Terrain-Dependent Intra-Cycle Leg Timing for Effective Locomotion on Granular Slopes

Bipedal robots are challenging to control because they operate close to instability, where small variations in foot-terrain contact can rapidly destabilize locomotion. On rigid terrain, bipedal robots mitigate this fragility by using well-established contact mechanics and control strategies. On flowable surfaces such as granular slopes, foot contact can induce large surface deformations and solid-fluid-like transitions, coupling terrain effects with robot dynamics, leading to underperformance or failure. This is partly due to the lack of reliable methods to represent the dynamics of flowable terrain, making it difficult to account for terrain effects in locomotion design. Here, we investigate how controlling terrain response can improve bipedal locomotion on granular slopes by studying the terradynamics of cleated feet, thin plates emanating from the foot soles. Systematic studies of a small-scale (1.4 kg) robophysical biped reveal that cleats with sparse and dense spacing lead to excessive terrain yielding and resistance, respectively, degrading performance and leading to failure. An intermediate cleat spacing distributes interaction forces to maintain substrate stresses near (or below) the yield threshold, enabling walking on granular slopes up to 30 degrees. Guided by these principles, we design a foot that actively adjusts cleat depth and accommodates both rigid and granular terrain. We also demonstrate that the principles of effective foot-terrain interaction translate to a larger (15 kg) autonomous biped. Our study presents an alternative to conventional body-centric robot control approaches, which regulate terrain-induced effects through body motion, by instead regulating terrain interactions through limb-centric approach.

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/