Daniel I. Goldman

Undulatory Swimming in Sand: Subsurface Locomotion of the Sandfish Lizard

Swimming Through Sand Although composed of solid particles, sand can behave like a fluid. If you had to swim through sand, how would you do it? Would you use your arms and legs for propulsion or would you make your body as compact as possible and try to wiggle and slither your way through? Maladen et al. (p. 314) used x-ray imaging to study the motion of sandfish lizards as they burrowed into sand. The sandfish lizard does not use its limbs, but instead flattens them against its body and uses large-amplitude traveling wave oscillation of its body to propel itself. Modeling can explain the motion the lizard uses to propel itself through a medium that is neither liquid nor solid. X-ray imaging reveals the undulatory motion of a sandfish lizard through a granular fluid. The desert-dwelling sandfish (Scincus scincus) moves within dry sand, a material that displays solid and fluidlike behavior. High-speed x-ray imaging shows that below the surface, the lizard no longer uses limbs for propulsion but generates thrust to overcome drag by propagating an undulatory traveling wave down the body. Although viscous hydrodynamics can predict swimming speed in fluids such as water, an equivalent theory for granular drag is not available. To predict sandfish swimming speed, we developed an empirical model by measuring granular drag force on a small cylinder oriented at different angles relative to the displacement direction and summing these forces over the animal movement profile. The agreement between model and experiment implies that the noninertial swimming occurs in a frictional fluid.

Active tails enhance arboreal acrobatics in geckos.

Geckos are natures elite climbers. Their remarkable climbing feats have been attributed to specialized feet with hairy toes that uncurl and peel in milliseconds. Here, we report that the secret to the geckos arboreal acrobatics includes an active tail. We examine the tails role during rapid climbing, aerial descent, and gliding. We show that a geckos tail functions as an emergency fifth leg to prevent falling during rapid climbing. A response initiated by slipping causes the tail tip to push against the vertical surface, thereby preventing pitch-back of the head and upper body. When pitch-back cannot be prevented, geckos avoid falling by placing their tail in a posture similar to a bicycles kickstand. Should a gecko fall with its back to the ground, a swing of its tail induces the most rapid, zero-angular momentum air-righting response yet measured. Once righted to a sprawled gliding posture, circular tail movements control yaw and pitch as the gecko descends. Our results suggest that large, active tails can function as effective control appendages. These results have provided biological inspiration for the design of an active tail on a climbing robot, and we anticipate their use in small, unmanned gliding vehicles and multisegment spacecraft.

Distributed mechanical feedback in arthropods and robots simplifies control of rapid running on challenging terrain

Terrestrial arthropods negotiate demanding terrain more effectively than any search-and-rescue robot. Slow, precise stepping using distributed neural feedback is one strategy for dealing with challenging terrain. Alternatively, arthropods could simplify control on demanding surfaces by rapid running that uses kinetic energy to bridge gaps between footholds. We demonstrate that this is achieved using distributed mechanical feedback, resulting from passive contacts along legs positioned by pre-programmed trajectories favorable to their attachment mechanisms. We used wire-mesh experimental surfaces to determine how a decrease in foothold probability affects speed and stability. Spiders and insects attained high running speeds on simulated terrain with 90% of the surface contact area removed. Cockroaches maintained high speeds even with their tarsi ablated, by generating horizontally oriented leg trajectories. Spiders with more vertically directed leg placement used leg spines, which resulted in more effective distributed contact by interlocking with asperities during leg extension, but collapsing during flexion, preventing entanglement. Ghost crabs, which naturally lack leg spines, showed increased mobility on wire mesh after the addition of artificial, collapsible spines. A bioinspired robot, RHex, was redesigned to maximize effective distributed leg contact, by changing leg orientation and adding directional spines. These changes improved RHexs agility on challenging surfaces without adding sensors or changing the control system.

Robotics in scansorial environments

We review a large multidisciplinary effort to develop a family of autonomous robots capable of rapid, agile maneuvers in and around natural and artificial vertical terrains such as walls, cliffs, caves, trees and rubble. Our robot designs are inspired by (but not direct copies of) biological climbers such as cockroaches, geckos, and squirrels. We are incorporating advanced materials (e.g., synthetic gecko hairs) into these designs and fabricating them using state of the art rapid prototyping techniques (e.g., shape deposition manufacturing) that permit multiple iterations of design and testing with an effective integration path for the novel materials and components. We are developing novel motion control techniques to support dexterous climbing behaviors that are inspired by neuroethological studies of animals and descended from earlier frameworks that have proven analytically tractable and empirically sound. Our near term behavioral targets call for vertical climbing on soft (e.g., bark) or rough surfaces and for ascents on smooth, hard steep inclines (e.g., 60 degree slopes on metal or glass sheets) at one body length per second.

Scaling and Dynamics of Sphere and Disk Impact into Granular Media

Direct measurements of the acceleration of spheres and disks impacting granular media reveal simple power law scalings along with complex dynamics which bear the signatures of both fluid and solid behavior. The penetration depth scales linearly with impact velocity while the collision duration is constant for sufficiently large impact velocity. Both quantities exhibit power law dependence on sphere diameter and density, and gravitational acceleration. The acceleration during impact is characterized by two jumps: a rapid, velocity-dependent increase upon initial contact and a similarly sharp depth-dependent decrease as the impacting object comes to rest. Examination of the measured forces on the sphere in the vicinity of these features leads to an experimentally based granular force model for collision. We discuss our findings in the context of recently proposed phenomenological models that capture qualitative dynamical features of impact but fail both quantitatively and in their inability to capture significant acceleration fluctuations that occur during penetration and which depend on the impacted material.

Sensitive dependence of the motion of a legged robot on granular media

Legged locomotion on flowing ground (e.g., granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bioinspired legged robots display speed relative to body size on hard ground comparable with high-performing organisms like cockroaches but suffer significant performance loss on flowing materials like sand. In laboratory experiments, we study the performance (speed) of a small (2.3 kg) 6-legged robot, SandBot, as it runs on a bed of granular media (1-mm poppy seeds). For an alternating tripod gait on the granular bed, standard gait control parameters achieve speeds at best 2 orders of magnitude smaller than the 2 body lengths/s (≈60 cm/s) for motion on hard ground. However, empirical adjustment of these control parameters away from the hard ground settings restores good performance, yielding top speeds of 30 cm/s. Robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough. We propose a kinematic model of the rotary walking mode based on generic features of penetration and slip of a curved limb in granular media. The model captures the dependence of robot speed on limb frequency and the transition between walking and swimming modes but highlights the need for a deeper understanding of the physics of granular media.

Dynamics of Drag and Force Distributions for Projectile Impact in a Granular Medium

Our experiments and molecular dynamics simulations on a projectile penetrating a two-dimensional granular medium reveal that the mean deceleration of the projectile is constant and proportional to the impact velocity. Thus, the time taken for a projectile to decelerate to a stop is independent of its impact velocity. The simulations show that the probability distribution function of forces on grains is time independent during a projectiles deceleration in the medium. At all times the force distribution function decreases exponentially for large forces.

Mechanical models of sandfish locomotion reveal principles of high performance subsurface sand-swimming

We integrate biological experiment, empirical theory, numerical simulation and a physical model to reveal principles of undulatory locomotion in granular media. High-speed X-ray imaging of the sandfish lizard, Scincus scincus, in 3 mm glass particles shows that it swims within the medium without using its limbs by propagating a single-period travelling sinusoidal wave down its body, resulting in a wave efficiency, η, the ratio of its average forward speed to the wave speed, of approximately 0.5. A resistive force theory (RFT) that balances granular thrust and drag forces along the body predicts η close to the observed value. We test this prediction against two other more detailed modelling approaches: a numerical model of the sandfish coupled to a discrete particle simulation of the granular medium, and an undulatory robot that swims within granular media. Using these models and analytical solutions of the RFT, we vary the ratio of undulation amplitude to wavelength (A/λ) and demonstrate an optimal condition for sand-swimming, which for a given A results from the competition between η and λ. The RFT, in agreement with the simulated and physical models, predicts that for a single-period sinusoidal wave, maximal speed occurs for A/λ ≈ 0.2, the same kinematics used by the sandfish.

Stationary state volume fluctuations in a granular medium

A statistical description of static granular material requires ergodic sampling of the phase space spanned by the different configurations of the particles. We periodically fluidize a column of glass beads and find that the sequence of volume fractions phi of postfluidized states is history independent and Gaussian distributed about a stationary state. The standard deviation of phi exhibits, as a function of phi, a minimum corresponding to a maximum in the number of statistically independent regions. Measurements of the fluctuations enable us to determine the compactivity X , a temperaturelike state variable introduced in the statistical theory of Edwards and Oakeshott [Physica A 157, 1080 (1989)].

Design of a Bio-inspired Dynamical Vertical Climbing Robot

This paper reviews a template for dynamical climbing originating in biology, explores its stability properties in a numerical model, and presents emperical data from a physical prototype as evidence of the feasibility of adapting the dynamics of the template to robot that runs vertically upward. The recently proposed pendulous climbing model abstracts remarkable similarities in dynamic wall scaling behavior exhibited by radically different animal species. The present paper’s first contribution summarizes a numerical study of this model to hypothesize that these animals’ apparently wasteful commitments to lateral oscillations may be justified by a significant gain in the dynamical stability and, hence, the robustness of their resulting climbing capability. The paper’s second contribution documents the design and offers preliminary empirical data arising from a physical instantiation of this model. Notwithstanding the substantial differences between the proposed bio-inspired template and this physical manifestation, initial data suggest the mechanical climber may be capable of reproducing both the motions and ground reaction forces characteristic of dynamical climbing animals. Even without proper tuning, the robot’s steady state trajectories manifest a substantial exchange of kinetic and potential energy, resulting in vertical speeds of 0.30 m/s (0.75 bl/s) and claiming its place as the first bio-inspired dynamical legged climbing platform.