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∘ (maximum ∼0.7∘); 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 Fx and (b) vertical force Fz over 7s .
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 ( κ=+1/R ), (b) straight leg ( κ=0 ), and (c) reversed-C leg ( κ=−1/R ). Here, θ is measured from vertical and 2R is the chord length.
Parameter
Value
Experiment
Leg geometry
straight, C-shaped, reversed-C
Angular velocity, ω
0.2rads−1
Chord length, 2R / width, w
76.2 / 25.4mm
Discretization
Domain / soil-bed size
0.30×0.30 / 0.29×0.165m
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 ω=0.2rads−1 [ 22 ] . Rows show the straight, C-shaped, and reversed-C legs; columns show Fx and Fz .
Leg
Fz NRMSD (%)
Fx 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 ∣θ∣<85∘ , normalized by the experimental peak-to-peak range (NRMSD, %).
Figure 7: Straight-leg force histories for ϕ=25∘ , 31∘ , 37∘ , and 43∘ at ω=0.2rads−1 : (a) Fx and (b) Fz .
Parameter
Values
Friction angle, ϕ
25∘ – 45∘ , 2∘ steps (11 cases)
Rotation rate, ω
0.2rads−1 (matches the benchmark)
Off-grid test angles
14 cases, ϕ∈[25.9∘,43.6∘]
Sweep range, θ
[−135∘,135∘]
Table 3: Straight-leg simulation library and off-grid test cases.
Figure 8: Observation vector d : (a) force histories resampled at 40 angles over ∣θ∣<85∘ ; (b) concatenated Fx and Fz segments.
Figure 9: GP forward map: fitted mean surfaces of (a) Fx(ϕ,θ) , and (b) Fz(ϕ,θ) .
Figure 10: GP forward-map held-out relative error, ∣FGP−Ftrue∣/Ftrue , 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) Fx and (b) Fz . 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 ϕ⋆=33.97∘ : (a) walker traces and burn-in cutoff; (b) posterior density and retained samples, with recovered ϕ=33.95∘ .
Figure 12: Matched-model recovery for 14 off-grid cases. Error bar of each recovered friction angle shows 68% credible intervals. Absolute error =ϕrecovered−ϕtrue ; the dashed line marks zero error.
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.
Payload forces must be accommodated during locomotion, while leash forces can specify desired motion. We investigate whether a shared three-dimensional force estimate in newtons, inferred from proprioceptive history under sustained loading, can support both tasks. An estimator and locomotion policy are jointly trained with supervised force and velocity outputs and learned latent context. The estimated force conditions locomotion and additionally generates planar-velocity and yaw-rate commands for leash guidance through an analytical map. In sustained-force simulation sweeps, temporal means of componentwise force root mean square error range from 1.44 to 2.83,N. Compared with a domain-randomized baseline, the framework reduces velocity-tracking and base-orientation error scores by 21.6% and 46.5%, respectively, and increases mean survival from 68.29% to 94.60% in separate sustained-force tests. Unitree Go1 experiments demonstrate stationary vertical and horizontal force estimation, locomotion with an 8.5,kg payload whose weight exceeds the 70,N training force limit, and leash guidance using the same force-estimation interface.
Run Wang, Xu Yang, Alapati Tuerxun +1
Department of Automation, Tsinghua University, Beijing, China
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.
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