Colloquium: Biophysical principles of undulatory self-propulsion in granular media

Biological locomotion, movement within environments through self-deformation, encompasses a range of time and length scales in an organism. These include the electrophysiology of the nervous system, the dynamics of muscle activation, the mechanics of the skeletal system, and the interaction mechanics of such structures within natural environments like water, air, sand, and mud. Unlike the many studies of cellular and molecular scale biophysical processes, movement of entire organisms (like flies, lizards, and snakes) is less explored. Further, while movement in fluids like air and water is also well studied, little is known in detail of the mechanics that organisms use to move on and within flowable terrestrial materials such as granular media, ensembles of small particles that collectively display solid, fluid, and gaslike behaviors. This Colloquium reviews recent progress to understand principles of biomechanics and granular physics responsible for locomotion of the sandfish, a small desert-dwelling lizard that “swims” within sand using undulation of its body. Kinematic and muscle activity measurements of sand swimming using high speed x-ray imaging and electromyography are discussed. This locomotion problem poses an interesting challenge: namely, that equations that govern the interaction of the lizard with its environment do not yet exist. Therefore, complementary modeling approaches are also described: resistive force theory for granular media, multiparticle simulation modeling, and robotic physical modeling. The models reproduce biomechanical and neuromechanical aspects of sand swimming and give insight into how effective locomotion arises from the coupling of the body movement and flow of the granular medium. The argument is given that biophysical study of movement provides exciting opportunities to investigate emergent aspects of living systems that might not depend sensitively on biological details.

Figure 1

A diversity of desert reptiles “swim” within dry granular media. (a) The Eastern sandfish Scincus mitranus. From Alexey Sergeev. (b) The desert plated lizard Gerrhosaurus skoogi. From Paul Freed, freedsphotography.com. (c) The three-toed snake skink Ophiomorus tridactylus. From Barbod Safaei Mahroo, CalPhotos. (d) The wedge-snouted skink Chalcides sepsoides. From Gabriel Martínez del Mármol Marín. (e) The Mojave shovel-nosed snake Chionactis occipitalis. From Perrin Schiebel. (f) The California legless lizard Anniella pulchra. From William Flaxington. (g) The Egyptian sandfish lizard Scincus scincus. From Sarah Sharpe. Scale bars in upper right of all panels denote approximately 1 cm.

Figure 2

Measuring drag forces in dry granular media. (a) A robotic arm moves a small horizontal cylinder (diameter 1.6 cm) within a fluidized bed of granular media (0.3 mm diameter glass particles shown); the air flow is used to prepare the substrate and is turned off during tests ([82]). Average drag forces during horizontal drag of the cylinder with axis oriented perpendicular to the drag direction through 0.3 mm diameter glass particles at (b) different depths below the surface and (c) at different drag speeds, at $\varphi =0.58$ and $\varphi =0.62$. The drag force from the support rod has been subtracted. (b), (c) Modified from [106].

Figure 3

(a) Experimental and visualization apparatus used to study sand swimming of the sandfish lizard, including fluidized bed to control the $\varphi$ of the GM, and x-ray source and image intensifier detector to visualize movement within the GM. Adapted from [106]. (b) X-ray snapshot of the sandfish. Black spots are lead markers attached to the back and limbs of the animal.

Figure 4

Kinematics of undulatory sand swimming. (a) Time series of midline tracks during above-ground walking, burial, and subsurface swimming. Snapshots indicate typical body posture in these regions. Adapted from [106]. (b) Space-time plot of deviation of the body from a straight line during subsurface swimming, normalized by body length $L$. Black snapshots show tracked animal midline position and are well fit to a single period traveling sinusoidal wave. Adapted from [86].

Figure 5

Independence of swimming kinematics on substrate properties. (a) Spatial kinematics of the undulatory wave during sand swimming in 0.3 mm diameter glass particles. LP bars indicate loosely packed GM at $\varphi =0.58$, and CP bars indicate more closely packed GM at $\varphi =0.62$. The inset shows a tracked midline of an animal and quantities of interest. (b) Average forward swimming speed normalized by $\lambda$ vs undulation frequency $f$ in slightly polydisperse 0.3 mm diameter glass particles of $\varphi =0.58$ (triangles) and $\varphi =0.62$ (circles) and in 3 mm diameter glass particles (inset).

Figure 6

Wave efficiencies $\eta$ for a diversity of swimmers and crawlers. Data are collected from the following sources: [115], [52], [53], [39], [45], [86], [69], and [106].

Figure 7

The discrete element method (DEM) simulation of GM. (a) Particle-particle interaction demonstrating virtual overlap which determines normal $F_n$ and tangential $F_s$ forces for a two-particle collision. The vectors $v_1$ and $v_2$ refer to incoming velocities of the two particles and $\delta$ the virtual overlap. (b) Schematic of apparatus to validate the DEM simulation. (c) Average force vs orientation angle of a horizontal cylinder (the same cylinder as used in Fig. 2) during constant speed drag in experiment and simulation.

Figure 8

Numerically simulated sandfish model of the animal. (a) Close-up view of the numerical sandfish with tapered body cross section (approximating that of the animal) in 3 mm simulated glass particles (particles above the sandfish model are rendered transparent). The inset shows the numerical sandfish with uniform body cross section, used to compare to RFT. The body length and height of the model sandfish were 12 and 1.6 cm, respectively. The mass of each segment was proportional to the cube of its width and the total weight of the simulated sandfish was 14 g. (b) Motor connections of a section of the simulated sandfish ($i=1$ refers to the head). $b$ indicates the width (maximum along the model) and height of the segments in the nontapered section of the animal model. (c) 3D view of a simulated sandfish at three different instants while swimming within a container of particles rendered semitransparent for visualization. The time taken by the simulated tapered head sandfish to swim across the container is approximately 3.5 s ($f=2\mathrm{Hz}$). (d) Midline kinematics of simulated tapered sandfish with $f=2.5\mathrm{Hz}$. Adapted from [85].

Figure 9

Swimming within a localized granular fluid. (a) The granular temperature calculated from particles within cells with dimensions of 0.3 cm ($W$) by 0.3 cm ($L$) by 1.6 cm ($H$). (b) Flow field in which color represents the instantaneous speed of particles (averaged within the same cells as in the temperature calculation). $A/\lambda =0.22$, $f=4\mathrm{Hz}$. (c) The peak power required to swim within the granular frictional fluid vs frequency for a tapered and uniform body shape. Adapted from [32].

Figure 10

Schematic diagram showing forces on an element of an undulatory locomotor, used for granular resistive force theory (RFT) calculations. In the granular RFT, the object has a square cross-sectional shape similar to that in the DEM simulation and forces are computed by integrating stresses over surface elements. (Lower panels) Empirical granular drag force relations used in the granular resistive force theory. Circles and triangles indicate $\varphi =0.62$ and $\varphi =0.58$, respectively. The dashed curve indicates forces for drag of a slender object in a low Reynolds number fluid, scaled to match in magnitude $F_{\parallel }$ for $\varphi =0.62$. Adapted from [86].

Figure 11

(a) Sandfish average forward swimming speed vs undulation frequency in 3 mm diameter glass particles compared to DEM simulation and RFT predictions with $A/\lambda =0.22$. Solid circles and triangles denote closely and loosely packed states, respectively. (b) Wave efficiency $\eta$ for animal, DEM simulation, and RFT. Ranges in DEM and RFT predictions indicate a tapered and uniform body. Adapted from [85].

Figure 12

Swimming in a robot physical model of the sandfish in experiment and DEM simulation. (a) The robot resting on a bed of 6 mm plastic spheres. The robot consists of six servomotors controlled to execute an approximate sinusoidal wave with fixed $A/\lambda$. (b) Speed (scaled by wavelength) vs $f$ in experiment (circles) and DEM simulation (triangles). Inset: Wave efficiency (calculated in DEM simulation) increases with increasing smoothness. (c) Speed in body lengths per cycle vs $A/\lambda$ in experiment (circles) and DEM simulation (triangles). Frequency is 0.5 Hz. Adapted from [85].

Figure 13

Optimal swimming in sand and its origin. (a) RFT predictions of speed in body lengths per cycle (“speed” curve) and weight specific mechanical cost of transport (CoT) vs $A/\lambda$. The vertical shaded region shows a probability distribution of $A/\lambda$ measured in the sandfish by [86] and [106], where the darker shading indicates more animals were observed operating with the corresponding spatial form. The average spatial form is $A/\lambda =0.2\pm 0.04$. (b) Predictions of $\eta$ (colored curve) vs $A/\lambda$ and $\lambda /L$ (black curve) vs $A/\lambda$ from RFT.

Figure 14

A wave of muscle activity accompanies the wave of body bending during subsurface swimming. (a) Diagrammatic representation of some superficial axial musculature in the sandfish. (b) Electromyography (EMG) setup with representation of the location of the EMG electrode implantation sites (circles) and the opaque markers (squares) attached to the exterior right dorsal side. (c) Top images show dorsal views of the sandfish body position during burial and subsurface movements. Each image corresponds to the onset of the 0.7 EMG burst (circles). Below, EMG recordings are shown at 0.5, 0.7, 0.9, and 1.1 SVL (snout-vent length) locations. Colored stems indicate the onset of the EMG burst. Adapted from [106].

Figure 15

The timing of muscle activation relative to body curvature during sand swimming. (a) The trace of a sandfish during sand swimming. Opaque markers (circles) are attached to the exterior midline to facilitate tracking. Colored dots represent implanted electrodes on the right side of the body. (b) Diagram of the model. $V$ arrows represent velocity and $F$ arrows represent forces from the medium. The inset shows the signs of the torque ($\tau$), the curvature ($\kappa$), and the rate of change of curvature ($\dot{\kappa }$) at approximately 0.6 body length. Negative $\tau$ corresponds to no muscle activation on the right side of the body (thick line). $\psi$ indicates the angle between the segment axis and its velocity. (c), (d) EMG recordings at 0.9 and 0.3 SVL, respectively, during sand swimming. The shaded regions indicate muscle activation. The line shows the measured angle between consecutive markers [see (a)]. Arrows indicate the difference in time between the onset of muscle activation and maximal convexity, the “neuromechanical phase lags.” (e), (f) The curvature, torque, and the predicted muscle activation (gray shaded regions) from the RFT model at two representative points indicated by black dots in (b). (g) The predicted onset and offset of muscle activation from the model compared to EMG measurements from the sandfish experiment (black error bars indicate standard deviations). The shaded areas indicate the periods of negative $\dot{\kappa }$. $A/\lambda =0.22$. Adapted from [33].