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

TL;DR

This work introduces a physics-informed machine learning framework, Inverse Resistive Force Theory (I-RFT), which integrates the Granular Resistive Force Theory model with Gaussian Processes to infer terrain properties from proprioceptively measured contact forces under arbitrary gait trajectories by embedding the granular force model within the learning process.

Abstract

For robots to navigate safely and efficiently on soft, granular terrains, it is crucial to gather information about the terrain's mechanical properties, which directly affect locomotion performance. Recent research has developed robotic legs that can accurately sense ground reaction forces during locomotion. However, existing tests of granular property estimation often rely on specific foot trajectories, such as vertical penetration or horizontal shear, limiting their applicability during natural locomotion. To address this limitation, we introduce a physics-informed machine learning framework, Inverse Resistive Force Theory (I-RFT), which integrates the Granular Resistive Force Theory model with Gaussian Processes to infer terrain properties from proprioceptively measured contact forces under arbitrary gait trajectories. By embedding the granular force model within the learning process, I-RFT preserves physical consistency while enabling generalization across diverse motion primitives. Experimental results demonstrate that I-RFT accurately estimates terrain properties across multiple gait trajectories and toe shapes. Moreover, we show that the quantified uncertainty over the terrain resistance stress map could enable robots to optimize foot design and gait trajectories for efficient information gathering. This approach establishes a new foundation for data-efficient characterization of complex granular environments and opens new avenues for locomotion strategies that actively adapt gait for autonomous terrain exploration.

Figures (5)

• Figure 1: Configuration-dependent robot–terrain interaction forces motivate inverse stress-map inference. (A, D) Two robotic platforms with variations in leg geometry and trajectory: a direct-drive robotic leg (A), and a C-legged hexapedal robot (D). (B, C) Example interaction forces of the direct-drive robotic leg under rectangular and cubic trajectories, respectively. (E, F) Example forces generated by the C-shaped robot leg during clockwise and counterclockwise rotation. For the same terrain, the measured interaction forces vary significantly with leg morphology and trajectory, demonstrating the configuration-dependent and the difficulty of directly inferring intrinsic terrain properties. (G, H) Stress map of the granular terrain in $x$ and $z$ directions, respectively. These intrinsic terrain properties are configuration-independent and are inferred using the proposed I-RFT framework.
• Figure 2: Forward Resistive Force Theory (RFT) formulation. (A) C-Toe discretized into 10 surface segments (labeled with index 0 to 9) for RFT evaluation. (B) Zoomed-in diagram illustrating local geometric parameters defining segment orientation, $\beta$, and motion direction, $\gamma$. (C) Instantaneous segment angles across the contact surface.. (D) Forward mapping: each segment samples a location in the terrain stress map and contributes to the net force through weighted superposition.
• Figure 3: Numerical validation of I-RFT across toe geometries and trajectories for the four configurations: (A) I-Toe, Gait 1; (B) I-Toe, Gait 2; (C) C-Toe, Gait 3; (D) C-Toe, Gait 2. (i) Experimental configurations: blue curve is toe trajectory; red curve indicates toe geometry; dashed line denotes sand surface. (ii) Sampling coverage in the ($\beta$, $\gamma$) domain induced by each configuration. (iii) Reconstructed stress maps, ($\alpha_x$, $\alpha_z$). (iv) Reconstruction error relative to ground truth.
• Figure 4: Experimental setup and trajectory configurations. (A) Direct-drive robotic leg mounted on the fluidized granular testbed. (B, C) I-Toe and C-Toe geometries installed on the direct-drive robotic leg. (D) A diagram illustrating gait 1, a rectangular trajectory. (E) A diagram illustrating gait 2, a cubic trajectory. Blue curve represents toe endpoint trajectory; orange dots represents spline control points. Green line represents sand surface level.
• Figure 5: Experimental evaluation of I-RFT. (A–B) I-Toe; (C–D) C-Toe. (i) Measured sand resistive force (blue), reference RFT prediction using scaled canonical stress maps (orange), and I-RFT reconstruction (red). (ii) Reconstructed stress maps from experimental data: (a) $\alpha_x$, (b) $\alpha_z$.