A Foot Resistive Force Model for Legged Locomotion on Muddy Terrains

Xunjie Chen; Liuyin Wang; Xinyan Huang; Jerry Shan; Yantao Shen; Jingang Yi

cs.ROApr 13, 2026

A Foot Resistive Force Model for Legged Locomotion on Muddy Terrains

Authors: Xunjie Chen, Liuyin Wang, Xinyan Huang, Jerry Shan, Yantao Shen, Jingang Yi

Organizations: Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854 USA · Department of Electrical and Biomedical Engineering, University of Nevada, Reno, NV 89557 USA

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.

Figures & tables

Exps.	Description	$W$ ( $\%$ )	Penetration angle	Plate shape
E1	VI	$25$	$0^{\circ}$	S, SC, T
E2	HI	$25$	$0^{\circ}$	S, SC, T.
E3	VI	$15,25,35$	$0,22.5,45,67.5,90^{\circ}$	S
E4	HI	$15,25,35$	$0,22.5,45,67.5,90^{\circ}$	S

TABLE I: List of planar intrusion experiments (HI/VI: horizontal/vertical intrusion;

W

: water content)

RMSE	Flat			Semi-cylindrical			Semi-spherical
RMSE	$F_{z}$ ( $\%$ )	$F_{x}$ ( $\%$ )	$F_{y}$ ( $\%$ )	$F_{z}$ ( $\%$ )	$F_{x}$ ( $\%$ )	$F_{y}$ ( $\%$ )	$F_{z}$ ( $\%$ )	$F_{x}$ ( $\%$ )	$F_{y}$ ( $\%$ )
Slow	${9.03\pm 1.10}$	${8.72\pm 1.84}$	${12.50\pm 3.90}$	${5.11\pm 1.09}$	${7.88\pm 4.72}$	$9.60\pm 2.20$	${4.59\pm 0.37}$	${12.32\pm 1.36}$	${13.10\pm 2.20}$
Medium	$7.69\pm 2.07$	$9.80\pm 1.48$	$12.80\pm 3.90$	$6.81\pm 1.16$	$7.64\pm 1.84$	${11.20\pm 3.10}$	$5.10\pm 0.87$	$11.60\pm 6.88$	$12.30\pm 5.40$
Fast	$9.21\pm 1.67$	$8.88\pm 2.20$	$18.09\pm 3.30$	$4.19\pm 0.77$	$6.08\pm 1.08$	$12.10\pm 3.10$	$9.79\pm 0.83$	$15.72\pm 4.71$	$10.60\pm 6.50$

TABLE II: Comparison of the relative RMSEs (mean and standard deviation) of three foot shapes at three different velocities.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Water content	$\alpha$ (MPa)	$n$	$\lambda$	$\eta_{\mathrm{m}}$ (MPa/s)	$\eta_{\infty}$ (MPa/s)	$\dot{\gamma}_{c}$ (s -1 )	$\tau_{\mathrm{build}}$ (s)	$\tau_{\mathrm{leak}}$ (s)	$\varepsilon$	$\sigma_{Y}$ (MPa)
$W=25\%$	0.024	0.53	1.9	0.067	0.014	0.14	0.32	0.17	0.016	0.0071
$W=35\%$	0.013	0.20	3.2	0.27	0.0031	0.042	0.49	0.089	0.054	0.011
$W=45\%$	0.010	0.18	2.6	0.31	0.0026	0.027	0.58	0.052	0.040	0.0086

Appendix

Table S1: List of model parameters for the vertical intrusion;

W

: water content

Water content	$\alpha$ (MPa)	$n$	$\lambda$	$\eta_{\mathrm{m}}$ (MPa/s)	$\eta_{\infty}$ (MPa/s)	$\dot{\gamma}_{c}$ (s -1 )	$\tau_{\mathrm{build}}$ (s)	$\tau_{\mathrm{leak}}$ (s)	$\varepsilon$	$\sigma_{Y}$ (MPa)
$W=25\%$	0.018	0.12	1.8	0.46	0.010	0.31	0.63	0.99	0.027	0.013
$W=35\%$	0.011	0.32	17	0.41	0.021	0.019	0.25	0.67	0.14	0.0094
$W=45\%$	0.0078	0.95	8.7	0.63	0.065	0.028	0.21	0.022	0.12	0.0061

Appendix

Table S2: List of model parameters for the horizontal intrusion;

W

: water content

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Simulating Robotic Locomotion in Sand: Resistive Force Theory in an Open-Source Physics Engine

Ryan Walker Brown, Laura K. Treers, Kathryn A. Daltorio

College of Engineering, Department of Mechanical and Aerospace Engineering at Case Western Reserve University, Cleveland, OH 44106 · College of Engineering and Mathematical Sciences, Department of Mechanical Engineering at the University of Vermont, Burlington, VT 05405

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Recent advancements in Resistive Force Theory (RFT) enable approximation of ground reaction forces for locomotion in sand without the computational expense of modeling interactions with individual grains. However, these tools have been absent in 3D physics engines commonly used for robot simulation. We explore if resistive force approximations are sufficient, when integrated with standard dynamics calculations, to provide a stable substrate for a freely walking robot. To determine this, we implement 3D Granular Resistive Force Theory (3D RFT) in a physics simulation engine, MuJoCo. We verify simulations in multiple scenarios to demonstrate that key trends due to end effector shape, speed, and loading are preserved. Our implementation predicts both walking distance and foot sinkage of a 12-Degree of Freedom hexapod robot within 7% of experiments in sand. While RFT has inherent approximations, the open source tool described here has potential to help develop new and improved robot designs to traverse granular media substrates.

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