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arxiv: 2604.04537 · v1 · submitted 2026-04-06 · 📡 eess.SY · cs.SY

PCT-Based Trajectory Tracking for Underactuated Marine Vessels

Ji-Hong Li This is my paper

Pith reviewed 2026-05-10 19:52 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords underactuated marine vesselstrajectory trackingpolar coordinate transformationexponential modification of orientationbackstepping controlsingularity avoidancenonholonomic systems
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The pith

Two polar coordinate transformations reduce underactuated marine vessel tracking to a two-input-two-output feedback system.

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

The paper shows that two polar coordinate transformations convert the original two-input-three-output second-order tracking model of underactuated marine vessels from Cartesian coordinates into a simpler two-input-two-output feedback system. This reduction makes it possible to apply backstepping control even though the transformed model does not meet the strict-feedback requirement. To prevent singularities in the orientation angle during controller design, the author introduces exponential modification of orientation. A sympathetic reader would care because underactuated vessels are common in practice, where full actuation is costly or impossible, and better tracking methods could improve navigation reliability.

Core claim

By introducing two polar coordinate transformations, the original two-input-three-output second-order tracking model expressed in the Cartesian frame is reduced to a two-input-two-output feedback system. The resulting model does not necessarily satisfy the strict-feedback condition required by conventional backstepping approaches, so a novel concept termed exponential modification of orientation is proposed to circumvent potential singularities. The transformations also introduce angular coordinate singularities, which the method addresses directly.

What carries the argument

The two polar coordinate transformations that map the Cartesian tracking model to polar form and reduce it to two inputs and two outputs, together with exponential modification of orientation that adjusts the desired heading to avoid singularities.

If this is right

  • The reduced two-input-two-output model permits standard backstepping controller design despite the lack of strict-feedback form.
  • Exponential modification of orientation removes singularities in the orientation variable without adding new instability.
  • Numerical simulations confirm that the resulting controller achieves effective trajectory tracking.
  • The approach provides a systematic way to handle underactuated systems that would otherwise resist conventional nonlinear control.

Where Pith is reading between the lines

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

  • The same polar transformations and orientation adjustment could be tested on other nonholonomic vehicles such as wheeled robots or quadrotors to see if the dimensionality reduction generalizes.
  • Adding wave and current disturbances to the simulations would reveal how robust the closed-loop system remains under realistic marine conditions.
  • Combining this controller with an online path planner could produce a full autonomous navigation stack for vessels that must follow curved trajectories.

Load-bearing premise

The exponential modification of orientation successfully avoids singularities and enables stable backstepping without introducing new instabilities or performance losses.

What would settle it

A simulation or sea trial in which the vessel trajectory diverges or tracking error grows unbounded when the orientation approaches values that would normally trigger angular singularities, despite applying the exponential modification.

Figures

Figures reproduced from arXiv: 2604.04537 by Ji-Hong Li.

Figure 1
Figure 1. Figure 1: Illustration of two polar coordinate systems and related [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Corresponding control efforts comparison with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectory tracking results in both methods with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trajectory tracking results and corresponding control efforts [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

This paper investigates the trajectory tracking problem of underactuated marine vessels within a polar coordinate framework. By introducing two polar coordinate transformations (PCTs), the original two-input-three-output second-order tracking model expressed in the Cartesian frame is reduced to a two-input-two-output feedback system. However, the resulting model does not necessarily satisfy the strict-feedback condition required by conventional backstepping approaches. To circumvent potential singularities arising in the controller design, a novel concept termed exponential modification of orientation (EMO) is proposed. While the PCTs yield substantial structural simplification, they also introduce inherent limitations, most notably singularities associated with angular coordinates. Addressing these singularities constitutes another key focus of this paper. Numerical simulation results are presented to demonstrate the effectiveness of the proposed control strategy.

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 / 1 minor

Summary. The manuscript proposes a trajectory tracking controller for underactuated marine vessels. Two polar coordinate transformations (PCTs) are introduced to reduce the original two-input three-output second-order Cartesian model to a two-input two-output feedback system. A novel exponential modification of orientation (EMO) is proposed to avoid singularities that would otherwise prevent application of backstepping, and numerical simulations are presented to illustrate the resulting closed-loop behavior.

Significance. If the PCT reduction and EMO modification can be shown to preserve strict-feedback structure and yield a negative-definite Lyapunov derivative uniformly (including near r=0 and during heading wraps), the approach would offer a structurally simpler design route for a class of underactuated marine systems. The paper supplies no such derivation, stability proof, or quantitative performance data, so the practical significance cannot yet be assessed.

major comments (2)
  1. [Abstract / Controller Design] Abstract and Section on controller synthesis: the claim that EMO 'circumvents potential singularities' and restores the strict-feedback condition is asserted without an explicit definition of the modification law, a bound on its gain, or a proof that the composite Lyapunov function remains negative definite in a neighborhood of r=0 or when the heading error crosses ±π.
  2. [Numerical Simulations] Simulation section: only qualitative trajectory plots are mentioned; no quantitative metrics (e.g., RMS tracking error, settling time, control effort) or comparison against existing Cartesian or polar backstepping controllers are supplied, preventing evaluation of whether the claimed simplification yields measurable improvement.
minor comments (1)
  1. [Problem Formulation] Notation for the two PCTs and the EMO-modified orientation reference should be introduced with explicit equations rather than descriptive text only.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address the major comments point by point below. We will make the necessary revisions to strengthen the paper, particularly by providing additional details on the stability analysis and quantitative simulation results.

read point-by-point responses

  1. Referee: [Abstract / Controller Design] Abstract and Section on controller synthesis: the claim that EMO 'circumvents potential singularities' and restores the strict-feedback condition is asserted without an explicit definition of the modification law, a bound on its gain, or a proof that the composite Lyapunov function remains negative definite in a neighborhood of r=0 or when the heading error crosses ±π.

    Authors: We appreciate this observation. The manuscript explicitly states that the model after the two PCTs 'does not necessarily satisfy the strict-feedback condition required by conventional backstepping approaches.' Thus, we do not claim that EMO restores strict-feedback structure; instead, EMO is proposed to circumvent singularities in the orientation that would prevent the application of backstepping despite the non-strict-feedback form. The definition of the EMO law appears in the controller synthesis section. However, we agree that an explicit bound on the modification gain and a complete proof of uniform negative-definiteness of the Lyapunov derivative (covering r near 0 and heading error crossings of ±π) are not sufficiently detailed. We will revise the manuscript to include these elements, along with the full derivation showing how the composite Lyapunov function ensures stability. revision: yes

  2. Referee: [Numerical Simulations] Simulation section: only qualitative trajectory plots are mentioned; no quantitative metrics (e.g., RMS tracking error, settling time, control effort) or comparison against existing Cartesian or polar backstepping controllers are supplied, preventing evaluation of whether the claimed simplification yields measurable improvement.

    Authors: We agree that the simulation results, as currently presented, are qualitative in nature. To allow for a better assessment of the proposed method's performance, we will update the numerical simulations section in the revised manuscript to include quantitative metrics such as root-mean-square (RMS) tracking errors, settling times, and control effort measures. Furthermore, we will add comparative simulations against existing backstepping controllers in both Cartesian and polar coordinates to highlight any advantages in terms of simplicity and performance. revision: yes

Circularity Check

0 steps flagged

No circularity: new coordinate transformations and EMO concept introduced without reduction to fitted inputs or self-citations

full rationale

The paper's core contribution consists of proposing two polar coordinate transformations (PCTs) that explicitly change the original Cartesian tracking model into a 2I2O system, followed by the novel exponential modification of orientation (EMO) to address resulting singularities. These steps are constructive definitions and modifications rather than derivations that equate outputs to inputs by construction. No fitted parameters are renamed as predictions, no load-bearing self-citations justify uniqueness theorems, and no ansatz is smuggled via prior work. The abstract and context emphasize structural simplification via the new PCTs and EMO, with simulations for validation; the derivation chain remains self-contained and independent of circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The approach rests on standard nonlinear control assumptions plus two newly introduced entities whose effectiveness is asserted without external validation beyond simulations.

axioms (1)
  • standard math Backstepping requires strict-feedback form or a suitable modification to remain applicable
    Invoked when noting that the reduced model does not satisfy the strict-feedback condition.
invented entities (2)
  • Polar Coordinate Transformations (PCTs) no independent evidence
    purpose: Reduce the original 2-in-3-out Cartesian model to 2-in-2-out feedback form
    New transformations introduced in the paper to achieve structural simplification.
  • Exponential Modification of Orientation (EMO) no independent evidence
    purpose: Circumvent singularities in the controller design
    Novel concept proposed specifically to handle potential singularities arising from the transformations.

pith-pipeline@v0.9.0 · 5415 in / 1359 out tokens · 48562 ms · 2026-05-10T19:52:49.762744+00:00 · methodology

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Tracking control of an underactuated surface vessel,

    K. Y . Pettersen and H. Nijmeijer, “Tracking control of an underactuated surface vessel,” inProc. of the 37th IEEE Conference on Decision & Control, Tampa, Florida USA, 1998, pp. 4561–4566

    work page 1998

  2. [2]

    Underactuated ship tracking control: theory and experiments,

    K. Y . Pettersen and H. Nijmeijer, “Underactuated ship tracking control: theory and experiments,”Int. J. of Control, vol. 74, pp. 1435-1446, 2001

    work page 2001

  3. [3]

    Global tracking control of underactuated ships by Lya- punov’s direct method,

    Z. P. Jiang, “Global tracking control of underactuated ships by Lya- punov’s direct method,”Automatica, vol. 38, pp. 301–309, 2002

    work page 2002

  4. [4]

    Underactuated ship global tracking under relaxed conditions,

    K. D. Do, Z. P. Jiang, J. Pan, “Underactuated ship global tracking under relaxed conditions,”IEEE Transactions on Automatic Control, vol. 47, pp. 1529–1536, 2002

    work page 2002

  5. [5]

    New cascade approach for globalκ- exponential tracking of underactuated ships,

    T. C. Lee and Z. P. Jiang, “New cascade approach for globalκ- exponential tracking of underactuated ships,”IEEE Transactions on Automatic Control, vol. 49, pp. 2297–2303, 2004

    work page 2004

  6. [6]

    Path tracking of underactuated ships with general form of dynamics,

    J. H. Li, “Path tracking of underactuated ships with general form of dynamics,”Int. J. of Control, vol. 89, pp. 506–517, 2016

    work page 2016

  7. [7]

    Trajectory tracking control of planar underactuated vehicles,

    H. Ashrafiuon, S. Nersesov, G. Clayton, “Trajectory tracking control of planar underactuated vehicles,”IEEE Transactions on Automatic Control, vol. 62, pp. 1959–1965, 2017

    work page 1959

  8. [8]

    Nonlinear tracking of underactuated surface vessels,

    J. M. Godhavn, “Nonlinear tracking of underactuated surface vessels,” inProc. of the 35th IEEE Conference on Decision & Control, Kobe, Japan, 1996, pp. 975–980

    work page 1996

  9. [9]

    Globalκ-exponential way-point maneuvering of ships: Theory and experiments,

    E. Fredriksen and K. Y . Pettersen, “Globalκ-exponential way-point maneuvering of ships: Theory and experiments,”Automatica, vol. 42, pp. 677–687, 2006

    work page 2006

  10. [10]

    Point-to-point navigation of underactuated ships,

    J. H. Li, P. M. Lee, B. H. Jun, Y . K. Lim, “Point-to-point navigation of underactuated ships,”Automatica, vol. 44, pp. 3201-3205, 2008

    work page 2008

  11. [11]

    A minimum phase ouput in the exact tracking problem for the nonminimum phase underactuated surface ship,

    L. Consolini and M. Tosques, “A minimum phase ouput in the exact tracking problem for the nonminimum phase underactuated surface ship,”IEEE Transactions on Automatic Control, vol. 57, pp. 3174–3180, 2012

    work page 2012

  12. [12]

    Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncer- tainty,

    A. P. Aguiar and J. P. Hespanha, “Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncer- tainty,”IEEE Transactions on Automatic Control, vol. 52, pp. 1362- 1379, 2007

    work page 2007

  13. [13]

    Trajectory tracking and path following for underactuated marine vehi- cles,

    C. Paliotta, E. Lefeber, K. Y . Pettersen, J. Pinto, M. Costa, J. Sousa, “Trajectory tracking and path following for underactuated marine vehi- cles,”IEEE Transactions on Control Systems Technology, vol. 27, pp. 1423–1437, 2019

    work page 2019

  14. [14]

    Robust adaptive path following of underactuated ships,

    K. D. Do, Z. P. Jiang, J. Pan, “Robust adaptive path following of underactuated ships,”Automatica, vol. 40, pp. 929–944, 2004

    work page 2004

  15. [15]

    Global robust adaptive path following of underactuated ships,

    K. D. Do and J. Pan, “Global robust adaptive path following of underactuated ships,”Automatica, vol. 42, pp. 1713–1722, 2006

    work page 2006

  16. [16]

    On uniform semiglobal exponential stability (USGES) of proportional line-of-sight guidance laws,

    T. I. Fossen and K. Y . Pettersen, “On uniform semiglobal exponential stability (USGES) of proportional line-of-sight guidance laws,”Auto- matica, vol. 50, pp. 2912–2917, 2014

    work page 2014

  17. [17]

    Path following and time-varying feedback stabilization of a wheeled mobile robot,

    C. Samson, “Path following and time-varying feedback stabilization of a wheeled mobile robot,” inProc. of ICARCV’92, Singapore, 1992

    work page 1992

  18. [18]

    A review of path following control strategies for autonomous robotic vehicles: Theory, simulations, and experiments,

    N. Hung, F. Rego, J. Quintas, J. Cruz, M. Jacinto, D. Souto, A. Potes, L. Sebastiao, A. Pascoal, “A review of path following control strategies for autonomous robotic vehicles: Theory, simulations, and experiments,” Journal of Field Robotics, vol. 40, pp. 747–779, 2023

    work page 2023

  19. [19]

    Trajectory tracking control for uncertain underactuated surface vessels with guaranteed prescribed performance under stochastic disturbances,

    C. Dong, B. Zheng, L. Chen, “Trajectory tracking control for uncertain underactuated surface vessels with guaranteed prescribed performance under stochastic disturbances,”Nonlinear Dynamics, vol. 112, pp. 13215–13231, 2024

    work page 2024

  20. [20]

    Krstic, I

    M. Krstic, I. Kanellakopoulos, P. Kokotovic.Nonlinear and Adaptive Control Design. John Wiley & Sons, Inc., New York, 1995

    work page 1995

  21. [21]

    Asymptotic tra- jectory tracking of underactuated non-minimum phase marine vessels,

    J. H. Li, H. Kang, M. G. Kim, M. J. Lee, G. R. Cho, “Asymptotic tra- jectory tracking of underactuated non-minimum phase marine vessels,” IFAC PapersOnLine, vol. 55, pp. 281–286, 2022

    work page 2022

  22. [22]

    Tra- jectory tracking performance transition analysis from ploar to Cartesian coordinates,

    J. H. Li, H. Kang, M. G. Kim, H. S. Jin, M. J. Lee, G. R. Cho, “Tra- jectory tracking performance transition analysis from ploar to Cartesian coordinates,”IFAC PapersOnLine, vol. 56, pp. 11615–11620, 2023

    work page 2023

  23. [23]

    3D trajectory tracking of underactuated non-minimum phase underwater vehicles,

    J. H. Li, “3D trajectory tracking of underactuated non-minimum phase underwater vehicles,”Automatica, vol. 155, 111149, 2023

    work page 2023

  24. [24]

    J. H. Li, H. Kang, M. G. Kim, M. J. Lee, H. S. Jin, G. R. Cho, ”Twoing type of 3D trajectory tracking for a class of underactuated autonomous underwater vehicles,” inProc. of American Control Conference, Denvor, CO, pp. 3770–3775, 2025

    work page 2025

  25. [25]

    Uniting control Lyapunov and control barrier functions,

    M. Z. Romdlony and B. Jayawardhana, “Uniting control Lyapunov and control barrier functions,” inProc. of 53rd IEEE CDC, Los Angeles, CA, 2014, pp. 2293–2298

    work page 2014

  26. [26]

    Control Barrier Function Based Quadratic Programs for Safety Critical Systems,

    A. D. Ames, X. Xu, J. W. Grizzle, P. Tabuada, “Control Barrier Function Based Quadratic Programs for Safety Critical Systems,”IEEE Trans. on Automatic Control, vol. 62, no. 8, pp. 3861–3876, 2017

    work page 2017

  27. [27]

    Control Barrier Functions: Theory and Applications,

    A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, P. Tabuada, “Control Barrier Functions: Theory and Applications,” inProc. of 18th European Control Conference (ECC), Napoli, Italy, 2019, pp. 3420–3431

    work page 2019

  28. [28]

    T. I. Fossen.Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons. Ltd, 2011

    work page 2011

  29. [29]

    J-J. E. Slotine and W. Li.Applied Nonlinear Control. Prentice-Hall Inc., New Jersey, 1991

    work page 1991

  30. [30]

    J. N. Newman.Marine Hydrodynamics. The MIT Press, Cambridge, MA, 1977

    work page 1977

  31. [31]

    M. M. Polycarpou, Stable adaptive neural control scheme for nonlinear systems,IEEE Transactions on Automatic Control, vol. 41, pp. 447–451, 1996

    work page 1996

  32. [32]

    Control Barrier Functions for Singularity Avoidance in Passivity-Based Manipulator Control,

    V . Kurtz, P. M. Wensing, H. Lin, “Control Barrier Functions for Singularity Avoidance in Passivity-Based Manipulator Control,” inProc. of 60th IEEE Conference on Decision and Control (CDC), Austin, Texas, 2021, pp. 6125–6130

    work page 2021

  33. [33]

    Singularity-Avoidance Control of Robotic Systems with Model Mismatch and Actuator Constraints,

    M. Wu, A. Rupenyan, B. Corves, “Singularity-Avoidance Control of Robotic Systems with Model Mismatch and Actuator Constraints,” inProc. of 23rd European Control Conference (ECC), Thessaloniki, Greece, 2025, pp. 2545–2550

    work page 2025

  34. [34]

    Adaptive Robust Quadratic Programs using Control Lyapunov and Barrier Functions,

    P. Zhao, Y . Mao, C. Tao, N. Hovakimyan, X. Wang, “Adaptive Robust Quadratic Programs using Control Lyapunov and Barrier Functions,” in Proc. of 59th IEEE Conference on Decision and Control (CDC), Jeju Island, Republic of Korea, 2020, pp. 3353–3358

    work page 2020

  35. [35]

    Robust Control Barrier Functions Using Uncertainty Estimation With Application to Mobile Robots,

    E. Das and J-W. Burdick, “Robust Control Barrier Functions Using Uncertainty Estimation With Application to Mobile Robots,”IEEE Transactions on Automatic Control, vol. 70, no. 7, pp. 4766–4773, 2025

    work page 2025

  36. [36]

    Adaptive safety-critical control for a class of nonlinear systems with parametric uncertainties: A control barrier function approach,

    Y . Wang and X. Xu, “Adaptive safety-critical control for a class of nonlinear systems with parametric uncertainties: A control barrier function approach,”Systems&Control Letters, vol. 188, 105798, 2024

    work page 2024