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arxiv: 2604.12006 · v1 · submitted 2026-04-13 · 💻 cs.RO

A Foot Resistive Force Model for Legged Locomotion on Muddy Terrains

Xunjie Chen , Liuyin Wang , Xinyan Huang , Jerry Shan , Yantao Shen , Jingang Yi This is my paper

Pith reviewed 2026-05-10 15:31 UTC · model grok-4.3

classification 💻 cs.RO
keywords resistive force modellegged locomotionmuddy terrainvisco-elasticitythixotropyretractive suctionmorphing footrheological modeling
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The pith

A uniform resistive force model captures mud's visco-elasticity, thixotropy and suction to predict foot forces accurately and guide a morphing foot design.

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

The paper introduces a resistive force model for how robot feet interact with mud. The model uses one consistent set of equations to represent the mud's stretching, time-dependent flow, and suction pull-back. Experiments show this lets the authors build a morphing foot that improves both how far robots can walk and how much energy they use. The same model is also positioned as a foundation for simulation and control software on muddy ground.

Core claim

A single uniform resistive force model can describe foot-mud interactions by simultaneously incorporating visco-elastic deformation, thixotropic time-dependent viscosity changes, and retractive suction forces, delivering both physical insight and quantitative force predictions that support the design of a morphing foot shown in tests to raise mobility and lower energy consumption.

What carries the argument

The uniform resistive force model, a single formulation that integrates rheological behaviors of mud to compute interaction forces.

If this is right

  • The morphing foot increases both distance traveled and energy efficiency during walking on mud.
  • The model supplies the physical basis needed for data-driven simulators of legged robots on yielding terrain.
  • Locomotion controllers can use the model's force predictions to adapt gait and foot shape in real time.
  • The same equations apply to multiple rheological effects without switching between separate sub-models.

Where Pith is reading between the lines

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

  • The model might extend to other soft terrains such as wet sand if the same uniform structure holds.
  • Robots equipped with this foot could maintain longer operation times in flood or disaster zones.
  • Control algorithms could close the loop around the force predictions to prevent sinking or slipping.

Load-bearing premise

A single set of equations can represent visco-elasticity, thixotropy and suction across different mud types and yield accurate forces without needing separate tuning for each terrain condition.

What would settle it

Measure foot forces on a new mud sample whose viscosity or water content differs from the training set; if prediction errors exceed those of existing models without parameter changes, the uniform formulation claim fails.

Figures

Figures reproduced from arXiv: 2604.12006 by Jerry Shan, Jingang Yi, Liuyin Wang, Xinyan Huang, Xunjie Chen, Yantao Shen.

Figure 3
Figure 3. Figure 3: (a) A simple flat-plate intrusion into mud. (b) The schematic of the [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sequential snapshots of the foot intrusion and retraction over mud. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The general 3D configuration of an arbitrary robotic foot consisting [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic diagrams for 3D resultant force calculation for semi [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: A 3D model of the morphing foot showing three core components. [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Experimental setup for vertical and horizontal penetration tests for mud rheology behavior calibrations using a robotic manipulator. (b) Robotic foot [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of the vertical and horizontal intrusion E3 and E4 using a square plate. (a)-(b) Vertical and horizontal stress versus intrusion displacement. Blue [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of model predictions and experimental results for three representative feet (top row: semi-cylindrical; middle row: semi-spherical; bottom [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalized 3D resistive stresses under various conditions. (a) Different foot shapes. (b) Different intrusion velocities. (c) Different water content. [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Average mud resistive stress for four different foot shapes within one walking gait. (a) Stress [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: (a) Comparisons of the maximum suction force and energy [PITH_FULL_IMAGE:figures/full_fig_p007_13.png] view at source ↗
read the original abstract

Legged robots face significant challenges in moving and navigating on deformable and highly yielding terrain such as mud. We present a resistive force model for legged foot-mud interactions. The model captures rheological behaviors such as visco-elasticity, thixotropy of the mud suspension and retractive suction. One attractive property of this new model lies in its effective, uniform formulation to provide underlying physical interpretation and accurate resistive force predictions. We further take advantage of the resistive force model to design a new morphing robotic foot for effective and efficient legged locomotion. We conduct extensive experiments to validate the force model, and the results demonstrate that the morphing foot enhances not only the locomotion mobility but also energy-efficiency of walking in mud. The new resistive force model can be further used to develop data-driven simulation and locomotion control of legged robots on muddy terrains.

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

2 major / 1 minor

Summary. The paper presents a resistive force model for legged robot foot interactions with muddy terrains. It claims to capture visco-elasticity, thixotropy of the mud suspension, and retractive suction via a single uniform formulation that supplies physical interpretation and accurate force predictions. The model is used to design a morphing robotic foot, with extensive experiments asserted to demonstrate gains in locomotion mobility and energy efficiency on mud; the model is also positioned for use in data-driven simulation and control.

Significance. If the uniform formulation can be shown to deliver accurate predictions across mud conditions without post-hoc parameter retuning and with parameters derived from measurable rheological properties, the work would be significant for legged robotics on highly deformable terrains. It would provide a physically interpretable alternative to purely empirical models, supporting better simulation, planning, and hardware design. The morphing foot concept adds a concrete application for energy-efficient locomotion.

major comments (2)
  1. [Abstract] Abstract: The manuscript asserts that the model provides an 'effective, uniform formulation' for 'accurate resistive force predictions' and 'underlying physical interpretation,' yet supplies no equations, derivation, or parameter-identification procedure. This prevents verification of whether the formulation is truly uniform and free of terrain-specific adjustments or whether it reduces to fitted parameters by construction.
  2. [Abstract] Abstract and validation description: The claim of validation through 'extensive experiments' showing improved mobility and energy-efficiency metrics is unsupported by any quantitative error measures (e.g., force-prediction RMSE), baseline comparisons, cross-mud generalization tests, or details on how model parameters were obtained from independent mud-property measurements rather than the force data being predicted.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it specified the number of mud conditions tested, the range of foot velocities or sinkage depths, and whether any parameter retuning occurred between conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below and have made revisions to improve clarity, particularly in the abstract.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The manuscript asserts that the model provides an 'effective, uniform formulation' for 'accurate resistive force predictions' and 'underlying physical interpretation,' yet supplies no equations, derivation, or parameter-identification procedure. This prevents verification of whether the formulation is truly uniform and free of terrain-specific adjustments or whether it reduces to fitted parameters by construction.

    Authors: The abstract serves as a high-level summary and therefore omits detailed equations. The full manuscript presents the uniform formulation in Section 3, where a single set of equations integrates visco-elastic, thixotropic, and retractive suction terms derived from rheological principles. Parameter identification is described in Section 4 using independent rheometer measurements of mud samples rather than direct fitting to force data, with no terrain-specific retuning. To address the concern, we have revised the abstract to include a concise outline of the formulation and identification approach. revision: yes

  2. Referee: [Abstract] Abstract and validation description: The claim of validation through 'extensive experiments' showing improved mobility and energy-efficiency metrics is unsupported by any quantitative error measures (e.g., force-prediction RMSE), baseline comparisons, cross-mud generalization tests, or details on how model parameters were obtained from independent mud-property measurements rather than the force data being predicted.

    Authors: Space constraints in the abstract limited inclusion of quantitative details. The manuscript's Section 5 reports force-prediction RMSE values, baseline comparisons against prior resistive force models, generalization tests across multiple mud consistencies, and explicit confirmation that parameters derive from separate rheometer-based mud-property measurements. We have revised the abstract to summarize these quantitative validation elements and the independent parameter procedure. revision: yes

Circularity Check

0 steps flagged

No derivation chain shown; model introduced as capturing observed behaviors without equations or fitting

full rationale

The provided abstract and context present the resistive force model as derived from rheological observations (visco-elasticity, thixotropy, retractive suction) in a uniform formulation for physical interpretation and predictions. No equations, parameter-fitting procedures, or self-citations are exhibited that would allow any claimed prediction to reduce to its inputs by construction. Validation is described via experiments, but the derivation remains self-contained against external benchmarks with no load-bearing self-referential steps. This is the common honest non-finding for papers that introduce models without showing internal reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Model rests on the domain assumption that mud suspension exhibits the listed rheological behaviors in a manner amenable to a single uniform force equation; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Mud exhibits visco-elasticity, thixotropy, and retractive suction during foot interaction
    Invoked as the physical basis for the resistive force model in the abstract

pith-pipeline@v0.9.0 · 5458 in / 1242 out tokens · 45008 ms · 2026-05-10T15:31:36.841841+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    Adaptation of flipper-mud interactions enables effective terrestrial locomotion on muddy substrates,

    S. Liu, B. Huang, and F. Qian, “Adaptation of flipper-mud interactions enables effective terrestrial locomotion on muddy substrates,”IEEE Robot. Automat. Lett., vol. 8, no. 12, pp. 7978–7985, 2023

    work page 2023

  2. [2]

    A reduced-order mud reaction force model for robotic foot-mud interactions,

    X. Chen, J. Yi, and J. Shan, “A reduced-order mud reaction force model for robotic foot-mud interactions,” in Proc. IEEE/ASME Int. Conf. Adv. Intelli. Mechatronics, Boston, MA, USA, 2024, pp. 1341–1346

    work page 2024

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    A terradynamics of legged locomotion on granular media,

    C. Li, T. Zhang, and D. I. Goldman, “A terradynamics of legged locomotion on granular media,”Science, vol. 339, no. 6126, pp. 1408–1412, 2013

    work page 2013

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    A dynamic foot–terrain interaction model for biped walking on granular media,

    X. Chen, X. Huang, J. Yi, J. W. Shan, and T. Liu, “A dynamic foot–terrain interaction model for biped walking on granular media,”IEEE/ASME Trans. Mechatronics, 2025

    work page 2025

  5. [5]

    An insight on mud behavior upon stepping,

    S. Godon, A. Ristolainen, and M. Kruusmaa, “An insight on mud behavior upon stepping,”IEEE Robot. Automat. Lett., vol. 7, no. 4, pp. 11 039–11 046, 2022. 12

    work page 2022