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arxiv: 2606.04790 · v1 · pith:AREMUJ6Cnew · submitted 2026-06-03 · 📡 eess.SY · cs.SY· math.OC

A model-free approach to control barrier functions for higher-order systems

Lukas Lanza , Johannes K\"ohler , Dario Dennst\"adt , Thomas Berger , Karl Worthmann This is my paper

Pith reviewed 2026-06-28 05:17 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords control barrier functionsfunnel controlmodel-free controlrelative degreesafety-critical systemsnonlinear controlrobotic manipulator
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The pith

Control barrier functions can be designed model-free for nonlinear systems of arbitrary relative degree by using funnel control to identify safe inputs.

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

The paper tries to extend model-free control barrier functions, previously limited to relative-degree-one systems via prescribed performance or funnel methods, to systems of any relative degree. A direct carry-over of the relative-degree-one results does not work, so the authors develop new funnel-control characterizations that isolate a usable subset of controls satisfying the barrier condition. This is done without access to the system dynamics or full state measurements. The result matters because many physical systems, such as robotic arms, have relative degree greater than one and accurate models are often unavailable. Applicability is shown on a seven-degree-of-freedom manipulator.

Core claim

We extend the model-free CBF design to nonlinear systems of arbitrary relative degree. We utilize novel techniques from funnel control to characterize a subset of the controls satisfying a CBF condition without requiring a dynamic model or state measurement.

What carries the argument

Novel funnel-control techniques that isolate a non-empty subset of admissible controls meeting the CBF condition for relative degree greater than one.

If this is right

  • Safe controls become available for relative-degree-two systems such as robotic manipulators without dynamics knowledge.
  • The method applies to general nonlinear systems of any relative degree.
  • Modular safety filters can be realized in settings where only partial state information is available.
  • The approach is demonstrated to function on a seven-degree-of-freedom robot arm.

Where Pith is reading between the lines

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

  • The same funnel-based subset characterization might be combined with other performance-oriented controllers to obtain joint safety and transient guarantees.
  • Extension to output-feedback or observer-based settings would be a natural next step for partially observed plants.
  • Numerical checks on the size of the admissible control set for increasing relative degree would quantify practical conservatism.

Load-bearing premise

The funnel-control techniques can be combined with the CBF condition to produce a non-empty set of admissible controls for relative degree greater than one without a dynamic model or full state measurement.

What would settle it

Finding a concrete system of relative degree two for which the derived funnel-based condition admits no control input would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.04790 by Dario Dennst\"adt, Johannes K\"ohler, Karl Worthmann, Lukas Lanza, Thomas Berger.

Figure 1
Figure 1. Figure 1: Evolution of the tracking error y − yref within the perfor￾mance boundary ψ. We consider the following conditions for the reference trajectory and the performance functions. Assumption 4. The reference trajectory yref belongs to the set Yref := Wr,∞(R≥0; R m). The performance function ψ is an element of the set Ψ :=  ψ ∈ W1,∞(R≥0; R) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Domains of the auxiliary variables β1, β2 from (12) for scalar ξ1, ξ2, yref ≡ 0, and ψ1 = ψ2 ≡ 1. The blank area −1 ≤ ξ1 ≤ 1 is a snapshot of D1 for a fixed time t, and the crosshatched area is a snapshot of D2, i.e., |ξ2 + ξ1 1−ξ 2 1 | < 1. We define the set D := {(t, ξ) ∈ Dr | kβr(t, ξ)k < 1} ⊂ Dr (13) and obtain the inclusion D ⊂ C for the safe set C from (5). Thus, ensuring that (t, y(t), y˙(t), . . . … view at source ↗
Figure 3
Figure 3. Figure 3: Initial and target configurations of the Kinova Gen3 ma￾nipulator used in simulations. the methods from [23] and [11] choose a desired velocity vd, e.g., based on the gradient of the constraint h, and then use a velocity controller to ensure an exponential decrease in the following Lyapunov-like function for the velocity error V (y, y, v ˙ d(t)) = ( ˙y − vd(t))⊤D(y)( ˙y − vd(t)). If the velocity controller… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

Control barrier functions (CBFs) are a widely applied modular tool to ensure safe operation of nonlinear dynamical control systems. However, for their construction accurate knowledge of the system dynamics is typically needed. This requirement was recently alleviated for relative-degree-one systems using techniques from prescribed performance control (PPC) or funnel control (FC). This article extends the model-free CBF design to nonlinear systems of arbitrary relative degree. Moreover, we show with a simple example that a straightforward extension of existing results for relative-degree-one systems fails. Instead, we utilize novel techniques from funnel control to characterize a subset of the controls satisfying a CBF condition without requiring a dynamic model or state measurement. Finally, we demonstrate the applicability of our results on a seven degrees of freedom robotic manipulator with relative degree two.

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 manuscript extends model-free CBF design from relative-degree-one systems to nonlinear systems of arbitrary relative degree. It first shows via a simple example that a direct extension of existing PPC/FC-based results fails, then uses novel funnel-control techniques to characterize a subset of admissible controls that satisfy a CBF condition from output measurements alone, without a dynamic model or full state. The approach is validated on a 7DOF robotic manipulator with relative degree two.

Significance. If the new characterization produces a non-empty admissible control set for relative degree >1 while remaining strictly model-free and output-based, the result would be a useful advance for safety-critical control of high-relative-degree systems (e.g., manipulators, vehicles) where accurate dynamics are unavailable. The explicit counter-example to the naive extension and the concrete robotic demonstration are concrete strengths.

major comments (2)
  1. [§4] §4 (characterization of admissible controls): the derivation that the funnel-control-based subset is non-empty for relative degree r>1 must be shown to avoid any implicit dependence on Lie derivatives of order greater than one or on unmeasured states; the current argument appears to rest on output-only measurements but the non-emptiness guarantee for the general class is not yet load-bearing.
  2. [§5] §5 (manipulator example): the specific performance functions, the exact form of the CBF inequality enforced, and the quantitative safety margins (e.g., minimum distance to the unsafe set) are not reported; without these it is impossible to verify that the model-free construction actually satisfies the higher-order CBF condition on the hardware.
minor comments (2)
  1. [§3] Notation for the funnel functions and the admissible-control set should be introduced with a single consistent symbol table to avoid confusion between the relative-degree-one and higher-order cases.
  2. [Abstract] The abstract states that the direct extension 'fails' but does not indicate the precise failure mode (empty set, loss of model-freeness, or violation of the CBF inequality); a one-sentence clarification would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive assessment of the manuscript's contributions. We address each major comment below and will incorporate clarifications and additional details in the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (characterization of admissible controls): the derivation that the funnel-control-based subset is non-empty for relative degree r>1 must be shown to avoid any implicit dependence on Lie derivatives of order greater than one or on unmeasured states; the current argument appears to rest on output-only measurements but the non-emptiness guarantee for the general class is not yet load-bearing.

    Authors: We appreciate this point. The derivation in §4 constructs the admissible control subset exclusively via the output-based funnel error transformation and the associated performance functions; no Lie derivatives of order greater than one or unmeasured states enter the argument. Non-emptiness follows directly from the existence of a funnel controller for relative-degree-r systems that keeps the transformed error within the prescribed bounds, thereby satisfying the CBF condition by construction. To address the concern that this guarantee is not yet sufficiently load-bearing, we will add an explicit remark and a short appendix paragraph restating the output-only nature of the construction and confirming the absence of higher-order derivatives. revision: partial

  2. Referee: [§5] §5 (manipulator example): the specific performance functions, the exact form of the CBF inequality enforced, and the quantitative safety margins (e.g., minimum distance to the unsafe set) are not reported; without these it is impossible to verify that the model-free construction actually satisfies the higher-order CBF condition on the hardware.

    Authors: We agree that these implementation details are required for full verification. In the revised manuscript we will report the exact performance functions employed for the 7DOF manipulator, the precise CBF inequality that is enforced at each step, and quantitative safety margins including the minimum observed distance to the unsafe set throughout the hardware experiments. revision: yes

Circularity Check

0 steps flagged

No circularity: model-free CBF extension for higher relative degree relies on external funnel-control techniques

full rationale

The paper's central step is to characterize a non-empty admissible control set for relative-degree >1 systems using novel funnel-control techniques, without model or full state. The abstract explicitly states that a direct extension of the rel-degree-1 case fails and therefore invokes external funnel-control methods to produce the characterization. No equation or claim reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the load-bearing novelty is presented as an independent combination of established PPC/FC results with the CBF inequality. The derivation is therefore self-contained against external benchmarks in the funnel-control literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the existence of a non-empty admissible control set derived from funnel-control performance bounds; the abstract does not introduce new free parameters, axioms, or invented entities beyond the standard CBF and funnel-control machinery.

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