Safe Quadrotor Navigation using Composite Control Barrier Functions

Kostas Alexis; Martin Jacquet; Marvin Harms

REVIEW 4 major objections 5 minor 30 references

Safe Quadrotor Navigation using Composite Control Barrier Functions

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A single composite control barrier function, built by soft-minimizing all distance constraints, can guarantee quadrotor collision avoidance everywhere except on a zero-volume set of singular states.

desk verdict Strong empirical work and a promising idea, but the central safety-filter equation is internally inconsistent with the defined CBF, so the formal guarantee is unsupported as written. read the letter →

arxiv 2502.04101 v1 pith:HYAWGMSQ submitted 2025-02-06 cs.RO

Marvin Harms , Martin Jacquet , Kostas Alexis This is my paper · ORCID

classification cs.RO

keywords compositecontrolbarrierfunctionquadrotorsafetyfiltercollisionavoidancerecursivefeasibilityzero-volumeinfeasiblesetLiDARnavigationquadraticprogramthird-ordernonlineardynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to make safety-filter collision avoidance for quadrotors scale to cluttered environments by compressing all position-obstacle constraints into one composite control barrier function $h_1$ acting on a third-order model of the vehicle. A quadratic-program safety filter then corrects a nominal controller so that the collision constraint and the thrust-rate constraint are jointly enforced. The central claim is that this stacked filter is recursively feasible except on a zero-volume set $X_{\mathrm{singular}}$ of singular configurations, making infeasibility negligible rather than a practical obstruction. If the claim holds, the result is a last-resort collision-avoidance layer that can run at high rate with thousands of constraints; the authors support it with indoor and outdoor flights against both naive and adversarial reference commands.

What carries the argument

The central object is the composite control barrier function $h_1$ in (15), a smooth soft-minimum of saturated third-order distance barriers $\nu_{i,2}$. It carries the argument by turning an arbitrary number of 'keep distance $\epsilon$ from point $x_i$' constraints into a single scalar constraint whose Lie-derivative regressor $L_g h_1$ has the interpretation of a virtual obstacle $\hat{x}(x) = \sum_i \lambda_i \cosh^{-2}(\nu_{i,2}(x)/\gamma) x_i$. The feasibility proof then reduces to analysing when this virtual obstacle is colinear with the thrust direction $Re_3$, which produces the null-set $X_{\mathrm{singular}}$.

What would settle it

Evaluate $L_f h_1 + \alpha_1 h_1$ and $L_g h_1$ over a grid of states near obstacles using the paper's parameters ($\kappa=20$, $\gamma=40$, $\alpha_1=1$, $p_0=-3$, $p_1=-2$) and search for a state where $L_g h_1 = 0$ while $L_f h_1 + \alpha_1 h_1 < 0$; any such state disproves CBF validity at the operating $\kappa$. In simulation with model (9), one can also drive the closed-loop filter toward the singular configuration $(x - \hat{x}) \times Re_3 = 0$ and check whether the QP (23) becomes infeasible.

Watch

Extended reading notes

Core claim

The paper proposes the safety filter in (23), which minimally corrects the geometric reference controller (22) by solving a quadratic program subject to the composite CBF constraint (16) and the thrust-rate CBF constraint (19). The composite function $h_1(x) = -\frac{\gamma}{\kappa} \log \sum_i e^{-\kappa \tanh(\nu_{i,2}(x)/\gamma)}$ aggregates all obstacle constraints, where $\nu_{i,2}$ is built from the squared distance to obstacle $i$ through the third-order chain (14). The proof that the infeasible set is negligible rests on the observation that conflicting constraints require the virtual obstacle $\hat{x}(x)$ -- a weighted average of obstacle positions -- to lie exactly on the thrust axis of the vehicle, and that this singular configuration has zero volume and can be left instantaneously by a suitable rotation command. The paper reports that evaluating $h_1$ and its derivatives for $10^4$ obstacles takes 3.4 ms on a laptop CPU, and the hardware experiments show the original distance constraints remaining satisfied over entire missions.

Load-bearing premise

The entire safety guarantee rests on assuming that the composite function $h_1$ in (15) qualifies as a control barrier function at the actual parameter value $\kappa=20$; the paper justifies this with a limit argument as $\kappa \to \infty$ and does not check the necessary and sufficient condition (8) at $\kappa=20$.

Editorial extensions

If this is right

The safety filter can run at 100 Hz on an onboard computer with hundreds of active obstacles, since evaluating the composite function and its derivatives for $10^4$ obstacles takes 3.4 ms on a laptop CPU.
Because the filter uses both attitude rates and thrust rate as inputs, it can avoid obstacles without a separate dynamic-inversion layer for the attitude dynamics.
The only states where joint feasibility can fail form a zero-volume set, so the filter is safe almost everywhere in the formal sense; at those states one constraint must be slackened.
Hardware flights, including references that actively steer into obstacles, show the original distance constraints $\nu_{i,0}$ remaining positive throughout the missions.
The approach works with an onboard LiDAR and a 20 cm voxel map, using the 400 closest occupied voxels as obstacle constraints.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

As an extension of the paper's own logic, the $\kappa \to \infty$ feasibility argument should be backed by a numerical check of condition (8) at the operating $\kappa=20$; without it, the formal invariance guarantee does not strictly cover the hardware results.
The zero-volume result suggests that, in continuous time with persistently exciting attitude commands, the probability of landing exactly in $X_{\mathrm{singular}}$ is zero; whether this survives sample-and-hold control at 100 Hz is a testable question the paper does not address.
The virtual-obstacle interpretation points toward a map-free variant in which a signed distance field is fed directly into the composite barrier, potentially reducing the constraint count while keeping the same filter structure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: the paper has a useful idea and real hardware results, but the central safety-filter equation is internally inconsistent with the CBF it claims to implement. As written, the QP (23) does not enforce invariance of the set defined by (15), so the "guaranteed everywhere except a null set" statement is unsupported. The fix looks straightforward, but it has to be done.

On the plus side, applying composite CBFs to a third-order thrust-augmented quadrotor model is new in this context, and the recursive-feasibility analysis (Proposition 1) is an interesting attempt to characterize the infeasible set. The scalability study (Table I) is clean and practically relevant, and the two hardware experiments—especially the adversarial forest run—are genuinely useful evidence that the method works in the real world. The paper also honestly reports h1 dipping below zero in flight, which is more than many CBF papers do.

The problems start with Eq. (16). Differentiating (15) gives Lf h1 = Σ_i w_i sech^2(ν_i,2/γ) Lf ν_i,2 with w_i the normalized softmin weight over tanh(ν_i,2/γ). The paper instead uses λ_i = e^{-κ(ν_i,2 - h1)} and omits the normalization, and the exponent is not tanh. On top of that, the bracket for Lf h1 reads L_f^3 ν_i,0 - p1 L_f^2 ν_i,0 + p1 p0 L_f ν_i,0, which drops the p0 L_f^2 term that appears when you compute L_f ν_i,2 from (14b). With p0=-3, p1=-2, that's a missing -3L_f^2 ν_i,0. There is also a scaling/sign issue in the τ component of Lg h1. These may be transcription errors, but they are load-bearing: the constraint implemented in the QP is not the CBF condition for (15). The paper also never verifies condition (8) for finite κ=20; Remark 1 only argues the κ→∞ limit. The QP (23) has no slack variable and no specified fallback on the null set, and Proposition 1's proof is a sketch, especially the claim about defining ∇r q across discontinuities of \hat{x}(x).

Who should read this? People working on CBF-based safety filters for multirotors will find the empirical results and scalability study worth their time. The formal analysis needs major repair, but the underlying approach is sensible and the fixes are identifiable.

Recommendation: send to peer review, but expect a major revision. The authors need to correct (16), verify the CBF condition for the finite κ they use, add a feasibility fallback, and tighten Proposition 1. As is, I wouldn't cite the guarantee, but I'd follow the experiments.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a safety filter for rate-controlled quadrotors based on a single composite control barrier function h1 formed by a softmin (with tanh saturation) over many distance-based obstacle constraints, together with a thrust constraint h2. The filter solves a QP (23) that minimally corrects a geometric reference controller. The authors analyze recursive feasibility of the two stacked constraints, claim the infeasible configurations form a zero-volume set, report scalability to 10^4 obstacles, and validate the approach in indoor and outdoor hardware experiments against naive and adversarial references.

Significance. If the central derivation were correct, the paper would make a useful engineering contribution: a single composite CBF that scales to thousands of constraints and a feasibility analysis for a third-order quadrotor model, with hardware demonstration. The scalability numbers (Table I) and the adversarial experiments are strong practical evidence. However, as written the key inequality (16) does not match the derivative of the proposed h1, and the finite-kappa CBF validity is not established; the formal claims are therefore not supported by the presented derivation.

major comments (4)

[Sec. IV-C, Eqs. (15)-(16)] Equation (16) is not the CBF condition for the h1 defined in Eq. (15). Differentiating (15) gives Lf h1 = sum_i [exp(-kappa tanh(nu_i,2/gamma)) / sum_j exp(-kappa tanh(nu_j,2/gamma))] sech^2(nu_i,2/gamma) Lf nu_i,2, and similarly for Lg h1, so the softmin weights are computed with the saturated values tanh(nu_i,2/gamma), not with nu_i,2, and the exponential weight is exp(-kappa(tanh(nu_i,2/gamma) - h1/gamma)), not e^{-kappa(nu_i,2 - h1)} as stated before Eq. (16). In addition, from Eq. (14b) the bracket in Lf h1 should be Lf^3 nu_0 - (p0+p1)Lf^2 nu_0 + p0 p1 Lf nu_0; the displayed expression Lf^3 nu_0 - p1 Lf^2 nu_0 + p1 p0 Lf nu_0 omits the -p0 Lf^2 nu_0 term. Since the proposed safety filter QP (23) uses Eq. (16), the implemented filter is not the one for which invariance is claimed. Please correct the transcription and supply a complete derivation, or state explicitly which h1 is actually used and prove the corresponding invariance condition.
[Sec. IV-C, after Eq. (16); Table II] For the finite value kappa=20 used in the experiments, the validity of h1 as a CBF is not established. Theorem 2 requires verifying condition (8) at all states where Lg h1 = 0. The paper only invokes the kappa-to-infinity argument in Remark 1, which concerns the limiting softmin of nu_i,2 and does not imply condition (8) for finite kappa. The text even states that 'a large value of kappa is generally chosen to avoid evaluation of (8)', which is not a proof. Since Theorem 2 is stated as an if-and-only-if condition, the invariance guarantee of the set {h1 >= 0} is unsupported for the actual parameters. Please verify condition (8) for the specific h1 in Eq. (15), either symbolically or numerically over the operating region, or provide a theorem that covers the finite-kappa case with the tanh saturation.
[Sec. IV-D, Proposition 1] The proof of Proposition 1 is a sketch rather than a rigorous zero-measure argument. It computes gradients of q = (x - xhat(x)) x R e3 with respect to r_i3 under kappa -> infinity and asserts that 'one can always choose Omega such that dq/dt != 0', but it does not show that the sublevel set has measure zero, and the phrase 'on either side in discontinuities of xhat(x)' is not a well-defined local argument. Please provide a complete proof, for example an explicit rank/submersion calculation or a measure-theoretic argument, and specify the exact definition of X_singular, including whether the case x = xhat(x) is included.
[Sec. IV-E, QP (23)] The QP (23) has no slack variables, and the paper does not specify a fallback controller for the set X_singular where the constraints conflict. The statement that 'one of the constraints (16) or (19) must be slackened' is not part of the proposed algorithm. As written, the safety filter has no defined behavior on X_singular, so the claim 'guaranteed everywhere except in the zero volume set X_singular' leaves the algorithm incomplete at exactly the states where the QP is infeasible. Please either add a explicit fallback strategy with a formal guarantee (for example, a bounded-violation slack formulation) or clearly state and justify the operational assumption that X_singular is never reached.

minor comments (5)

[Sec. V-C, Fig. 3] During experiment A the value of h1 crosses below zero; since a CBF safety filter is intended to enforce h1 >= 0 in the nominal model, please explain whether these crossings are due to the derivative mismatch in Eq. (16) or to disturbances and modeling errors, and quantify the maximum violation.
[Sec. IV-C, Eq. (17)] The term 'weighted average' for the virtual obstacle is imprecise because the weights appearing in Eq. (16) are not normalized; please specify the normalized weights or adjust the wording.
[Sec. IV-D, Proposition 1] The statement of Proposition 1 contains an incomplete clause ('…when.'); please complete the sentence and clarify the conditions under which X_singular is a null set.
[Sec. V-A, Table I] The reported runtimes cover only the composition of h1 and its Lie derivatives, not the solution of the QP (23); it would be helpful to report the full filter-loop time, since the claimed 3.4 ms is not the end-to-end computation.
[References] Reference [9] is missing author and title information; please complete the bibliographic entry.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the composite-CBF derivation rests on external theorems, the hardware self-citation is not load-bearing, and the unverified finite-kappa condition is a rigor gap rather than a circular step.

full rationale

The paper's derivation chain is not circular. The composite CBF construction (Eq. 15), the derivative identity (Eq. 7), and the validity condition (Theorem 2) are taken from the external reference [22] (Molnar and Ames), not from the authors' own prior work. The recursive-feasibility result (Proposition 1) is a new geometric sensitivity argument about the zero set of q = (x - xhat) x R e3; it does not assume the conclusion it is meant to prove, and it is explicitly derived for the 'idealized case (Remark 1)' from the same external theory. The main weakness flagged by a careful reading is that for the finite value kappa=20 used in experiments, the sufficient condition (8) is not verified; the text says 'we resort to the argument taken in Remark 1, i.e. that for one i, lim_{kappa -> infinity} h1(x) = gamma tanh(nu_{i,2}(x)/gamma), while nu_{i,2} is a CBF.' That is an appeal to a limiting argument to justify a finite-kappa design, and it is a correctness/rigor concern, not a circular reduction: the CBF candidate h1 is not defined in terms of the QP constraint (16), nor is any fitted parameter renamed as a prediction. The only self-citation is [16] for the physical quadrotor platform ('The experiments relied on a custom-built quadrotor from [16]'), which does not support any mathematical claim. The scalability numbers in Table I and the hardware experiments are external validation data rather than outputs forced by construction. The equation-level mismatch between the derivative of (15) and the regressor in (16) (e.g., the apparent dropped p0 Lf^2 nu_{i,0} term and the gamma/tanh exponent issue) would be a substantive bug if the implemented filter matches (16) literally, but a bug is not circularity: the central claim does not reduce to its inputs by definition. Score 0.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The central mathematical results rest on standard CBF theory from [17,19,22] and on the assumed validity of the finite-kappa composite barrier. The hand-tuned filter parameters (Table II) and the 400-voxel obstacle representation are practical choices that enter the experiments but are not fitted to data. No new physical entities are introduced; the virtual obstacle in (17) is a weighted average of real obstacle positions, not a new entity.

free parameters (9)

p0 = -3
Pole in the ECBF construction (14), chosen by hand; affects the decay rate of the barrier dynamics.
p1 = -2
Second pole in the ECBF construction (14), chosen by hand.
alpha1 = 1
Gain in the collision CBF condition (16), chosen by hand; larger values allow more aggressive approach.
gamma = 40
Saturation parameter in the tanh-based softmin composite (15), taken from [22], chosen by hand.
kappa = 20
Softmin sharpness in (15); the paper relies on large kappa to approximate the min CBF but uses a finite value without verifying the validity condition (8).
epsilon = 0.5
Safety radius around each obstacle point and the vehicle center in (13); chosen to account for vehicle size and voxel resolution, but conservativeness is not analyzed.
alpha2 = 5
Gain in the thrust CBF condition (19), chosen by hand.
epsilon_T = 7.5
Minimum thrust bound in (18); a physical lower limit on collective thrust, chosen by hand.
P (QP weight matrix) = not specified
The QP (23) objective weight matrix P is not given in the paper, leaving the filter's aggressiveness undefined.

assumptions (5)

standard math CBF theory: Theorem 1 (ECBF invariance) and Theorem 2 (softmin composite CBF condition) from [17,22] hold.
Used directly to construct the composite CBF h1; not re-proven in this paper.
domain assumption The quadrotor is exactly described by the rate-controlled third-order model (9), including thrust dynamics with no actuator delay or input magnitude bounds (except thrust lower bound).
The CBF design and feasibility analysis are computed on this model; real-world deviations are acknowledged as the cause of h1 crossing below zero.
domain assumption Obstacles can be represented as static points xi with a uniform margin epsilon, using the closest 400 occupied voxels at 20cm resolution, updated at 10Hz.
The collision constraints (13) treat each voxel as a point obstacle; conservativeness w.r.t. the actual vehicle footprint and voxel extent is not quantified.
ad hoc to paper The condition (8) of Theorem 2 holds for the finite kappa used in practice, based on the kappa-to-infinity argument in Remark 1.
The paper explicitly does not check (8) online and relies on the limiting argument; this is load-bearing for h1 being a valid CBF.
ad hoc to paper The QP (23) is feasible at all times encountered in operation, i.e. the state never enters X_singular (a zero-volume set) and the solver always returns a solution.
The paper shows X_singular is null but gives no fallback if the QP becomes infeasible on that set, so the implemented filter relies on never exactly hitting it.

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Cite this review

Pith. "Pith review of Safe Quadrotor Navigation using Composite Control Barrier Functions." pith.science (2026). https://pith.science/paper/HYAWGMSQ

@misc{pith2026250204101,
  author       = {Pith},
  title        = {Pith review of: Safe Quadrotor Navigation using Composite Control Barrier Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYAWGMSQ}},
  note         = {Machine review of arXiv:2502.04101}
}

read the original abstract

This paper introduces a safety filter to ensure collision avoidance for multirotor aerial robots. The proposed formalism leverages a single Composite Control Barrier Function from all position constraints acting on a third-order nonlinear representation of the robot's dynamics. We analyze the recursive feasibility of the safety filter under the composite constraint and demonstrate that the infeasible set is negligible. The proposed method allows computational scalability against thousands of constraints and, thus, complex scenes with numerous obstacles. We experimentally demonstrate its ability to guarantee the safety of a quadrotor with an onboard LiDAR, operating in both indoor and outdoor cluttered environments against both naive and adversarial nominal policies.

Figures

Figures reproduced from arXiv: 2502.04101 by the authors.

**Figure 1.** Left: Coordinate conventions utilized. All equations are expressed in North-East-Down (NED) and Front-Right-Down (FRD) frame. Right: Singular configuration example. A necessary requirement for conflicting constraints in (23) is that the virtual obstacle (orange) lies on the positive thrust axis (shown in red). Ω = −kR 1 2 [R t dR − R T Rd] ∧ + R T RdΩd, (10a) Td = −(Kxex + Kvev + mge3 − mx¨) T Re3, (10b) τ = kT (Td … view at source ↗

**Figure 1.** By studying the sensitivity of constraint (21) for the idealized case (Remark 1), we arrive at the following result: Proposition 1. The set of states Xsingular ⊂ X satisfying equation (21) is a zero-volume set in X for x ̸= ˆx(x) when. That is Xsingular is a null set. Proof. Assume we have a solution x˜ ∈ Xsingular, denoted by (x − xˆ(x˜)) × Re3 = q = 03×1 . For Xsingular to be a zero volume set, we require d dt q ̸… view at source ↗

**Figure 2.** Aggregated map and path of experiment A. An example of the obstacle map used by the safety filter is highlighted in green. The mission starts at the cyan circle on the right, receiving a constant velocity reference of 1m/s in the positive X direction, shown as a blue arrow. The norm of the intervention (cost of the QP (23)) is color-coded into the path, highlighting the areas where the safety filter becomes active. … view at source ↗

Figures from the paper (3 more)

**Figure 3.** Figure 3: Constraint values (top) and velocity error w.r.t. the reference (center) and filter-induced deviation (bottom) during experiment A. usafe = argmin u∈U (u − uref) T P(u − uref) (23a) s.t. Lgh1(x) Lgh2(x) u ≥ −Lf h1(x) − α1h1(x) b2(x) , (23b) where we simply stac…

**Figure 4.** Figure 4: Aggregated map and path of experiment B. The mission starts at the cyan circle on the right. The quadrotor receives an adversarial velocity reference that actively tries to collide with obstacles. The red arrows depict this reference velocity for some selected time ins…

**Figure 5.** Figure 5: Constraint values during experiment B. [ΩT , Test + τ∆t] T is sent to the autopilot, where ∆t is the time interval between control updates. The safety filter and reference controller run on the Orin NX CPU with an update rate of 100Hz with the parameters listed in Tab.…

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Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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