Modeling and Control of a Pneumatic Morphing Soft Quadrotor based on the SOFA Framework for Dynamic Soft Robotic Simulation

B. Vanderborght; F. Labra Caso; G. Nikolakopoulos; J. Haluska; P. Ferrentino; V. Sumathy

arxiv: 2605.21031 · v2 · pith:NGPU7NR5new · submitted 2026-05-20 · 💻 cs.RO

Modeling and Control of a Pneumatic Morphing Soft Quadrotor based on the SOFA Framework for Dynamic Soft Robotic Simulation

F. Labra Caso , V. Sumathy , P. Ferrentino , B. Vanderborght , J. Haluska , G. Nikolakopoulos This is my paper

Pith reviewed 2026-05-25 06:16 UTC · model grok-4.3

classification 💻 cs.RO

keywords soft roboticsquadrotorfinite element methodpneumatic actuationSOFA frameworkmorphingdynamic simulationcontrol

0 comments

The pith

A SOFA-based finite element method models pneumatic morphing soft quadrotor arms while preserving traditional quadrotor dynamics.

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

The paper develops a simulation framework for a quadrotor with soft pneumatic arms that can change shape. It applies finite element analysis inside the SOFA environment to represent the arms as meshes whose internal forces come from an elastic material law. The approach keeps the standard physical equations and control laws of rigid quadrotors unchanged. Pneumatic pressure is applied inside arm cavities and a proportional-integral controller is used to drive the arms to target positions. Simulation tests show the model can handle both periodic actuation and closed-loop position commands.

Core claim

The proposed SOFA based finite element method for the soft body modeling preserves the physical interpretability and control structure of traditional quadrotor dynamics, while capturing the complex, time-varying behavior of pneumatically actuated soft arms. In SOFA, the soft pneumatically actuated arms are discretized as a tetrahedral mesh following an elastic material law that produces internal forces adequate to the real dynamic behavior of the body. Pneumatic actuation governed by both periodic and error-based control signals is applied within the internal cavities to analyze the morphing capability. A proportional-integral controller is proposed to study the controlled dynamic behavior.

What carries the argument

SOFA-based finite element discretization of soft arms as tetrahedral meshes under an elastic material law that computes internal forces and accepts pneumatic actuation inputs.

Load-bearing premise

The tetrahedral mesh discretization following an elastic material law produces internal forces adequate to the real dynamic behavior of the pneumatic arms.

What would settle it

Quantitative comparison of simulated arm tip positions and deformation shapes against measurements from a physical pneumatic prototype driven by the same pressure signals.

Figures

Figures reproduced from arXiv: 2605.21031 by B. Vanderborght, F. Labra Caso, G. Nikolakopoulos, J. Haluska, P. Ferrentino, V. Sumathy.

**Figure 1.** Figure 1: Visualization of the presented morphing pneumatic quadrotor design in SOFA at initial state (b) and during [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Overview of the components in a SOFA scene [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Soft Morphing Quadrotor at initial state (a) and during actuation (b). The Soft Pneumatic Arms connect [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: (a) The different parts of the UAV arm modelled in SOFA. The pneumatic actuator (white), the pneumatic [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: Experimental setup of the periodic control for [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Pneumatic actuation effect under the passive [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 8.** Figure 8: Experimental setup of the pressure controller [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 10.** Figure 10: Distributed control signal i.e. individual cavity pressure for every quadrant experiment [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

read the original abstract

This article presents a novel SOFA based finite element method for the soft body modeling and the corresponding dynamic simulation and control of a pneumatic morphing soft quadrotor. The proposed modeling preserves the physical interpretability and control structure of traditional quadrotor dynamics, while capturing the complex, time-varying behavior of pneumatically actuated soft arms. In SOFA, the soft pneumatically actuated arms are discretized as a tetrahedral mesh following an elastic material law that produces internal forces adequate to the real dynamic behavior of the body. Pneumatic actuation governed by both periodic and error-based control signals is applied within the internal cavities to analyze the morphing capability. Finally, a proportional-integral controller is proposed to study the controlled dynamic behavior and morphing capabilities of the pneumatic arm, wherein the pneumatic actuation to the soft arm is controlled to achieve the desired target position. The simulation results show the effectiveness of the proposed novel modeling framework and the related controller design.

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

This applies the existing SOFA framework to a pneumatic morphing soft quadrotor but asserts model adequacy to real dynamics without any supporting data or validation.

read the letter

The paper takes the SOFA finite element simulator and applies it to a new robot concept: a quadrotor with pneumatic soft arms that can morph. The arms are modeled as tetrahedral meshes under elastic material laws, with pressure applied inside cavities. A standard PI controller then drives the arms toward target positions while the overall vehicle keeps its familiar quadrotor dynamics structure. Periodic and error-based actuation signals are used to explore the morphing behavior in simulation. The main positive is that the control architecture stays interpretable on top of the soft-body model, which could help researchers already comfortable with rigid quadrotor methods extend them to soft versions. The description of how the cavities are pressurized is concrete enough to serve as a setup example for similar simulations. The clear weakness is the total lack of quantitative grounding. The abstract states that the discretization produces internal forces adequate to real dynamic behavior, yet no error metrics, tip trajectories, force comparisons, parameter identification steps, or hardware data appear. All reported outcomes stay inside the assumed model, so the central claim about matching physical pneumatic arms is not tested. The citation pattern is ordinary for SOFA applications and raises no red flags. This work is mainly useful to the small group already running SOFA simulations who want an aerial soft-robotics example. Readers looking for validated models or new control results will not get much. I would not bring it to a reading group and would not cite it. It does not deserve peer review in this form because the modeling claim rests on an unexamined premise rather than evidence.

Referee Report

2 major / 1 minor

Summary. The paper presents a SOFA-based finite element method for modeling the soft pneumatically actuated arms of a morphing quadrotor as tetrahedral meshes obeying elastic material laws. Pneumatic actuation is applied via periodic and error-based signals inside the cavities; a standard PI controller is used to drive the arms to target positions. The central claim is that this discretization preserves the physical interpretability and control structure of rigid quadrotor dynamics while capturing the time-varying soft-body behavior, with simulation results asserted to demonstrate effectiveness.

Significance. If the tetrahedral-mesh model were shown to reproduce measured pneumatic-arm dynamics, the framework could supply a reusable simulation environment for controller synthesis on soft aerial robots. The retention of conventional quadrotor control structure is a modest conceptual advantage, but the work relies entirely on internal simulation outputs under an unvalidated constitutive law and supplies no quantitative metrics or comparisons.

major comments (2)

[Modeling description] Modeling description (abstract and § on SOFA discretization): the assertion that the tetrahedral mesh with elastic material law 'produces internal forces adequate to the real dynamic behavior of the pneumatic arms' is stated without any supporting evidence. No experimental tip trajectories, force measurements, or parameter-identification procedure from physical prototypes is described; all reported results remain internal to the assumed model.
[Simulation results] Simulation results paragraph (abstract): the statement that 'simulation results show the effectiveness of the proposed novel modeling framework and the related controller design' is unsupported by any quantitative metrics, error norms, convergence times, baseline comparisons (e.g., rigid-body model or alternative soft-body formulations), or statistical measures. This leaves the central claim of demonstrated effectiveness without measurable grounding.

minor comments (1)

[Modeling] Clarify the mesh resolution (number of tetrahedra, element size) and the specific elastic constitutive parameters employed, as these directly affect reproducibility of the reported trajectories.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Modeling description] Modeling description (abstract and § on SOFA discretization): the assertion that the tetrahedral mesh with elastic material law 'produces internal forces adequate to the real dynamic behavior of the pneumatic arms' is stated without any supporting evidence. No experimental tip trajectories, force measurements, or parameter-identification procedure from physical prototypes is described; all reported results remain internal to the assumed model.

Authors: We acknowledge that the manuscript provides no experimental data, tip trajectories, or parameter identification from physical prototypes to support the claim of adequacy to real dynamic behavior. The tetrahedral discretization follows standard elastic material laws implemented in SOFA, which are established for soft-body approximation in the framework. We will revise the abstract and modeling section to qualify the statement, noting that the model relies on these constitutive laws and that direct experimental validation lies outside the current simulation-focused scope. revision: partial
Referee: [Simulation results] Simulation results paragraph (abstract): the statement that 'simulation results show the effectiveness of the proposed novel modeling framework and the related controller design' is unsupported by any quantitative metrics, error norms, convergence times, baseline comparisons (e.g., rigid-body model or alternative soft-body formulations), or statistical measures. This leaves the central claim of demonstrated effectiveness without measurable grounding.

Authors: We agree that the current presentation of simulation results lacks quantitative metrics, error norms, convergence times, and baseline comparisons. In the revised manuscript we will expand the results section to include position-tracking error norms, settling times, and direct comparisons against a rigid-body quadrotor model under equivalent actuation, thereby supplying measurable support for the effectiveness claims. revision: yes

standing simulated objections not resolved

Absence of experimental validation against physical prototypes, as the work is confined to internal simulation outputs within the SOFA framework.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core modeling uses standard tetrahedral FEM discretization with elastic material laws inside the external SOFA framework, followed by a conventional PI controller. No equations, parameters, or predictions are shown to reduce by construction to quantities defined within the paper itself (e.g., no fitted inputs renamed as predictions, no self-definitional loops, and no load-bearing self-citations that substitute for independent justification). The assertion that internal forces are 'adequate to the real dynamic behavior' is an unsupported modeling premise rather than a derived result that collapses to the inputs. The derivation chain remains self-contained against external benchmarks and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the chosen elastic material law and tetrahedral discretization capture real pneumatic arm dynamics; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)

domain assumption The tetrahedral mesh discretization following an elastic material law produces internal forces adequate to the real dynamic behavior of the pneumatic arms.
This premise is invoked directly in the description of the soft-body modeling approach.

pith-pipeline@v0.9.0 · 5721 in / 1215 out tokens · 50427 ms · 2026-05-25T06:16:19.842987+00:00 · methodology

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Reference graph

Works this paper leans on

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Quasi-static fea model for a multi- material soft pneumatic actuator in sofa.IEEE Robotics and Automation Letters, 7(3):7391–7398, 2022

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Finite element dynamics of a concentric tube robot motion and in- teraction with environment using sofa-framework

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An open-source user-friendly interface for simulating magnetic soft robots using simulation open framework architec- ture (sofa).arXiv preprint arXiv:2508.10686, 2025

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Development of a ppo- reinforcement learned walking tripedal soft-legged robot using sofa.arXiv preprint arXiv:2504.09242, 2025

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