A Template-and-Anchor Approach to Stable Sagittal-Plane Legged Locomotion on Yielding Terrain

The template-and-anchor framework's value to hard-ground legged locomotion is exemplified by the descriptive and prescriptive power of the spring-loaded inverted pendulum (SLIP) template. We extend this framework to soft-ground locomotion, introducing a monopedal template consisting of a massive body and massive foot, with foot-ground interaction modeled by resistive force theory. An active generalized spring, described by a 6x6 stiffness matrix, couples the body and foot and provides a way to inject energy to counteract energy losses due to finite foot mass and ground deformation. Given leg spring parameters and a leg angle at touchdown, the search for periodic sagittal-plane gaits reduces to finding fixed points of a three-dimensional map that describes the hop-to-hop evolution of the monopod's centroidal momentum. Fixed points are stabilized by hop-to-hop adjustments of the leg angle at touchdown and the energy injected by the leg spring relative to their nominal values. Stabilized fixed points are characterized in terms of their basins of attraction and their sensitivity to variations in ground stiffness. We compare these theoretical and numerical results to experimental results and discuss their implications for practical legged robotic applications on yielding substrates.