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arxiv: 2605.20625 · v1 · pith:ZE4UJCHRnew · submitted 2026-05-20 · 📡 eess.SY · cs.MA· cs.RO· cs.SY

Time-To-Reach Separation and Safety Filtering for Safe, Fair, and Efficient Multi-Agent Coordination

Matthew Low , Jasmine Jerry Aloor , Victoria Marie Tuck , Pierluigi Nuzzo , Jason J. Choi This is my paper

Pith reviewed 2026-05-21 04:18 UTC · model grok-4.3

classification 📡 eess.SY cs.MAcs.ROcs.SY
keywords multi-agent coordinationtime-to-reachsafety filteringAdvanced Air MobilityHamilton-Jacobi reachabilityair corridor mergingcollision avoidancefairness
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The pith

A framework using minimum time-to-reach metrics coordinates aerial vehicles for safe, fair, and efficient corridor merging.

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

The paper develops a coordination method for aerial vehicles merging into air corridors by using minimum time-to-reach as a key metric to assign priorities and set temporal spacings. This temporal approach induces the necessary spatial separations between vehicles. A safety filter based on reachability analysis then ensures no collisions occur while making only small changes to the planned paths and preserving the assigned priorities. In simulations of busy merging scenarios, this leads to better safety, fairness, and efficiency than using pure time-optimal paths or safety filters that ignore priorities.

Core claim

Minimum time-to-reach serves as a unifying metric that enables arrival-consistent priority assignment, target TTR values for temporal separation inducing spatial separation, and a priority-consistent safety filtering layer using Hamilton-Jacobi reachability value functions to guarantee collision avoidance with minimal modification to reference guidance.

What carries the argument

Minimum time-to-reach (TTR) as a unifying metric for priority assignment and temporal separation, together with priority-consistent safety filtering via Hamilton-Jacobi reachability value functions.

Load-bearing premise

Target TTR values can be chosen so that temporal spacing reliably produces the needed spatial separation, and the safety filter modifies guidance minimally without creating new conflicts or violating fairness.

What would settle it

A simulation or flight test where the achieved spatial separations drop below safe distances or where the safety filter alters priorities or introduces fairness issues despite using the proposed TTR targets.

Figures

Figures reproduced from arXiv: 2605.20625 by Jasmine Jerry Aloor, Jason J. Choi, Matthew Low, Pierluigi Nuzzo, Victoria Marie Tuck.

Figure 1
Figure 1. Figure 1: Computed minimum reach-avoid time-to-reach (TTR) in colormap (red-blue), for vehicles navigating to the air corridor (blue), and the simulated [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Five vehicles initialized at states with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Trajectories for each combination of reference guidance and safety filters on a representative scenario with eight vehicles. Red indicates safety [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

Advanced Air Mobility (AAM) operations are expected to significantly increase aerial traffic in urban airspace, requiring autonomous traffic management systems to ensure collision-free operations in highly congested environments. In this paper, we propose a multi-agent coordination framework that uses minimum time-to-reach (TTR) as a unifying metric for priority assignment, temporal separation, and safety filtering. We focus on the problem of coordinating multiple aerial vehicles merging into an air corridor while maintaining safe separation between vehicles. Vehicles are assigned arrival-consistent priority based on TTR, and target TTR values are used to enforce temporal spacing that induces spatial separation. A priority-consistent safety filtering layer based on Hamilton-Jacobi reachability value functions ensures collision avoidance while minimally modifying the reference guidance. Simulation results in a highly congested corridor merging scenario show that the proposed method improves safety, fairness, and efficiency compared to time-optimal guidance and priority-agnostic safety filtering.

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 proposes a multi-agent coordination framework for Advanced Air Mobility (AAM) corridor merging that uses minimum time-to-reach (TTR) as a unifying metric for arrival-consistent priority assignment, selection of target TTR values to enforce temporal spacing (claimed to induce spatial separation), and a priority-consistent Hamilton-Jacobi reachability safety filter that minimally modifies reference guidance. Simulation results in a highly congested merging scenario are presented as demonstrating improvements in safety, fairness, and efficiency relative to time-optimal guidance and priority-agnostic safety filtering.

Significance. If the central claims hold, the work provides a coherent integration of priority, separation, and formal safety filtering via an established reachability tool, which could support scalable AAM traffic management. The reliance on HJ reachability for the safety layer supplies a formal foundation that is a clear strength relative to purely heuristic approaches.

major comments (2)
  1. [Simulation Results] Simulation results section: the headline improvements in safety, fairness, and efficiency are reported without error bars, statistical significance tests, or sensitivity analysis to the specific target TTR values chosen. This is load-bearing for the central claim because the reader's weakest assumption (that TTR targets reliably produce the required separation) cannot be evaluated without such details, and post-hoc scenario selection cannot be ruled out.
  2. [Proposed Framework] Method description (TTR-based separation): the assertion that target TTR values enforce temporal spacing which induces the required spatial separation lacks an analytic bound or sensitivity analysis under velocity heterogeneity. In corridor merges, spatial clearance depends on both time headway and instantaneous speed/heading; a fixed TTR delta does not guarantee minimum distance, and no evidence is given that the priority-consistent HJ filter remains inactive or non-disruptive once TTR targets are set. This assumption directly supports the reported simulation gains.
minor comments (2)
  1. [Abstract] The abstract would benefit from explicitly stating the number of agents simulated and the quantitative metrics (e.g., minimum separation distance, throughput) used to quantify fairness and efficiency.
  2. [Introduction] Notation for TTR and the distinction between minimum TTR (for priority) and target TTR (for separation) could be introduced earlier and used consistently to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments highlight important aspects for strengthening the presentation of our simulation results and the justification of the TTR-based separation mechanism. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Simulation Results] Simulation results section: the headline improvements in safety, fairness, and efficiency are reported without error bars, statistical significance tests, or sensitivity analysis to the specific target TTR values chosen. This is load-bearing for the central claim because the reader's weakest assumption (that TTR targets reliably produce the required separation) cannot be evaluated without such details, and post-hoc scenario selection cannot be ruled out.

    Authors: We agree that the current simulation presentation would be strengthened by additional statistical rigor and sensitivity analysis. In the revised manuscript we will report results aggregated over multiple Monte Carlo trials with randomized initial conditions and vehicle parameters, include error bars and statistical significance tests for the observed improvements in safety, fairness, and efficiency metrics, and add a sensitivity study varying the target TTR values around the nominal choices. These changes will directly address concerns about post-hoc selection and provide clearer evidence that the reported gains are robust. revision: yes

  2. Referee: [Proposed Framework] Method description (TTR-based separation): the assertion that target TTR values enforce temporal spacing which induces the required spatial separation lacks an analytic bound or sensitivity analysis under velocity heterogeneity. In corridor merges, spatial clearance depends on both time headway and instantaneous speed/heading; a fixed TTR delta does not guarantee minimum distance, and no evidence is given that the priority-consistent HJ filter remains inactive or non-disruptive once TTR targets are set. This assumption directly supports the reported simulation gains.

    Authors: We acknowledge that a purely analytic guarantee of minimum spatial separation from a fixed TTR delta is not provided under arbitrary velocity heterogeneity, and that explicit evidence on filter activity would be helpful. The priority-consistent HJ safety filter is designed to enforce collision avoidance regardless of the reference guidance, so the overall safety claim does not rest solely on the TTR targets. Nevertheless, to strengthen the exposition we will derive a conservative bound on induced spatial separation under bounded speed and heading variations representative of AAM vehicles, and we will add plots and statistics quantifying the frequency and magnitude of safety-filter interventions when TTR targets are applied. These additions will clarify the interplay between the nominal TTR guidance and the safety layer. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation relies on established external methods

full rationale

The paper proposes a coordination framework using minimum TTR for priority assignment and temporal spacing, combined with a priority-consistent safety filter based on Hamilton-Jacobi reachability value functions. No load-bearing derivation step reduces the claimed improvements in safety, fairness, or efficiency to quantities defined by the paper's own fitted parameters, self-citations, or definitional equivalences. The approach builds on standard reachability concepts and presents simulation results as empirical evidence rather than a closed tautology. The target TTR selection and filter behavior are design choices validated externally in the corridor scenario, with no quoted equations showing constructional equivalence to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on domain assumptions about TTR computability and the ability of temporal targets to induce spatial separation; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Vehicles can be assigned arrival-consistent priority based on minimum time-to-reach
    Invoked for priority assignment and used to define target TTR values for spacing.
  • domain assumption Hamilton-Jacobi reachability value functions can provide a priority-consistent safety filter that minimally modifies reference guidance
    Central to the safety filtering layer described in the abstract.

pith-pipeline@v0.9.0 · 5709 in / 1322 out tokens · 33846 ms · 2026-05-21T04:18:42.531527+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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    Relation between the paper passage and the cited Recognition theorem.

    Vehicles are assigned arrival-consistent priority based on TTR, and target TTR values are used to enforce temporal spacing that induces spatial separation. A priority-consistent safety filtering layer based on Hamilton-Jacobi reachability value functions ensures collision avoidance while minimally modifying the reference guidance.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    (2025, 10) Compendium to FAA aerospace forecast FY 2025-2045 emerging aviation entrants: Unmanned aircraft systems and Advanced air mobility

    FAA Aerospace Forecasts. (2025, 10) Compendium to FAA aerospace forecast FY 2025-2045 emerging aviation entrants: Unmanned aircraft systems and Advanced air mobility

    work page 2025

  2. [2]

    Air Traffic Controller Hiring

    work page

  3. [3]

    Design and Evaluation of a Corridors-in-the-Sky Concept: The Benefits and Feasibility of Adding Highly Structured Routes to a Mixed Equipage Environment,

    J. Homola, P. U. Lee, H. Lee, and C. Brasil, “Design and Evaluation of a Corridors-in-the-Sky Concept: The Benefits and Feasibility of Adding Highly Structured Routes to a Mixed Equipage Environment,” inAIAA GNC Conference, 2012

    work page 2012

  4. [4]

    The Advanced Air Mobility National Strategy: A Bold Policy Vision for 2025–2035,

    US Department of Transportation Advanced Air Mobility Interagency Working Group, “The Advanced Air Mobility National Strategy: A Bold Policy Vision for 2025–2035,” Tech. Rep., 2025

    work page 2025

  5. [5]

    SESAR Joint Undertaking|U-space CONOPS 4th Edition,

    EUROCONTROL, “SESAR Joint Undertaking|U-space CONOPS 4th Edition,” European Union, Tech. Rep., 2023

    work page 2023

  6. [6]

    A quasi-dynamic air traffic assignment model for mitigating air traffic complexity and congestion for high-density UAM operations,

    Z. Wang, D. Delahaye, J.-L. Farges, and S. Alam, “A quasi-dynamic air traffic assignment model for mitigating air traffic complexity and congestion for high-density UAM operations,”Transportation Research Part C: Emerging Technologies, 2023

    work page 2023

  7. [7]

    Decentralized Coordination of Autonomous Traffic Through Advanced Air Mobility Corridors,

    J. J. Aloor and H. Balakrishnan, “Decentralized Coordination of Autonomous Traffic Through Advanced Air Mobility Corridors,” in AIAA SCITECH 2026 Forum, 2026

    work page 2026

  8. [8]

    Traf- fic management protocols for advanced air mobility,

    C. Chin, V . Qin, K. Gopalakrishnan, and H. Balakrishnan, “Traf- fic management protocols for advanced air mobility,”Frontiers in Aerospace Engineering, May 2023

    work page 2023

  9. [9]

    Introduction to TCAS II, Version 7.1,

    FAA, “Introduction to TCAS II, Version 7.1,” 2011

    work page 2011

  10. [10]

    Next-generation airborne collision avoidance system,

    M. J. Kochenderfer, J. E. Holland, and J. P. Chryssanthacopoulos, “Next-generation airborne collision avoidance system,” 2012

    work page 2012

  11. [11]

    Tactical conflict resolution in urban airspace for unmanned aerial vehicles operations using attention-based deep reinforcement learning,

    M. Zhang, C. Yan, W. Dai, X. Xiang, and K. H. Low, “Tactical conflict resolution in urban airspace for unmanned aerial vehicles operations using attention-based deep reinforcement learning,”Green Energy and Intelligent Transportation, 2023

    work page 2023

  12. [12]

    Resolving Conflicting Constraints in Multi-Agent Reinforcement Learning with Layered Safety,

    J. J. Choi, J. J. Aloor, J. Li, M. G. Mendoza, H. Balakrishnan, and C. Tomlin, “Resolving Conflicting Constraints in Multi-Agent Reinforcement Learning with Layered Safety,” inRobotics: Science and Systems (RSS), 2025

    work page 2025

  13. [13]

    I. M. Mitchell,Application of level set methods to control and reachability problems in continuous and hybrid systems. Stanford University, 2002

    work page 2002

  14. [14]

    Efficiency and fairness in unmanned air traffic flow manage- ment,

    C. Chin, K. Gopalakrishnan, M. Egorov, A. Evans, and H. Balakr- ishnan, “Efficiency and fairness in unmanned air traffic flow manage- ment,”IEEE Trans. Intell. Transp. Syst., 2021

    work page 2021

  15. [15]

    Hamilton-Jacobi multi-time reachability,

    M. Doshi, M. Bhabra, M. Wiggert, C. J. Tomlin, and P. F. Lermusiaux, “Hamilton-Jacobi multi-time reachability,” inIEEE CDC, 2022

    work page 2022

  16. [16]

    TTR-Based Reward for Reinforcement Learn- ing with Implicit Model Priors,

    X. Lyu and M. Chen, “TTR-Based Reward for Reinforcement Learn- ing with Implicit Model Priors,” inIEEE/RSJ IROS, 2020

    work page 2020

  17. [17]

    Decentralized Control Synthesis for Air Traffic Management in Urban Air Mobility,

    S. Bharadwaj, S. Carr, N. Neogi, and U. Topcu, “Decentralized Control Synthesis for Air Traffic Management in Urban Air Mobility,”IEEE Trans. Control Netw. Syst., 2021

    work page 2021

  18. [18]

    HMARL-CBF–Hierarchical Multi-Agent Rein- forcement Learning with Control Barrier Functions for Safety-Critical Autonomous Systems,

    H. S. Ahmadet al., “HMARL-CBF–Hierarchical Multi-Agent Rein- forcement Learning with Control Barrier Functions for Safety-Critical Autonomous Systems,” inNeurIPS, 2025

    work page 2025

  19. [19]

    Combinatorial Safety-Critical Coordination of Multi-Agent Systems via Mixed- Integer Responsibility Allocation and Control Barrier Functions,

    J. Autenrieb, M. Spiller, H.-S. Shin, and N. Cho, “Combinatorial Safety-Critical Coordination of Multi-Agent Systems via Mixed- Integer Responsibility Allocation and Control Barrier Functions,” arXiv preprint arXiv:2603.05762, 2026

    work page arXiv 2026

  20. [20]

    Sastry,Nonlinear systems: analysis, stability, and control

    S. Sastry,Nonlinear systems: analysis, stability, and control. Springer, 2013

    work page 2013

  21. [21]

    Data-Driven Safety Filters: Hamilton-Jacobi Reachability, Control Barrier Functions, and Predictive Methods for Uncertain Systems,

    K. P. Wabersichet al., “Data-Driven Safety Filters: Hamilton-Jacobi Reachability, Control Barrier Functions, and Predictive Methods for Uncertain Systems,”IEEE CSM, 2023

    work page 2023

  22. [22]

    Aubin, A

    J. Aubin, A. M. Bayen, and P. Saint-Pierre,Viability theory: new directions. Springer Science & Business Media, 2011

    work page 2011

  23. [23]

    Bardi and I

    M. Bardi and I. C. Dolcetta,Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Springer, 1997

    work page 1997

  24. [24]

    Reach-avoid problems with time-varying dynamics, targets and constraints,

    J. F. Fisac, M. Chen, C. J. Tomlin, and S. S. Sastry, “Reach-avoid problems with time-varying dynamics, targets and constraints,” inInt. Conf. on Hybrid Sys.: Computation and Control (HSCC), 2015

    work page 2015

  25. [25]

    Deepreach: A deep learning approach to high-dimensional reachability,

    S. Bansal and C. J. Tomlin, “Deepreach: A deep learning approach to high-dimensional reachability,” inIEEE ICRA, 2021

    work page 2021

  26. [26]

    Control barrier function based quadratic programs for safety critical systems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,”IEEE TAC, 2016

    work page 2016

  27. [27]

    Robust control barrier–value functions for safety-critical control,

    J. J. Choi, D. Lee, K. Sreenath, C. J. Tomlin, and S. L. Herbert, “Robust control barrier–value functions for safety-critical control,” in IEEE CDC, 2021

    work page 2021

  28. [28]

    Discrete GCBF Proximal Policy Optimization for Multi-agent Safe Optimal Control,

    S. Zhang, O. So, M. Black, and C. Fan, “Discrete GCBF Proximal Policy Optimization for Multi-agent Safe Optimal Control,” inICLR, 2025

    work page 2025

  29. [29]

    Safety with agency: Human-centered safety filter with application to ai-assisted motorsports,

    D. D. Ohet al., “Safety with agency: Human-centered safety filter with application to ai-assisted motorsports,” inRSS, 2025

    work page 2025

  30. [30]

    A forward reachability perspective on robust control invariance and discount factors in reachability analysis,

    J. J. Choiet al., “A forward reachability perspective on robust control invariance and discount factors in reachability analysis,”arXiv preprint arXiv:2310.17180, 2023

    work page arXiv 2023

  31. [31]

    One- shot computation of reachable sets for differential games,

    I. Yang, S. Becker-Weimann, M. J. Bissell, and C. J. Tomlin, “One- shot computation of reachable sets for differential games,” inHSCC, 2013

    work page 2013

  32. [32]

    helperOC: Optimal Control Toolbox for Hamilton-Jacobi Reachability

    S. Bansal, M. Chen, S. Herbert, and C. J. Tomlin, “helperOC: Optimal Control Toolbox for Hamilton-Jacobi Reachability.”

    work page

  33. [33]

    A New Measure of Rank Correlation,

    M. Kendall, “A New Measure of Rank Correlation,”Biometrika, 1938

    work page 1938

  34. [34]

    Joby Aviation - Official Website,

    Joby Aviation, “Joby Aviation - Official Website,” 2026

    work page 2026

  35. [35]

    Midnight eVTOL - Aircraft Performance Specs,

    Archer Aviation, “Midnight eVTOL - Aircraft Performance Specs,” 2026

    work page 2026

  36. [36]

    Wisk Aero - Our Aircraft,

    Wisk Aero, “Wisk Aero - Our Aircraft,” 2026

    work page 2026

  37. [37]

    Aeronautical Information Manual (AIM), Chapter 7: Safety of Flight, Section 7: Near Midair Collision Reporting

    FAA, “Aeronautical Information Manual (AIM), Chapter 7: Safety of Flight, Section 7: Near Midair Collision Reporting.”

    work page

  38. [38]

    Safety Barrier Certificates for Collisions-Free Multirobot Systems,

    L. Wang, A. D. Ames, and M. Egerstedt, “Safety Barrier Certificates for Collisions-Free Multirobot Systems,”IEEE T-Ro, 2017

    work page 2017

  39. [39]

    OSQP: An Operator Splitting Solver for Quadratic Programs,

    B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP: An Operator Splitting Solver for Quadratic Programs,”Math. Prog. Comp., 2020. APPENDIX A. Parameters Used in Simulation Study TABLE IV PARAMETERS OFAIRTAXIDYNAMICS& AIRCORRIDOR Parameter Air taxi (Sim) Groundspeed Minimum speedv min 60 knot (30 m/s) Maximum speedv max 175 knot (90 m/s) N...

    work page 2020