result of the building sequence. Our system not only provides location estimations and deterministic protection levels, but also yields fine-grained maps. unknown ground-truth location, thereby providing a safety reference for downstream tasks. 7.4 Real-world results In addition to the public datasets, to verify the effectiveness of the protection levels estimated by… view at source ↗
Figure 9
Figure 9. Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Our wheeled mobile robot platform. The Velodyne VLP-16 LiDAR and CH110 IMU were used as the sensors. The real-time kinematic GNSS was equipped to provide ground truth [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Estimated protection levels and trajectories from our system and PALoc (σ 2 = 0.001), and the ground truth using different sequences with our wheeled mobile robot. The protection levels obtained by our system can more effectively cover the ground truth [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time cost of different steps in the update stage. The uncertainty resolving method we proposed has a relatively small computational load [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
read the original abstract

In safety-critical scenarios, the protection level of the autonomous navigation system is crucial for enabling mobile robots to perform safe tasks. However, existing studies on probabilistic navigation systems for robots usually perform offline accuracy evaluations using limited datasets and assume that the results can be applied to unknown real-world environments. As a result, current autonomous mobile robots often lack protection levels for online safety assessment. To fill this gap, we propose a safety-critical LiDAR-inertial odometry (LIO) that provides deterministic protection levels based on on-manifold deterministic state estimation. By adopting the unknown but bounded assumption, we derive a neat closed-form relationship between point cloud noise and the uncertainty of the estimation from the iterated closest point algorithm. Using this relationship, we design an on-manifold ellipsoidal set-membership filter and implement it within the LIO system. Leveraging the properties of the set-membership filter, our system offers the feasible sets of the estimated locations as the deterministic protection levels, serving as safety references for the robots' downstream autonomous operations. The experimental results show that our system can provide effective deterministic online safety references for diverse robots in various environments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a safety-critical LiDAR-inertial odometry (LIO) system that, under the unknown-but-bounded noise assumption, derives a closed-form relationship between point-cloud perturbations and the uncertainty of iterated closest point (ICP) pose estimates. This relationship is used to construct an on-manifold ellipsoidal set-membership filter whose feasible sets serve as deterministic protection levels for the estimated trajectory, intended to support online safety assessment without probabilistic assumptions or offline dataset calibration.

Significance. If the closed-form mapping is rigorously derived and conservative with respect to ICP's iterative correspondence updates, the work would provide a useful deterministic alternative to probabilistic protection-level methods in robotics. The on-manifold formulation and integration into a full LIO pipeline address a practical gap for safety-critical applications. The experimental claims on diverse robots and environments would be strengthened by the absence of free parameters in the core derivation, though this remains to be verified from the full equations.

major comments (2)
  1. [Section 3 (closed-form ICP uncertainty relationship)] The central derivation (Section 3) claims a closed-form relationship between bounded point-cloud noise and the feasible set of ICP estimates. Because standard ICP recomputes nearest-neighbor correspondences at each iteration, a perturbation inside the declared bound can flip associations on surfaces with varying density or grazing angles. The manuscript does not appear to enclose all possible re-associations or convergence points; if the mapping is obtained by linearizing around a nominal solution with fixed matches, the resulting ellipsoid can exclude the true pose while still respecting the noise bound. This directly undermines the guarantee of the protection levels supplied by the subsequent filter.
  2. [Section 4 (on-manifold set-membership filter)] In the on-manifold ellipsoidal set-membership filter (Section 4), the propagation step must account for the manifold retraction when correspondences change. If the linearization or ellipsoid update omits this or assumes fixed matches, the filter is not guaranteed to remain conservative. The paper should supply the explicit retraction formula and a proof that the propagated set contains all possible ICP outcomes under the bounded-noise axiom.
minor comments (2)
  1. [Abstract] The abstract would be clearer if it included the key closed-form expression or at least the main variables in the noise-to-uncertainty mapping.
  2. [Figures 4-7] Figure captions and axis labels should explicitly state whether the plotted protection levels are the full ellipsoidal sets or their projections; current captions are ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the requirements for rigorous conservatism in our deterministic protection-level approach. Below we respond point-by-point to the major comments and indicate planned revisions.

read point-by-point responses

  1. Referee: The central derivation (Section 3) claims a closed-form relationship between bounded point-cloud noise and the feasible set of ICP estimates. Because standard ICP recomputes nearest-neighbor correspondences at each iteration, a perturbation inside the declared bound can flip associations on surfaces with varying density or grazing angles. The manuscript does not appear to enclose all possible re-associations or convergence points; if the mapping is obtained by linearizing around a nominal solution with fixed matches, the resulting ellipsoid can exclude the true pose while still respecting the noise bound.

    Authors: We acknowledge that the closed-form mapping in Section 3 is obtained by linearizing the ICP objective around a nominal pose with correspondences fixed from the unperturbed cloud. This linearization yields an explicit ellipsoid relating bounded point-cloud perturbations to pose uncertainty. To ensure the mapping remains conservative under possible correspondence flips, the bound is derived from the worst-case residual within the noise set, which implicitly over-approximates the effect of association changes on surfaces of varying density. We will revise Section 3 to explicitly state this assumption, add a short proof sketch showing that the linearization error is absorbed into the ellipsoid radius, and include additional numerical checks on grazing-angle and density-varying surfaces demonstrating that the true pose remains inside the reported set. revision: partial

  2. Referee: In the on-manifold ellipsoidal set-membership filter (Section 4), the propagation step must account for the manifold retraction when correspondences change. If the linearization or ellipsoid update omits this or assumes fixed matches, the filter is not guaranteed to remain conservative. The paper should supply the explicit retraction formula and a proof that the propagated set contains all possible ICP outcomes under the bounded-noise axiom.

    Authors: We agree that the propagation step must preserve conservatism under the manifold structure. Section 4 employs the standard exponential-map retraction for the special Euclidean group to propagate the ellipsoidal set forward in time; the update equation already incorporates a bounded linearization error term that is added to the ellipsoid matrix. We will revise the manuscript to (i) write the explicit retraction formula used, (ii) provide the step-by-step derivation showing how the retraction is applied to the set-membership update, and (iii) include a short lemma proving that the propagated ellipsoid contains every possible ICP outcome consistent with the unknown-but-bounded noise model, even when correspondences are recomputed at each iteration. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivation chain

full rationale

The paper states it adopts the unknown-but-bounded noise assumption to derive a closed-form relationship between point-cloud perturbations and ICP pose uncertainty, then constructs an on-manifold ellipsoidal set-membership filter whose feasible sets serve as deterministic protection levels. No equations or steps in the provided text reduce this relationship to a fitted parameter, a self-citation chain, or a renaming of an input quantity by construction. The central claim is presented as following directly from set-membership theory under the stated axiom and standard ICP properties, remaining independent of the paper's own outputs. This is the normal case of a self-contained first-principles derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unknown-but-bounded noise model as the key domain assumption and on the validity of the closed-form ICP uncertainty relation; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Unknown but bounded assumption for point cloud noise
    Invoked to derive the closed-form relationship between noise and estimation uncertainty from ICP.

pith-pipeline@v0.9.0 · 5512 in / 1303 out tokens · 38617 ms · 2026-05-12T04:41:48.975924+00:00 · methodology

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