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arxiv: 2604.17776 · v1 · submitted 2026-04-20 · 📡 eess.SY · cs.SY· math.OC· math.PR

Trajectory-Based Optimization for Air Traffic Control in the Terminal Maneuvering Area

Yutian Pang , Daniel Delahaye , John-Paul Clarke This is my paper

Pith reviewed 2026-05-10 04:45 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OCmath.PR
keywords trajectory optimizationterminal maneuvering areaarrival sequencingspeed profilespath stretchlanding order policieswake turbulence separationair traffic control
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The pith

Trajectory optimization computes implementable speed profiles and path extensions for terminal arrivals that achieve required separations.

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

The paper develops a framework that directly optimizes aircraft trajectories in the terminal maneuvering area instead of reducing them to time-delay variables at nodes. It models each arrival path as a tangent leg, radius-to-fix turn, and final approach, then solves a nonlinear program to choose path stretch and segment speeds while respecting wake-turbulence minima. Monte Carlo trials on a simplified A80 TMA compare three landing-order policies and find that First-on-Final-First-Serve consistently produces lower total delay, less path extension, and lower fuel burn than First-Entry-First-Serve by using the geometric differences among arrival streams. An online rolling-horizon version commits each trajectory when the aircraft enters the area, enabling decisions fast enough for real-time use.

Core claim

The framework combines an analytic TMA path model with a nonlinear program that jointly optimizes path stretch and segment speeds under a weighted objective. Three landing-order policies are examined: First-Entry-First-Serve, First-on-Final-First-Serve, and FOFFS with Constrained Position Shifting up to k positions. Monte Carlo experiments on the simplified A80 TMA show that FOFFS consistently outperforms FEFS in delay, path stretch, and fuel burn by exploiting geometric asymmetries among arrival streams, while CPS further reduces separation violations at higher computational cost.

What carries the argument

Analytic TMA path model (tangent leg plus radius-to-fix turn plus final-approach segment) optimized by nonlinear programming for path stretch and segment speeds, with mixed-integer linear programming for landing-order selection under constrained position shifting.

If this is right

  • FOFFS reduces total delay, path stretch, and fuel burn relative to FEFS by using arrival-stream geometry.
  • Constrained position shifting up to a few spots further lowers separation violations and path stretch, though solver time grows rapidly.
  • Fuel-burn trends remain consistent across BADA 3 and OpenAP estimates.
  • Per-aircraft optimization finishes in near real time and supports rolling-horizon commitment upon TMA entry.

Where Pith is reading between the lines

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

  • The same path model and optimizer could be adapted to other TMAs whose geometry differs from the A80 simplification.
  • Pre-computed trajectories might lower controller workload by replacing some manual vectoring instructions.
  • Integration with existing ground-based automation would require mapping the analytic segments onto actual radar and navigation data.
  • Extending the objective to include noise or emissions would test whether the same policy ordering remains optimal.

Load-bearing premise

The analytic model of terminal paths is accurate enough that the optimized speed profiles and extensions will produce real trajectories that maintain required separations under the modeled wind perturbations and fleet mix.

What would settle it

A high-fidelity simulation or flight trial on the A80 TMA in which the computed trajectories are flown under measured winds and actual aircraft performance, then checked for whether landing separations match or exceed the required minima.

Figures

Figures reproduced from arXiv: 2604.17776 by Daniel Delahaye, John-Paul Clarke, Yutian Pang.

Figure 1
Figure 1. Figure 1: Schematic diagram of the Atlanta VORTAC 30 DME (TCP) of the Atlanta Large [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Arrival path geometry for north and south flows. Each path consists of a tangent [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Average path stretch ¯di vs. Aircraft Gate Arrival Rate across the 2 × 2 dis￾cretization grid (∆d rows, ∆s columns). FEFS (red dashed) always requires the largest extension, while the CPS family (k ≥ 1) monotonically lowers ¯di over FOFFS. Error bars show the ±1σ wind uncertainty from 10 wind samples per seed. rows in [PITH_FULL_IMAGE:figures/full_fig_p044_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Average separation violation vs. Aircraft Gate Arrival Rate. Below 40 AC/hr [PITH_FULL_IMAGE:figures/full_fig_p045_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average per-aircraft fuel burn vs. Aircraft Gate Arrival Rate, comparing BADA [PITH_FULL_IMAGE:figures/full_fig_p047_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average per-entry solver runtime vs. Aircraft Gate Arrival Rate. FEFS, FOFFS, [PITH_FULL_IMAGE:figures/full_fig_p049_6.png] view at source ↗
read the original abstract

We present a trajectory-based optimization framework for arrival sequencing and scheduling in the terminal maneuvering area (TMA). Unlike node-link scheduling models that reduce trajectories to time-delay variables, the proposed method computes implementable per-aircraft speed profiles and path extensions that achieve required landing separation through terminal air traffic control actions. The framework combines an analytic TMA path model, consisting of a tangent leg, a radius-to-fix turn, and a final-approach segment, with a nonlinear program (NLP) that jointly optimizes path stretch and segment speeds under a weighted objective. Three landing-order policies are examined: First-Entry-First-Serve (FEFS), First-on-Final-First-Serve (FOFFS), and FOFFS with Constrained Position Shifting (CPS) up to $k$ positions. CPS is implemented through a two-phase approach coupling mixed-integer linear programming (MILP) with NLP to select an optimized landing order before trajectory optimization. The aircraft population follows a realistic weight-class fleet mix with pair-specific wake-turbulence separation, and each scenario is perturbed by a Gaussian wind sample projected onto each segment to convert commanded airspeeds into ground speeds. An online rolling-horizon formulation commits each aircraft trajectory irrevocably upon entry, enabling real-time decision-making. Monte Carlo experiments on the simplified A80 TMA show that: (i) FOFFS consistently outperforms FEFS in delay, path stretch, and fuel burn by exploiting geometric asymmetries among arrival streams; (ii) CPS further reduces separation violations and path stretch, though with diminishing returns and rapidly increasing solver cost; (iii) fuel estimates from BADA 3 and OpenAP show consistent qualitative trends; and (iv) per-entry optimization completes in near real-time, supporting practical deployment.

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

3 major / 3 minor

Summary. The paper proposes a trajectory-based optimization framework for arrival sequencing and scheduling in the TMA. It replaces node-link time-delay models with an analytic path model (tangent leg + radius-to-fix turn + final approach) and an NLP that jointly optimizes path stretch and segment speeds to produce implementable per-aircraft speed profiles and extensions that satisfy wake-turbulence separations. Three policies are evaluated—FEFS, FOFFS, and FOFFS with CPS (via a two-phase MILP-NLP procedure)—under a realistic weight-class fleet mix and per-segment Gaussian wind perturbations. An online rolling-horizon formulation commits trajectories upon entry. Monte Carlo experiments on a simplified A80 TMA show FOFFS outperforming FEFS in delay, path stretch, and fuel burn, with CPS yielding further reductions in violations and stretch at increasing computational cost; fuel trends are consistent between BADA 3 and OpenAP.

Significance. If the analytic path model and wind perturbations prove representative, the framework supplies a concrete, real-time method for computing executable trajectories that exploit arrival-stream geometry, rather than abstract scheduling variables. The rolling-horizon commitment and near-real-time per-entry solves are practical strengths; the explicit comparison of geometrically aware policies (FOFFS vs. FEFS) and the two-phase CPS handling are technically useful contributions to TMA optimization literature.

major comments (3)
  1. [Monte Carlo experiments] Monte Carlo experiments section: performance metrics (delay, path stretch, fuel) are reported as averages without error bars, confidence intervals, or statistical significance tests despite stochastic wind sampling and fleet-mix variation; this leaves the claim of 'consistent' outperformance without quantitative support for variability.
  2. [Analytic TMA path model] Analytic TMA path model (tangent leg, radius-to-fix turn, final-approach segment): the model abstracts continuous wind fields, aircraft-specific turn performance, pilot/aircraft response lags, and ATC vectoring tolerances. No sensitivity analysis or validation against real flight data is provided to confirm that the optimized ground-speed trajectories remain feasible and separation-compliant once executed; this directly affects the central implementability claim.
  3. [CPS implementation] CPS implementation (two-phase MILP-NLP): the paper states that CPS up to k positions reduces separation violations and path stretch, yet provides no quantitative trade-off curves or solver-time scaling for realistic k values; the claim of 'diminishing returns' therefore lacks the supporting data needed to assess practical utility.
minor comments (3)
  1. [Wind perturbation model] Notation for segment speeds and ground-speed conversion under wind should be clarified with an explicit equation relating commanded airspeed, wind projection, and resulting ground speed.
  2. [NLP formulation] The objective-function weights are treated as free parameters; a brief sensitivity table or justification for the chosen values would strengthen reproducibility.
  3. [Figures] Figure captions for the A80 TMA geometry and sample trajectories should include scale and wind-vector annotations for clarity.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

  1. Referee: [Monte Carlo experiments] Monte Carlo experiments section: performance metrics (delay, path stretch, fuel) are reported as averages without error bars, confidence intervals, or statistical significance tests despite stochastic wind sampling and fleet-mix variation; this leaves the claim of 'consistent' outperformance without quantitative support for variability.

    Authors: We agree that measures of variability and statistical support are important to substantiate the performance claims. The Monte Carlo setup uses repeated runs with stochastic wind perturbations and fleet-mix sampling. In the revised manuscript we will add error bars (mean ± standard deviation) to all reported average metrics in the figures and tables. We will also include results from appropriate statistical tests (e.g., paired t-tests or Wilcoxon signed-rank tests) between policy pairs to quantify the significance of observed differences. revision: yes

  2. Referee: [Analytic TMA path model] Analytic TMA path model (tangent leg, radius-to-fix turn, final-approach segment): the model abstracts continuous wind fields, aircraft-specific turn performance, pilot/aircraft response lags, and ATC vectoring tolerances. No sensitivity analysis or validation against real flight data is provided to confirm that the optimized ground-speed trajectories remain feasible and separation-compliant once executed; this directly affects the central implementability claim.

    Authors: The analytic path model is intentionally simplified to keep the NLP tractable while retaining the essential TMA geometry. We will add a sensitivity-analysis subsection that perturbs key parameters (turn radius, wind variance, segment lengths) and reports the resulting changes in feasibility and separation compliance. Full validation against real flight data, however, is not feasible in the present study because it requires proprietary ATC and flight-recorder datasets to which we do not have access; we will expand the limitations discussion and identify empirical validation as future work. revision: partial

  3. Referee: [CPS implementation] CPS implementation (two-phase MILP-NLP): the paper states that CPS up to k positions reduces separation violations and path stretch, yet provides no quantitative trade-off curves or solver-time scaling for realistic k values; the claim of 'diminishing returns' therefore lacks the supporting data needed to assess practical utility.

    Authors: We acknowledge that quantitative trade-off data for different values of k are needed to assess practical utility. In the revised manuscript we will include additional experiments that report separation violations, path stretch, and solver run times for k ranging from 0 to 5. These results will be presented as trade-off curves, providing concrete support for the diminishing-returns statement and allowing readers to evaluate computational cost versus benefit. revision: yes

standing simulated objections not resolved
  • Comprehensive validation of the analytic TMA path model against real-world flight data and ATC execution, which would require access to proprietary operational datasets unavailable to the authors.

Circularity Check

0 steps flagged

No circularity in optimization-based trajectory computation

full rationale

The paper's derivation chain consists of defining an analytic TMA path model (tangent leg + radius-to-fix turn + final approach), formulating an NLP that optimizes path stretch and segment speeds subject to wake separation constraints and wind perturbations, and running Monte Carlo simulations to compare landing-order policies. All outputs (speed profiles, extensions, delay, fuel burn) are direct solutions to the stated optimization problems given explicit inputs such as separation minima, fleet mix, and Gaussian wind samples. No equations reduce performance metrics to definitions of the inputs, no fitted parameters are renamed as predictions, and no load-bearing self-citations or uniqueness theorems are invoked. The framework is therefore self-contained; the reported outperformance of FOFFS over FEFS is a computed experimental outcome rather than a tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim depends on the geometric path model being a faithful enough abstraction and on the NLP producing feasible, implementable trajectories. No new physical entities are introduced.

free parameters (2)
  • objective function weights
    Weights balancing delay, path stretch, and fuel burn are chosen to produce the reported trade-offs but are not derived from first principles.
  • CPS shift limit k
    The maximum position shift allowed in the constrained policy is a tunable parameter whose value affects both performance and solver cost.
axioms (2)
  • domain assumption The terminal maneuvering area geometry can be adequately represented by a tangent leg, radius-to-fix turn, and final-approach segment for the purpose of computing path stretch and speed profiles.
    This simplification is invoked to enable the analytic path model and subsequent NLP.
  • domain assumption Gaussian wind samples projected onto each segment sufficiently capture the conversion from commanded airspeed to ground speed for separation enforcement.
    Used to perturb scenarios in the Monte Carlo experiments.

pith-pipeline@v0.9.0 · 5629 in / 1575 out tokens · 58570 ms · 2026-05-10T04:45:27.458858+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

38 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    ForeFlight: Integrated Flight App for Pilots , howpublished =

  2. [2]

    AIAA guidance, navigation, and control conference and exhibit , pages=

    Scheduling aircraft landings under constrained position shifting , author=. AIAA guidance, navigation, and control conference and exhibit , pages=

  3. [3]

    European Journal of Operational Research , volume=

    Runway sequencing with holding patterns , author=. European Journal of Operational Research , volume=. 2008 , publisher=

    2008

  4. [4]

    Air transport management , pages=

    Airport systems planning, design, and management , author=. Air transport management , pages=. 2020 , publisher=

    2020

  5. [5]

    36th Annual Hawaii International Conference on System Sciences, 2003

    Runway operations planning: a two-stage solution methodology , author=. 36th Annual Hawaii International Conference on System Sciences, 2003. Proceedings of the , pages=. 2003 , organization=

    2003

  6. [6]

    4OR , volume=

    Airport runway scheduling , author=. 4OR , volume=. 2011 , publisher=

    2011

  7. [7]

    Rensselaer Polytechnic Inst., DSES Tech

    Statistical analysis of intervals between projected airport arrivals , author=. Rensselaer Polytechnic Inst., DSES Tech. Rept , pages=

  8. [8]

    1991 , number=

    Analysis of delay reducing and fuel saving sequencing and spacing algorithms for arrival traffic , author=. 1991 , number=

    1991

  9. [9]

    An optimization framework for terminal sequencing and scheduling: The single runway case , author=. Complex Systems Design & Management Asia: Smart Nations--Sustaining and Designing: Proceedings of the Second Asia-Pacific Conference on Complex Systems Design & Management, CSD&M Asia 2016 , pages=. 2016 , organization=

    2016

  10. [10]

    Optimization Letters , volume=

    A metaheuristic algorithm for efficient aircraft sequencing and scheduling in terminal maneuvering areas , author=. Optimization Letters , volume=. 2025 , publisher=

    2025

  11. [11]

    Proceedings of the Tenth USA/Europe Air Traffic Management Seminar, Chicago, Illinois , year=

    Automated integration of arrival/departure schedules , author=. Proceedings of the Tenth USA/Europe Air Traffic Management Seminar, Chicago, Illinois , year=

  12. [12]

    2023 IEEE Congress on Evolutionary Computation (CEC) , pages=

    Meta-Heuristics Approach for Arrival Sequencing and Delay Absorption Through Automated Vectoring , author=. 2023 IEEE Congress on Evolutionary Computation (CEC) , pages=. 2023 , organization=

    2023

  13. [13]

    Transportation Research Part C: Emerging Technologies , volume=

    Integrated sequencing and merging aircraft to parallel runways with automated conflict resolution and advanced avionics capabilities , author=. Transportation Research Part C: Emerging Technologies , volume=. 2017 , publisher=

    2017

  14. [14]

    Transportation Research Part C: Emerging Technologies , volume=

    An optimization approach for the terminal airspace scheduling problem , author=. Transportation Research Part C: Emerging Technologies , volume=. 2024 , publisher=

    2024

  15. [15]

    [1992] Proceedings IEEE/AIAA 11th Digital Avionics Systems Conference , pages=

    A systems approach to design: developing a usable automation tool for air traffic control , author=. [1992] Proceedings IEEE/AIAA 11th Digital Avionics Systems Conference , pages=. 1992 , organization=

    1992

  16. [16]

    9th AIAA Aviation Technology, Integration, and Operations Conference (ATIO) and Aircraft Noise and Emissions Reduction Symposium (ANERS) , pages=

    Contrast and comparison of metroplex operations: an air traffic management study of Atlanta, Los Angeles, New York, and Miami , author=. 9th AIAA Aviation Technology, Integration, and Operations Conference (ATIO) and Aircraft Noise and Emissions Reduction Symposium (ANERS) , pages=

  17. [17]

    Modelling and Simulation in Air Traffic Management , pages=

    The center-tracon automation system: Simulation and field testing , author=. Modelling and Simulation in Air Traffic Management , pages=. 1995 , publisher=

    1995

  18. [18]

    Transportation Research Part C: Emerging Technologies , volume=

    Machine learning-enhanced aircraft landing scheduling under uncertainties , author=. Transportation Research Part C: Emerging Technologies , volume=. 2024 , publisher=

    2024

  19. [19]

    2022 , month = aug, note =

    2022

  20. [20]

    Modeling the Impact of Communication and Human Uncertainties on Runway Capacity in Terminal Airspace

    Modeling the Impact of Communication and Human Uncertainties on Runway Capacity in Terminal Airspace , author=. arXiv preprint arXiv:2510.09943 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    Computers & Chemical Engineering , volume=

    Large-scale nonlinear programming using IPOPT: An integrating framework for enterprise-wide dynamic optimization , author=. Computers & Chemical Engineering , volume=. 2009 , publisher=

    2009

  22. [22]

    2014 , month = apr # " 3", url =

    2014

  23. [23]

    International journal of adaptive control and signal processing , volume=

    BADA: An advanced aircraft performance model for present and future ATM systems , author=. International journal of adaptive control and signal processing , volume=. 2010 , publisher=

    2010

  24. [24]

    , title =

    Nuic, A. , title =. 2004 , month = jul, url =

    2004

  25. [25]

    Proceedings of the International Conference on Research in Air Transportation , year=

    Fuel and emission benefits for continuous descent approaches at Schiphol , author=. Proceedings of the International Conference on Research in Air Transportation , year=

  26. [26]

    Journal of Aircraft , volume=

    Optimized profile descent arrivals at Los Angeles international airport , author=. Journal of Aircraft , volume=. 2013 , publisher=

    2013

  27. [27]

    Journal of Air Transportation , volume=

    Modeling of delayed deceleration approaches for community noise reduction , author=. Journal of Air Transportation , volume=. 2021 , publisher=

    2021

  28. [28]

    Aerospace , volume=

    OpenAP: An open-source aircraft performance model for air transportation studies and simulations , author=. Aerospace , volume=. 2020 , publisher=

    2020

  29. [29]

    2012 IEEE/AIAA 31st Digital Avionics Systems Conference (DASC) , pages=

    An integrated approach to air traffic management to achieve trajectory based operations , author=. 2012 IEEE/AIAA 31st Digital Avionics Systems Conference (DASC) , pages=. 2012 , organization=

    2012

  30. [30]

    IEEE Transactions on Aerospace and Electronic Systems , volume=

    Multiobjective 4D trajectory optimization for integrated avionics and air traffic management systems , author=. IEEE Transactions on Aerospace and Electronic Systems , volume=. 2018 , publisher=

    2018

  31. [31]

    9th AIAA Aviation Technology, Integration, and Operations Conference (ATIO) , pages=

    Towards optimal routing and scheduling of metroplex operations , author=. 9th AIAA Aviation Technology, Integration, and Operations Conference (ATIO) , pages=

  32. [32]

    Transportation science , volume=

    Scheduling aircraft landings---the static case , author=. Transportation science , volume=. 2000 , publisher=

    2000

  33. [33]

    Operations Research , volume=

    A dynamic programming approach for sequencing groups of identical jobs , author=. Operations Research , volume=. 1980 , publisher=

    1980

  34. [34]

    1976 , institution=

    The dynamic scheduling of aircraft in the near terminal area , author=. 1976 , institution=

    1976

  35. [35]

    2007 , school=

    Modeling and managing separation for noise abatement arrival procedures , author=. 2007 , school=

    2007

  36. [36]

    1997 , school=

    A systems analysis methodology for developing single event noise abatement procedures , author=. 1997 , school=

    1997

  37. [37]

    2008 , school=

    En route speed optimization for continuous descent arrival , author=. 2008 , school=

    2008

  38. [38]

    Ninth USA/Europe Air Traffic Management Research and Development Seminar (ATM2011) , year=

    Evaluation of continuous descent approach as a standard terminal airspace operation , author=. Ninth USA/Europe Air Traffic Management Research and Development Seminar (ATM2011) , year=