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arxiv: 2510.24457 · v2 · pith:FLE5W7MHnew · submitted 2025-10-28 · 💻 cs.RO · cs.SY· eess.SY

Flatness-based trajectory planning for 3D overhead cranes with friction compensation and collision avoidance

Jorge Vicente-Martinez (1) , Edgar Ramirez-Laboreo (1) ((1) Universidad de Zaragoza) This is my paper

Pith reviewed 2026-05-21 19:36 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords overhead cranesdifferential flatnesstrajectory planningfriction compensationcollision avoidance3D motionoptimal controlunderactuated systems
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The pith

Modeling nonlinear friction is required to generate fast safe trajectories for 3D overhead cranes that avoid collisions and actuator saturation.

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

The paper develops an optimal trajectory generation method for three-dimensional overhead cranes that exploits differential flatness to embed nonlinear dry friction and collision avoidance constraints for both payload and rope directly into the planning process. This setup permits aggressive motions while enforcing swing limits only at the terminal time. Comparative simulations show that trajectories computed without friction terms produce actuator saturation and collisions, while those that include friction succeed.

Core claim

By preserving differential flatness after the addition of nonlinear dry friction, the method recovers all states and inputs from chosen flat outputs and their derivatives, allowing an optimization problem to enforce friction dynamics, input limits, and geometric collision constraints without extra inversion steps; simulations confirm that omitting friction produces saturating torques and unsafe paths.

What carries the argument

Differential flatness of the crane model augmented with nonlinear dry friction, which expresses states and inputs algebraically from a small set of flat outputs so that path constraints become algebraic inequalities in the optimization.

If this is right

  • Crane trajectories can be planned that respect actuator limits while respecting friction and collision geometry.
  • Only terminal swing constraints are needed, enabling more aggressive intermediate motion than pointwise swing limits.
  • Neglecting friction in the planner reliably produces infeasible commands that saturate the motors or violate safety margins.
  • The same flatness-based optimizer can incorporate additional algebraic constraints such as rope length limits or obstacle geometries.

Where Pith is reading between the lines

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

  • The method could be tested on other underactuated mechanical systems whose flatness is preserved under similar friction models.
  • Parameter identification of the friction coefficients would become a prerequisite for reliable deployment.
  • Real-time replanning could use the same flat-output parameterization if friction parameters drift or obstacles move.

Load-bearing premise

The crane system stays differentially flat once nonlinear dry friction is included, so states and inputs can still be recovered directly from the flat outputs.

What would settle it

A closed-loop experiment or high-fidelity simulation in which a friction-compensated trajectory is executed on a physical or modeled crane and still produces actuator saturation or a detected collision.

Figures

Figures reproduced from arXiv: 2510.24457 by Edgar Ramirez-Laboreo (1) ((1) Universidad de Zaragoza), Jorge Vicente-Martinez (1).

Figure 1
Figure 1. Figure 1: Schematic diagram of the overhead crane. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Parameterization of the i-th obstacle model and visualization of the collision avoidance strategy. where the subscripts MIN and MAX denote the lower and upper bounds for each respective variable. These variables are obtained at each time from the state vector ¯x(t) using the inverse dynamics mapping (12)–(16). 2) Actuator Saturation: The control forces applied to the system are limited, hence, the followin… view at source ↗
Figure 3
Figure 3. Figure 3: Example of the trajectory optimization process. The figure shows [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Planned (target) vs. simulated trajectories for Scenario 1. Each [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectory generated on optimization (uff ) and simulated (uff + uf b) comparison for Scenario 1 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Planned vs. simulated trolley position and rope for NFM nominal [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Planned (target) vs. simulated trajectories for Scenario 2. Each [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Trajectory generated on optimization (uff ) and simulated (uff + uf b) comparison for Scenario 2 [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Analysis of the residual payload oscillation at stop time with [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 9
Figure 9. Figure 9: Randomly generated dry friction profiles for the robustness [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Analysis of the stopping time of the trolley and rope with respect [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
read the original abstract

This paper presents an optimal trajectory generation method for 3D overhead cranes by leveraging differential flatness. This framework enables the direct inclusion of complex physical and dynamic constraints, such as nonlinear friction and collision avoidance for both payload and rope. Our approach allows for aggressive movements by constraining payload swing only at the final point. A comparative simulation study validates our approach, demonstrating that neglecting dry friction leads to actuator saturation and collisions. The results show that friction modeling is a fundamental requirement for fast and safe crane trajectories.

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

1 major / 1 minor

Summary. The paper proposes an optimal trajectory generation method for 3D overhead cranes that leverages differential flatness to directly incorporate nonlinear dry friction compensation and collision avoidance constraints for both the payload and the rope. The framework permits aggressive motions by enforcing payload swing constraints only at the terminal point. A comparative simulation study is used to demonstrate that trajectories generated without friction modeling result in actuator saturation and collisions, supporting the conclusion that friction modeling is a fundamental requirement for fast and safe crane operation.

Significance. If the differential-flatness property is shown to survive the inclusion of discontinuous friction and the simulation results are quantitatively substantiated, the work would offer a practical advance in crane motion planning by enabling constraint-aware trajectory optimization without dynamic inversion. Explicit credit is due for attempting to embed both friction and geometric collision constraints inside a flatness-based planner rather than treating them as post-processing corrections.

major comments (1)
  1. [Dynamics and flatness property] No section or equation demonstrates the algebraic inversion formulas required to recover states and inputs from the flat outputs once nonlinear dry friction (e.g., a term proportional to sign(velocity)) is inserted into the dynamics. The sign discontinuity generally destroys the finite-order differentiability and algebraic invertibility presupposed by standard flatness-based constraint enforcement and collision-avoidance formulations.
minor comments (1)
  1. [Abstract] The abstract and provided material describe a comparative simulation study yet supply neither parameter values, quantitative performance metrics, nor any tabulated results, preventing independent assessment of the claimed improvements in saturation avoidance and collision-free behavior.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and for acknowledging the practical value of embedding friction and collision constraints directly within a flatness-based planner. We address the single major comment below and will revise the manuscript to provide the requested explicit derivations.

read point-by-point responses

  1. Referee: [Dynamics and flatness property] No section or equation demonstrates the algebraic inversion formulas required to recover states and inputs from the flat outputs once nonlinear dry friction (e.g., a term proportional to sign(velocity)) is inserted into the dynamics. The sign discontinuity generally destroys the finite-order differentiability and algebraic invertibility presupposed by standard flatness-based constraint enforcement and collision-avoidance formulations.

    Authors: We appreciate the referee highlighting the need for explicit algebraic inversion formulas. In the revised manuscript we will add a dedicated subsection that derives the state and input recovery expressions when the nonlinear dry-friction term (proportional to sign of trolley velocity) is included in the dynamics. The flat outputs remain the three-dimensional payload position together with the trolley horizontal coordinates; the required trolley force is recovered by direct algebraic rearrangement of the translational dynamics, subtracting the friction term evaluated at the current velocity sign. To preserve the required differentiability for constraint enforcement, trajectories are generated over intervals of constant velocity sign, with C^2 continuity imposed at the (finitely many) switching instants. Collision-avoidance inequalities, expressed solely in the flat outputs and their derivatives, are therefore unaffected. This explicit construction shows that the flatness property is retained in a piecewise sense and that the original optimization problem remains well-posed without post-hoc correction. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on established flatness property

full rationale

The paper applies the standard differential-flatness property of 3D overhead-crane dynamics to generate trajectories that embed nonlinear friction and collision constraints. No equation or section reduces a claimed result to a fitted parameter or to a self-citation that itself depends on the target claim. The comparison between friction-inclusive and friction-neglected planners is performed by direct simulation of the same flat-output parameterization, which is independent of the paper's own fitted values. The derivation chain therefore remains self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the method relies on the standard assumption of differential flatness for the crane model.

pith-pipeline@v0.9.0 · 5624 in / 998 out tokens · 97028 ms · 2026-05-21T19:36:18.568269+00:00 · methodology

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Reference graph

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