Reactive velocity control reduces energetic cost of jumping with a virtual leg spring on simulated granular media

Abstract

Robots capable of dynamic locomotion behaviors and high-bandwidth sensing with their limbs have a high cost of transport, especially when locomoting over highly dissipative substrates such as sand. We formulate the problem of reducing the energetic cost of locomotion by a Minitaur robot on sand, reacting to robot state variables in the inertial world frame without modeling the ground online. Using a bulkbehavior model of high-velocity intrusions into dry granular media, we simulated single jumps by a onelegged hopper using a Raibert-style compression-extension virtual leg spring. We compose this controller with a controller that added damping to the leg spring in proportion to the intrusion velocity of the robot's foot into the simulated sand while the robot is pushing off in the second half of stance. This has the effect of both reducing the torque exerted by the motors because the added virtual "active damping" force acts in opposition to the virtual leg spring force, and reducing the transfer of energy from the robot to the sand by slowing the intrusion velocity of the foot. Varying the simulated robot's initial conditions and the simulated ground parameters, we gained a consistent 20% energy savings by adding active damping with no cost in apex height. For more information, see the Kod*lab website: kodlab.seas.upenn.edu

Virtual Energy Management for Physical Energy Savings in a Legged Robot Hopping on Granular Media

“…This equation describes the "power function" of the ground, or the rate at which energy is physically transferred from the foot to the ground. See Roberts and Koditschek (2018) for a more detailed derivation.…”

“…In previous work (Roberts and Koditschek, 2018;Roberts and Koditschek, 2019) we introduced a reactive controller for sand locomotion that uses a new variant of active damping to reduce the energetic cost of transport (see Section 2.2) and performed initial tests using a one-legged robot in simulation and emulation. In Roberts and Koditschek (2018) we introduced the reactive controller and showed in simulation that it reduced the energy necessary to jump on granular media relative to its base controller when varying forces from the ground and initial conditions. In Roberts and Koditschek (2019) we built a robotic platform that emulates the forces exerted by a simplified granular media model and tested a physical robot jumping with the active damping controller, varying the robot's jump height.…”

“…1) Mathematical analysis showing that the energy transferred to the ground when using active damping is strictly less than the energy transferred to the ground when using a comparison controller (Section 2.2.2; Figure 1) 2) Simulations comparing the effects of a wider range of parameters (foot size, extension stiffness, ground force functions, and active damping coefficient) than in Roberts and Koditschek (2018) and discrete element model simulations suggesting that active damping gives energetic savings for legged locomotion on sand (Section 2.3; Figures 2, 3) 3) Experiments on physical prepared granular media comparing the nominal and active damping controllers, corroborating the results of the simulations and analysis (Section 2.4; Figures 4, 5)…”

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

“…For legged locomotors on yielding terrain, ground deformation due to foot penetration constitutes an irrecoverable energy loss, so minimizing foot penetration depth is relevant to energy-efficient locomotion. Recent research ( [1,2,3,4]) has addressed jumping from rest on yielding terrain, i.e., the stanceto-flight transition. Here, conversely, we address the flightto-stance transition, focusing specifically on the challenge of minimizing foot penetration depth.…”

“…Similarly, Hubicki et al [2] demonstrated that jumping on granular media could be improved by incorporating a dynamic model of the ground reaction force (GRF) into the robot dynamics used by optimization-based motion planning algorithms. Although previous studies examined jumping from rest (e.g., [1], [2]) and cyclic hopping (e.g., [3], [4]) on yielding substrates, this paper focuses specifically on minimum-penetration-depth landing, which to the best of our knowledge has yet to be addressed.…”