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Avoiding Deadlocks Is Not Enough: Analysis and Resolution of Blocked Airplanes

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper shows that two airplanes enter parallel 'blocking' flight exactly when their cruising headings are mirror images inside a narrow angular window — a condition far less restrictive than deadlock — and gives a communication-free…

desk verdict A clean CBF-based analysis of transient parallel-flight blocking, with a clever decentralized resolution scheme; the formal results are for an idealized surrogate, so the transfer to real DAA systems remains the one big caveat. read the letter →

arxiv 2411.16911 v2 pith:245UVF7A submitted 2024-11-25 eess.SY cs.SY

Shuhao Qi , Zengjie Zhang , Zhiyong Sun , Sofie Haesaert This is my paper · ORCID
classification eess.SYcs.SY MSC 93A1493C85
keywords blockingmodetwo-airplaneencounterdetect-and-avoidsystemscontrolbarrierfunctionssafetyfilterdeadlockintention-awareresolutionadaptivepriority
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

Two aircraft, each guided by a collision-avoidance safety filter, can get stuck flying parallel to each other for a long stretch instead of passing — a finite-time failure the paper calls blocking mode, distinct from the endlessly-stuck deadlock studied in multi-robot work. The paper builds a tractable model (a cruising controller toward a target plus a control-barrier-function safety filter) and derives an exact condition for blocking: the two cruising headings must lie on mirrored sides of the line joining the aircraft, inside an angular window that widens as the aircraft draw closer. It then shows this condition is much easier to satisfy than deadlock's, quantifies how long a blocking episode lasts, and proposes a resolution that provably breaks the lock using only local information and no communication. If the characterization is right, blocking — not deadlock — is the encounter failure that aviation safety logic should be designed against.

What carries the argument

The load-bearing mechanism is the explicit solution of the CBF safety filter (Theorem 1): if a cruising heading is unsafe, meaning its normalized deviation from the bearing angle $\beta_j^i$ lies inside $(-\Delta, \Delta)$, the filter rotates the aircraft to the nearest boundary of the safe set, $\beta_j^i \pm \Delta$, and otherwise leaves the cruising heading untouched. The window half-width $\Delta \in [0, \pi/2]$ is $0$ in free flight and grows toward $\pi/2$ as the aircraft approach the safety radius, so the geometry of the encounter is fully captured by two angles and one window. Theorem 2 then reduces blocking to a mirror condition: both aircraft block exactly when one cruising angle sits in the positive part of the window and the other in the negative part, so the two filters pick symmetric boundaries and freeze the bearing. The companion mechanism is the self-unblocking convergence of Corollary 2, by which each cruising angle $\phi_i$ is driven toward the bearing angle $\beta_j^i$ during blocking, so an episode ends when one aircraft's target, the aircraft itself, and the other aircraft become collinear, and the duration bounds of Eq. (9) follow from that geometry.

What would settle it

Replay the symmetric encounter of Corollary 1 in a high-fidelity DAA simulation with lookup-table logic and realistic turn rates: if the aircraft fail to enter sustained parallel flight, or if blocking appears in configurations that violate the Theorem 2 mirror condition, the surrogate transfer fails. A cheaper in-model test is to add any nonzero heading-tracking lag and check whether blocking still occurs exactly when the mirror condition holds.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: both aircraft are in blocking mode if and only if, for one of the two mirror sides $s \in \{-1, 1\}$, the normalized cruising-angle deviations satisfy $s\measuredangle(\phi_i - \beta_j^i) \in [0, \Delta)$ and $-s\measuredangle(\phi_j - \beta_i^j) \in [0, \Delta)$, where $\Delta$ is the half-width of the unsafe heading window around the bearing line. Because each safety filter minimally corrects to the nearest safe heading (Theorem 1), mirrored cruising angles make both aircraft steer to symmetric boundaries, so their relative velocity is parallel to their relative position and the bearing freezes. The paper argues this is why blocking is a live risk: with uniformly distributed other-aircraft positions and headings the blocking condition has probability $1/8$ while deadlock has probability $0$, since deadlock requires exact head-on collinearity and blocking only requires mirrored deviations. It also proves the lock breaks itself once one cruising angle converges to the bearing angle, bounds the blocking duration via Eq. (9), and shows the proposed resolution also removes the deadlock cases that blocking can deteriorate into.

Load-bearing premise

The proofs treat each airplane as a single-integrator point that tracks heading commands instantly and a safety filter that always corrects to the nearest safe heading, and the paper assumes this surrogate faithfully emulates real detect-and-avoid logic — if actual DAA systems or finite turn dynamics break that nearest-boundary rule, the blocking conditions and duration bounds need not transfer.

Editorial extensions

If this is right

  • The same mirror trap should appear in any avoidance logic that picks the minimal safe deviation: the paper demonstrates parallel flight under velocity-obstacle and potential-field controllers as well, so blocking is a property of the controller class, not of the CBF filter chosen for analysis.
  • Blocking is far more probable than deadlock: in the paper's uniform geometric model the blocking probability is $1/8$ against $0$ for deadlock, so encounter-test metrics that only check for deadlock will miss the dominant failure.
  • Blocking duration is predictable from geometry before the lock fully develops, via the bounds $T_{lb} \le T \le T_{ub}$ in Eq. (9), which the paper uses to assess impact and to drive the priority decision.
  • A provably safe, communication-free resolution exists: temporarily retargeting the selected aircraft to the other's position instantly triggers the self-unblocking condition, and the adaptive priority rule chooses which aircraft to move based on independently computed duration estimates.
  • Resolving blocking also eliminates deadlock: the deadlock configurations that can develop out of blocking satisfy the blocking condition, so the resolution strategy covers both phenomena.

Reading between the lines

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

  • Editorial extension: the mirror condition predicts that blocking frequency rises as the unsafe window $\Delta$ widens, i.e., as encounters begin at closer separations or persist longer; this is a quantitative, testable prediction the paper does not extract, and it could be checked by sweeping initial separation in the same Monte Carlo setup.
  • Editorial extension: the nearest-safe-heading assumption is the one piece that links the elegant $1/8$-type probabilities to real DAA logic; a direct replay of the Theorem 2 configurations against an actual lookup-table DAA implementation would either confirm the surrogate or delimit where blocking analysis must be redone.
  • Editorial extension: the intention-revealing maneuver assumes the opponent, once its filter deactivates, honestly cruises toward its target; a deceptive or uncooperative opponent could feed a wrong triangulation, and a robustness analysis of the estimator would clarify how much trust the no-communication protocol requires.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the 'blocking mode' — prolonged parallel flight in two-airplane encounters — reported in [4]. It models each airplane as a single-integrator with constant speed controlled by a cruising law and a decentralized CBF safety filter, gives an explicit solution for the filter (Theorem 1), derives a necessary-and-sufficient condition for entry to blocking (Theorem 2), a self-unblocking condition (Corollary 2), duration bounds (Eq. 9), and compares blocking with deadlock by a uniform-geometry probability argument (Section V-B). It then proposes an intention-aware, communication-free resolution strategy with adaptive priority and validates it by simulations.

Significance. The paper identifies a practically important phenomenon that is not deadlock, and it does so with a tractable analytical model. The explicit solution in Theorem 1 and the characterization in Theorem 2 are genuine contributions, and the proposed resolution strategy is nontrivial and evaluated in Monte Carlo simulations. The main value, however, is conditional: the formal results are proven for an idealized CBF surrogate with instantaneous heading tracking, and the empirical claim that blocking is 'significantly less restrictive than deadlock' rests on a stylized probability model. If these gaps are closed, the paper would be a solid addition to the DAA and multi-robot safety literature.

major comments (4)
  1. [Section III-B / Theorem 2 / Eq. (9)] Section III-B states that the CBF-based safety filter 'emulates the functionality of DAA systems,' but the formal results that follow — the nearest-safe-heading solution in Eq. (8), the blocking conditions in Theorem 2, and the duration bounds in Eq. (9) — are all specific to the surrogate's minimal-deviation correction. Real ACAS-like DAA systems, including the NLR system in [4], use dynamic programming and lookup tables; the paper does not show that their advisory headings are the nearest-boundary corrections of Eq. (8), nor that the blocking set has the same geometry. Without this link, the central claims about airplane encounters are conditional on an untested equivalence; the simulations in Section VII use the same surrogate and therefore do not provide external validation.
  2. [Theorem 2 / Section V-A] The necessary-and-sufficient statement of Theorem 2 is not quite well-posed at the boundary. Eq. (8) leaves the output undefined when phi_i = beta^i_j (the '±' case), and Theorem 2 allows s*angle(...) in [0, Delta), which includes zero. The proof in Appendix C silently chooses one of the two boundary headings, but the actual choice is made later by the preferred direction lambda_i introduced in Section V-A. Thus the theorem should either state that it holds for almost all initial conditions, or it should incorporate lambda_i into its statement and proof, especially because the deadlock analysis in Section V-A depends on lambda_i.
  3. [Section V-B] The comparison that 'the probability of a blocking event is 1/8, while the probability of deadlock is 0' is asserted without a derivation and is an artifact of the chosen sample space. If the other airplane's position is distributed uniformly along a circle (a one-dimensional set), arcs have positive probability and a single point has probability zero, so the comparison essentially restates that deadlock requires a measure-zero coincidence on that circle. Under a two-dimensional position distribution, both events would have probability zero. The authors should give an explicit sample space, a precise event definition, and a calculation of 1/8; otherwise the headline claim that blocking is 'significantly less restrictive' is not quantitatively supported.
  4. [Section VII] Section VII validates the analytical model only against simulations of the same single-integrator/CBF model that was analyzed. The motivating NLR encounter [4] is not replayed, and no comparison is made with the actual ACAS-Xu lookup-table behavior or with the velocity-obstacle and potential-field experiments mentioned in Appendix A. As a result, the paper's external-validity claims rest on a single assertion in Section III-B. A replay of the reported blocking encounter, or at least a parameter study that varies the tracking gain and the safety-filter geometry, would make the central claims testable.
minor comments (5)
  1. [Section II, Eq. (2)] The second row of the unicycle model should be v*sin(theta_i), not v*cos(theta_i).
  2. [Sections V-A and VI-B] There are typographical errors: 'Theoreom' should be 'Theorem,' and 'Sloving' in Algorithm 2 should be 'Solving.'
  3. [Section III-C, Definition 3] Definition 3 defines blocking by a constant bearing angle, which includes the pre-parallel closing phase during which the aircraft are still approaching each other; the text should state explicitly that this phase is part of the modeled blocking mode, since the motivating description in Fig. 1 emphasizes prolonged parallel flight.
  4. [Section V-B, Fig. 7] The colored curves and colored arcs in Fig. 7 are not tied to any equation in the text; adding the corresponding geometric expressions would make the probability calculation reproducible.
  5. [Appendix C] The equality beta^1_2 - beta^2_1 = pi holds only modulo 2*pi; writing it without the angular-normalization qualifier invites confusion in the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No derivation-level circularity: The blocking conditions, duration bounds, and resolution strategy are derived from the stated CBF safety-filter model, without fitted inputs or load-bearing self-citation.

full rationale

The paper's central characterization (Theorem 2) is a derived iff statement connecting the observable blocking definition (Definition 3: safety filter active and constant bearing) to an angular condition on the two cruising angles; the proof in Appendix C uses the explicit filter solution (Eq. 8) and the bearing derivative formula, so the condition is not a restatement of the definition by construction. Corollary 1 follows from Theorem 2, and Corollary 2 with the duration bounds (Eq. 9) is obtained from the geometry of the model in Appendix F; no parameter is fitted to a target output and then renamed a prediction. The only author-overlap citation is [33], used for the standard unicycle model, which does not supply any premise for the blocking theorems. Section V-B's probability comparison (1/8 vs. 0) is a measure computation under an explicit uniform-geometry assumption, not a fit. Section VII validates the resolution strategy on simulations of the same surrogate model; this is an external-validity support gap (the surrogate-to-DAA equivalence in Section III-B is asserted rather than established), but it does not make any derivation step circular. Consequently, no circular step is identified.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to empirical data in the derivation. The analysis rests on modeling assumptions: constant-speed horizontal single-integrator dynamics, instantaneous heading tracking, a CBF safety filter as a surrogate for DAA, and a preferred-turn direction introduced to resolve the safety-filter ambiguity. The one hand-chosen quantity listed is the CBF gain alpha, set to 3 in simulations.

free parameters (1)
  • CBF gain alpha = 3 in simulations; otherwise symbolic
    Controls the size of the unsafe heading interval Delta; chosen by hand in the experiments, not calibrated to encounter data. The blocking conditions are stated for any alpha, but the claimed frequency of blocking depends on the resulting interval.
assumptions (5)
  • domain assumption Heading tracking is high-gain so theta_i converges to theta*_i almost instantaneously, reducing the unicycle model to a single integrator.
    Invoked in Section III before the explicit solution and all blocking theorems; if the heading loop is not fast, the distance and angular conditions in Theorem 2 need not hold.
  • domain assumption The CBF-based safety filter with half-responsibility separation (Eq. 7) is an adequate surrogate for real detect-and-avoid systems.
    Used to make the system analytically tractable in Section III-B; the appendices show similar blocking for velocity obstacles and potential fields, but the theorems and duration formulas are proven only for this filter.
  • domain assumption Airplanes fly in a horizontal plane at the same constant speed v and only heading is controlled.
    Problem setting in Section II; excludes speed, vertical rate, and asymmetric-speed encounters where the blocking geometry could differ.
  • ad hoc to paper A preferred direction lambda_i in {-1,1} disambiguates the safety filter output when the cruising angle equals the bearing angle; deadlock requires lambda_1 = -lambda_2.
    Introduced in Section V-A solely to complete the piecewise heading rule and to create deadlock and livelock scenarios; it is not derived from the dynamics.
  • domain assumption Targets can be estimated by triangulating two non-collinear cruising poses observed during an interactive maneuver.
    Underpins Algorithm 2 in Section VI-B; requires the other airplane to reveal cruising heading at least twice, which the paper asserts but does not formally prove.

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Cite this review

Pith. "Pith review of Avoiding Deadlocks Is Not Enough: Analysis and Resolution of Blocked Airplanes." pith.science (2026). https://pith.science/paper/245UVF7A

@misc{pith2026241116911,
  author       = {Pith},
  title        = {Pith review of: Avoiding Deadlocks Is Not Enough: Analysis and Resolution of Blocked Airplanes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/245UVF7A}},
  note         = {Machine review of arXiv:2411.16911}
}
read the original abstract

This paper is devoted to the analysis and resolution of a pathological phenomenon in airplane encounters called blocking mode. As autonomy in airplane systems increases, a pathological phenomenon can be observed in two-aircraft encounter scenarios, where airplanes stick together and fly in parallel for an extended period. This parallel flight results in a temporary blocking that significantly delays progress. In contrast to widely studied deadlocks in multi-robot systems, such transient blocking is often overlooked in existing literature. Since such prolonged parallel flying places high-speed airplanes at elevated risks of near-miss collisions, encounter conflicts must be resolved as quickly as possible in the context of aviation. We develop a mathematical model for a two-airplane encounter system that replicates this blocking phenomenon. Using this model, we analyze the conditions under which blocking occurs, quantify the duration of the blocking period, and demonstrate that the blocking condition is significantly less restrictive than that of deadlock. Based on these analytical insights, we propose an intention-aware strategy with an adaptive priority mechanism that enables efficient resolution of ongoing blocking phenomena while also incidentally eliminating deadlocks. Notably, the developed strategy does not rely on central coordination and communications that can be unreliable in harsh situations. The analytical findings and the proposed resolution strategy are validated through extensive simulations.

Figures

Figures reproduced from arXiv: 2411.16911 by the authors.

Figure 1
Figure 1. Blocking phenomenon in a two-airplane system. Two [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Encounter scenario of a two-airplane system. The solid [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Three modes of an airplane in an encounter. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (8 more)
Figure 3
Figure 3. Figure 3: The process of an encounter with a blocking mode. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 5
Figure 5. Figure 5: Sketch of explicit solution for the safety filter. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: Visualization of blocking and deadlock conditions. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 6
Figure 6. Figure 6: Deadlock and livelock phenomenon in simulation. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 9
Figure 9. Figure 9: Encounter simulations involving four airplanes. Circle [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: Geometric interpretation of the safety filter (7). [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Duration approximation for three options. [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Geometric sketch of the blocking duration. Lines [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

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Forward citations

Cited by 1 Pith paper

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    On triangulation algorithms in large scale camera network systems,

    A. Masiero and A. Cenedese, “On triangulation algorithms in large scale camera network systems,” in 2012 American Control Conference (ACC), pp. 4096–4101, 2012. APPENDIX A. Blocking induced by different controllers Aside from the CBF-based safety filter, two widely used collis...

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    Like the safety filter, velocity command is normally selected to minimize deviation from the preferred velocity [21] outside the velocity obstacle, min 1 2 ∥ui − ˜ ui∥2 s.t

    Velocity obstacle for Ai is the set of all velocities for Ai that will result in a collision with Aj over a time horizon τ, assuming Aj maintains its current velocity: VOi|j = {ui | ∃t ∈ [0, τ], ∥pj + ujt − (pi + uit)∥ ≤r} , where i, j= {1, 2} and i ̸= j. Like the safety filte...

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    The target exerts an attractive force, Fatt, that pulls the robot toward the target, while the other agent exerts a repulsive force, Frep, that pushes the agent away

    Potential Field treats each agent as a particle moving in an artificial potential field. The target exerts an attractive force, Fatt, that pulls the robot toward the target, while the other agent exerts a repulsive force, Frep, that pushes the agent away. Both forces depend on...

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    To make ˙βj i = 0, θ2 = ∡(β2 − s∆) such that −s∡(ϕ2 − β1 2 ) ∈ [0, ∆)

    Suppose A2 is in blocking mode such that ∥∡(ϕ2 − β1 2 )∥ ∈[0, ∆). To make ˙βj i = 0, θ2 = ∡(β2 − s∆) such that −s∡(ϕ2 − β1 2 ) ∈ [0, ∆)

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    To make ˙βj i = 0,θ2 = ∡(β2 1 + s∆) = θ1 such that ϕ2 = ∡(β2 1 + s∆)

    Suppose A2 is not in blocking mode such that ∥∡(ϕ2 − β1 2 )∥ /∈ [0, ∆). To make ˙βj i = 0,θ2 = ∡(β2 1 + s∆) = θ1 such that ϕ2 = ∡(β2 1 + s∆). 11 (a) Maintain blocking (b) Unblocking by Ai (c) Unblocking by Aj Fig. 11: Duration approximation for three options. Therefore, the ne...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.