Inverse Resistive Force Theory: Estimating granular terrain properties with arbitrary gait trajectories

For legged robots to navigate safely and efficiently on soft, granular terrains, it is crucial to assess the terrain's mechanical properties, which directly affect locomotion performance. Existing approaches for granular property estimation, such as penetration and shear tests, often rely on specific foot trajectories, limiting their applicability for accurate terrain sensing during natural locomotion. To address this limitation, we introduce a physics-informed machine learning framework, Inverse Resistive Force Theory (I-RFT), which integrates the 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. Experiments demonstrate that I-RFT accurately estimates terrain properties across different gait trajectories and toe shapes, and more importantly, enables robots to actively optimize gait trajectories for efficient information gathering. This approach establishes a new foundation for data-efficient characterization of complex granular environments.