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Trajectory-Based Co-Optimization of Arrival Scheduling and Descent Path Design in the Terminal Maneuvering Area

Yutian Pang, John-Paul Clarke

arXiv:2609.03234v1eess.SY

TL;DR

The paper addresses the separation between arrival scheduling and individual descent-procedure design by jointly selecting lateral extensions and vertical descent parameters. It verifies candidate trajectories with wind-aware planning and 6DOF simulation, then schedules them under wake constraints. Co-optimized continuous descent saves about 15% of fleet fuel below saturation in free descent and 9–10% on published flows, while delayed deceleration saves about 23% and 20–21%, respectively.

  • Problem

    Arrival schedulers assign landing times while descent studies optimize one aircraft with the schedule fixed, leaving their joint fuel and trajectory decisions unresolved.

  • Method

    A four-dimensional scheduler jointly commits landing order, lateral extension, glideslope-capture distance, and flap-trigger speeds using wind-aware planning and 6DOF verification.

  • Results

    About 15% and 23% of fleet fuel are saved below saturation by co-optimized continuous descent and delayed deceleration, falling to 9–10% and 20–21% on published flows.

  • Takeaways & Limitations

    Published altitude floors remove 31% of the design lattice, while observed aircraft-specific wind effects largely cancel at fleet level and explain at most 4% of fleet-fuel variance.

  • Takeaways & Limitations

    The evaluation assumes each aircraft observes wind at the gate and models a vector-to-final path instead of the published downwind and upstream step-down restrictions.

Abstract

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Terminal arrival scheduling and descent procedure design are studied in two largely separate literatures. Scheduling models reduce each aircraft to a travel time and deliver target landing times, and fuel-efficient descent procedures are designed one aircraft at a time with the schedule taken as given, although both decide where an arriving aircraft absorbs delay before final approach. Existing formulations therefore cannot trade a slower, earlier-configuring descent against level track miles, a schedule that is efficient in time can be expensive in fuel, and autonomous or reduced-crew operations will need a single trajectory plan that ground automation and the flight management system both accept. To close this gap, we propose a four-dimensional terminal arrival scheduler that selects each aircraft's lateral path extension, glideslope-capture distance, and flap-deployment trigger speeds in one decision. We evaluate every candidate idle-thrust descent offline with a wind-aware backward plan and a six-degree-of-freedom forward simulation that returns descent time, fuel burn, minimum track length, and stabilized-approach feasibility, and a rolling-horizon scheduler commits one verified descent and one extension per aircraft under wake-separation constraints and observed entry winds. Two Atlanta terminal airspace case studies quantify the benefit. We show that co-optimized continuous descents save about 15\% of fleet fuel below saturation in a free-descent environment and that delayed deceleration saves about 23\%, while on the six published Runway 8L arrival flows the charted altitude floors remove 31\% of the design lattice and reduce the savings to 9--10\% and 20--21\%, respectively. We also find that wind moves single-aircraft descent fuel by 34--81\% yet explains at most 4\% of fleet fuel variance, because aircraft-specific wind effects average out across a scenario.

1. Introduction

Terminal arrival management must connect landing-time scheduling with flyable, fuel-aware descent trajectories. The paper proposes a unified four-dimensional framework that jointly commits path extension and descent design, then evaluates it in Atlanta case studies.

  • Motivation: Terminal-area operations merge six arrival flows under wake-separation constraints, while vectoring and path stretching create workload and fuel inefficiency.Radar vectoring requires rapid controller decisions and time-critical crew execution; terminal-area inefficiency accounts for 1.5–4.5% of flight fuel.
  • Contribution: The framework jointly commits landing order, track miles, glideslope-capture distance, and flap-trigger schedules, with safety, fuel, and delay resolved in strict priority order.Candidates are evaluated using wind-aware backward planning and six-degree-of-freedom simulation, with stabilized-approach criteria as hard constraints.
  • Results: Wind changes individual-arrival fuel by 34–81% but explains at most 4% of fleet-fuel variability because aircraft-specific effects largely cancel.The study models each aircraft’s observed entry wind explicitly before trajectory commitment.

2. Literature Review

Prior research largely separates runway scheduling, terminal delay absorption, and individual descent design. The paper positions its contribution as a trajectory-computing co-optimization that exposes geometric and configuration decisions absent from common abstractions.

  • Runway and arrival scheduling: Runway scheduling literature optimizes landing order and timing using dynamic programming, mixed-integer programming, branch-and-bound, and heuristics.Arrival scheduling repeatedly balances throughput against fairness and wake-separation effects.
  • Terminal scheduling: Terminal-network models assign time over nodes or route segments but do not guarantee that a given aircraft can fly the assigned trajectory.They also lose the geometric coupling between continuous base-leg extension and arrival time.
  • Delay absorption: Prior delay-absorption studies compare slower cruise or descent with extra level distance, identifying descent-speed reduction as the cheapest strategy across examined aircraft types.These analyses study the fuel price of delay absorption without embedding the procedure choice in fleet scheduling.
  • Descent design: Continuous- and delayed-deceleration research establishes fuel, noise, and flyability effects, while later work optimizes flap triggers and capture distance for individual aircraft.The companion study evaluates candidate designs in 6DOF simulation under wind uncertainty and stabilized-approach constraints.
  • Research gap: Existing coupled studies keep scheduling primary and adapt descent within assigned time windows, rather than making aircraft configuration a scheduling decision.The present framework instead uses a verified lattice of descent designs and jointly selects vertical and lateral decisions.
  • Contribution: The proposed formulation addresses the gap by jointly committing vertical design and lateral extension, verifying candidates in 6DOF, modeling observed wind, and comparing enforced versus relaxed published structure.This preserves analytic path geometry while replacing commanded speeds with descent designs.

3. Methodology

The methodology combines an analytic lateral path model with a finite lattice of verified idle-thrust descent designs. A scheduler selects sequencing, extension, and descent configuration under traffic, aircraft, wind, and approach constraints.

  • Trajectory formulation: Each aircraft extends its base leg upstream of the FAF, while its calibrated-airspeed history comes from an idle-thrust descent design rather than commanded segment speeds.The formulation couples lateral arrival-time control with vertical procedure design.
  • Descent design: The vertical design lattice varies glideslope-capture distance and flap-deployment trigger speeds within aircraft placard limits.Each candidate returns descent time, fuel burn, minimum track length, and stabilized-approach feasibility.
  • Scheduling and experiments: The scheduler uses FOFFS sequencing, constrained position shifting, and per-aircraft joint commitment to select verified trajectories.Case studies compare free descent with the six published Runway 8L flows and enforce or relax charted altitude floors.
  • Traffic and fleet: Traffic is generated from four independent shifted-Poisson feeder streams with a 90 s entry separation buffer and pair-specific wake minima.The simulated fleet contains two Large and two Heavy simulator-calibrated airframe types.
  • Lateral geometry: The lateral path comprises a tangent leg, an RF arc, and a runway-aligned extension, with total distance mapped smoothly from the extension variable.Monotonicity makes the path-distance map invertible and lets minimum descent track length constrain lateral extension.

3.3. Wind Uncertainty Modeling

Wind uncertainty is represented through an along-track climatology observed by each aircraft at the metering gate. The observed wind re-anchors descent planning before deterministic trajectory commitment.

  • Wind model: Along-track wind is sampled at 10,000 ft from a truncated-normal climatology and propagated through altitude with a power-law profile.The work uses σw = 10 kt and a profile anchored at 10,000 ft that vanishes at runway elevation.
  • Scenario construction: The climatology is discretized into evenly spaced wind nodes with probability-proportional weights, producing deterministic quadrature with common random numbers.Each aircraft independently draws a wind node, and the wind vector is projected onto its tangent-leg heading.
  • Information structure: Each aircraft observes its wind node at TRACON entry, after which the flight-management system plans the descent using that wind.Conditional on observed wind, the commitment problem is deterministic and the two-stage stochastic program factorizes into per-node problems.

3.4. Descent Architectures and the Vertical Design Space

The paper compares two idle-thrust descent architectures and parameterizes each aircraft’s vertical design through capture distance and flap-trigger offsets. Both use the same 3.0° final and stabilized-approach gates, isolating deceleration scheduling as the architectural difference.

  • Descent Architectures: Each arrival uses either a continuous-descent CDA or delayed-deceleration DDA architecture from a common 10,000-ft, 240-KCAS, clean entry gate.Both architectures use idle thrust and share the same final approach geometry.
  • Descent Architectures: CDA completes landing configuration before glideslope capture, whereas DDA remains clean and fast deeper into the arrival before deploying landing flap on final.CDA absorbs flap changes on shallow deceleration segments; DDA delays deceleration.
  • Comparison Basis: Both architectures use the same 3.0° final and capture-distance grid, so their comparison changes only the deceleration schedule.A steeper final would confound the comparison with a glide-path-angle effect.
  • Vertical Design Space: The vertical design vector contains glideslope-capture distance and flap-deployment trigger offsets, with trigger speeds clipped, quantized, and constrained by placard windows.A monotone minimum cascade converts group trigger speeds into per-detent speeds.
  • Feasibility Constraints: Stabilized-approach feasibility requires landing flap at 1,000 ft AGL, CAS within the specified Vref bounds, and peak longitudinal deceleration no greater than 0.12 g.These criteria are encoded by a binary stabilization indicator.

3.5. Simulation-in-the-Loop Vertical Trajectory Optimization

Candidate descents are evaluated offline with a wind-aware backward plan followed by six-degree-of-freedom forward simulation. The resulting descent metrics support fuel-aware lateral extension decisions while preserving feasibility at observed winds.

  • Trajectory Evaluation: A wind-aware backward calculation constructs each candidate’s idle-thrust vertical plan from the threshold upstream.Distance kinematics are corrected for the observed wind profile.
  • Trajectory Evaluation: The backward plan allocates deceleration distance consistently with the wind the aircraft will encounter, then continues the clean idle segment to the entry gate.Its along-track length defines the minimum descent track distance.
  • Trajectory Evaluation: The TASAT six-degree-of-freedom forward simulation returns descent time to the FAF, descent fuel burn, and stabilized-approach feasibility.Level-cruise calibration supplies entry-gate fuel-per-distance and true-airspeed quantities.
  • Fuel and Path Coupling: Level flight at the entry altitude is the most expensive way to cover distance, so fuel-optimal commitments absorb track miles inside descent when earlier-configuring designs permit.Every evaluation is cached, leaving the fleet layer to perform table lookups.

3.6. Fleet Co-Optimization Problem and Solution

The fleet problem jointly selects landing order, vertical design, and base-leg extension under separation, stabilization, and descent-feasibility constraints. A rolling-horizon forward pass commits one aircraft at a time, using descent retardation before level path stretch.

  • Optimization Model: The co-optimization selects a landing-order permutation, each aircraft’s lattice-based vertical design, and its base-leg extension.The architecture is fixed per experiment rather than optimized.
  • Optimization Model: The objective prioritizes separation safety, then fleet fuel, then earliest arrival through a strict penalty hierarchy.The delay term breaks fuel ties and protects later slots.
  • Constraints: The constraints impose soft wake/runway separation, hard stabilized-approach feasibility at observed wind, descent-feasibility floors, and lattice and box bounds.Slack records saturation of available delay authority, while no holding variable is included.
  • Solution Procedure: The rolling-horizon algorithm fixes the order and commits each aircraft’s design and extension in one online forward pass using cached evaluations.FOFFS orders aircraft by nominal FAF arrival time; FEFS provides an entry-time fairness reference.
  • Solution Procedure: The scheduler first uses descent retardation through slower, earlier-configuring designs and adds level path stretch only when the design menu cannot absorb the required delay.Fuel ties are broken by earliest arrival.

3.7. Structural Properties of the Commitment Problem

Structural results make each per-aircraft commitment exactly solvable over a finite design lattice and continuous extension interval. The analysis also formalizes the lexicographic priorities and the wait-and-see interpretation under observed winds.

  • Monotonicity: For each design, FAF arrival time and total fuel increase continuously and strictly with feasible extension, with fuel increasing at a fixed price per added second.Changing the vertical design instead buys time at the descent-evaluation cost.
  • Exact Commitment: Each stabilized design has a unique smallest feasible extension, and enumerating designs with bisection selects the exact per-aircraft optimum.The procedure enumerates at most the finite lattice and bisects twice per design.
  • Feasibility Boundary: The stabilization filter is the structural failure point: an empty stabilized menu makes the commitment infeasible.Both case studies retained at least one stabilized design at every wind-quadrature cell.
  • Priority Structure: Under the stated separation condition on weights, the weighted objective is equivalent to lexicographically minimizing separation slack, fuel, and arrival time.This removes numerical weight tuning from the reported results.
  • Stochastic Interpretation: Reported fleet metrics estimate the wait-and-see value because each decision is selected after its corresponding wind node is observed.The wind vector consists of independently drawn per-aircraft wind nodes.

3.8. FOFFS with Constrained Position Shifting

FOFFS-CPS constrains landing-rank changes to a local window and uses dynamic programming to balance separation slack, landing time, and wake effects. A capped Pareto frontier keeps the computation tractable but makes Phase 1 approximate.

  • Constrained Position Shifting: FOFFS-CPS permits each aircraft’s landing rank to deviate by at most k positions from the FOFFS order, with k ∈ {1, 2, 3}.The policy first chooses the order, then executes the trajectory scheduler under that fixed order.
  • Constrained Position Shifting: The window dynamic program composes landing times from cached per-aircraft nominal arrival-time envelopes [E_j, L_j].The fastest stabilized design determines E_j, while the slowest design at d_max determines L_j; excess delay accumulates as slack.
  • Constrained Position Shifting: Each dynamic-programming state tracks the local occupancy pattern and last-placed aircraft, yielding a reachable state count linear in fleet size for fixed k.Successors are limited to the 2k + 1 ranks in the FOFFS window, which avoids unrestricted permutation search.
  • Constrained Position Shifting: The algorithm carries a Pareto frontier over accumulated slack, accumulated time, and last landing time rather than a single scalar value.A successor is discarded when an incumbent dominates it in all three components.
  • Constrained Position Shifting: Capping each state at 24 incomparable frontier items makes Phase 1 a bounded-width approximation rather than an exact solve.On 30–61-aircraft synthetic instances, the capped program matched the uncapped order in about three quarters of cases and lost at most 54 s of total slack otherwise.

4. Case Study 1: Free-Descent Trajectory-Based Operations

Case Study 1 evaluates joint descent and lateral-path decisions without published transition-leg restrictions, isolating the value of full 4D freedom. Across demand, wind, and noise analyses, delayed deceleration generally delivers larger fuel savings, while congestion and procedure choices shape the remaining trade-offs.

  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: Case Study 1 removes published transition-leg crossing restrictions so aircraft performance, placard schedules, and stabilization criteria determine descent feasibility.The setting is intended to isolate full 4D trajectory freedom and bound the procedure-constrained case.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: The design lattice combines capture distances of 10.0, 11.5, and 12.48 nmi with five normalized flap-trigger offsets.Both CDA and DDA share the grid because the common 3.0° final gives them the same capture geometry.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: The DDA design cloud lies below the CDA cloud for every type, and both architectures favor the smallest shared capture distance, 10 nmi.Every lattice design passed stabilization, and the feasible set remained nonempty across the ±25 kt wind envelope.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: The optimized CDA saves 17–23% versus the Baseline, while the optimized DDA saves 24–37% at the zero-wind node.At matched glideslope, DDA’s advantage over optimized CDA ranges from 8% for the B737-800 and B767-400ER to 18% for the A319 and A340-300.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: 58–65% of aircraft commit their unconstrained fuel-optimal design, while the remainder trade descent design against separation slots or surplus distance.A fast clean design can require positive extension on short corner geometries, and surplus track miles can make earlier configuration cheaper than level stretch.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: At 60 AC/hr under plain FOFFS, DDA saves 13.9% and CDA saves 8.7%, down from approximately 23% and 15.5% at low demand.Congestion forces level path stretch across policies, diluting descent-side savings.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: At 60 AC/hr with CPS3, savings are 15.1% for CDA and 20.6% for DDA, while mean slack falls from 182–190 s to 44–48 s.CPS3 also reduces mean violating aircraft from about 3.0 to 1.1–1.4 per scenario.
  • 4. Case Study 1: Free-Descent Trajectory-Based Operations: A 20 kt tailwind-to-headwind change raises individual CDA Baseline fuel by 34–81%, but wind contributes at most 4% of fleet fuel variance.Aircraft-specific wind effects average out across scenarios, although the wind-optimal design can change repeatedly by airframe.

5. Case Study 2: Published Arrival Flows to Runway 8L

Case Study 2 tests trajectory co-optimization on six published Runway 8L arrival flows with asymmetric geometry and charted altitude restrictions. The restrictions prune feasible descent designs and reduce, but do not eliminate, fleet fuel savings across demand levels.

  • Effect of the Charted Floors on the Design Space: 31% of the design lattice is removed by charted altitude floors, all from the JAAJJ 5,000 ft restriction eliminating the 10.0 nmi capture row.With floors enforced, the fuel-optimal capture is 11.5 nmi; without them, it is 10.0 nmi.
  • Flow Geometry and Fleet Results: Near-side flows absorb more delay through extensions, while far-side extensions cost 1.9 track nmi per nautical mile and add to existing surplus track miles.Mean extensions are 7.6–10.9 nmi near-side versus 1.5–2.6 nmi far-side, making geometry a major fuel determinant.
  • Fleet Results Across Demand: 8.8–10.0% CDA and 17.7–21.2% DDA fleet fuel savings occur below saturation against the Baseline on the published flows.These fleet savings are lower than design-level gains because every arm pays surplus track-mile costs imposed by the published vectoring geometry.
  • Fleet Results Across Demand: Saturation begins between 36 and 42 AC/hr nominal, after which congestion forces level extension and erodes plain FOFFS savings.CPS2 recovers part of the savings and most of the safety margin, while CPS3 reduces slack further but gives back fuel under lexicographic priorities.
  • Case-Study Comparison: Across the two case studies, published floors reduce below-saturation fleet savings from about 15.5% to 9.6% for CDA and from 23% to 20% for DDA.At the design level, floors remove 31% of the lattice and reduce optimized-CDA savings from 17–23% to 6.5–15.5%.

6. Discussion

The discussion shows that the dominant co-optimization decision changes with the binding constraint: descent design matters most below saturation, while sequencing and extension placement matter more at saturation. Published-flow restrictions and modeling assumptions limit the reported benefits and define key extensions.

  • The Dominant Decision at Each Demand Level: Below wake capacity, descent design carries almost the entire benefit, while savings remain flat because separation slack has not appeared.At saturation, congestion-forced level extension dilutes descent savings and shifts remaining leverage toward sequencing and extension placement.
  • The Dominant Decision at Each Demand Level: A scheduler with two positions of repair has more leverage after runway saturation, while bounded position shifting preserves descent-side savings.The two interventions complement each other rather than one replacing the other.
  • Fuel, Noise, and Procedure Comparisons: Delayed deceleration saves 8–18% relative to optimized continuous descent on a matched 3.0° final, with the advantage tracking clean-to-approach speed range.The comparison is reported for B737-800, B767-400ER, A319, and A340-300 aircraft classes.
  • Wind Uncertainty at the Aircraft and Fleet Levels: Wind changes individual descent fuel substantially but contributes little fleet-level variation because aircraft-specific effects average out across scenarios.The wind-aware plan re-anchors each descent at the observed metering-gate wind; demand realization instead shifts the whole scenario.
  • Verification and Scope: Stabilized approach remained a hard constraint at every observed-wind node, although wide-body DDA lost designs at the strongest tailwind nodes.The favorable wind-observable setting avoids the harder chance-constraint problem faced by a static wind-blind design.
  • Limitations and Future Studies: The study models downstream vectors as a tangent leg, RF arc, and centerline extension, omitting charted downwind and upstream flow-specific step-downs.Including those restrictions would raise far-side fuel, reduce delay authority, and prune more of the design lattice.
  • Limitations and Future Studies: Uniform demand across six flows, homogeneous CDA or DDA fleets, and omitted same-flow non-overtaking precedence constrain realism and leave mixed-fleet spacing unresolved.Real demand is heavier from the north and west, while clean and configured aircraft can differ markedly in speed on final approach.

7. Conclusion

The paper presents a four-dimensional, simulation-in-the-loop scheduler that jointly commits descent design and lateral extension for each aircraft. Atlanta case studies show fuel savings, quantify the cost of published altitude floors, and demonstrate direct verification of executable trajectory commitments.

  • Conclusion: The framework jointly commits each aircraft’s vertical descent design and lateral path extension using a wind-aware backward plan and six-degree-of-freedom simulation.The evaluated design lattice returns descent time, fuel burn, minimum track length, and stabilized-approach feasibility.
  • Conclusion: Free-descent experiments save about 15% of fleet fuel with co-optimized continuous descent and about 23% with delayed deceleration below saturation.Bounded two-position shifting keeps these savings nearly flat through saturation while reducing mean separation shortfall by roughly fourfold.
  • Conclusion: Published Runway 8L altitude floors remove 31% of the design lattice, reduce continuous-descent savings to 9–10%, and leave delayed-deceleration savings at 20–21%.Relaxing the floors prices them at 1.6–6.7% of fleet fuel depending on architecture and demand.
  • Conclusion: Each commitment includes glideslope-capture distance, flap-trigger schedule, and extension distance, all verified against stabilized-approach criteria at observed wind.The committed object is intended for direct flight-management-system execution and ground-automation checking without a crew in the loop.

Appendix A. Monotonicity Lemmas

Appendix A establishes monotonicity of lateral track distance, FAF arrival time, and total fuel with respect to extension distance. These properties support efficient one-dimensional feasibility and optimization searches.

  • Monotonicity Lemmas: The lateral track-distance function is continuously differentiable and strictly increasing over the feasible extension interval.Its sampled derivative lies between 0.28 and 1.98 for the studied geometries and published flows.
  • Monotonicity Lemmas: For a fixed descent design, FAF arrival time and total fuel are continuous and strictly increasing in lateral extension.The proof uses increasing surplus distance and positive time and fuel slopes at the observed wind node.

Appendix B. Structural Propositions

Appendix B proves structural properties that make the per-aircraft and weighted selection problems exact and interpretable. It also clarifies that reported stochastic metrics use wind-observed wait-and-see decisions rather than wind-blind commitments.

  • Proposition 1: The stabilized per-aircraft menu is finite, and the fuel-minimal feasible extension for each design is uniquely found by bisection.Enumerating stabilized designs and bisecting twice per design solves the per-aircraft problem exactly.
  • Proposition 2: The weighted per-aircraft objective coincides with lexicographic minimization of slack, fuel, and arrival time when the weight inequalities exceed attainable value gaps.Finiteness guarantees positive smallest nonzero gaps for slack and fuel.
  • Proposition 3: Reported fleet metrics estimate the wait-and-see value Ew[J⋆(w)] because each decision is chosen after its aircraft-specific wind node is observed.This value is bounded below the expected cost of any feasible wind-blind design rule.
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