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

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.

Small-scale Geotechnical Testing Using a Six-axis Robot

“…Advanced actuating systems capable of generating motion and force in multiple directions are needed for problems routinely encountered in geotechnical engineering (e.g., foundation subjected to combined loading). Examples of problems that lie at interfaces with other disciplines include soil excavation (Maciejewski and Jarzebowski, 2002), soil-wheel interaction for off-road vehicle mobility (Wong and Reece, 1967;Wu et al, 1984;Khot et al, 2007;, and the locomotion of legged and wheeled robots on earth and other planets (Bauer et al, 2005;Li et al, 2013;Lynch et al, 2020). Physical modeling completed to date has largely rested on the use of customized multi-degree-of-freedom actuation systems O'Loughlin et al, 2018).…”

Real-time Optimal Landing Control of the MIT Mini Cheetah

“…A common approach for landing actuated systems is to control contact point impedances with some additional feedforward reaction forces. Lynch et al describes the socalled 'soft landing problem', exploring control methods to minimize foot penetration depth into a surface from a fall with prismatic actuation, but discussion was limited to single impacts on a body without rigid contacts [5]. Similar to quadrupeds, [6] explores actuated landing gear for rotorcraft and finding optimal impedances to mitigate impacts, but considers only horizontal landings.…”

Balance Recovery Based on Whole-Body Control Using Joint Torque Feedback for Quadrupedal Robots

“…Instead of virtual components, using the measured or estimated GRF improved the compliant behavior for soft interaction [20,21]. Furthermore, an impedance control method capable of reducing the penetration depth for minimizing energy loss was proposed [22]. However, general impedance controllers had to be assigned to each leg.…”

“…3 Balance Recovery Using Feed-Forward Force 3.1 Moment-Based Impedance Control. Impedance control laws have been employed in legged robots to mitigate the landing impact and ensure smooth interaction with the ground [19][20][21][22]. Figure 4 shows a comparison between the (a) conventional and (b) proposed impedance control methods.…”