  Enabling robots to walk and run on yielding terrain is increasingly 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 permanent 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 dimensionless soft landing problem. Through
simulations, we find that optimal impedance reduces penetration depth nearly as
much as the open-loop force profile, while remaining robust to model
uncertainty. Through simulations and experiments, we find that the solution
space is rich, exhibiting qualitatively different relationships between impact
velocity and the optimal impedance for small and large dimensionless impact
velocities. Lastly, we discuss the relevance of this work to
minimum-cost-of-transport locomotion for several actuator design choices.
