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 model, 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.

翻译：轮式或履带式应用场景中，腿部机器人运动受限于基面可变形特性与现有规划控制工具（多假设平坦刚性基面）之间的根本矛盾。本研究聚焦塑性地形变形对弹簧腿单足弹跳机器人的跳间能量动力学影响，该机器人采用开关柔顺能量注入控制器驱动。基于这一刻意简化的机器人-地形耦合模型，我们推导出跳间能量回归映射，并通过物理实验与仿真验证了该映射在真实机器人在可变形基面上弹跳时的有效性。该映射的动力学特性（不动点、特征值、吸引域）揭示了可变形地形上高效、响应与鲁棒运动的本质规律：在控制器参数空间中识别出常不动点曲面，表明可通过调节控制参数实现目标步态能量水平下的效率或响应性优化；同时确定了能量映射不动点全局稳定的条件，并进一步刻画了条件不满足时不动点的吸引域特征。最后，本文讨论该跳间能量映射对可变形地形上高效、敏捷、鲁棒腿部运动规划、控制与状态估计的应用启示。