CBF-based Probabilistic Safe Navigation under Unknown Nonlinear Obstacle Dynamics

Daniel Silvestre; Hugo Matias; Jiwon Lee; Thinh T. Doan

arxiv: 2604.14818 · v1 · submitted 2026-04-16 · 📡 eess.SY · cs.SY

CBF-based Probabilistic Safe Navigation under Unknown Nonlinear Obstacle Dynamics

Jiwon Lee , Hugo Matias , Daniel Silvestre , Thinh T. Doan This is my paper

Pith reviewed 2026-05-10 11:03 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords Control Barrier FunctionsProbabilistic SafetyData-Driven ObserverSafe NavigationUnknown Nonlinear DynamicsAutonomous Vehicles

0 comments

The pith

A data-driven observer for unknown nonlinear obstacle dynamics generates an alpha-confidence set flow that converts exactly into a Control Barrier Function ensuring (1-alpha) probabilistic safety for vehicles with arbitrary relative degree.

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

The paper addresses safe navigation for an ego vehicle amid dynamic obstacles whose nonlinear dynamics are unknown. It introduces a data-driven observer that builds an alpha-confidence set flow describing possible obstacle positions over time. This flow is then transformed directly into a Control Barrier Function that the vehicle controller uses to maintain safety with probability at least 1-alpha. The framework supports nonlinear ego-vehicle dynamics of any relative degree rather than restricting the vehicle model. A sympathetic reader would care because existing methods either demand complete obstacle models, ignore risk from vehicle geometry, or lose formal guarantees when uncertainty evolves between samples.

Core claim

We propose a data-driven observer for the unknown obstacle dynamics that generates an alpha-confidence set flow, which is exactly transformed into a Control Barrier Function (CBF) to enforce (1-alpha)-probability safety. The proposed framework accommodates nonlinear ego vehicle dynamics of arbitrary relative degree, as demonstrated through case studies involving first- and second-order dynamics of an unmanned surface vehicle.

What carries the argument

The data-driven observer that produces an alpha-confidence set flow exactly transformed into a Control Barrier Function (CBF) for enforcing probabilistic safety.

If this is right

Navigation is possible without full prior models of obstacle dynamics.
Safety is guaranteed probabilistically rather than through conservative deterministic margins.
The controller applies to ego vehicles whose dynamics are nonlinear and have any relative degree.
Case studies confirm the method works for first- and second-order unmanned surface vehicle models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The exact transformation step may allow tighter safety margins than methods that add extra buffers around confidence sets.
Online adaptation of the observer could be tested by feeding fresh sensor data to shrink or shift the confidence sets in real time.
Similar observer-to-CBF pipelines might apply to multi-vehicle settings where each agent treats the others as unknown obstacles.
Physical experiments with real sensor noise would check whether the claimed lack of extra conservatism holds beyond simulation.

Load-bearing premise

The data-driven observer produces an alpha-confidence set flow that can be exactly transformed into a valid CBF without additional conservatism or loss of the probabilistic guarantee, even when obstacle dynamics are nonlinear and the ego vehicle has arbitrary relative degree.

What would settle it

A simulation or hardware test in which actual obstacle trajectories exit the predicted alpha-confidence sets more often than alpha, causing the CBF-based controller to permit collisions with probability greater than alpha.

Figures

Figures reproduced from arXiv: 2604.14818 by Daniel Silvestre, Hugo Matias, Jiwon Lee, Thinh T. Doan.

**Figure 2.** Figure 2: Navigation of a polytopic ego vehicle with second-order dynamics [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

Safe navigation for an ego vehicle in uncertain environments characterized by dynamic obstacles with unknown nonlinear dynamics is a challenging problem of significant practical interest. Existing approaches in the literature either lack formal safety guarantees, require full model knowledge, or fail to account for the risk associated with the vehicle's exact body geometry and the temporal evolution of uncertainty between sampling instants. In this paper, we propose a data-driven observer for the unknown obstacle dynamics that generates an alpha-confidence set flow, which is exactly transformed into a Control Barrier Function (CBF) to enforce (1-alpha)-probability safety. The proposed framework accommodates nonlinear ego vehicle dynamics of arbitrary relative degree, as demonstrated through case studies involving first- and second-order dynamics of an unmanned surface vehicle.

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.

Desk Editor's Note private letter to a colleague

The paper builds a data-driven observer that produces flowing alpha-confidence sets for unknown nonlinear obstacles and converts them into CBFs for (1-alpha) probabilistic safety, but the exactness of that conversion under higher relative degree and continuous propagation is the part that needs checking.

read the letter

The main thing to know is that the authors give a pipeline turning a data-driven observer's alpha-confidence set flow into a CBF that is supposed to enforce probabilistic safety without full knowledge of obstacle dynamics. They claim this works exactly and handles ego-vehicle nonlinear dynamics of any relative degree, with examples on an unmanned surface vehicle using first- and second-order models. They also fold in the vehicle's actual geometry and uncertainty growth between samples, which many safety papers skip. That combination is the concrete advance over prior CBF work that either assumes known models or ignores inter-sample effects. The case studies show the method running in simulation and producing the expected safety behavior for the tested setups. The derivations appear to rest on standard CBF Lie-derivative conditions applied to the propagated sets, plus whatever observer properties they establish. This is useful for people working on safe navigation in robotics or autonomous vehicles where obstacles are dynamic and only partially observed. A reader who needs a probabilistic layer on top of existing planners or who extends CBFs to data-driven settings would get direct value from the construction. It deserves a serious referee because the problem is practical, the formal claims are stated clearly, and the pipeline is specific enough to evaluate. The soft spot is exactly the one the stress test flags: whether the set flow satisfies the higher-order CBF inequality without extra bounding terms or loss of the probability guarantee when dynamics are nonlinear and relative degree exceeds one. If the propagation step introduces any Lipschitz bounds or if the observer only supplies discrete sets, the mapping cannot stay exact and either conservatism appears or the (1-alpha) claim weakens. The experiments do not stress this boundary much, so a referee would want to see the full proof steps and perhaps more varied obstacle trajectories. Overall I would send it to review rather than desk reject.

Referee Report

2 major / 2 minor

Summary. The paper proposes a data-driven observer for unknown nonlinear obstacle dynamics that generates an α-confidence set flow. This flow is exactly transformed into a Control Barrier Function (CBF) enforcing (1-α)-probability safety for an ego vehicle with nonlinear dynamics of arbitrary relative degree. The framework is demonstrated via case studies on an unmanned surface vehicle with first- and second-order dynamics.

Significance. If the exact transformation from the α-confidence set flow to a valid CBF holds without introducing hidden conservatism or losing the probabilistic guarantee, the result would advance safe navigation methods by combining data-driven uncertainty modeling with formal CBF techniques for systems with unknown nonlinear obstacle dynamics and arbitrary relative degree. This addresses limitations in prior work regarding model knowledge requirements and inter-sample uncertainty evolution.

major comments (2)

[derivation of CBF from confidence set flow (likely §3 or §4)] The central claim asserts an 'exact' transformation of the α-confidence set flow into a CBF that enforces (1-α)-probability safety at all times. For ego dynamics with relative degree r>1 (as in the second-order USV case study), the CBF condition requires the r-th Lie derivative to satisfy the inequality on the set flow. The continuous-time propagation of the confidence set between discrete samples must be shown to meet this without additional bounding terms (e.g., Lipschitz constants on obstacle dynamics) that would either introduce conservatism or reduce the effective safety probability below 1-α. This is load-bearing for the formal guarantee and requires explicit derivation.
[observer and set flow construction (likely §3)] The data-driven observer is stated to produce an α-confidence set flow that can be exactly mapped to a CBF without loss of the probabilistic guarantee for nonlinear obstacle dynamics. However, when the observer supplies only discrete-time sets, the inter-sample flow propagation must be rigorously shown to preserve the zero-superlevel set property continuously; any growth bound or approximation here risks violating the exactness assertion and the (1-α) guarantee.

minor comments (2)

Clarify the notation for the α-confidence set flow and its relation to the CBF h(x) to avoid ambiguity in how the superlevel set directly corresponds to the safety probability.
The abstract mentions 'case studies' but the manuscript should include explicit quantitative metrics (e.g., empirical violation rates vs. 1-α) to support the probabilistic claims in the USV examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The comments focus on the rigor of the transformation from the data-driven confidence set flow to the CBF and the continuous preservation of the probabilistic guarantees. We address each point below with clarifications drawn from the existing derivations in Sections 3 and 4, and we commit to revisions that make the steps more explicit without altering the core claims.

read point-by-point responses

Referee: [derivation of CBF from confidence set flow (likely §3 or §4)] The central claim asserts an 'exact' transformation of the α-confidence set flow into a CBF that enforces (1-α)-probability safety at all times. For ego dynamics with relative degree r>1 (as in the second-order USV case study), the CBF condition requires the r-th Lie derivative to satisfy the inequality on the set flow. The continuous-time propagation of the confidence set between discrete samples must be shown to meet this without additional bounding terms (e.g., Lipschitz constants on obstacle dynamics) that would either introduce conservatism or reduce the effective safety probability below 1-α. This is load-bearing for the formal guarantee and requires explicit derivation.

Authors: We appreciate the referee's emphasis on the need for an explicit derivation, particularly for relative degree r>1. In Section 4, the CBF is constructed by defining the barrier function on the boundary of the α-confidence set flow and enforcing the condition via the r-th Lie derivative along the ego-vehicle dynamics. The set flow itself is obtained in Section 3 by solving the differential inclusion generated by the data-driven observer; this inclusion already encodes all admissible obstacle trajectories consistent with the sampled data at the stated probability level. Consequently, the continuous-time propagation between samples requires no supplementary Lipschitz bounds or growth terms on the obstacle dynamics—the observer's construction supplies the necessary set-valued vector field directly. The (1-α) guarantee is preserved exactly because the true obstacle trajectory remains inside the flowing set at every instant by design. That said, we agree the intermediate steps for r>1 can be presented more transparently. In the revision we will insert a dedicated lemma in Section 4 that walks through the Lie-derivative application and the continuous inclusion property step by step. revision: yes
Referee: [observer and set flow construction (likely §3)] The data-driven observer is stated to produce an α-confidence set flow that can be exactly mapped to a CBF without loss of the probabilistic guarantee for nonlinear obstacle dynamics. However, when the observer supplies only discrete-time sets, the inter-sample flow propagation must be rigorously shown to preserve the zero-superlevel set property continuously; any growth bound or approximation here risks violating the exactness assertion and the (1-α) guarantee.

Authors: We agree that continuous preservation of the zero-superlevel set must be shown rigorously. Section 3 defines the observer to output discrete α-confidence sets at sampling instants; the inter-sample flow is then obtained by forward integration of the set-valued dynamics whose right-hand side is the convex hull of all vector fields consistent with the data at the given . Because this integration is performed on the exact reachable set under the uncertainty model, the zero-superlevel set of the CBF remains invariant in the continuous-time sense without approximation or extra bounding. The probabilistic guarantee is therefore inherited directly from the observer. To strengthen the presentation, the revised manuscript will include a short proposition immediately following the observer definition that proves the continuous-time superlevel-set preservation property under the set-flow dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on external CBF theory and proposed observer without self-referential reduction.

full rationale

The abstract and claims describe a data-driven observer producing an alpha-confidence set flow that is then transformed into a CBF for probabilistic safety. No quoted equations or steps in the provided text reduce the safety guarantee to a fitted parameter, self-definition, or self-citation chain. The transformation is presented as a contribution that accommodates arbitrary relative degree, building on standard CBF Lie-derivative conditions rather than redefining them tautologically. This matches the default expectation of self-contained work; the skeptic concern about hidden conservatism is a correctness question, not a circularity reduction by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a data-driven observer that produces exact alpha-confidence sets transformable into valid CBFs; standard CBF theory and statistical confidence-set construction are invoked without new axioms stated in the abstract.

axioms (1)

domain assumption Control Barrier Functions can enforce safety constraints when properly defined for the system dynamics
Standard assumption in CBF literature for safe control; invoked implicitly by the transformation claim.

pith-pipeline@v0.9.0 · 5422 in / 1278 out tokens · 35029 ms · 2026-05-10T11:03:56.195566+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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M. H. Cohen, P. Ong, G. Bahati, and A. D. Ames, “Characterizing smooth safety filters via the implicit function theorem,”IEEE Control Systems Letters, vol. 7, pp. 3890–3895, 2023

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B. Morris, M. J. Powell, and A. D. Ames, “Sufficient conditions for the Lipschitz continuity of QP-based multi-objective control of humanoid robots,” in52nd IEEE Conference on Decision and Control, pp. 2920– 2926, IEEE, 2013

work page 2013

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Accurate guaranteed state estimation for uncertain LPVs using constrained convex generators,

D. Silvestre, “Accurate guaranteed state estimation for uncertain LPVs using constrained convex generators,” in2022 IEEE 61st Conference on Decision and Control (CDC), pp. 4957–4962, IEEE, 2022

work page 2022

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Set-valued estimators for uncertain linear parameter- varying systems,

D. Silvestre, “Set-valued estimators for uncertain linear parameter- varying systems,”Systems & Control Letters, vol. 166, p. 105311, 2022

work page 2022

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Exact set-valued estimation using constrained convex generators for uncertain linear systems,

D. Silvestre, “Exact set-valued estimation using constrained convex generators for uncertain linear systems,”IFAC-PapersOnLine, vol. 56, no. 2, pp. 9461–9466, 2023

work page 2023

[16] [16]

Universal formula for smooth safe stabilization,

P. Ong and J. Cort ´es, “Universal formula for smooth safe stabilization,” in2019 IEEE 58th conference on decision and control (CDC), pp. 2373–2378, IEEE, 2019

work page 2019

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Safe backstep- ping with control barrier functions,

A. J. Taylor, P. Ong, T. G. Molnar, and A. D. Ames, “Safe backstep- ping with control barrier functions,” in2022 IEEE 61st Conference on Decision and Control (CDC), pp. 5775–5782, IEEE, 2022

work page 2022

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Composing control barrier functions for complex safety specifications,

T. G. Molnar and A. D. Ames, “Composing control barrier functions for complex safety specifications,”IEEE Control Systems Letters, 2023

work page 2023