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

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 stiffness parameters 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 (6)

• Figure 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.
• Figure 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.
• Figure 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 \ref{['eqn:quasi_static']} compared with load-cell ground-truth measurements.
• Figure 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 $m_\mathrm{a}a$.
• Figure 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.