Simple Trajectory Smoothing for UAV Reference Path Planning Based on Decoupling, Spatial Modeling and Linear Programming

Mogens Plessen

arxiv: 2603.21713 · v3 · submitted 2026-03-23 · 📡 eess.SY · cs.SY

Simple Trajectory Smoothing for UAV Reference Path Planning Based on Decoupling, Spatial Modeling and Linear Programming

Mogens Plessen This is my paper

Pith reviewed 2026-05-15 01:19 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords UAV path planningtrajectory smoothingDubins airplanedecouplinglinear programmingspatial modelingflight controlreference path

0 comments

The pith

Decoupling the Dubins airplane model reduces UAV trajectory smoothing to algebraic controls for flight-path angle and speed plus one small linear program for roll angle.

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

The paper presents a smoothing method for UAV reference paths that starts from the Dubins airplane dynamics and applies a decoupling step to separate the controls. This separation produces direct algebraic formulas for flight-path angle and speed, leaving only roll angle to be handled by solving a compact linear program. Two versions are shown, one tracking a reference centerline and one adding a path-shaping constraint, both exploiting spatial modeling to cut dimensionality. The overall procedure stays simple enough for direct implementation and is demonstrated in simulation, with sketched extensions to aerobatic flight and 3D terrain following.

Core claim

The central claim is that a decoupling step applied to the Dubins airplane model yields algebraic control laws for flight-path angle and speed, so that reference-path smoothing reduces to solving a small linear program only for roll-angle commands; the same structure supports both centerline tracking and an added path-shaping constraint while preserving low computational cost through spatial dimensionality reduction.

What carries the argument

The decoupling step that algebraically separates flight-path angle and speed control from roll-angle control inside a spatial model of the Dubins airplane dynamics, with only the roll component solved via linear programming.

If this is right

The method admits two explicit variants: pure centerline tracking and tracking with an added path-shaping constraint.
Spatial modeling yields a natural dimensionality reduction that keeps the linear program small.
An extension to aerobatic flight is possible by accepting a model approximation while retaining the same overall structure.
The identical decoupling and linear-program structure can be reused for tractor path planning over 3D terrain.

Where Pith is reading between the lines

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

Because only one small linear program is required, the approach could run on embedded hardware with limited compute.
If the Dubins structure is a reasonable local approximation, similar algebraic decoupling may apply to other under-actuated vehicle models.
The separation of algebraic and optimized channels suggests a natural way to insert wind or disturbance estimators without enlarging the program.

Load-bearing premise

The Dubins airplane model must accurately represent the UAV dynamics without external disturbances or model mismatches.

What would settle it

Flight tests or high-fidelity simulations in which the algebraic commands for flight-path angle and speed produce large trajectory errors while the linear program for roll angle still converges would falsify the decoupling claim.

Figures

Figures reproduced from arXiv: 2603.21713 by Mogens Plessen.

**Figure 2.** Figure 2: Illustration of the effects of LP1 and LP2 for an edgy reference smoothing example. V. DISCUSSION, LIMITATIONS AND OUTLOOK The Dubins airplane model is a simplifying model [3]. Limitations include (i) not accounting for wind influences, (ii) time-delays in inner low-level control loops that regulate flight-path angle, roll angle and speed, and (iii) aerodynamic [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of 10 numerical experiments. For Ex. 1 a path generated by the method from [3] served as reference. For the other examples waypoints [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of a detail discussed in Sect. III-B. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: A very unfavorable reference can still be tracked. See Sect. V. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Detailed view of examples in Fig. 3 and Fig. 5 in the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

read the original abstract

A method for trajectory smoothing for UAV reference path planning is presented. It is derived based on the dynamics of a Dubins airplane model, and involves a decoupling step, spatial modeling and linear programming. The decoupling step enables algebraic control laws for flight-path angle and speed control. Only for roll angle control an optimization step is applied, involving the solution of a small linear program. Two variations are discussed. They differ by reference centerline tracking and the introduction of a path shaping constraint. The benefit of natural dimensionality reduction for spatial modeling is discussed. The simplicity of the overall method is highlighted. An extension to aerobatic flight is outlined, which comes at the cost of a model approximation, however at the gain of maintaining the general model structure. An extension of the method to tractor path planning along 3D terrain is discussed. The method is validated in simulations.

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.

Desk Editor's Note private letter to a colleague

The decoupling step turns most UAV smoothing into algebra and leaves only a small LP for roll angle.

read the letter

The main point is that this decoupling approach for Dubins airplane models lets you derive algebraic laws for flight-path angle and speed, so the only optimization is a small linear program for the roll angle. That keeps things simple and reduces the computational load. The paper derives the method from the model dynamics, uses spatial modeling to cut dimensions, and presents two variations: one for centerline tracking and one with a path shaping constraint. It discusses the natural dimensionality reduction and outlines extensions to aerobatic flight and 3D terrain planning. The simulations validate the approach under the stated conditions, and the derivations hold up without contradictions. It does a good job highlighting the simplicity while maintaining the model structure. The explicit control laws and small LP size are clear advantages for practical implementation. A soft spot is the assumption that the Dubins model captures the UAV dynamics well enough without disturbances. Real flights would likely need robustness checks against wind or mismatches, which aren't covered here. The aerobatic extension involves an approximation whose effects could be quantified more. Validation is limited to simulations, so that's another area for expansion. This work is aimed at UAV path planning practitioners and researchers who want lighter methods than full nonlinear optimization. Readers interested in hybrid algebraic and optimization techniques for control will find it useful. It has solid grounding in the math and enough new combination to merit serious peer review. I would recommend sending it to referees, perhaps asking for added comparisons or disturbance tests.

Referee Report

1 major / 2 minor

Summary. The paper presents a trajectory smoothing method for UAV reference path planning derived from the Dubins airplane kinematic model. It uses a decoupling transformation to obtain algebraic control laws for flight-path angle and speed, restricting optimization to a small linear program for roll angle. Two variants are discussed (differing in centerline tracking and path-shaping constraints), along with extensions to aerobatic flight (with model approximation) and tractor path planning on 3D terrain. The approach emphasizes simplicity and natural dimensionality reduction via spatial modeling, and is validated through simulations.

Significance. If the derived controls and LP formulation perform as claimed, the method offers a lightweight, structure-exploiting alternative to full nonlinear trajectory optimization for UAVs. The algebraic laws for two channels and restriction of optimization to a small LP could enable real-time use, while the spatial reformulation provides dimensionality reduction that is valuable in path-planning contexts.

major comments (1)

Simulation section: the validation is stated to consist of simulations consistent with the derived laws, but the manuscript provides no quantitative error metrics, trajectory deviation statistics, or comparisons against baseline smoothers (e.g., spline or optimization-based methods). Without these, the claims of simplicity and practical benefit remain difficult to assess.

minor comments (2)

Notation: the decoupling transformation and spatial variables should be introduced with explicit definitions and a table of symbols to improve readability for readers unfamiliar with the Dubins airplane model.
Figures: trajectory plots would benefit from overlaid reference paths, smoothed outputs, and control inputs to visually demonstrate the effect of the LP step versus the algebraic laws.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary, and recommendation of minor revision. We address the single major comment below and will update the manuscript accordingly.

read point-by-point responses

Referee: Simulation section: the validation is stated to consist of simulations consistent with the derived laws, but the manuscript provides no quantitative error metrics, trajectory deviation statistics, or comparisons against baseline smoothers (e.g., spline or optimization-based methods). Without these, the claims of simplicity and practical benefit remain difficult to assess.

Authors: We agree that quantitative metrics and baseline comparisons would strengthen the validation and make the claims of simplicity and practical benefit easier to evaluate. Although the original simulations were intended to illustrate consistency with the derived algebraic laws and the small LP, we will revise the simulation section to include RMS position and attitude errors, maximum deviation statistics, and direct comparisons against cubic-spline smoothing as well as a standard nonlinear trajectory optimizer. These additions will be placed in the updated Section V and will quantify both accuracy and computational effort. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from Dubins model; no circular steps

full rationale

The paper explicitly starts from the Dubins airplane kinematic model, introduces a decoupling transformation that yields algebraic laws for flight-path angle and speed, reformulates the problem spatially, and restricts optimization to a small LP on roll angle. All steps are derived forward from the stated dynamics without any fitted parameter being relabeled as a prediction, without self-definitional loops, and without load-bearing self-citations that close the argument. The two variations (centerline tracking vs. path-shaping constraint) and the outlined extensions remain within the same model structure. No equation reduces to its own input by construction, and the simulation results are presented as consistency checks rather than independent validation of a circular claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim relies on the standard Dubins airplane model assumptions and the validity of the decoupling step for enabling algebraic controls.

axioms (1)

domain assumption Dynamics of a Dubins airplane model
The method is derived based on the dynamics of a Dubins airplane model as stated in the abstract.

pith-pipeline@v0.9.0 · 5443 in / 1206 out tokens · 67444 ms · 2026-05-15T01:19:10.100906+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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[1] [1]

On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,

L. E. Dubins, “On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents,”American Journal of Mathematics, vol. 79, no. 3, pp. 497–516, 1957

work page 1957

[2] [2]

A seventy-year review of dubins robot path planning: basics, developments and opportunities,

X. Zhou, L. Li, X. Zhang, S. Jin, X. Xu, D. Shi, and H. Wang, “A seventy-year review of dubins robot path planning: basics, developments and opportunities,” Industrial Robot: The International Journal of Robotics Research and Application, pp. 1–23, 2026

work page 2026

[3] [3]

Implementing dubins airplane paths on fixed-wing UA Vs,

T. McLain, R. W. Beard, and M. Owen, “Implementing dubins airplane paths on fixed-wing UA Vs,” 2014

work page 2014

[4] [4]

Time-optimal paths for a dubins airplane,

H. Chitsaz and S. M. LaValle, “Time-optimal paths for a dubins airplane,” inIEEE Conference on Decision and Control (CDC), 2007, pp. 2379–2384

work page 2007

[5] [5]

Autonomous and efficient large-scale snow avalanche monitoring with an unmanned aerial system (uas),

J. Lim, E. Hafner-Aeschbacher, F. Achermann, R. Girod, D. Rohr, N. Lawrance, Y . B ¨uhler, and R. Siegwart, “Autonomous and efficient large-scale snow avalanche monitoring with an unmanned aerial system (uas),”Nat- ural Hazards and Earth System Sciences, vol. 26, no. 1, pp. 411–431, 2026

work page 2026

[6] [6]

Comprehensive review of path planning techniques for unmanned aerial vehicles (UA Vs),

P. Kumar, K. Pal, and M. C. Govil, “Comprehensive review of path planning techniques for unmanned aerial vehicles (UA Vs),”ACM Computing Surveys, vol. 58, no. 3, pp. 1–44, 2025

work page 2025

[7] [7]

Model predictive control for trajectory tracking of un- manned aerial vehicles using robot operating system,

M. Kamel, T. Stastny, K. Alexis, and R. Siegwart, “Model predictive control for trajectory tracking of un- manned aerial vehicles using robot operating system,” in Robot Operating System (ROS) The Complete Reference (Volume 2). Springer, 2017, pp. 3–39

work page 2017

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Min- imal 3D dubins path with bounded curvature and pitch angle,

P. V ´aˇna, A. A. Neto, J. Faigl, and D. G. Macharet, “Min- imal 3D dubins path with bounded curvature and pitch angle,” inIEEE International Conference on Robotics and Automation (ICRA), 2020, pp. 8497–8503

work page 2020

[9] [9]

Path tracking control of trailer-like mobile robot,

M. Sampei, T. Tamura, T. Itoh, and M. Nakamichi, “Path tracking control of trailer-like mobile robot,” in Proceedings IROS’91: IEEE/RSJ International Workshop on Intelligent Robots and Systems’ 91, 1991, pp. 193– 198

work page 1991

[10] [10]

Trajectory tracking for unicycle-type and two-steering-wheels mobile robots,

A. Micaelli and C. Samson, “Trajectory tracking for unicycle-type and two-steering-wheels mobile robots,” Ph.D. dissertation, Inria, 1993

work page 1993

[11] [11]

M. P. Do Carmo,Differential geometry of curves and surfaces: revised and updated second edition. Courier Dover Publications, 2016

work page 2016

[12] [12]

Op- timal trajectory generation for dynamic street scenarios in a frenet frame,

M. Werling, J. Ziegler, S. Kammel, and S. Thrun, “Op- timal trajectory generation for dynamic street scenarios in a frenet frame,” inIEEE International Conference on Robotics and Automation (ICRA), 2010, pp. 987–993

work page 2010

[13] [13]

A universal formulation for path-parametric planning and control,

J. Arrizabalaga, Z. ˇS´ır, Z. Manchester, and M. Ryll, “A universal formulation for path-parametric planning and control,”arXiv preprint arXiv:2410.04664, 2024

work page arXiv 2024

[14] [14]

Euclidean and non-euclidean trajectory optimization approaches for quadrotor racing,

T. Fork and F. Borrelli, “Euclidean and non-euclidean trajectory optimization approaches for quadrotor racing,” arXiv preprint arXiv:2309.07262, 2023

work page arXiv 2023

[15] [15]

A gravity-referenced moving frame for vehicle path following applications in 3D,

J. C. H. Ram ´ırez and M. Nahon, “A gravity-referenced moving frame for vehicle path following applications in 3D,”IEEE Robotics and Automation Letters, vol. 6, no. 3, pp. 4393–4400, 2021

work page 2021

[16] [16]

An algorithm for planning coverage of an area with obstacles with a heterogeneous group of drones using a genetic algorithm and parame- terized polygon decomposition,

K. Yakunin, Y . Kuchin, E. Muhamedijeva, A. Symagulov, and R. I. Mukhamediev, “An algorithm for planning coverage of an area with obstacles with a heterogeneous group of drones using a genetic algorithm and parame- terized polygon decomposition,”Drones, vol. 9, no. 9, p. 658, 2025

work page 2025

[17] [17]

Path planning for spot spraying with UA Vs combining tsp and area coverages,

M. Plessen, “Path planning for spot spraying with UA Vs combining tsp and area coverages,”Smart Agricultural Technology, vol. 11, p. 100965, 2025

work page 2025

[18] [18]

Path planning using 3D du- bins curve for unmanned aerial vehicles,

Y . Lin and S. Saripalli, “Path planning using 3D du- bins curve for unmanned aerial vehicles,” inIEEE In- ternational Conference on Unmanned Aircraft Systems (ICUAS), 2014, pp. 296–304

work page 2014

[19] [19]

Smoothing of headland path edges and headland-to-mainfield lane transitions based on a spatial domain transformation and linear programming,

M. Plessen, “Smoothing of headland path edges and headland-to-mainfield lane transitions based on a spatial domain transformation and linear programming,”Biosys- tems Engineering, vol. 257, p. 104229, 2025

work page 2025

[20] [20]

From 2D to 3D terrain-following area coverage path planning,

——, “From 2D to 3D terrain-following area coverage path planning,”arXiv preprint arXiv:2601.00614, 2026

work page arXiv 2026

[21] [21]

Trajectory planning of automated vehi- cles in tube-like road segments,

M. G. Plessen, “Trajectory planning of automated vehi- cles in tube-like road segments,” inIEEE International Conference on Intelligent Transportation Systems (ITSC), 2017, pp. 1–6

work page 2017

[22] [22]

Efficient un- manned aerial vehicle formation rendezvous trajectory planning using dubins path and sequential convex pro- gramming,

Z. Wang, L. Liu, T. Long, and G. Xu, “Efficient un- manned aerial vehicle formation rendezvous trajectory planning using dubins path and sequential convex pro- gramming,”Engineering Optimization, vol. 51, no. 8, pp. 1412–1429, 2019

work page 2019

[23] [23]

UA V formation flight based on nonlinear model predictive control,

Z. Chao, S.-L. Zhou, L. Ming, and W.-G. Zhang, “UA V formation flight based on nonlinear model predictive control,”Mathematical Problems in Engineering, vol. 2012, no. 1, p. 261367, 2012

work page 2012

[24] [24]

Multi-automated vehicle coordination using decoupled prioritized path planning for multi-lane one-and bi- directional traffic flow control,

M. G. Plessen, D. Bernardini, H. Esen, and A. Bemporad, “Multi-automated vehicle coordination using decoupled prioritized path planning for multi-lane one-and bi- directional traffic flow control,” inIEEE Conference on Decision and Control (CDC), 2016, pp. 1582–1588. APPENDIX (a) Ex. 2,LP2. (b) Ex. 4,LP2. Note how the resulting path stays to the right-han...

work page 2016