Hopping on Deformable Terrain with State-Based Switching: Dynamics and Implications for Robot Design

Legged robot locomotion is hindered by a mismatch between applications where legs can outperform wheels or treads, most of which feature deformable substrates, and existing tools for planning and control, most of which assume flat, rigid substrates. While locomotion on any soft substrate can be challenging (e.g., maintaining upright posture, coordinating multiple legs), we focus on vertically-constrained single-leg hopping because it is arguably the simplest setting for studying the interplay between the depth-dependent substrate yield threshold and the forces applied during stance—a fundamental feature of legged locomotion on deformable terrain. We derive a Poincaré map that captures the hop-to-hop energy dynamics of a monopod hopping on deformable terrain. The map reveals complex boundaries in the design-control parameter space between periodic and decaying gaits, as well as families of transient responses and basins of attraction associated with each gait. These results emphasize the value of long, compliant legs and large, lightweight feet and indicate how the hop-to-hop energy dynamics can be leveraged to estimate terrain properties and to plan transitions between gaits in order to maximize efficiency, convergence rate, and robustness to terrain uncertainty.