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Sidewinding with minimal slip: Snake and robot ascent of sandy slopes

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Science  10 Oct 2014:
Vol. 346, Issue 6206, pp. 224-229
DOI: 10.1126/science.1255718

Abstract

Limbless organisms such as snakes can navigate nearly all terrain. In particular, desert-dwelling sidewinder rattlesnakes (Crotalus cerastes) operate effectively on inclined granular media (such as sand dunes) that induce failure in field-tested limbless robots through slipping and pitching. Our laboratory experiments reveal that as granular incline angle increases, sidewinder rattlesnakes increase the length of their body in contact with the sand. Implementing this strategy in a physical robot model of the snake enables the device to ascend sandy slopes close to the angle of maximum slope stability. Plate drag experiments demonstrate that granular yield stresses decrease with increasing incline angle. Together, these three approaches demonstrate how sidewinding with contact-length control mitigates failure on granular media.

What's that coming over the hill—is it a robot?

Crossing a slope can be difficult, particularly if it is made of sand. Sidewinder rattlesnakes manage to climb sandy hills by adjusting the length of their body in contact with the sand. Marvi et al. designed robots based on this idea to determine what affects climbing ability on sandy slopes (see the Perspective by Socha). Based on the behavior of the robots, the authors performed further animal studies, and used an iterative approach to improve the robots' capabilities and to better understand animal motion.

Science, this issue p. 224; see also p. 160

The majority of terrestrial mobile robots are restricted to laboratory environments, in part because such robots are designed to roll on hard flat surfaces. It is difficult to systematically improve such terrestrial robots because we lack understanding of the physics of interaction with complex natural substrates such as sand, dirt, and tree bark. We are thus limited in our ability to computationally explore designs for potential all-terrain vehicles; in contrast, many of the recent developments in aerial and aquatic vehicles have been enabled by sophisticated computational-dynamics tools that allow such systems to be designed in silico (1).

Compared with human-made devices, organisms such as snakes, lizards, and insects move effectively in nearly all natural environments. In recent years, scientists and engineers have sought to systematically discover biological principles of movement and implement these in robots (2). This “bioinspired robotics” approach (3) has proven fruitful to design laboratory robots with new capabilities (new gaits, morphologies, and control schemes), including rapid running (2, 4), slithering (5), flying (6), and swimming in sand (7). Fewer studies have transferred biological principles into robust field-ready devices (4, 8) capable of operating in, and interacting with, natural terrain.

Limbless locomotors such as snakes are excellent systems to study to advance real-world all-terrain mobility. Snakes are masters of most terrains: They can move rapidly on land (9, 10) and through water (11), burrow and swim through sand and soil (12), slither through tiny spaces (13), climb complex surfaces (14), and even glide through the air (15). Relative to legged locomotion, limbless locomotion is less studied, and thus broad principles that govern multi-environment movement are lacking. Recently developed limbless robotic platforms (5), based generally on the snake body plan, are appealing for multifunctional robotics study because they are also capable of a variety of modes of locomotion. These robots can traverse confined spaces, climb trees and pipes, and potentially dive through loose material. However, the gaits that carry these robots across firm ground can be stymied by loose granular materials, collections of particles that display solidlike features but flow beyond critical (yield) stresses.

A peculiar gait called sidewinding is observed in a variety of phylogenetically diverse species of snakes (9, 16). Animals with excellent sidewinding ability—such as the North American desert-dwelling sidewinder rattlesnake (Crotalus cerastes) in Fig. 1, A and C—are often found in environments that are either composed of granular materials (such as sand dunes) or are smooth, such as hardpan desert. Sidewinding is the translation of a limbless system through lifting of body segments while others remain in static (in the world frame, locally rolling/peeling in the frame of the body) contact with the ground. Such interactions minimize shear forces at contact, and thus, sidewinding is thought to be an adaptation to yielding or slippery surfaces (16). Biological measurements (17, 18) indicate that relative to lateral undulation, rectilinear locomotion, and concertina motion, sidewinding confers energetic advantages on hard surfaces, mainly through lack of slip at the points of contact. But although sidewinding motion of biological snakes has been extensively studied on hard ground, (16, 17, 19, 20), only one study has reported kinematics on granular media (9).

Fig. 1 Sidewinding locomotion on granular media.

(A) A Sidewinder rattlesnake (C. cerastes) locomoting on a granular substrate of 20° inclination. (B) The CMU modular snake robot traversing granular media at ≈20° inclination. The arrow indicates the direction of motion. (C and D) Two superimposed snapshots of (C) a sidewinder and (D) the robot moving on level sand. The initial position is denoted by “1,” and the successive position is denoted by “2.” The time between snapshots in (C) and (D) is 0.2 and 3.1 s, respectively. The vertical bars beneath the images indicate vertical displacement along the body at the two times. (E) Two rendered snapshots of a snake sidewinding on level sand, with red indicating the initial time and blue a successive time; time between images is 0.2 s. The 3D configurations were reconstructed according to shape data collected from a sidewinder rattlesnake moving on level sand. Centerline projections of the 3D snake data are shown on three orthogonal planes. The z coordinates of the snake are scaled up four times to increase visibility in the yz and xz projections. The horizontal (xy) and vertical (yz) waves travel in the posterior direction of the snake with respect to a coordinate system rigidly attached to the animal (as shown by the green arrows). (F) Snapshots of the CMU snake robot executing a sidewinding gait on level sand and projections of the resultant body shapes onto three orthogonal planes. Red and blue indicate the initial and successive images, respectively. Time between images is 6.3 s. The horizontal and vertical waves travel in the posterior direction of the robot with respect to a body-fixed coordinate system (as shown by the green arrows).

An initial robotic version of the sidewinding gait has proven useful to enable snake robots to maneuver over flat and bumpy terrain (21). In a robotic sidewinding mode, the device maintains two to three static (locally rolling/peeling) contacts with the substrate at any moment (22). During this motion, individual segments of the robot are progressively laid into ground contact, peeled up into arch segments, and then transferred forward to become new ground contact segments (16, 22). However, when in field tests a limbless robot [which we will refer to as the Carnegie Mellon University (CMU) robot (Fig. 1, B and D)] was confronted with even modest granular inclines (≈10°) far from the maximum angle of stability (≈30°), it failed to climb (either slipping or rolling downhill).

We posit that the study of success and failure modes in biological snakes can improve the mechanics and control of limbless robot performance, whereas study of success and failure modes of robots can give insight into important mechanisms that enable locomotion on loose material and perhaps explain why sidewinding has evolved in such organisms. This idea builds on recent work using biologically inspired robots as “physical models” of the organisms; such models have revealed principles that govern movement in biological systems, as well as new insights into low-dimensional dynamical systems [for example, (8) and references therein].

To discover principles of sidewinding in loose terrain, we challenged C. cerastes (n = 6 snakes, mass = 98 ± 18 g, body length, tip-to-tail, L = 48 ± 6 cm) (23) to climb loose granular inclines (table S2) (28) of varied incline angle, in a custom trackway housed at Zoo Atlanta (fig. S2A). The data comprised 54 trials: six snakes, three inclinations, three trials each. Before each experiment, air flow through a porous rigid floor fluidized the granular media and left the surface of the sand in a loosely packed state with a smooth surface (movie S4); details of this technique are described in (24). Three high-speed cameras tracked three-dimensional (3D) kinematics of 8 to 10 marked points on the body. For systematic testing, we chose three incline angles, θ = 0°, 10°, and 20°; the maximum angle of stability θ0 of the loosely packed material, where local perturbations resulted in system-wide surface flows, was θ0 = 27°.

As shown in Fig. 1, A and C, and fig. S1A and as previously noted (9, 16), on level granular media the snake generated two contacts (as highlighted by dashed lines in Fig. 1A) with the substrate at each instant, similar to the movement on hard ground (16, 17). As shown in Fig. 1E, the sidewinding pattern could be approximated by posteriorly propagating waves in both horizontal (xy) and vertical (yz) planes, with a phase offset of Δϕ = 1.51 ± 0.17 rad. The position of contact points moved from head to tail, leaving pairs of straight parallel lines on the substrate. The angle between these lines and the direction of motion (fig. S4D) was α = 33 ± 8° on granular media. The length of each track line was the same as snake body length; the spacing of tracks was thus determined by the spacing of contact points on the snake’s body (fig. S4C) (16). As previously observed on hard ground (17), speed increased with frequency (defined here as the number of cycles per second) (fig. 2C, inset). The snake body intruded into sand by 1.4 ± 1.3 cm and created a hill of material at each contact point; details on penetration depth measurement are provided in (23).

Fig. 2 Sidewinding motion at different granular incline angles θ.

(A) Traces of the sidewinder rattlesnake (C. cerastes) sidewinding on level sand; green dashed lines indicate tracks that an animal leaves on sand. (B) Displacement versus time of the snake at θ = 0° (top) and 20° (bottom); colored lines correspond to colored dots on the snake in (A). Regions of each curve that have zero or near zero slope correspond to static contact with the granular medium. (C) Contact length l normalized by the body length L as a function of θ for sidewinder rattlesnakes. The data shown encompass six sidewinder rattlesnakes, with three trials per animal at each condition. (Inset) Snake forward speed v versus wave frequency f. Green circles, light blue triangles, and red upside down triangles correspond to θ = 0°, 10°, and 20°, respectively. Data are mean ± SD. (D) Illustration of the CMU snake robot sidewinding on level sand; green dashed lines indicate the tracks the robot leaves on the sand. (E) Displacement versus time of the robot at inclinations of 0° (top) and 20° (bottom); colored lines correspond to the colored dots on the robot in (D). The robot wave frequency in both of these plots was f = 0.31 Hz, and its normalized contact length was 0.53 ± 0.03 and 0.45 ± 0.05 for θ = 0° and 20°, respectively. (F) Minimum (black), optimum (red), and maximum (blue) normalized contact length l/L for successful robot climbs as a function of θ. The gray region below the minimum l/L corresponds to pitching failure. The pink region between the minimum and optimum l/L indicates slipping (but still maintains forward progress) due to insufficient contact, and the blue region above the maximum l/L indicates slipping failure due to insufficient lifting. Dashed lines are estimated boundaries of regions of different performance. The robot wave frequency for this plot was f = 0.08 Hz. l/L at several other wave frequencies are plotted in Fig. S6B. Data denote mean ± SD.

We next examined the behavior of the animals as θ increased. A sidewinder climbing an inclination of θ = 20° is shown in fig. S1A and movie S1. Unlike the robot, the snakes were able to ascend the granular inclines without any axial or lateral slip (movie S1), indicated by the horizontal regions in Fig. 2B. In addition, increasing θ did not significantly change the penetration depth [analysis of variance (ANOVA), F2,51 = 0.33, P = 0.72], the angle between direction of motion and track lines (ANOVA, F2,51 = 0.06, P = 0.81), or the number of contacts. Instead, we observed that as θ increased, the length of the contact regions relative to the body length increased (ANOVA, F2,51 = 48.99, P < 0.0001); we refer to this quantity as l/L. To determine this quantity, we analyzed the 3D kinematics data to find the length of snake body in static (peeling) contact with sand in each cycle; details on contact length measurement are provided in (23). As shown in Fig. 2C, l/L increased by 41% as θ increased from 0° to 20°. At all θ, portions that were not in contact were lifted clear of the substrate. Wave frequency and climbing speed decreased with increasing θ (fig. S4, A and B) with effective step length St [defined as the distance the head moves forward in each period (fig. S4C)] remaining constant over all frequencies and θ. Further kinematic details of the movement are provided in the supplementary text.

To determine whether the ability to climb effectively on sandy inclines is common in closely related snakes, we examined the locomotor behavior of 13 species of pit vipers [subfamily Crotalinae (table S1)] in the collection of Zoo Atlanta on granular media. None of these species performed sidewinding in any conditions, instead using either lateral undulation or concertina, sometimes combined with rectilinear, gaits (table S1). On level sand, many species failed to achieve forward movement, and on sand slopes at θ = 10°, all but a single species failed to move uphill (Fig. 3C and movie S3). Yielding of the sand was accentuated by the failure of these species to lift portions of their body above the substrate (movie S3).

Fig. 3 Sensitivity of locomotor performance on granular media.

(A) CMU snake robot speed versus l/L for inclination angles θ = 0° (green circles), 5° (light blue rectangles), 10° (dark blue triangles), 15° (purple diamonds), and 20° (red upside down triangles). Failure regimes due to pitching and slipping are circled in black. Three trials were performed at each condition. Data indicate mean ± SD. (B) Superimposed frames showing pitching and slipping failure modes in the robot ascending θ = 20° and 10° inclines, respectively. Uphill direction is vertically aligned with the page. t0 and tF represent the time at which each body configuration is captured. The time between two images in the pitching and slipping failure modes is 1.6 and 6.3 s, respectively. (C) Slipping failure of nonsidewinding snakes. Superimposed images show failed lateral undulation on level sand by Sistrurus miliarius (left, length 47 cm) and failed concertina locomotion on level sand by Mixcoatlus melanurus (right, length 47 cm). The time between two images of S. miliarius and M. melanurus is 0.5 and 9 s, respectively.

We hypothesize that the sidewinder rattlesnake’s ability to move effectively on loose substrates (compared with the robot and the other closely related snakes) is made possible by neuromechanical control that generates appropriate contact length as sandy inclines become more susceptible to flow. To test this hypothesis, we next used the CMU robot as a physical model to study locomotor performance as a function of l/L for different θ (23). In particular, the CMU robot’s ability to deform in arbitrary modes [using 17 modules (23), section 1.3], including a combination of traveling waves that generate sidewinding, makes it an attractive model on which to test the generality and efficacy of the contact length mechanism. And despite the differences in weight and size of the snake robot relative to the organism—as demonstrated in previous studies of fish, turtles, and cockroaches (8, 25, 26) (and shown below)—principles of small organism locomotor bio and neuromechanics can be deduced through systematic variation of parameters in larger-scale physical models.

To generate sidewinding in the robot, we controlled its modules (and joints) to generate two posteriorly directed traveling waves in the horizontal and vertical planes with a 1/4 period phase offset, Δϕ = 1.57 rad (comparable with that of the animal). These waves were generated so that dorsal and ventral surfaces maintained their respective orientations throughout the motion. This produced a pattern of undulation and lifting similar to that observed in the biological snakes (Fig. 1, E and F). When the traveling waves are approximated by using sinusoids, the resultant shape of the robot resembles an elliptical helix whose minor axis is perpendicular to the ground and major axis is aligned with the direction of motion. Similar to the biological snake, during a gait cycle each module (segment) traced an approximately elliptical trajectory (Fig. 1F). All experiments were performed at constant wavelength (λ = 0.5L) and at frequencies sufficiently low as to avoid motor angular position and speed error. On level hard ground, a minor-to-major-axis aspect ratio of 0.9 produced steady motion at a speed of 0.03 m/s (at frequency f = 0.08 Hz). On hard surfaces at θ = 20°, decreasing the aspect ratio to 0.7 allowed the robot to climb without pitching or rolling down the hill and at comparable speeds as on level hard ground (22).

The waves that generated effective robot sidewinding on solid inclines were ineffective on granular media (movie S2). To improve the performance, we decreased the aspect ratio of the elliptical core, which increased the contact length in a manner analogous to the behavior observed in the biological snakes (fig. S6A). The minimum l/L needed for successful climbing (defined as ascending at positive forward speed without pitching; example traces are provided in Fig. 2D and tracked markers in Fig. 2E) increased by ≈70% as θ increased from 0° to 20°. Within the successful regions, we were further able to optimize the contact lengths to minimize slip and slightly improve the speed as compared with moving with the lower-bound of l/L. Details on robot contact length measurements are provided in (23). There was no dependence on frequency, indicating that body inertial forces were minimal and that granular frictional forces determined the resistive forces (27). Thus, despite the many differences between the biological and robot sidewinding, comparable performance was achieved through use of a similar strategy of increasing contact length with increases in θ. The elliptical helical gait is only one way to generate the traveling wave that propels the sidewinding in both hard ground and sand. Experimenting with helices of other cross-sections (such as oval shapes) led to similar results. Also, the maximum penetration depth d at the minimum l/L for successful climbing (at f = 0.08 Hz) decreased by ≈30% as θ increased from 0° to 20° (ANOVA, F4, 10 = 9.03, P = 0.007).

We next systematically studied how robot performance changed with l/L (Fig. 3A). As shown in Figs. 2F and 3A, the robot was able to ascend effectively for a given θ only within a range of l/L. The range narrowed as θ increased, indicating that for shallow slope angles, sidewinding performance was robust to variations in contact length, whereas for higher slope angles, the control effort must increase to target a narrower range of contact length. On inclinations of <15°, the robot exhibited high performance (speed) in a wide range of l/L until too large l/L produced slipping that decreased forward speed; this was related to an inability of the robot to lift itself sufficiently to avoid unnecessary ground contact (similar to the slipping failures observed in other snake species). At the other extreme, small l/L resulted in an insufficient supporting region, causing the robot to pitch down the slope. In the pink region between minimum and optimum l/L in Fig. 2F, the robot slipped due to insufficient contact but still made forward progress. For large θ, even within the band of effective ascent, performance was degraded because of downhill slip, which decreased the effective step length (fig. S6D).

The success of the robot and its similar performance to the biological sidewinder suggest the following picture of sidewinding on sandy slopes. First, the animal targets a neuromechanical control scheme consisting of two independently controlled and appropriately phased orthogonal waves (Fig. 1, E and F). Such a scheme satisfies the definition of a “template”—that is, a behavior “contains the smallest number of variables and parameters that exhibits a behavior of interest” (4). Discovery of templates is useful because they provide an organizing principle for the enormous number of degrees of freedom inherent in all organisms. Second, the robot experiments suggest that the two-wave template dynamics can be simply modified on sandy slopes so that the animal targets a pattern of l/L and body-segment lifting that minimizes slip and pitching.

Understanding the ground reaction forces responsible for the relevant interactions (for example, optimal l/L without slip) requires a model of the interaction of objects with granular media. However, there is no fundamental theory for bio- and robotically relevant interactions in granular media and a wider class of materials (mud, rubble, or leaf litter). In particular, despite recent discoveries in terradynamics (28), drag (27), and impact (29) on level dry granular media [and theoretical approaches to characterize granular rheology (30)], none of these studies have investigated drag forces on granular inclines.

We therefore studied the physics of transient granular yield forces as a function of θ. For context, studies of steady-state drag on level granular surfaces reveal that force is proportional to plate width, w, and depth squared, d2 (31). To make the first measurements of drag force F dependence on θ, d, and w, we performed drag force measurements on granular inclines (23). We inserted flat plates of different widths (w = 1.5 to 6 cm) to different depths (d = 1 to 2 cm) into the granular medium perpendicular to the slope plane and displaced the plate downhill 15 cm. Depths were chosen to bound the range of penetration depths observed in the animal experiments. Because we were interested in the yield behavior of the granular slope in the quasi-static limit (we assume inertial effects are small at speeds at which the snakes generally operate), we performed drag at slow speeds and studied the force increases under small displacements δ from rest.

As shown in Fig. 4B, force increased substantially for small displacements until it reached a saturation regime, after which there was a slower increase in force associated with the buildup of a granular pile in front of the plate. We refer to the regime of rapid rise as the “stiffness” of the sand (before large yielding) and denote it k (F/δ near δ = 0). We estimated k by fitting a line to the first 17% of data (corresponding to a displacement 0.5 cm, before material yielded noticeably) presented in Fig. 4B, and we plot this in Fig. 4C. For fixed d, k decreased by ≈50% as θ increased from 0° to 20° (Fig. 4C). For a fixed θ, k increased quadratically with d and nearly linearly with w. This scaling indicates that in the limit of shallow drag and small displacements δ, Fk δ with kw0.8 d2cos{[π/(2θ0)] θ}, where the phenomenological cos term reflects the rapid decrease in force as θ approaches θ0 [details on curve fitting are provided in (23)]. For θ = 0° and fixed δ, the dependence on w is similar to that previously observed (31), even though our depths are shallower.

Fig. 4 Granular incline drag force experiments.

(A) Schematic of drag force apparatus. (B) Drag force F versus horizontal displacement δ for different plate widths w and penetration depths d. The dashed line indicates the region used to calculate sand stiffness, k. (C) The fitted k versus inclination angle θ for different w and d. (Inset) The sand stiffness data collapse when normalized by w0.8d2; the thick black line is the fit function 0.12 cos([π/(2θ0)] θ), with θ0 = 0.47 rad (θ0 = 27°). Triangles, circles, and squares represent w = 1.5, 3, and 6 cm, respectively. Black, red, and blue colors illustrate d = 1, 1.5, and 2 cm, respectively. Data denote mean ± SD.

The drag force measurements support our hypothesis of contact length control: As θ increases, effective sidewinding can be maintained by increasing the contact length to offset the decrease in yield forces. This allows the locomotor to maintain stresses below the yield stress, minimizing slip. Insights from the drag measurements also indicate why the appropriate amount of body lift and contact to be within the range of l/L is important. If lift is too small, other segments of the locomotor encounter drag forces that must then be offset by either increases in l/L, or potentially increasing intrusion depth. However, increases in l/L would decrease lift, resulting in greater drag. Increasing contact length relative to increasing intrusion depth has obvious benefits because intrusion into granular media requires yielding material, and the force to do so increases with depth [the challenges associated with penetration are described in (32)]. The energetic cost to vary l/L is small in comparison.

The contact length modulation strategy has benefits in terms of locomotor control: Once the animal or robot is moving using a sidewinding template (again, two independently controlled waves with phase difference of approximately π/2), slip and pitch mitigation in flowable substrates can be effected with relatively simple modulations of the basic template wave pattern. This control has features in common with our previous biological (33) and robotic (24, 26) studies of relatively slow-legged movement on the surface of granular media, which also demonstrated that use of the solid features of granular media had benefits in certain locomotor regimes. We predict that other locomotors that move on granular surfaces could target movement patterns whose modulation can be used to achieve minimal-slip.

As expected, there were differences in the details between the sidewinding in the two systems. We view these differences, necessarily present because of the relative simplicity of the robot compared with the organism, as consequences of the different mechanisms that are used (biological complexity versus relative robot simplicity) as “anchors” of the templates, in the terminology discussed in (8). For example, the CMU robot has no compliance in its joints, many fewer degrees of freedom, and larger range of rotation (–90° to 90°) compared with those of biological snakes. Another interesting difference between a snake and the robot is that although both of them slowed as θ increased (figs. S4A and S6C), the snake’s decrease in speed was correlated with a decrease in frequency (indicating possibly either muscle limitations or active control to decrease inertial forces), whereas the robot speed decrease was largely determined by a decrease in spacing between tracks, as shown in fig. S6D. This decrease in effective step length was related to slipping of the robot at the highest θ. Comparative study of the anchoring mechanics is useful to learn about which lower-level mechanisms in the control hierarchy are critical, both to generate template dynamics as well as to understand neuromechanical control targets for the anchors.

Supplementary Materials

www.sciencemag.org/content/346/6206/224/suppl/DC1

Materials and Methods

Figs. S1 to S6

Tables S1 and S2

References (3438)

Movies S1 to S5

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: We thank M. Tesch, E. Cappo, J. Rembisz, and L. Li from Robotics Institute at Carnegie Mellon University for helping with the robot experiments; J. Brock, D. Brothers, R. Hill, L. Wyrwich, and B. Lock from Zoo Atlanta for helping with snake experiments; A. Young and K. Young for assistance with collecting snakes; D. Dimenichi, R. Chrystal, and J. Shieh from Georgia Institute of Technology for helping with construction and video analysis; T. Nowak and C. Hobbs for photography; V. Linevich for CAD drawings; P. Umbanhowar and A. Zangwill for helpful discussion; the National Science Foundation (NSF) (CMMI-1000389, PHY-0848894, PHY-1205878, and PHY-1150760); Army Research Office under grants W911NF-11-1-0514 and W911NF1310092; the Army Research Lab Micro Autonomous Systems and Technology Collaborative Technology Alliance under grant W911NF-08-2-0004; and the Elizabeth Smithgall Watts endowment, for financial support. D.I.G., H.C., and D.H. also acknowledge the Army Research Office and NSF Physics of Living Systems for supporting the Locomotion Systems Science Workshop in Arlington, Virginia, May 2012. The data collected for this study will be made available through the SMARTech data repository at Georgia Institute of Technology.
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