The Soft-Landing Problem: Minimizing Energy Loss by a Legged Robot Impacting Yielding Terrain

Enabling robots to walk and run on yielding terrain is vital to endeavors ranging from disaster response to extraterrestrial exploration. While dynamic legged locomotion on rigid ground is challenging enough, yielding terrain presents additional challenges such as ground deformation which dissipates energy. In this paper, we examine the soft-landing problem: given some impact momentum, bring the robot to rest while minimizing foot penetration depth. To gain insight into properties of penetration depth-minimizing control policies, we formulate a constrained optimal control problem and obtain a bang-bang open-loop force profile. Motivated by examples from biology and recent advances in legged robotics, we also examine impedance-control solutions to the soft-landing problem. Through simulations and experiments, we find that optimal impedance reduces penetration depth nearly as much as the open-loop force profile, while remaining robust to model uncertainty. Lastly, we discuss the relevance of this work to minimum-cost-of-transport locomotion for several actuator design choices.