Inferring Soil Friction Angle from Robot Foot-Ground Force Histories: A Bayesian Inverse Approach to Proprioceptive Soil Sensing

Dawei Xu; Zhijie Wang

cs.ROSep 29, 2026

Inferring Soil Friction Angle from Robot Foot-Ground Force Histories: A Bayesian Inverse Approach to Proprioceptive Soil Sensing

Authors: Dawei Xu, Zhijie Wang

Organizations: Department of Civil and Environmental Engineering Washington State University Pullman, WA 99164, USA

Abstract

Foot-ground interaction signals recorded by quadruped robots may enable spatially distributed, in situ characterization of soil strength. As a first step, we test whether the internal friction angle $φ$ of cohesionless soil can be identified from the force history of a simplified rotating leg. A two-dimensional continuum model implemented with the material point method, benchmarked against measured rotating-leg force histories, generates the training data, and two Gaussian-process surrogates support Bayesian inversion of the full histories. In matched-model experiments, the framework recovers 14 off-grid friction angles with a median absolute error of approximately $0.1^\circ$ (maximum $\sim 0.7^\circ$ ); the reported credible intervals contain the true value in every case. These results establish that $φ$ is identifiable when the forward model is correctly specified, and support further development of proprioceptive soil sensing for spatially variable terrain, with applications from physics-grounded world models for robot training to post-wildfire slope assessment.

Figures & tables

Figure 1: Physics-grounded real-to-simulation-to-real framework for legged-robot terrain understanding: (a,b) a wildfire-burned site and field deployment on it; (c) laboratory characterization of physical terrain properties; (d) the proposed inversion of robot force histories for soil-strength inference; and (e) inferred physical properties parameterize high-fidelity granular terrain models for robot training and redeployment (generated using the Genesis platform [ 7 ] ). The loop connects real-world robot–terrain interaction with physics-grounded world modeling, enabling continual refinement of terrain representation and robot behavior.

Figure 2: Proprioceptive foot-force estimates for a Unitree Go2 walking on sand: (a) fore–aft force

F_{x}

and (b) vertical force

F_{z}

over

7\,\mathrm{s}

Figure 3: Stride-aligned mean stance forces from multiple stance pulses (

68

pulses across four feet).

Figure 4: Rotating legs in sand with different geometries: (a) C-leg (

\kappa=+1/R

), (b) straight leg (

\kappa=0

), and (c) reversed-C leg (

\kappa=-1/R

). Here,

\theta

is measured from vertical and

2R

is the chord length.

Experiment
Parameter	Value
Leg geometry	straight, C-shaped, reversed-C
Angular velocity, $\omega$	$0.2~\mathrm{rad\,s^{-1}}$
Chord length, $2R$ / width, $w$	$76.2$ / $25.4~\mathrm{mm}$
Discretization
Domain / soil-bed size	$0.30\times 0.30$ / $0.29\times 0.165~\mathrm{m}$

Table 1: Rotating-leg benchmark parameters.

Figure 5: Reversed-C leg in sand at a matched sweep angle. (a) experiment [ 22 ] , where the leg is partially obscured by sand and the red arrow marks the resultant contact force vector; (b) simulated material points colored by velocity magnitude (blue: low, red: high).

Figure 6: Comparison of simulated and measured rotating-leg force histories at

\omega=0.2~\mathrm{rad\,s^{-1}}

[ 22 ] . Rows show the straight, C-shaped, and reversed-C legs; columns show

F_{x}

and

F_{z}

Leg	$F_{z}$ NRMSD (%)	$F_{x}$ NRMSD (%)
Straight	24.8	14.9
C-shaped	17.6	15.0
Reversed-C	18.7	14.9

Table 2: MPM force-history deviations from experiment over

|\theta|<85^{\circ}

, normalized by the experimental peak-to-peak range (NRMSD, %).

Figure 7: Straight-leg force histories for

\phi=25^{\circ}

31^{\circ}

37^{\circ}

, and

43^{\circ}

\omega=0.2~\mathrm{rad\,s^{-1}}

: (a)

F_{x}

and (b)

F_{z}

Parameter	Values
Friction angle, $\phi$	$25^{\circ}$ – $45^{\circ}$ , $2^{\circ}$ steps (11 cases)
Rotation rate, $\omega$	$0.2~\mathrm{rad\,s^{-1}}$ (matches the benchmark)
Off-grid test angles	14 cases, $\phi\in[25.9^{\circ},43.6^{\circ}]$
Sweep range, $\theta$	$[-135^{\circ},135^{\circ}]$

Table 3: Straight-leg simulation library and off-grid test cases.

Figure 8: Observation vector

\mathbf{d}

: (a) force histories resampled at 40 angles over

|\theta|<85^{\circ}

; (b) concatenated

F_{x}

and

F_{z}

segments.

Figure 9: GP forward map: fitted mean surfaces of (a) $F_{x}(\phi,\theta)$ , and (b) $F_{z}(\phi,\theta)$ .

Figure 10: GP forward-map held-out relative error,

|F^{\mathrm{GP}}-F^{\mathrm{true}}|/F^{\mathrm{true}}

, between the GP mean and the true simulated value for each of the

14

off-grid test cases described in Section 4.2 , evaluated at representative leg angles: (a)

F_{x}

and (b)

F_{z}

. The insets show representative simulated force histories over the full leg-angle range, with vertical dashed lines indicating the angles selected for error evaluation.

Figure 11: MCMC sampling for $\phi^{\star}=33.97^{\circ}$ : (a) walker traces and burn-in cutoff; (b) posterior density and retained samples, with recovered $\phi=33.95^{\circ}$ .

Figure 12: Matched-model recovery for 14 off-grid cases. Error bar of each recovered friction angle shows

68\%

credible intervals. Absolute error

=\phi_{\mathrm{recovered}}-\phi_{\mathrm{true}}

; the dashed line marks zero error.

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QuietWalk: Physics-Informed Reinforcement Learning for Ground Reaction Force-Aware Humanoid Locomotion Under Diverse Footwear

Hanze Hu, Luying Feng, Silu Chen, Tianjiang Zheng+5

Ground Reaction Forces Inverse Dynamics Physics-Informed Learning

Open paper AskPDF arXiv

Humanoid robots operating in human-centered environments (e.g., homes, hospitals, and offices) must mitigate foot--ground impact transients, as impact-induced vibration and noise degrade user experience and repeated impacts accelerate hardware wear. However, existing low-noise locomotion training often relies on kinematic proxy objectives or fragile force sensors, and footwear-induced changes in contact dynamics introduce distribution shifts that hinder policy generalization.We present QuietWalk, a physics-informed reinforcement learning framework for ground-reaction-force-aware humanoid locomotion under diverse footwear conditions. QuietWalk employs an inverse-dynamics-constrained physics-informed neural network (PINN) to estimate per-foot vertical ground reaction forces (GRFs) from proprioceptive signals, and integrates the frozen predictor into the RL training loop to penalize predicted impact forces without requiring force sensors at deployment.On a held-out real-robot dataset, enforcing inverse-dynamics consistency reduces vertical GRF prediction errors by 82%-86% compared with a purely supervised predictor and improves the coefficient of determination from 0.39/0.67 to 0.99/0.99 for the left/right feet. On hardware at 1.2 m/s (barefoot; averaged over four floor materials), QuietWalk reduces mean A-weighted noise level by 7.17 dB and peak noise level by 4.98 dB under a consistent recording setup. Cross-footwear experiments (barefoot, skate shoes, athletic sneakers, and high heels) across multiple surfaces further demonstrate robust adaptation to footwear-induced contact variations.

Abstract

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