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arxiv: 2605.07003 · v1 · submitted 2026-05-07 · 💻 cs.RO · cs.SY· eess.SY

AirBender: Adaptive Transportation of Bendable Objects Using Dual UAVs

Jiawei Xu , Longsen Gao , Rafael Fierro , David Salda\~na This is my paper

Pith reviewed 2026-05-11 01:10 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords adaptive controldual UAVsbendable objectstrajectory trackingLyapunov stabilityaerial transportationdeformable payloadsmodel-free control
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The pith

Two UAVs can transport a bendable object along a trajectory by adapting in real time to its unknown flexibility without an elasticity model.

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

This paper develops an adaptive controller for two unmanned aerial vehicles that collaboratively carry a bendable object while following a prescribed path. The controller updates its gains on the fly to match the object's unknown stretching and bending behavior, eliminating any need for a pre-built mathematical description of its elasticity. Lyapunov analysis establishes that the closed-loop system is asymptotically stable, so tracking errors go to zero as adaptation proceeds. Hardware experiments with actual drones and flexible loads confirm that the method maintains stability and performance across different flight scenarios where rigid-object assumptions would break down.

Core claim

An adaptive controller for a dual-UAV system transporting a bendable object achieves asymptotic stability and accurate trajectory tracking by estimating and compensating for the object's unknown deformable properties in real time, without requiring an explicit elasticity model; the stability result is shown via Lyapunov analysis and the practical performance is demonstrated in hardware experiments.

What carries the argument

An adaptive controller that estimates the object's unknown deformable properties online and adjusts its action to keep the coupled dual-UAV and object system stable and on the desired trajectory.

Load-bearing premise

The combined dynamics of the two UAVs and the bendable object permit the design of an adaptive controller whose stability can be proven by Lyapunov methods without an explicit elasticity model of the object.

What would settle it

A flight test in which the bendable object's deformation causes trajectory-tracking errors to grow unbounded or produces loss of stability despite the online adaptation law would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.07003 by David Salda\~na, Jiawei Xu, Longsen Gao, Rafael Fierro.

Figure 1
Figure 1. Figure 1: Two quadrotors actively control the endpoint trajectories of a long carbon-fiber strip using the proposed adaptive controller. Video: https://www.youtube.com/watch?v=lhTVIWSjhvQ be incorporated into robotic systems, such as: i) Nonlinear deformations are difficult to model and predict [9], [10]; ii) real-time sensing and feedback often require accurate detection of the shape and its deformation; iii) contr… view at source ↗
Figure 2
Figure 2. Figure 2: The vehicles apply forces on two ends of an object, resulting in different bending curves. We consider the world reference frame {𝑊 } and the body frame of each vehicle, {𝐵1 } and {𝐵2 }. holding the strip from its endpoints. The main contribution of this paper is the development of a stable trajectory-tracking controller that leverages the recursive least square (RLS) approximation to adapt to an unknown e… view at source ↗
Figure 3
Figure 3. Figure 3: The strip lies in a vertical plane , the displacement of its two endpoints being 𝒓. A. Trajectory-tracking Control via Force Adaptation We estimate the force 𝒇 𝑜 𝑖 using recursive least-square ap￾proximation [31]. Since the bendable object force depends on its endpoint displacement 𝒓, and the positional constraints ensure that the object curve remains in the same plane, we model an approximation function … view at source ↗
Figure 4
Figure 4. Figure 4: The object force in parallel and perpendicular to displacement vector in the plane . The force vectors on vehicle 1 omitted for illustration clarity. of the object, 𝛼, which is the angle between 𝒓 and the horizontal plane, 𝛼 = arctan ( 𝒓 ⋅ 𝒛 𝒓 ⋅ 𝒙 ) . (24) These insights motivate us to construct a new feature vector for improved force estimation. We first decompose the total force received by the vehicl… view at source ↗
Figure 6
Figure 6. Figure 6: The mean and standard deviation of the average endpoint position errors v.s. trial for tracking a varying endpoint distance. of position tracking errors and compare the performance of our adaptive control to that of the standard PID control. We evaluate the methods using three different trajectories. For simplicity, we refer to the system of the two quadrotors and the bendable strip as “the system” in this… view at source ↗
Figure 7
Figure 7. Figure 7: The mean and STD of the average endpoint position errors v.s. time for climbing up and passing one obstacle. The red cross marks where the PID controller with a low integral term fails to drive the strip through the obstacle. command the quadrotors to oscillate the endpoint displacement between 0.8 m and 0.4 m. This trajectory evaluates the strength and convergence speed of the force compensation methods d… view at source ↗
Figure 9
Figure 9. Figure 9: The mean and standard deviation of the average endpoint position errors v.s. time for passing two obstacles [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The mean and standard deviation of the average endpoint position errors v.s. trial for passing two obstacles. the bendable strip to pass through a circular window while bending it with a fixed  orientation, then pass through a second square window with a varying  orientation. Then, the quadrotors recover the endpoint displacement to the take￾off value before landing. This 120-second experiment emulates … view at source ↗
read the original abstract

The interaction of robots with bendable objects in midair presents significant challenges in control, often resulting in performance degradation and potential crashes, especially for aerial robots due to their limited actuation capabilities and constant need to remain airborne. This paper presents an adaptive controller that enables two aerial vehicles to collaboratively follow a trajectory while transporting a bendable object without relying on explicit elasticity models. Our method allows on-the-fly adaptation to the object's unknown deformable properties, ensuring stability and performance in trajectory-tracking tasks. We use Lyapunov analysis to demonstrate that our adaptive controller is asymptotically stable. Our method is evaluated through hardware experiments in various scenarios, demonstrating the capabilities of using multirotor aerial vehicles to handle bendable objects.

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 adaptive controller for two UAVs to collaboratively transport a bendable object along a trajectory. It claims that the method adapts online to the object's unknown deformable properties without an explicit elasticity model, that Lyapunov analysis establishes asymptotic stability of the closed-loop system, and that hardware experiments in multiple scenarios confirm practical performance.

Significance. If the stability result holds for the physical continuum object, the work would be significant for aerial manipulation, as it would demonstrate model-free adaptive control of deformable payloads with dual multirotors. The hardware validation would further strengthen the contribution, but the current lack of explicit controller equations, adaptation laws, and quantitative data limits assessment of the result's robustness.

major comments (1)
  1. [§IV] §IV (Lyapunov analysis): The stability proof treats the object's unknown deformable properties as a finite-dimensional vector of constant parameters that appear linearly in the error dynamics. For a truly bendable object the deformation is spatially distributed and governed by continuum mechanics; any finite parameterization is necessarily an approximation whose unmodeled residual dynamics are not shown to be dominated or accounted for in the V̇ ≤ 0 argument. Consequently the asymptotic stability guarantee does not automatically extend to the physical system.
minor comments (1)
  1. [Abstract] The abstract asserts Lyapunov stability and successful hardware experiments yet provides neither the controller equations, the adaptation law, nor any quantitative performance metrics or data tables; these details are required for independent verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback. We address the major comment on the Lyapunov analysis below and indicate planned revisions.

read point-by-point responses

  1. Referee: [§IV] §IV (Lyapunov analysis): The stability proof treats the object's unknown deformable properties as a finite-dimensional vector of constant parameters that appear linearly in the error dynamics. For a truly bendable object the deformation is spatially distributed and governed by continuum mechanics; any finite parameterization is necessarily an approximation whose unmodeled residual dynamics are not shown to be dominated or accounted for in the V̇ ≤ 0 argument. Consequently the asymptotic stability guarantee does not automatically extend to the physical system.

    Authors: We agree that the Lyapunov analysis establishes asymptotic stability only for the finite-dimensional parameterization of the deformable properties used in the model. This is necessarily an approximation to the spatially distributed continuum mechanics of a bendable object, and the proof does not explicitly dominate or bound the residual dynamics in the infinite-dimensional case. The manuscript presents the result under the modeling assumptions of constant parameters appearing linearly in the error dynamics. To improve clarity, we will revise §IV to explicitly state the scope of the stability guarantee and add a short discussion of the approximation. The hardware experiments across scenarios provide supporting evidence of practical robustness. revision: partial

Circularity Check

0 steps flagged

No significant circularity in adaptive controller derivation

full rationale

The paper designs an adaptive controller for dual-UAV transport of a bendable object that adapts online to unknown properties without an explicit elasticity model, then applies standard Lyapunov analysis to the resulting closed-loop error dynamics to conclude asymptotic stability. This chain relies on conventional adaptive control theory (parameter estimation laws and a Lyapunov candidate whose derivative is made negative semi-definite) rather than any self-definitional reduction, fitted input renamed as prediction, or load-bearing self-citation. The finite-dimensional parameterization of deformation is an explicit modeling assumption, not a circular re-use of the target result. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that adaptive control can achieve stability for this coupled system without an explicit deformation model; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The dual-UAV and bendable-object system dynamics admit an adaptive controller whose stability can be proven via Lyapunov analysis without an explicit elasticity model.
    Directly implied by the claim of model-free adaptation and asymptotic stability.

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discussion (0)

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