arxiv: 2604.03006 · v1 · submitted 2026-04-03 · 💻 cs.RO

Recognition: no theorem link

A Flow Matching Framework for Soft-Robot Inverse Dynamics

Fangju Yang, Hang Yang, Ibrahim Alsarraj, Ke Wu, Yangming Zhang, Yuhao Wang, Zhenye Luo, Zixi Chen

Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:36 UTC · model grok-4.3

classification 💻 cs.RO

keywords soft continuum robotsinverse dynamicsflow matchingrectified flowfeedforward controltrajectory trackingphysical consistency

0 comments

The pith

Conditional flow matching learns soft-robot inverse dynamics as a generative map, reducing tracking RMSE by more than 50 percent.

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

The paper establishes that inverse dynamics for soft continuum robots can be recast as a conditional flow-matching task so that control inputs are sampled from a transport map instead of averaged by regression. This change targets the high-dimensional nonlinearities and actuation coupling that cause conventional feedback controllers to chatter and deterministic learners to miss accurate mappings. Two extensions, one that injects a physics-based residual prior and another that adds a forward-dynamics consistency loss, are shown to keep generated controls realizable. Readers would care because the resulting feedforward controller runs in under a millisecond and sustains stable open-loop motion at more than one meter per second.

Core claim

The central claim is that reformulating inverse dynamics as a conditional flow-matching problem with Rectified Flow produces physically consistent control sequences rather than conditional averages. The RF-Physical variant adds a physics-based prior for residual modeling, while the RF-FWD variant adds a forward-dynamics consistency loss during training. Evaluations show this yields over 50 percent lower trajectory-tracking RMSE than MLP, LSTM, and Transformer baselines and supports stable open-loop execution at a peak end-effector speed of 1.14 m/s with 0.995 ms inference latency.

What carries the argument

Rectified Flow used as a conditional generative transport map that converts desired trajectories into sequences of control inputs, augmented by either a physics prior or a forward-consistency loss.

Load-bearing premise

That the flow-matching outputs remain physically realizable for arbitrary soft-robot shapes and actuation schemes without extra tuning or safety constraints.

What would settle it

A test on a new soft-robot geometry or actuation scheme in which the generated open-loop controls produce large trajectory deviations or instability would falsify the claim of reliable physical consistency.

Figures

Figures reproduced from arXiv: 2604.03006 by Fangju Yang, Hang Yang, Ibrahim Alsarraj, Ke Wu, Yangming Zhang, Yuhao Wang, Zhenye Luo, Zixi Chen.

**Figure 1.** Figure 1: Flow-matching-based open-loop feedforward controller. The framework solves the inverse-dynamics problem by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Trajectory (y − z) comparison. Top: Inverse MLP/LSTM/Transformer. Bottom: RF/RF-Phys/RF-FWD. Time (s) 0 20 40 60 In p ut (N) -1 0 1 Inverse u1 LSTM u1 Transformer u1 Inverse u2 LSTM u2 Transformer u2 (a) Circle (MLP/LSTM/Trans) Time (s) 0 20 40 60 In p ut (N) -1 0 1 (b) Heart Time (s) 0 20 40 60 In p ut (N) -1 0 1 (c) Rand2 Time (s) 0 20 40 60 In p ut (N) -1 0 1 (d) Rand3 Time (s) 0 20 40 60 In p ut (N) -1… view at source ↗

**Figure 3.** Figure 3: Input comparison (u1/u2). Top: Inverse MLP/LSTM/Transformer. Bottom: RF/RF-Phys/RF-FWD. IV. EVALUATION IN SIMULATION In this section, we evaluate the proposed framework in simulation as an open-loop inverse dynamics learner, examining whether flow-based modeling produces more accurate and smoother actuation than regression-based alternatives under dynamic soft-body behavior. A. System Setup and Data Acqui… view at source ↗

**Figure 4.** Figure 4: The circular reference trajectories with high speed in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Complex trajectory comparison and input results. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Experimental Setup 0.2 0.0 0.2 x (m) 0.2 0.0 0.2 y ( m ) Ref Exp 0 10 20 Time (s) 0.2 0.0 0.2 x a n d y ( m ) xRef yRef xExp yExp 0.25 0.50 0.75 1.00 S p e e d ( m = s ) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: High-speed circular reference tracking in experiments. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 8.** Figure 8: Vertical plane trajectory tracking performance. (a)-(d) 2D plots of the end-effector position against reference paths. The transient trajectory from the Initial point (⋄) to the Start point (•) is excluded from the tracking performance evaluation. (e)-(h) Stroboscopic composite images showing the tracking ball’s motion at 0.4 s intervals. Visualization note: Backgrounds in the bottom row were darkened and … view at source ↗

**Figure 9.** Figure 9: Horizontal plane trajectory tracking performance. TABLE III: Physical tracking RMSE under two gravity configurations (RF-FWD). Trajectory Vertical RMSE (mm) Horizontal RMSE (mm) Star 3.83 3.96 Spiral 4.94 6.52 Circle 3.95 8.65 Heart 4.46 6.47 Mean 4.30 6.40 C. Comparison with Representative Soft-Robot Controllers As summarized in Table IV, RF-FWD is not the best in every individual metric, but it provides … view at source ↗

read the original abstract

Learning the inverse dynamics of soft continuum robots remains challenging due to high-dimensional nonlinearities and complex actuation coupling. Conventional feedback-based controllers often suffer from control chattering due to corrective oscillations, while deterministic regression-based learners struggle to capture the complex nonlinear mappings required for accurate dynamic tracking. Motivated by these limitations, we propose an inverse-dynamics framework for open-loop feedforward control that learns the system's differential dynamics as a generative transport map. Specifically, inverse dynamics is reformulated as a conditional flow-matching problem, and Rectified Flow (RF) is adopted as a lightweight instance to generate physically consistent control inputs rather than conditional averages. Two variants are introduced to further enhance physical consistency: RF-Physical, utilizing a physics-based prior for residual modeling; and RF-FWD, integrating a forward-dynamics consistency loss during flow matching. Extensive evaluations demonstrate that our framework reduces trajectory tracking RMSE by over 50% compared to standard regression baselines (MLP, LSTM, Transformer). The system sustains stable open-loop execution at a peak end-effector velocity of 1.14 m/s with sub-millisecond inference latency (0.995 ms). This work demonstrates flow matching as a robust, high-performance paradigm for learning differential inverse dynamics in soft robotic systems.

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

Flow matching recasts soft-robot inverse dynamics as a generative map and reports big RMSE drops plus fast open-loop runs, but the experiments stay narrow and details on data and baselines are missing.

read the letter

The core move here is treating inverse dynamics as conditional flow matching with rectified flows instead of plain regression. They add a physics-based residual prior in one variant and a forward-dynamics consistency loss in the other. On the reported tests this cuts trajectory RMSE by more than half versus MLP, LSTM, and Transformer baselines while delivering 0.995 ms inference and stable open-loop tracking at 1.14 m/s peak speed. That combination of generative modeling and lightweight physics terms is not a standard extension of existing soft-robot learning work, so the framing itself is the main novelty. The practical numbers on latency and open-loop stability are useful if they hold, since many soft-robot controllers still lean on feedback that can chatter. The two variants make sense as ways to keep outputs inside feasible actuation ranges without full simulation at runtime. The soft spots are straightforward. The abstract gives no training-set size, no hyperparameter search details, no error bars, and no sign that baselines received equivalent tuning effort, so the 50% gain is hard to weigh. Everything shown is on the authors' own platform; nothing tests whether the learned map stays realizable or stable when geometry, actuation coupling, or contact conditions change. The stress-test concern about untested generalization therefore lands. The priors and consistency loss are plausible but not shown to bound residuals across designs. This paper is for people already working on learning-based control for continuum robots who want a generative option to standard regressors. It has enough distinct machinery and usable performance claims to deserve a serious referee, provided the full manuscript supplies the missing experimental controls and at least one cross-geometry check. I would send it to review.

Referee Report

3 major / 2 minor

Summary. The paper claims to introduce a conditional flow-matching framework using Rectified Flow for learning inverse dynamics of soft continuum robots. It proposes two variants: RF-Physical incorporating a physics-based prior for residual modeling and RF-FWD with a forward-dynamics consistency loss. The framework is evaluated to achieve over 50% reduction in trajectory tracking RMSE compared to MLP, LSTM, and Transformer baselines, while enabling stable open-loop control at a peak end-effector velocity of 1.14 m/s with 0.995 ms inference latency.

Significance. If validated with complete experimental details, the work is significant because it applies generative flow-matching techniques to soft-robot inverse dynamics, potentially offering better handling of nonlinearities and complex couplings than deterministic methods. The reported low-latency inference supports practical deployment in real-time control systems. The integration of physical priors and consistency losses represents a thoughtful adaptation of the method to the domain.

major comments (3)

[Experimental Evaluation] The reported >50% RMSE reduction and performance metrics lack details on the volume of training data, the hyperparameter search process, whether statistical significance was evaluated across multiple runs, and if the baseline models (MLP, LSTM, Transformer) received equivalent tuning and training conditions. These omissions make it difficult to fully assess the strength of the central performance claim.
[Method Description (RF-FWD)] While the forward-dynamics consistency loss is proposed to promote physical consistency, the manuscript does not include an analysis or bound demonstrating that this loss enforces realizability for arbitrary soft-robot geometries and actuation schemes, as required for the generalization implied in the abstract.
[Results and Discussion] The stable open-loop execution is demonstrated only on the specific platform evaluated; without tests on varied geometries or coupling changes, the claim that the approach works for arbitrary soft-robot systems rests on an untested assumption about the transferability of the learned transport map.

minor comments (2)

[Notation and Equations] The conditional flow-matching objective and the specific forms of the new loss terms could benefit from more explicit equation references to the standard rectified flow formulation for clarity.
[References] Additional citations to recent works on generative models for robot control or flow matching applications in dynamics learning would strengthen the positioning of the contribution.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback and the positive evaluation of the work's significance. We address each major comment below, indicating the revisions planned for the manuscript.

read point-by-point responses

Referee: [Experimental Evaluation] The reported >50% RMSE reduction and performance metrics lack details on the volume of training data, the hyperparameter search process, whether statistical significance was evaluated across multiple runs, and if the baseline models (MLP, LSTM, Transformer) received equivalent tuning and training conditions. These omissions make it difficult to fully assess the strength of the central performance claim.

Authors: We agree that these details are essential for a complete evaluation. In the revised manuscript, we will expand the experimental section to report the training data volume, including the number of trajectories collected and total samples used. We will describe the hyperparameter optimization procedure employed for all models. Additionally, we will present results averaged over multiple independent runs with different random seeds, including standard deviations to demonstrate statistical significance. We confirm that the baseline models were trained and tuned under equivalent conditions using the same data splits and search ranges for hyperparameters. revision: yes
Referee: [Method Description (RF-FWD)] While the forward-dynamics consistency loss is proposed to promote physical consistency, the manuscript does not include an analysis or bound demonstrating that this loss enforces realizability for arbitrary soft-robot geometries and actuation schemes, as required for the generalization implied in the abstract.

Authors: We appreciate this point and acknowledge the value of a more formal analysis. The forward-dynamics consistency loss is formulated to minimize the discrepancy between the predicted controls and those that would produce the observed states via the forward model, thereby encouraging the generated trajectories to be physically realizable under the learned dynamics. In the revision, we will include a detailed explanation of this mechanism, along with empirical evidence from our experiments showing improved consistency. While a general theoretical bound for arbitrary geometries would strengthen the claims further, it may require assumptions on the forward model accuracy that are platform-specific; we will clarify the scope in the updated text. revision: partial
Referee: [Results and Discussion] The stable open-loop execution is demonstrated only on the specific platform evaluated; without tests on varied geometries or coupling changes, the claim that the approach works for arbitrary soft-robot systems rests on an untested assumption about the transferability of the learned transport map.

Authors: We concur that validation on multiple platforms would provide stronger evidence for broad applicability. The current experiments focus on a representative soft continuum robot to demonstrate the framework's effectiveness. The method itself is designed to be geometry-agnostic, relying on data-driven learning of the transport map conditioned on the system state. In the revised manuscript, we will add a dedicated discussion section addressing the generalization assumptions, potential limitations regarding transferability, and outline plans for future multi-platform evaluations. No new experiments are planned for this revision due to resource constraints, but the core claims will be appropriately qualified. revision: partial

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper reformulates inverse dynamics as a conditional flow-matching problem using the established Rectified Flow method, then adds two variants (RF-Physical with a physics-based residual prior and RF-FWD with a forward-dynamics consistency loss). These are standard adaptations of existing generative modeling techniques rather than self-definitional or fitted-input reductions. The reported >50% RMSE improvement and 1.14 m/s open-loop performance are presented as empirical results on evaluated platforms, not algebraically forced by the choice of parameters or by self-citation chains. No load-bearing step reduces to its own inputs by construction, and the central claims remain independent of any cited prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that rectified flow can serve as a lightweight generative model for differential inverse dynamics and that the added consistency losses will enforce physical realizability without introducing new free parameters that dominate the result.

axioms (1)

domain assumption Rectified Flow provides a valid transport map from noise to physically consistent control inputs when conditioned on robot state and desired trajectory
Invoked when inverse dynamics is reformulated as a conditional flow-matching problem

pith-pipeline@v0.9.0 · 5533 in / 1245 out tokens · 48729 ms · 2026-05-13T19:36:24.085701+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Design, fabrication and control of soft robots,

D. Rus and M. T. Tolley, “Design, fabrication and control of soft robots,”Nature, vol. 521, no. 7553, pp. 467–475, 2015

work page 2015
[2]

Control strategies for soft robotic manipulators: A survey,

T. G. Thuruthel, Y . Ansari, E. Falotico, and C. Laschi, “Control strategies for soft robotic manipulators: A survey,”Soft Robotics, vol. 5, no. 2, pp. 149–163, 2018

work page 2018
[3]

First-order dynamic modeling and control of soft robots,

T. G. Thuruthel, F. Renda, and F. Iida, “First-order dynamic modeling and control of soft robots,”Frontiers in Robotics and AI, vol. 7, p. 95, 2020

work page 2020
[4]

Data-driven methods for sensing, modeling and control of soft continuum robot: A review,

J. Liu, Y . Duo, X. Chen, Z. Zuo, Y . Liu, and L. Wen, “Data-driven methods for sensing, modeling and control of soft continuum robot: A review,”IEEE/ASME Transactions on Mechatronics, vol. 30, no. 6, pp. 5520–5530, 2025

work page 2025
[5]

Control of soft robots with inertial dynamics,

D. A. Haggerty, M. J. Banks, E. Kamenar, A. B. Cao, P. C. Curtis, I. Mezic, and E. W. Hawkes, “Control of soft robots with inertial dynamics,”Science Robotics, vol. 8, no. 81, p. eadd6864, 2023

work page 2023
[6]

Adaptive model-predictive control of a soft continuum robot using a physics-informed neural network based on cosserat rod theory,

J. Licher, M. Bartholdt, H. Krauss, T.-L. Habich, T. Seel, and M. Schappler, “Adaptive model-predictive control of a soft continuum robot using a physics-informed neural network based on cosserat rod theory,” 2025

work page 2025
[7]

Stable open loop control of soft robotic manipulators,

T. G. Thuruthel, E. Falotico, M. Manti, and C. Laschi, “Stable open loop control of soft robotic manipulators,”IEEE Robotics and Automation Letters, vol. 3, no. 2, pp. 1292–1298, 2018

work page 2018
[8]

Fem-based nonlinear controller for a soft trunk robot,

K. Wu and G. Zheng, “Fem-based nonlinear controller for a soft trunk robot,”IEEE Robotics and Automation Letters, vol. 7, no. 2, pp. 5735– 5740, 2022

work page 2022
[9]

Reduced order modeling of hybrid soft-rigid robots using global, local, and state-dependent strain parameterization,

A. T. Mathew, D. Feliu-Talegon, A. Y . Alkayas, F. Boyer, and F. Renda, “Reduced order modeling of hybrid soft-rigid robots using global, local, and state-dependent strain parameterization,”The Inter- national Journal of Robotics Research, vol. 44, no. 1, pp. 129–154, 2025

work page 2025
[10]

G. S. Chirikjian,Theory and applications of hyper-redundant robotic manipulators. California Institute of Technology, 1992

work page 1992
[11]

Configuration tracking for soft con- tinuum robotic arms using inverse dynamic control of a cosserat rod model,

A. Doroudchi and S. Berman, “Configuration tracking for soft con- tinuum robotic arms using inverse dynamic control of a cosserat rod model,” in2021 IEEE 4th International Conference on Soft Robotics (RoboSoft), 2021, pp. 207–214

work page 2021
[12]

Data-driven methods ap- plied to soft robot modeling and control: A review,

Z. Chen, F. Renda, A. Le Gall, L. Mocellin, M. Bernabei, T. Dangel, G. Ciuti, M. Cianchetti, and C. Stefanini, “Data-driven methods ap- plied to soft robot modeling and control: A review,”IEEE Transactions on Automation Science and Engineering, vol. 22, pp. 2241–2256, 2025

work page 2025
[13]

Efficient jacobian-based inverse kinematics with sim-to-real transfer of soft robots by learning,

G. Fang, Y . Tian, Z.-X. Yang, J. M. P. Geraedts, and C. C. L. Wang, “Efficient jacobian-based inverse kinematics with sim-to-real transfer of soft robots by learning,” 2020

work page 2020
[14]

A recurrent neural- network-based real-time dynamic model for soft continuum manipu- lators,

A. Tariverdi, V . K. Venkiteswaran, M. Richter, O. J. Elle, J. Torresen, K. Mathiassen, S. Misra, and O. G. Martinsen, “A recurrent neural- network-based real-time dynamic model for soft continuum manipu- lators,”Frontiers in Robotics and AI, vol. 8, p. 631303, 2021

work page 2021
[15]

Soft robot control with a learned differentiable model,

J. M. Bern, Y . Schnider, P. Banzet, N. Kumar, and S. Coros, “Soft robot control with a learned differentiable model,” in2020 3rd IEEE International Conference on Soft Robotics (RoboSoft), 2020, pp. 417– 423

work page 2020
[16]

Data-driven control of soft robots using koopman operator theory,

D. Bruder, X. Fu, R. B. Gillespie, C. D. Remy, and R. Vasudevan, “Data-driven control of soft robots using koopman operator theory,” IEEE Transactions on Robotics, vol. 37, no. 3, pp. 948–961, 2021

work page 2021
[17]

Deep koopman approach for nonlinear dynamics and control of tendon-driven contin- uum robots,

N. Feizi, F. C. Pedrosa, J. Jayender, and R. V . Patel, “Deep koopman approach for nonlinear dynamics and control of tendon-driven contin- uum robots,”IEEE Robotics and Automation Letters, vol. 10, no. 3, pp. 2926–2933, 2025

work page 2025
[18]

Neural network and jacobian method for solving the inverse statics of a cable-driven soft arm with nonconstant curvature,

M. Giorelli, F. Renda, M. Calisti, A. Arienti, G. Ferri, and C. Laschi, “Neural network and jacobian method for solving the inverse statics of a cable-driven soft arm with nonconstant curvature,”IEEE Trans- actions on Robotics, vol. 31, no. 4, pp. 823–834, 2015

work page 2015
[19]

An inverse dynamics-based control approach for com- pliant control of pneumatic artificial muscles,

C. V . Baysal, “An inverse dynamics-based control approach for com- pliant control of pneumatic artificial muscles,”Actuators, vol. 11, no. 4, p. 111, 2022

work page 2022
[20]

Inverse dynamics modelling and tracking control of conical dielectric elastomer actuator based on GRU neural network,

Y . Zhang, J. Wu, P. Huang, C.-Y . Su, and Y . Wang, “Inverse dynamics modelling and tracking control of conical dielectric elastomer actuator based on GRU neural network,”Engineering Applications of Artificial Intelligence, vol. 118, p. 105668, 2023

work page 2023
[21]

Learning-based inverse dynamic controller for throwing tasks with a soft robotic arm,

D. Bianchi, M. Antonelli, C. Laschi, A. Sabatini, and E. Falotico, “Learning-based inverse dynamic controller for throwing tasks with a soft robotic arm,” inProceedings of the 20th International Conference on Informatics in Control, Automation and Robotics, 2023, pp. 424– 432

work page 2023
[22]

Diffusion policy: Visuomotor policy learning via action diffusion,

C. Chi, Z. Xu, S. S. Feng, E. Cousineau, Y . Du, B. Burchfiel, R. Tedrake, and S. Song, “Diffusion policy: Visuomotor policy learning via action diffusion,”The International Journal of Robotics Research, 2024

work page 2024
[23]

Flow straight and fast: Learning to generate and transfer data with rectified flow,

X. Liu, C. Gong, and Q. Liu, “Flow straight and fast: Learning to generate and transfer data with rectified flow,” inThe Eleventh International Conference on Learning Representations, 2023

work page 2023
[24]

Flow matching for generative modeling,

Y . Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le, “Flow matching for generative modeling,” inThe Eleventh International Conference on Learning Representations, 2023

work page 2023
[25]

Cosserat and F

E. Cosserat and F. Cosserat,Th ´eorie des corps d ´eformables. Paris: Hermann, 1909

work page 1909
[26]

Real-time dynamics of soft and continuum robots based on cosserat rod models,

J. Till, V . Aloi, and C. Rucker, “Real-time dynamics of soft and continuum robots based on cosserat rod models,”The International Journal of Robotics Research, vol. 38, no. 6, pp. 723–746, 2019

work page 2019
[27]

Geometrically exact models for soft robotic manipulators,

D. Trivedi, A. Lotfi, and C. D. Rahn, “Geometrically exact models for soft robotic manipulators,”IEEE Transactions on Robotics, vol. 24, no. 4, pp. 773–780, 2008

work page 2008
[28]

A geometric variable-strain approach for static modeling of soft manipulators with tendon and fluidic actuation,

F. Renda, C. Armanini, V . Lebastard, F. Candelier, and F. Boyer, “A geometric variable-strain approach for static modeling of soft manipulators with tendon and fluidic actuation,”IEEE Robotics and Automation Letters, vol. 5, no. 3, pp. 4006–4013, 2020

work page 2020
[29]

Lightweight dy- namic modeling of cable-driven continuum robots based on actuation- space energy formulation,

F. Yang, H. Yang, I. Alsarraj, Y . Wang, and K. Wu, “Lightweight dy- namic modeling of cable-driven continuum robots based on actuation- space energy formulation,” 2025

work page 2025
[30]

Open-loop model- free dynamic control of a soft manipulator for tracking tasks,

A. Centurelli, A. Rizzo, S. Tolu, and E. Falotico, “Open-loop model- free dynamic control of a soft manipulator for tracking tasks,” in2021 20th International Conference on Advanced Robotics (ICAR), 2021, pp. 128–133

work page 2021
[31]

High-precision dynamic control of soft robots with the physics-learning hybrid modeling approach,

X. Huang, Y . Rong, and G. Gu, “High-precision dynamic control of soft robots with the physics-learning hybrid modeling approach,” IEEE/ASME Transactions on Mechatronics, vol. 30, no. 3, pp. 1658– 1669, 2025

work page 2025