Soft landing of a legged robot on yielding terrain

Many robotics applications where legged robots are preferable to wheeled or treaded robots feature yielding terrain. While legged locomotion on rigid ground is non-trivial, yielding terrain presents additional challenges such as permanent ground deformation, and dissipative effects. Regardless of substrate, successful legged locomotion entails starting, maintaining, and stopping a gait. We specifically address the challenge of terminating the hopping gait of a simple single-leg robot on soft ground while minimizing foot penetration depth, a locomotion primitive we call a “soft landing.” Using analytical methods from optimal control theory, we find that the penetration-minimizing feedforward control program is a bang-bang controller, followed by a third constant-force segment. While this control is optimal given our formulation of the soft landing problem, it is sensitive to model uncertainty and timing errors. Consequently, we also compare the robustness to terrain uncertainty of the optimal feedforward control to a biologically-inspired virtual impedance controller.