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. In this study, we focus on the ramifications of plastic terrain deformation on the hop-to-hop energy dynamics of a spring-legged monopedal hopping robot animated by a switched-compliance energy injection controller. From this deliberately simple robot-terrain template, we derive a hop-to-hop energy return map, and we use physical experiments and simulations to validate the hop-to-hop energy map for a real robot hopping on a real deformable substrate. The dynamical properties (fixed points, eigenvalues, basins of attraction) of this map provide insights into efficient, responsive, and robust locomotion on deformable terrain. Specifically, we identify constant-fixed-point surfaces in a controller parameter space that suggest it is possible to tune control parameters for efficiency or responsiveness while targeting a desired gait energy level. We also identify conditions under which fixed points of the energy map are globally stable, and we further characterize the basins of attraction of fixed points when these conditions are not satisfied. We conclude by discussing the implications of this hop-to-hop energy map for planning, control, and estimation for efficient, agile, and robust legged locomotion on deformable terrain.

Although they take many forms, legged robots rely upon springs to achieve high speed, dynamic locomotion. In this paper we examine the effect of adding parallel springs to robots that rely on virtual compliance. Specifically, we consider the trade-off between energetic efficiency and leg versatility that comes while using Parallel Elastic Actuators (PEAs). To do this, we vary the ratio of physical to virtual compliance for legged systems using a) a modified SLIP model, b) a single legged hopping robot, and c) a multibody simulation of the quadruped robot LLAMA. In each case we show that having a small physical compliance significantly improves the efficiency while also maintaining the robot's versatility.