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

TL;DR

This paper investigates minimizing energy loss during soft landings of legged robots on yielding terrain by formulating optimal control and impedance-control solutions, demonstrating their effectiveness through simulations and experiments.

Contribution

It introduces a constrained optimal control framework and impedance-control strategies for soft landing, providing insights into their robustness and relationship with impact velocity.

Findings

Optimal impedance nearly matches open-loop force profile in reducing penetration.

Robustness of impedance control to model uncertainties.

Different relationships between impact velocity and optimal impedance for various velocities.

Abstract

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.