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

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

Force Dynamics in Weakly Vibrated Granular Packings

The oscillatory force F_b^ac on the bottom of a rigid, vertically vibrated, grain filled column, reveals rich granular dynamics, even when the peak acceleration of the vibrations is signicantly less than the gravitational acceleration at the earth's surface. For loose packings or high frequencies, F_b^ac 's dynamics are dominated by grain motion. For moderate driving conditions in more compact samples, grain motion is virtually absent, but F_b^ac nevertheless exhibits strongly nonlinear and hysteretic behavior, evidencing a granular regime dominated by nontrivial force-network dynamics.Comment: 4 pages, 5 figure

Nearest pattern interaction and global pattern formation

We studied the effect of nearest pattern interaction on a globally pattern formation in a 2-dimensional space, where patterns are to grow initially from a noise in the presence of periodic supply of energy. Although our approach is general, we found that this study is relevant in particular to the pattern formation on a periodically vibrated granular layer, as it gives a unified perspective of the experimentally observed pattern dynamics such as oscillon and stripe formations, skew-varicose and crossroll instabilities, and also a kink formation and decoration

Model of coarsening and vortex formation in vibrated granular rods

Neicu and Kudrolli observed experimentally spontaneous formation of the long-range orientational order and large-scale vortices in a system of vibrated macroscopic rods. We propose a phenomenological theory of this phenomenon, based on a coupled system of equations for local rods density and tilt. The density evolution is described by modified Cahn-Hilliard equation, while the tilt is described by the Ginzburg-Landau type equation. Our analysis shows that, in accordance to the Cahn-Hilliard dynamics, the islands of the ordered phase appear spontaneously and grow due to coarsening. The generic vortex solutions of the Ginzburg-Landau equation for the tilt correspond to the vortical motion of the rods around the cores which are located near the centers of the islands.Comment: 4 pages, 5 figures, submitted to Phys. Rev. Let

Stationary and Oscillatory Spatial Patterns Induced by Global Periodic Switching

We propose a new mechanism for pattern formation based on the global alternation of two dynamics neither of which exhibits patterns. When driven by either one of the separate dynamics, the system goes to a spatially homogeneous state associated with that dynamics. However, when the two dynamics are globally alternated sufficiently rapidly, the system exhibits stationary spatial patterns. Somewhat slower switching leads to oscillatory patterns. We support our findings by numerical simulations and discuss the results in terms of the symmetries of the system and the ratio of two relevant characteristic times, the switching period and the relaxation time to a homogeneous state in each separate dynamics.Comment: REVTEX preprint: 12 pages including 1 (B&W) + 3 (COLOR) figures (to appear in Physical Review Letters

Square to stripe transition and superlattice patterns in vertically oscillated granular layers

We investigated the physical mechanism for the pattern transition from square lattice to stripes, which appears in vertically oscillating granular layers. We present a continuum model to show that the transition depends on the competition between inertial force and local saturation of transport. By introducing multiple free-flight times, this model further enables us to analyze the formation of superlattices as well as hexagonal lattice

Phase Changes in an Inelastic Hard Disk System with a Heat Bath under Weak Gravity for Granular Fluidization

We performed numerical simulations on a two-dimensional inelastic hard disk system under gravity with a heat bath to study the dynamics of granular fluidization. Upon increasing the temperature of the heat bath, we found that three phases, namely, the condensed phase, locally fluidized phase, and granular turbulent phase, can be distinguished using the maximum packing fraction and the excitation ratio, or the ratio of the kinetic energy to the potential energy.It is shown that the system behavior in each phase is very different from that of an ordinary vibrating bed.Comment: 4 pages, including 5 figure

Efficient, Responsive, and Robust Hopping on Deformable Terrain

Legged robot locomotion is hindered by a mismatch between applications where legs can outperform wheels or treads, most of which feature deformable substrates, and existing tools for planning and control, most of which assume flat, rigid substrates. In this study we focus on the ramifications of plastic terrain deformation on the hop-to-hop energy dynamics of a spring-legged monopedal hopping robot animated by a switched-compliance energy injection controller. From this deliberately simple robot-terrain model, we derive a hop-to-hop energy return map, and we use physical experiments and simulations to validate the hop-to-hop energy map for a real robot hopping on a real deformable substrate. The dynamical properties (fixed points, eigenvalues, basins of attraction) of this map provide insights into efficient, responsive, and robust locomotion on deformable terrain. Specifically, we identify constant-fixed-point surfaces in a controller parameter space that suggest it is possible to tune control parameters for efficiency or responsiveness while targeting a desired gait energy level. We also identify conditions under which fixed points of the energy map are globally stable, and we further characterize the basins of attraction of fixed points when these conditions are not satisfied. We conclude by discussing the implications of this hop-to-hop energy map for planning, control, and estimation for efficient, agile, and robust legged locomotion on deformable terrain.Comment: 17 pages, 13 figures, submitted to IEEE Transactions on Robotic

Kink-induced transport and segregation in oscillated granular layers

We use experiments and molecular dynamics simulations of vertically oscillated granular layers to study horizontal particle segregation induced by a kink (a boundary between domains oscillating out of phase). Counter-rotating convection rolls carry the larger particles in a bidisperse layer along the granular surface to a kink, where they become trapped. The convection originates from avalanches that occur inside the layer, along the interface between solidified and fluidized grains. The position of a kink can be controlled by modulation of the container frequency, making possible systematic harvesting of the larger particles.Comment: 4 pages, 5 figures. to appear in Phys. Rev. Let

Convective Motion in a Vibrated Granular Layer

Experimental results are presented for a vertically shaken granular layer. In the range of accelerations explored, the layer develops a convective motion in the form of one or more rolls. The velocity of the grains near the wall has been measured. It grows linearly with the acceleration, then the growing rate slows down. A rescaling with the amplitude of the wall velocity and the height of the granular layer makes all data collapse in a single curve. This can provide insights on the mechanism driving the motion.Comment: 10 pages, 5 figures submitted to Phys. Rev. Let