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arxiv: 2510.11413 · v2 · submitted 2025-10-13 · 📡 eess.SY · cs.SY

Trajectory control of a suspended load with non-stopping flying carriers

Sofia Girardello , Giulia Michieletto , Angelo Cenedese , Antonio Franchi , Chiara Gabellieri This is my paper

Pith reviewed 2026-05-18 07:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords cooperative aerial transportsuspended load controlnon-stopping carrierswrench feedback controlinternal force optimizationgrasp matrixtrajectory trackingaerial robotics
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The pith

A control framework lets flying carriers transport a suspended load while keeping continuous motion.

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

This paper establishes a closed-loop method for multiple flying carriers to move a suspended load along a chosen path without any carrier ever stopping. A feedback wrench controller computes the total force and torque needed at the load to follow the desired pose. An optimization layer then uses the redundancy in how forces are distributed among carriers to adjust internal forces so that each carrier's velocity stays nonzero. The net wrench on the load itself remains exactly the one required for tracking. A reader would care because stopping carriers can cause slack cables or instability in practical transport tasks, and this approach removes that requirement while preserving accurate load motion.

Core claim

The work presents the first closed-loop control framework for cooperative payload transportation with non-stopping flying carriers. It includes a feedback wrench-controller that actively regulates the load's pose by computing the wrench required for tracking its desired pose trajectory. Building upon grasp-matrix formulation and internal force redundancy, an optimization layer dynamically shapes internal-force parameters to guarantee persistent carrier motion while not altering the desired load wrench. The desired non-stopping carrier trajectories are computed using the system's kinematics and desired cable forces.

What carries the argument

The optimization layer that reshapes internal-force parameters using grasp-matrix formulation and force redundancy to enforce nonzero carrier velocities without changing the net load wrench.

If this is right

  • Carrier trajectories are generated directly from kinematics and the cable forces needed for the load.
  • The load pose tracks its reference while internal adjustments keep all carriers moving.
  • Numerical simulations confirm both persistent motion and successful load tracking.
  • The net wrench delivered to the load stays identical to the one computed by the wrench controller.

Where Pith is reading between the lines

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

  • The same internal-force shaping could be tested on physical quadrotors to measure how well non-stopping holds under wind or model error.
  • Extending the redundancy count might allow the method to handle more carriers or different cable lengths without redesign.
  • If the optimization remains feasible across changing load masses, the framework could support variable-payload scenarios.

Load-bearing premise

The optimization layer can always find internal-force parameters that guarantee persistent carrier motion for any desired load trajectory without changing the net wrench applied to the load.

What would settle it

Run the controller on a chosen load trajectory and check whether any carrier velocity drops to zero or the load pose error grows beyond the level seen without the non-stopping constraint.

Figures

Figures reproduced from arXiv: 2510.11413 by Angelo Cenedese, Antonio Franchi, Chiara Gabellieri, Giulia Michieletto, Sofia Girardello.

Figure 1
Figure 1. Figure 1: Trajectory tracking of a suspended load using non-stopping [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overall control scheme including the outer-loop load-wrench controller (green block), the inner-loop carrier-trajectory generator [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Components of the desired load trajectory. The load is [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental validation of the proposed model using three [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Load tracking errors without velocity optimization. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Carrier velocity norms without optimization. Carrier veloci [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 5
Figure 5. Figure 5: Carrier trajectories (colored) without velocity optimization. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 9
Figure 9. Figure 9: Load tracking errors with velocity optimization. [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Carrier velocity norms with optimization. Carrier velocities [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Top-view of the trajectories. The load position (black) starts [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
read the original abstract

This work presents the first closed-loop control framework for cooperative payload transportation with non-stopping flying carriers. The proposed method includes a feedback wrench-controller that actively regulates the load's pose by computing the wrench required for tracking its desired pose trajectory. Building upon grasp-matrix formulation and internal force redundancy, an optimization layer dynamically shapes internal-force parameters to guarantee persistent carrier motion, while not altering the desired load wrench. The desired non-stopping carrier's trajectories are computed using the system's kinematics and desired cable forces. Numerical simulations demonstrate that the method successfully prevents the carriers from stopping, while achieving a successful tracking of the desired load trajectory.

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 / 2 minor

Summary. The manuscript presents a closed-loop control framework for cooperative transportation of a suspended load by multiple flying carriers that ensures carriers maintain non-zero velocities. It combines a feedback wrench-controller that computes the wrench needed to track a desired load pose trajectory with an optimization layer that dynamically selects internal-force parameters in the nullspace of the grasp matrix to enforce persistent carrier motion while exactly preserving the net load wrench. Desired carrier trajectories are obtained from system kinematics and cable forces, and the approach is illustrated with numerical simulations that show successful load tracking without carrier stops.

Significance. If the optimization layer is shown to be feasible for arbitrary trajectories, the work would contribute to practical aerial payload systems by mitigating risks associated with carrier stopping, such as cable slack or reduced controllability. The approach extends standard grasp-matrix and redundancy-resolution techniques from multi-robot manipulation literature with a dynamic shaping of internal forces; the numerical demonstrations provide preliminary support, though additional validation would be needed to establish broader applicability.

major comments (1)
  1. [Optimization layer (as described in Abstract and method sections)] The central claim that the optimization layer 'guarantees persistent carrier motion, while not altering the desired load wrench' (Abstract) rests on the unproven assumption that the feasible set for internal-force parameters is always non-empty. The non-zero velocity inequalities imposed on the nullspace vectors may become infeasible for certain wrench directions, cable lengths, or carrier configurations, in which case the layer cannot simultaneously satisfy both the non-stopping requirement and exact wrench preservation. A feasibility analysis, sufficient conditions, or explicit handling of infeasible cases is required to support the guarantee.
minor comments (2)
  1. [Numerical simulations] Numerical simulations are presented without error bars, disturbance rejection tests, or baseline comparisons (e.g., methods that permit stopping), which limits assessment of robustness and practical advantage.
  2. [Method formulation] Notation for the grasp matrix, internal forces, and wrench mapping should include explicit definitions and citations to foundational references in the first use.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive review. The comment on the optimization layer highlights an important theoretical aspect that strengthens the manuscript. We address it directly below and commit to revisions that enhance the rigor of the claims without altering the core contribution.

read point-by-point responses

  1. Referee: [Optimization layer (as described in Abstract and method sections)] The central claim that the optimization layer 'guarantees persistent carrier motion, while not altering the desired load wrench' (Abstract) rests on the unproven assumption that the feasible set for internal-force parameters is always non-empty. The non-zero velocity inequalities imposed on the nullspace vectors may become infeasible for certain wrench directions, cable lengths, or carrier configurations, in which case the layer cannot simultaneously satisfy both the non-stopping requirement and exact wrench preservation. A feasibility analysis, sufficient conditions, or explicit handling of infeasible cases is required to support the guarantee.

    Authors: We agree that the current formulation assumes feasibility of the internal-force optimization without providing a formal analysis of when the feasible set remains non-empty. The optimization selects nullspace parameters to satisfy non-zero velocity inequalities while exactly preserving the net wrench, but this can indeed become infeasible for certain wrench directions or configurations. In the revised manuscript we will add a dedicated subsection deriving sufficient conditions for feasibility based on the rank and geometry of the nullspace of the grasp matrix together with bounds on cable lengths and admissible wrench directions. We will also include a brief discussion of practical handling for rare infeasible cases, such as a temporary relaxation of the velocity lower bound or a smooth transition to a stopping-permitted mode, while preserving closed-loop stability. These additions will be supported by additional numerical examples that probe boundary cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper proposes a control architecture that separates net load wrench (from feedback controller) from internal forces via the grasp matrix nullspace, then uses an optimization layer to select internal-force parameters enforcing non-zero carrier velocities. This separation is a standard property of the grasp matrix formulation in cooperative manipulation literature and does not reduce to a self-definition or fitted prediction. No equations in the provided abstract or description equate a claimed result to its own inputs by construction, nor do they rely on load-bearing self-citations that themselves assume the target outcome. The feasibility of the optimization for arbitrary trajectories is an unproven assumption rather than a tautological step, placing any concern under correctness rather than circularity. The derivation remains self-contained as a proposed controller design.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard multi-robot grasp-matrix assumptions and the existence of internal-force redundancy that can be shaped without affecting net wrench; no new entities are postulated and no free parameters are explicitly fitted in the abstract.

axioms (2)
  • domain assumption The system possesses sufficient internal force redundancy to allow parameter shaping that enforces non-zero carrier velocities while preserving the desired net wrench on the load.
    Invoked when the optimization layer is said to 'guarantee persistent carrier motion, while not altering the desired load wrench.'
  • domain assumption Carrier trajectories can be computed from system kinematics and desired cable forces without violating actuator limits or cable tension constraints.
    Stated in the sentence 'The desired non-stopping carrier's trajectories are computed using the system's kinematics and desired cable forces.'

pith-pipeline@v0.9.0 · 5638 in / 1497 out tokens · 36413 ms · 2026-05-18T07:58:37.231010+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Review of aerial transportation of suspended-cable payloads with quadrotors,

    J. Estevez, G. Garate, J. M. Lopez-Guede, and M. Larrea, “Review of aerial transportation of suspended-cable payloads with quadrotors,” Drones, vol. 8, no. 2, p. 35, 2024

    work page 2024

  2. [2]

    Flying robots,

    S. Leutenegger, C. H ¨urzeler, A. K. Stowers, K. Alexis, M. W. Achtelik, D. Lentink, P. Y . Oh, and R. Siegwart, “Flying robots,”Springer Handbook of Robotics, pp. 623–670, 2016

    work page 2016

  3. [3]

    Conceptual design of fixed wing- vtol uav for aed transport,

    W. Saengphet and C. Thumthae, “Conceptual design of fixed wing- vtol uav for aed transport,” inThe 7th TSME International Conference on Mechanical Engineering, pp. 1–16, 2016

    work page 2016

  4. [4]

    Design and control of an ultra-low-cost logistic delivery fixed-wing uav,

    Y . Zhang, Q. Zhao, P. Mao, Q. Bai, F. Li, and S. Pavlova, “Design and control of an ultra-low-cost logistic delivery fixed-wing uav,”Applied Sciences, vol. 14, no. 11, p. 4358, 2024

    work page 2024

  5. [5]

    An online trajectory planning approach for a quadrotor uav with a slung payload,

    B. Xian, S. Wang, and S. Yang, “An online trajectory planning approach for a quadrotor uav with a slung payload,”IEEE Transactions on Industrial Electronics, vol. 67, no. 8, pp. 6669–6678, 2020

    work page 2020

  6. [6]

    Slung load transporta- tion with a single aerial vehicle and disturbance removal,

    P. Pereira, M. Herzog, and D. Dimarogonas, “Slung load transporta- tion with a single aerial vehicle and disturbance removal,” in2016 24th Mediterranean Conference on Control and Automation (MED), pp. 671–676, IEEE, 2016

    work page 2016

  7. [7]

    Trajectory generation and control of a quadrotor with a cable-suspended load—a differentially- flat hybrid system,

    K. Sreenath, N. Michael, and V . Kumar, “Trajectory generation and control of a quadrotor with a cable-suspended load—a differentially- flat hybrid system,” in2013 IEEE International Conference on Robotics and Automation (ICRA), pp. 4888–4895, IEEE, 2013

    work page 2013

  8. [8]

    Cooperative manipulation and transportation with aerial robots,

    N. Michael, J. Fink, and V . Kumar, “Cooperative manipulation and transportation with aerial robots,”Autonomous Robots, vol. 30, pp. 73– 86, 2011

    work page 2011

  9. [9]

    Cooperative transportation of cable-suspended slender payload using two quadrotors,

    T. Chen and J. Shan, “Cooperative transportation of cable-suspended slender payload using two quadrotors,” in2019 IEEE International Conference on Unmanned Systems (ICUS), pp. 432–437, IEEE, 2019

    work page 2019

  10. [10]

    Cooperative transportation of a payload using quadrotors: A reconfigurable cable-driven parallel robot,

    C. Masone, H. B ¨ulthoff, and P. Stegagno, “Cooperative transportation of a payload using quadrotors: A reconfigurable cable-driven parallel robot,” in2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1623–1630, IEEE, 2016

    work page 2016

  11. [11]

    Cooperative trans- portation of a payload using quadrotors: A reconfigurable cable- driven parallel robot,

    C. Masone, H. H. B ¨ulthoff, and P. Stegagno, “Cooperative trans- portation of a payload using quadrotors: A reconfigurable cable- driven parallel robot,” in2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1623–1630, IEEE, 2016

    work page 2016

  12. [12]

    Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing,

    G. Li, R. Ge, and G. Loianno, “Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing,”IEEE Robotics and Automation Letters, vol. 6, no. 3, pp. 5316–5323, 2021

    work page 2021

  13. [13]

    Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty,

    C. Gabellieri, M. Tognon, D. Sanalitro, and A. Franchi, “Equilibria, stability, and sensitivity for the aerial suspended beam robotic system subject to parameter uncertainty,”IEEE Transactions on Robotics, vol. 39, no. 5, pp. 3977–3993, 2023

    work page 2023

  14. [14]

    On the existence of static equilibria of a cable-suspended load with non-stopping flying carriers,

    C. Gabellieri and A. Franchi, “On the existence of static equilibria of a cable-suspended load with non-stopping flying carriers,” in2024 International Conference on Unmanned Aircraft Systems (ICUAS), p. 638–644, IEEE, June 2024

    work page 2024

  15. [15]

    Cyclic Nullspace Coordination: Perpetual Flight of Aerial Carriers for Static Suspension

    C. Gabellieri and A. Franchi, “Coordinated trajectories for non- stop flying carriers holding a cable-suspended load,”arXiv preprint arXiv:2503.03481, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  16. [16]

    Crazyswarm: A large nano-quadcopter swarm,

    J. A. Preiss*, W. H ¨onig*, G. S. Sukhatme, and N. Ayanian, “Crazyswarm: A large nano-quadcopter swarm,” in2017 ICRA, pp. 3299–3304, 2017

    work page 2017