From Impact to Insight: Dynamics-Aware Proprioceptive Terrain Sensing on Granular Media

Yifeng Zhang; Yue Wu; Jake Futterman; Jacob Meseha; Eduardo Rosales; Irie Cooper; J. Diego Caporale; Feifei Qian

cs.ROApr 2, 2026

From Impact to Insight: Dynamics-Aware Proprioceptive Terrain Sensing on Granular Media

Authors: Yifeng Zhang, Yue Wu, Jake Futterman, Jacob Meseha, Eduardo Rosales, Irie Cooper, J. Diego Caporale, Feifei Qian

Organizations: Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA

Abstract

Robots that traverse natural terrain must interpret contact forces generated under highly dynamic conditions. However, most terrain characterization approaches rely on quasi-static assumptions that neglect velocity- and acceleration-dependent effects arising during impact and rapid stance transitions. In this work, we investigate granular terrain interaction during high-speed hopping and develop a physics-based framework for dynamic terrain characterization using proprioceptive sensing alone. Through controlled hopping experiments with systematically varied impact speed and leg compliance, our measurements reveal that quasi-static based assumptions lead to large discrepancies in granular terrain property estimation during high-speed hopping, particularly upon touchdown and controller-induced stiffness transitions. Velocity-dependent drag alone cannot explain these discrepancies. Instead, acceleration-dependent added-mass effects - associated with grain entrainment beneath the foot - dominate transient force responses. We integrate this force decomposition with a momentum-observer-based estimator that compensates for rigid-body inertia and gravity, and introduce an acceleration-aware weighted regression to account for increased force variance during high-acceleration events. Together, these methods enable consistent recovery of granular resistance across locomotion conditions, closely matching linear-actuator ground truth. Our results demonstrate that accurate terrain inference during high-speed locomotion requires explicit treatment of acceleration-dependent granular effects, and provide a foundation for robots to characterize complex deformable terrain during dynamic exploration of terrestrial and planetary environments.

Figures & tables

Fig. 1: Locomotion regimes on deformable terrain and their implications for proprioceptive sensing . (a) A quadruped robot navigating natural deformable terrain with spatially-varying strength and mechanics. (b) Dynamic leg-terrain interaction from a representative stride. (c) Locomotion regimes spanning quasi-static gaits (higher sensing fidelity) to dynamic gaits (higher speed), highlighting the trade-off between terrain inference accuracy and locomotion performance.

Fig. 2: (a) Robotic hopper. (b) Load cell instrumented linear actuator for ground truth measurements. (c) Hopper free body diagram. (d) SLIP controller state machine. (e) State estimation pipeline.

Fig. 3: Representative dynamic hopping stride on granular medium. (a) Snapshots of the hopper during a representative stride: initial compression after touchdown, maximum compression, deeper penetration induced by the stiffer extension phase, and post-liftoff. (b) Virtual leg length, defined as the body–foot height difference, measured by motor encoder. TD: touchdown; CE: compression–extension transition; LO: liftoff. (c, d) Body and foot height and velocity, respectively, measured via motion capture (MoCap). (e) Body and foot acceleration measured by onboard IMUs. (f) Quasi-static proprioceptive estimate ( 1 ) compared with load-cell ground-truth measurements.

Fig. 4: Terrain force decomposition under high-speed intrusion. (a) Force–depth with linear fit. (b) Intrusion speed–depth. (c) Depth–speed force map from constant-speed linear-actuator intrusions; gray: representative trials, cyan: hopper trajectory projected for prediction. (d) Measured force vs map prediction. (e) Foot acceleration. Positive represents downward direction. (f) $F_{\mathrm{res}}$ and added-mass term $m_{\mathrm{a}}a$ .

Fig. 5: Kinematic state estimation and MO-based force inference. (a-d) Body and foot height (a,b) and velocity (c,d) estimated via the onboard-sensing-based Kalman filter, compared with MoCap measurements. (e) Toe force estimated by the momentum observer using Kalman-filter-estimated kinematics. (f) MO force vs. KF-estimated depth, compared with load-cell force vs. MoCap depth.

Fig. 6: Granular resistance estimation across three treatments. (a) Representative depth–force curve comparing w/o MO, w/o GD (QS; OLS fit) versus w/ MO, w/o GD (MO-corrected; OLS fit). (b) Representative MO-based fit comparing w/ MO, w/o GD (OLS) versus w/ MO, w/ GD (acceleration-aware WLS; points weighted in the regression). (c,d) Estimated resistance for w/o MO, w/o GD (gray), w/ MO, w/o GD (light blue), and w/ MO, w/ GD (dark blue), shown versus (c) touchdown speed and (d) compression stiffness. In (c), $k_{c}=3.75\,\mathrm{N/cm}$ ; in (d), $v_{\mathrm{TD}}\approx 1.0\,\mathrm{m/s}$ . Points are means over $n=5$ trials; error bars indicate $\pm$ SEM. Dashed lines in (c,d) denote $k_{\mathrm{GT}}$ , the linear-actuator ground-truth granular resistance.

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Humanoid locomotion on granular terrain remains a significant challenge due to its complex foot-terrain interaction dynamics that are difficult to model. Existing approaches either ignore granular contact dynamics or incorporate simplified normal force models with heuristic tangential components. In this work, we present a physics-grounded granular contact model based on three-dimensional resistive force theory (3D RFT) and efficiently simulate granular terrain for reinforcement learning (RL) training. Unlike traditional rigid contact models and simplified granular contact models with ad-hoc heuristics, our contact solver produces physically accurate granular intrusion dynamics without resorting to heuristics. It captures realistic penetration and tangential drag during training, enabling the policy to learn behaviors that transfer reliably to real-world granular terrain where rigid contact models fail. To adapt to varying terrain conditions, we train a terrain-adaptive locomotion controller via teacher-student RL, using a variational autoencoder to encode terrain information into a compact latent representation. Simulation studies using material point method (MPM) with NVIDIA Newton demonstrate that our method generalizes to unseen granular terrains, achieves a significantly higher success rate than baselines, and demonstrates zero-shot terrain identification and adaptation. We further validate our approach through extensive hardware experiments across diverse real-world granular terrains including basalt, dry sand, and beach sand. To the best of our knowledge, this is the first demonstration of agile humanoid locomotion on real-world granular terrain. Project page: https://humanoid-gm-locomotion.github.io/HUMANOID-GM/

Abstract

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