Inverse Resistive Force Theory (I-RFT): Learning granular properties through robot-terrain physical interactions

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.

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/

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.