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Perrin slithering a passive in of dynamics role the reveals diffraction Mechanical opoe hmevs h nygi hrdb l ibes elon- limbless, all by shared only heterogeneities The terrain themselves. complex on propel rely to often animals which terrestrial , in like difficult particularly is heteroge- environments in neous strategies movement Understanding obsta- 16). or 15, (7, turbulence cles) (e.g., and interactions unplanned eels, handle worms, snakes nematode like about known organisms is body-undulating little 14), how (9, studied of carefully been have dynamics collisions passive system. the nervous the on from input rapidly perturbations rely additional unexpected without contrast, reject can rapidly In to 14) structures 13). mechanical (9, (12, interactions self-deforms animals and novel moving terrain to the response about in information collect sensory to uses animal input the movements) deliberate careful, associ- with (typically ated former of the will determination During pattern. consequent We self-deformation information and the terrain between ways. surrounding relationship different the (11), the about loop as in defined open heterogeneities these and consider closed with a dealing between each spectrum a on classified and unex- schemes control collisions. and unplanned biological accommodate discontinuously interactions how bodyplans unclear because change it part can making in pectedly, in environments frontier is these a This in remains (7–9). terrain understand- studies heterogeneous (4–6)], locomotion sand environments in dry homogeneous movement and ground, in ing hard locomotion flat fluids, in [open made integration such been in progress has theory, While (1–4). control physics dis- soft-matter biomechanics, from and neurobiology, insights including integrating requires ciplines environments natural in D snake in transit dynam- robust sensing. passive allowing minimal how by with environments reveal locomotors heterogeneous limbless results benefit Our can animals. may ics the strategy by control used similar reproduced a be suggesting dynamics patterns, body-buckling observed using and the model patterns A mus- activation self- posts. snake cle described the their previously grab change incorporating or control not open-loop avoid particles. did to animals subatomic pattern the deformation of indicated rem- diffraction manner patterns matter-wave a Force in the array the of force-sensitive by iniscent reoriented of were array snakes regular In a The a across posts. sand. through move homogeneous and to substrate on snakes uniform challenged quickly we move experiments, to laboratory wave desert traveling animals a tail limbless studied We how contacts. about snake, terrestrial obstacle known unplanned is most with little of contend result, a physics As loco- unknown effective motion. for postures the strategies of body understanding and frustrates complex materials snakes The some environ- heterogeneities. of body terrestrial with elongate the contacts most on drag via overcome inhabit to for thrust snakes (received generating 2018 ments, like 19, December Levine animals Herbert Member Limbless Board Editorial by accepted and 2018) MA, 21, Cambridge, May University, review Harvard Mahadevan, L. by Edited a colo hsc,GogaIsiueo ehooy tat,G 30332 GA Atlanta, Technology, of Institute Georgia Physics, of School hl h taeisue ylme nml ocnedwith contend to animals limbed by used strategies the While is control locomotion (10), neuromechanics organismal In lg,adnuooyt icvrpicpe fmovement of principles discover physi- to mechanics, neurology environmental and of ology, roles the isentangling | locomotion hc ssaseetpdhead-to- stereotyped a uses which occipitalis, Chionactis | neuromechanics a enfrM Rieser M. 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D.I.G. and and L.C., A.M.H., J.M.R., P.E.S., contributions: Author environments heterogeneous in snakes undulating laterally of ics change, performance. confound shape and terrain, shapes between complex pre- relationship the These was of (16). understanding deformation density obstacles local the use subtle, to to cipitated and related was (21), waveform heterogeneities the of variety that a found encompassing habitats) range of geographical snakes diverse generalist a of with studies Previous (those robust unexpected systems. facilitate limbed with in could as contend mechanics transit (20). passive organisms terrain whether by limbless novel and evidenced a how collisions in as unclear move task is to nontrivial challenged It snakes a of is failure terrain the complex and trunk the by introduced propulsion. anisotropy generate (19)—to frictional integument or the (18), of structure sand such of bends body piles the the by as used of subsequently and curves obstacles created those discrete planar (16), heterogeneities—whether primarily against which press commonly in trunk the motion is This slithering (17). seen undulation lateral is vertebrates gate https://github.com/PerrinESchiebel/MatlabSnakeModel owo orsodnesol eadesd mi:[email protected]. 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PHYSIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY by working with a relatively simple system, a desert-dwelling of a sand substrate interspersed with sparse heterogeneities such snake that relies on a stereotyped self-deformation pattern to as small plants and twigs (Fig. 1B). This desert-dwelling species move within its habitat, composed largely of homogeneous sand uses a stereotyped traveling wave when moving quickly (30 to but containing sparse obstacles (SI Appendix, Fig. S1). In the 80 centimeters per second, 0.8 to 2.1 body lengths per second) absence of the ability to interrogate the motor control system on the surface of sand (18) (Movie S1). in freely moving snakes (unlike the increasing number of tools Fig. 1C is an archetypal example of the sinuous waveform in microscopic swimmers, e.g., ref. 22), we explored the animals’ used by this species when moving on spatially uniform and yield- response to unexpected terrain interactions using a “scattering” ing substrates like sand or the sand-mimic substrate, high-pile approach (23)—studying the kinematic and dynamic outcomes carpet, used in this study to increase the rate of data collec- of collisions with heterogeneities. Determining principles of body tion (SI Appendix). This traveling wave initiates near the head coordination in complex terrains will help improve mobility of and passes posteriorly with little variation along the arclength, snake-like robots (20, 24). s, of either maximum curvature, κm, or wavelength (Fig. 1D). Principal component analysis (PCA) (inspired by ref. 25) is a Results and Discussion method of dimensionality reduction in which one calculates the Biological Model. The shovel-nosed snake, Chionactis occipitalis eigenvectors of the curvature covariance matrix. The resulting (Fig. 1A), transits open desert between the cover of larger flora principal components (PCs) form an orthogonal basis describ- to forage or escape threats (18). This intervening terrain consists ing the variation of κ along the body. PCA revealed that κ was well-approximated by a serpenoid curve (26) κ(s, t) ≈ 2π −1 κmsin( L ξs + ωt) with κm = 25.2 ± 3.0 m and wavenumber ξ = 2.0 ± 0.3 35 (Fig. 1 E and F) (27) (the waveform changes in time A D according to ωt; however, because in our system inertia is domi- κ nated by damping, we will not include ω in analysis and ω = 1 in rostral (m all models). -1

) We previously discovered that targeting this serpenoid wave-

s 0 form conferred benefits to the snake’s movement on the surface 1cm of homogeneous sand (28), providing rationale for the conserved appearance of the waveform across individuals and trials. Given BCB 15cm C these locomotor benefits we next examined how the wave pattern

-35 changed upon unexpected collisions. Would the animal actively

caudal alter the waveform to either avoid or use reaction forces from the 0 time(s) 1 posts or could it use passive dynamics to transit the array without track E PC active changes to the serpenoid wave? 5cm 1 We modeled the sand and sparse heterogeneities of the desert terrain as a row of six rigid, force-sensitive posts (Materials and Methods) embedded in the carpet substrate (Fig. 2 A and B). We

PC2 challenged C. occipitalis (N = 8) to travel across the substrate rostral s caudal and through the post array. High-speed video captured kinemat- z 14000 plant x 40 ics and custom MATLAB software digitized the snake midlines F for analysis (27, 29) (SI Appendix). We obscured the spectacle 1 ) -1 scales of the snakes using nontoxic face paint (Snazaroo Clas- (m Frame No. sic; SI Appendix) to focus on control modalities where the animal 2 time (s) time (s) reacted to collisions rather than avoided them. This behavior is 0 00 likely relevant to this species as they do not appear to rely on vision during fast movement. This also prevented reaction of the animal to cues external to the experiment like movement of the researchers. We focused our analysis on trials in which the snake initially

-40 1 traveled parallel to zˆ, passed through the array, then contin- -40 0 (m-1) 40 1 ued along a straight trajectory (Fig. 2 B and C). The pattern of alternating body bends was preserved throughout (Fig. 2 C Fig. 1. Stereotyped waveform of a desert snake. (A) C. occipitalis, the and D). We measured an average decrease in speed of 84 ± 27% shovel-nosed snake, at rest on sand. Total length L = 37.3 ± 1.9 cm, mass from before initial contact to once the snake was entirely clear is 19 ± 3 g, and body width is 0.80 ± 0.04 cm (N = 8 individuals). (B) Pic- of the array (SI Appendix, Fig. S4). While significant (P < 0.001, ture of desert terrain with C. occipitalis tracks. The dashed line lies to the Wilcoxon signed rank test), the decrease was similar to that mea- right of the sinuous track in the sand left by the snake. This snake was mov- sured when no posts were present (83 ± 19% P < 0.001) and ing from bottom to top, changing direction as it encountered plant matter. the animals were never observed stopping or turning back after (C) Example snake midlines colored by time; 100 x, z coordinates along the body on the sand-mimic substrate in the laboratory with no obstacles contact. present. (D) Space–time plot of curvature, κ (SI Appendix), of the trajec- After transiting the array, many snakes were deflected from tory in C. Vertical axis is arclength along the midline, s. Diagonal bands their original heading (Fig. 3A). We characterized the pattern of are indicative of a traveling wave initiated near the head and passed pos- reorientations by finding the angle with respect to zˆ, θ, of each teriorly with limited variation. (E) PCA of κ on the sand-mimic substrate trajectory (Fig. 2C) and calculating the probability density of all without posts (N = 7, n = 47 trials, 90 equidistant measurements along s, trials combined. 15,398 frames). The first two components, PC1 and PC2, captured 86% of ◦ 2 When no posts were present, θ was at most 25 (Fig. 3A, Inset). the variance and were well-fit by sinusoids (dashed gray lines; PC1 R = 0.98 In contrast, trajectories which passed through the posts were and PC R2 = 0.99) approximately π out of phase (0.45π ± 0.04π). (F) κ can 2 2 spread over −57.2◦ to 56.1◦; the animals were diffracted by be approximated by κ(s, t) ≈ α (t)PC (s) + α (t)PC (s). A plot of α (t) ver- 1 1 2 2 1 the interaction like waves passing through narrow apertures sus α2(t), colored by frame number, revealed trajectories moved clockwise along a circular path which, combined with the PCs, produces a traveling (Fig. 3B). We observed three central peaks in the distribution, sinusoidal wave. reminiscent of the interference pattern observed in matter-wave

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1808675116 Schiebel et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 pc–ieposof plots Space–time (D) undulation. in single trials a distance radial in a traveled between distance found the trial to each in respect are points r with tail data measured all to angle, of angle, Scattering head post, polar central time. from the denoting snake averaging color by the with hun- characterized of instant One midlines each 124). the at and plotted tracing (120 points a individuals data in two dred placed from posts trajectories are snake rectangle digitized dashed the along line Within straight carpet. scale. the to not is is substrate Diagram Methods). and cap- Snapshots (Materials deflections (B) from camera post obtained post from the were calculated by kinematics were tured forces Snake while s. camera cameras per testbed overhead frames Two the 200 board. flat at radius a video to captured with glued carpet posts high-pile force-sensitive track: mental Six terrain. desert r for model laboratory 2. Fig. eomto atr pndtcigacliint vi the avoid to collision self- a its used detecting change sand- those not obstacle. upon the did with pattern snake in compared the deformation force that nonnegligible implying drag RFT, were locomotion, of from for post S7 4.9, calculated the Fig. ratio, Appendix, that from (SI the times sand to versus 5.2 mea- mimic similar force was median is posts locomotion The which the Appendix). diverse by (SI force predict (30) sured resistive to media granular used using in previously sand (RFT), across theory movement during segments measured; were mN 3 above forces 70% comparing test took it time italis the to comparable timescales over persisted S5 Fig. Appendix, (SI method measurement in and B the Methods ) to present robust and were were (Materials distributions features strapping-predicted These diffraction. posts. of row cible al. et Schiebel A s post and and epeitdtefre xeine by experienced forces the predicted We posts, the from forces Reaction = B otastteary(i.4 (Fig. array the transit to r madoergdfiuilwr iil fxdt h experi- the to affixed rigidly were fiducial rigid one and mm 3.2 s D). iga fthe of Diagram (A) terrain. multicomponent model with Interaction h ahdylo ieidctstelcto ntebd fthe of body the on location the indicates line yellow dashed The C. + v o .v T. ≈ 0m pr fasaemvn rmbto otp Gray top. to bottom from moving snake a of apart ms 60 o sadmninesui fdsac qa oteaverage the to equal distance of unit dimensionless a is T ˆ x 5cm tacne-ocne distance center-to-center a at camera testbed posts ftesaetasttm otettltime total the to time transit snake the of

caudal rostral D posts posts 0 camera post C B and i.S7D). Fig. Appendix, SI 12 (s) F C; post (s) B P and d .9 indrank signed 0.29, = Fg 4 (Fig. = .occipitalis C. (m 3m.(C mm. 23 -1 .Tu,forces Thus, C). ) 80% r 0 s 165 cm A fboot- of .occip- C. Example ) and κ (s) 85 cm o the for 1 θ body was , B), aso ahidvda’ rjcoybfr h ra a within was array calculated of the center We before the post. trajectory so central individual’s shifted were each Data of individuals. mass eight using collected tories 3. Fig. 5C). (Fig. unchanged was localized; simultaneously, spatially was hfe uhta h vrg ieto fmto ftefis hr fall of third first the of motion of direction with average aligned trials the that such shifted of density Probability at board, obstacle the the bypass to to parallel strategy changed traveled ( animal head the point the which until extrema planning est path visual further on providing rely not collision, made do movement. the snakes during snakes avoid and these that to trials evidence wall. these effort solid during observable a obscured with no array not post was the Vision replacing by elicited we deformation obstacles, large with interactions during deformed body the (e.g., subtle S2 were array Movie post the wave, the to with changes forward interaction However, during the (31). waveform snake of the symmetry of reorientation the causing broke which piles changes against shape pushes it way same the animal with in sand. the of posts consistent that the the is with suggesting and forces interacts definitive propulsion, animal transverse a undulatory of the make noninertial fur- pattern to between While necessary this acting be 4D). statement, forces would (Fig. the substrate substrate of sand-mimic of sand distribution study a Inter- similar ther with collision post. a interaction the upon primarily predicted from strategy using alter RFT forward not estingly, themselves did (i.e., propel animals kine- with contact to antialigned the changing at forces that observed motion applying cluded not or of posts direction were the the snakes “grab” the to matics As “forward/back” than 4D). rather “left/right” (Fig. pushed snake the is, that revealed ,adterapaac eeddo h esrmn ehdand method S5 measurement Fig. the Appendix, (SI on used depended size appearance they bin their (NS); significant and present not only ods), were were and bootstrapping curve 25% using this estimated in SD in the We than features less bootstrapping. small were 10,000-iteration the a that of from note one) estimated is quantile 84.1th curve and the under (integral sity (n trials array distributions. See included. were arc the (n within experiment array the A oi S3 Movie pncliinwt h altebd bcld ttenear- the at “buckled” body the wall the with collision Upon passive caused posts the from forcing that hypothesized We angle, orientation force the of Measurement 1v aeil n Meth- and (Materials distributions bootstrapping-predicted the of w ude fytretrajec- fifty-three hundred Two (A) pattern. diffraction Mechanical o T F n i.2 Fig. and .Ti us fhg uvtr Fg 5B) (Fig. curvature high of pulse This 5A). Fig. and post = 44, 6 60 0 -60 ˆ z a ototnoine epniua to perpendicular oriented often most was N h attotid eeue ocalculate to used were two-thirds last The . 0.02 No Posts = θ ) h oi uv stenraie rbblt den- probability normalized the is curve solid The 8). NS esrdwe opsswr rsn.Til were Trials present. were posts no when measured = C NS 194, and θ κ o ahtiluigdt nteac (Inset arc. the in data using trial each for m N i.S6 Fig. Appendix, SI .Teeoe oudrtn how understand to Therefore, D). 6 60 0 -60 0

= pdf A 0.06 ftenx xrm,measured extrema, next the of ) nytil with trials Only 8). and B θ .(B) B). h hddae ste15.9th the is area shaded The . -30 NSLts Articles Latest PNAS θ θ rbblt est from density probability F o xml individual example for = ,w con- we ±180), ≥ θ θ ( 0dt points data 50 F ° ) θ 30 Fg 4A), (Fig. F θ si the in as d 2 arising | fthe of f6 of 3 z ˆ ) ,

PHYSIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY 20 F was passed down the body at the wave speed. In both the B 2 x A model and the biological snake, interactions with each post were 0 independent.

(mN) F 1 We initiated the model trajectories at 700 locations encom- -20 passing all possible interactions (SI Appendix, Fig. S8A). Like z C. occipitalis, the model trajectories were spread by the array interaction (Fig. 6B), either continuing approximately along zˆ z or at an angle θ = ±22.0 ± 7.0◦ (Fig. 6C). There was a rela- tionship between the initial conditions and the scattering angle, x 01time (s) suggesting this system is deterministic (SI Appendix, Fig. S8D). 2.5 experiment The location and relative prominence of the peaks de- C D RFT pended on the geometric parameters. We defined Rbuckle = 0.015 −1 −1 L (κabsrpost) and Dbuckle = smaxrpost, where smax = 2ξ was the largest possible value of sbuckle. In the model we used the same Density

Probability d = 23 0 0 mm as in the snake experiment but increased post 0 0.5 1 -180-90 0 90 180 radius to rpost = 10 mm as the model was infinitesimally (| |>3mN)/ (transit) ( ) time F time θF ° thin. We used Rbuckle = 4.1 and Dbuckle = 20 in the model. These parameters were comparable to the average snake val- Fig. 4. Pattern of forces during transit through array. (A) θF is the angle ~ ues of Rbuckle = 3.5 ± 0.4 [mean and range estimated using between Fpost and ˆz. Reaction forces F1,2 were measured using post deflec- tions (Materials and Methods). Diagram is not to scale. (B) Example post both the wall trials and an anesthetized snake (38)] and forces in ˆx (top) and ˆz (bottom). The animal in this trial contacted a post Dbuckle = 31.8 ± 3.1. This combination of parameters yielded the ~ ~ on both its right (orange, F1) and left (red, F2). (C) Compare the force con- same distribution of post contacts as in experiment (Fig. 6C, tact time—the total time in a trial the force magnitude |F| was above a Inset and SI Appendix, Fig. S10A). threshold of 3 mN—to the transit time—the time between when the first Inspired by the effect of slit width on diffraction patterns, we tracked point first reaches the array to the last tracked point exiting (n = used the purely deterministic model to examine θq , the spread ≥ 233, N = 8). A value of 1 indicates the snake experienced force 3 mN of the distribution, as Rbuckle and d varied. We found as Rbuckle for the duration of the time it was passing the posts. (D) θ measured in F increased θq decreased (Fig. 6D). Increasing Rbuckle meant that experiment (black) and predicted by RFT (gray) (n = 194, N = 8, 28497 total more segments were involved in buckling to bypass the post, measurements). effectively increasing sbuckle and requiring less angular deflection of the trajectory (as seen in Fig. 7B). Similar to diffraction of fluids or subatomic particles, θq Undulating and Buckling Model. We previously observed “mechan- decreased as d increased (Fig. 6E). When freely moving, the ical diffraction” patterns in an open-loop snake robot which was ◦ maximum angle the body made relative to zˆ was φo = 45.6 . At rigidly rotated to certain θ during interaction with posts (32). We d = 23 mm the maximum angle the body could make between hypothesized that mechanical diffraction in the animals could ◦ the posts was φp = 29.6 and θ was determined primarily by the arise from adherence to a serpenoid self-deformation pattern wave phase at contact (SI Appendix, Fig. S8D, Top). In contrast, which passively buckled in response to external forcing by the ◦ when d = 50 mm φp = 66 , the model never contacted more than posts. one post in a trial, and the impact location on the post was influ- We developed a model to explore the outcome of our hypoth- ential (SI Appendix, Fig. S8D, Bottom). We rationalized that for esized strategy. Previous models of elongate, undulatory animals small spacings there was a greater probability the body had to calculated the acceleration of body segments using internal reorient to match φp regardless of the initial contact location. forces, such as muscle activation and viscosity of the viscera, and As φp became larger than φo the wave always “fit between” the external forces, like friction of the scales, discrete posts, or fluids (19, 33, 34). Rather than use a dynamical model of our system, we chose to develop a purely geometric model which allowed us to focus on the contribution and consequences of the hypothesized AB-1 C strategy. t(s)=-0.03 0 0.1 -70 (m )0 70 0 s (m) 0.08 We bypassed the complexity of the interaction between the κ 80 spatially extended body, the substrate, and the post by assum- m,1 v T ing forces between the body and the substrate were such that o /3 κ rostral m,2 )

the snake propagated with no slip and the body was buckled by -1 the post the minimum amount necessary to prevent overlap. The s (m position of each segment was prescribed at each time based on v T

o /6 m,1

these assumptions (35) (SI Appendix). κ During terrestrial lateral undulation snakes self-deform using waves of unilateral activation of the epaxial muscles (36). Thus, t(s) 0 we assumed passive body buckling would occur at those loca- caudal -0.3 0 040 -0.35 0 tions on the waveform where the shape change would further time (s) κ (m-1) shorten active muscle segments (37). We dictated the model κ m,2 changed at a “preferred buckling” location, sbuckle, determined Fig. 5. Localized deformations arose from collisions. (A) Three snapshots by this muscle activation pattern (Fig. 6A). from experiment at the times indicated and the digitized midlines at all times, indicated by color. Dashed orange circles indicate where κ and κ When the model snake contacted a post, curvature at sbuckle m,1 m,2 in C were measured. (B) Space–time plot of κ for the trial in A. The snout was set to the predetermined absolute maximum value of κabs = 50 m−1 (nominal amplitude 25 m−1) over as many segments first contacts the wall at time t = 0 indicated by the dashed orange line. (C) Maximum curvature of the bend closest to the wall, κ , versus that of the necessary to solve the constraint (Movie S4), leading to a pulse m,1 second closest, κm,2, as illustrated in A. Lines and gray area are the mean of high curvature like that observed in the snake wall-collision and SD of the nominal κm measured when the snake is not in contact with trials (Fig. 6B, Inset). Given the assumption that the body was the wall. κm,1 increased above the average value during contact with the passively deformed by external forces, the modified waveform obstacle to a value related to s measured at first contact, indicated by color.

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PHYSIOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY been closed-loop tracking of joint trajectories (43). However, Bootstrapping. We used bootstrapping to estimate the importance of the future work is needed to determine when this strategy is not features observed in the θ histograms. Using θ measured in 194 trials we appropriate (e.g., terrains with more dense obstacles or even generated 10,000 distributions using random resampling with replacement. certain waveform/obstacle combinations). We counted how many times the three central peaks and two central valleys Finally, we comment on our results in the context of the were present in the distribution by searching for features of prominence at least 0.004 (smallest prominence in the real distribution = 0.0052) occurring emerging field of active matter and collisions which do not on the interval between its neighboring features. Peak locations, deter- conserve momentum (44). Our biological mechanical diffrac- mined by their locations in the distribution were [−30, 10)◦,(−10, 15)◦, and tion mimics phenomena found in subatomic systems; this is (15, 35]◦ and valleys were (−20, 5)◦ and (5, 20)◦. The same procedure was part of a growing realization that active collisions are a fer- used to count the three peaks in the distribution without posts with peak tile source of interesting dynamics from mechanical diffraction locations at [−30, −5)◦,(−5, 10)◦, and (10, 35]◦ and valleys were (−10, 5)◦ in a robot (32) and scattering of microorganisms off walls and (5, 15)◦. (45, 46) to shape-induced reorientation of a cockroach (8). Force-Sensitive Posts. The theoretical prediction for force, F, as a function We posit that a framework which takes inspiration and tools 4 3Eπrpost from diverse systems will broaden understanding of principles of peg tip deflection, δ, is F = δ. Lpost = 7 cm, a = 2.81 cm is the Lpost 2a3(9 −5) of heterogeneous interaction in self-propelled systems across a scales. height of applied force measured from base, Young’s modulus E = 5.7 ± 0.6 MPa. Peg tip deflection δ was measured from the high-speed video. Post deformations were small compared with snake length scales (3-mm deflec- Materials and Methods tion at tip for a load of 0.050 N at a = 2.7 cm). Uncertainty due to variation All codes and datasets are available at smartech.gatech.edu. in a was < 10% (SI Appendix, Fig. S3C). Relative movement between the arena and the camera was corrected using the fiducial post. Snake Experiments. All C. occipitalis were collected in accordance with scien- tific collection permits (nos. SP790952, SP625775, and SP666119) approved ACKNOWLEDGMENTS. We thank Joseph R. Mendelson III, Sarah Bowling, by the Arizona Game and Fish Department. All snake experiments were con- and the reviewers, whose input improved this manuscript. This work was supported by NSF Grants PoLS PHY-1205878, PHY-1150760, and CMMI- ducted under the Georgia Institute of Technology IACUC protocols A14066 1361778; Army Research Office Grant W911NF-11-1-0514; US Department and A14067. Snakes were set in the arena and once they began moving of Defense, National Defense Science and Engineering Graduate Fellowship were not contacted during a trial. Temperature in the testing and holding 32 CFR 168a and Georgia Tech Southeast Center for Mathematics and Biol- ◦ area was maintained at 27.2 ± 0.6 C. All reported values are mean± SD ogy (to P.E.S.); Dunn Family Professorship (D.I.G.); and a Defense Advanced unless otherwise indicated. Research Projects Agency Young Faculty Award.

1. Berman GJ (2018) Measuring behavior across scales. BMC Biol 16:23. 26. Hirose S (1993) Biologically Inspired Robots: Serpentile Locomotors and Manipulators 2. Alexander RM (2003) Principles of (Princeton Univ Press, Princeton). (Oxford Univ Press, Oxford). 3. Cowan NJ, et al. (2014) Feedback control as a framework for understanding tradeoffs 27. Schiebel PE, et al. (2019) Data from “Tracked data for Chionactis occipitalis through in biology. Integr Comp Biol 54:223–237. a post array.” SMARTech. Available at hdl.handle.net/1853/60847. Deposited January 4. Goldman DI (2014) Colloquium: Biophysical principles of undulatory self-propulsion 25, 2019. in granular media. Rev Mod Phys 86:943–958. 28. Schiebel P, et al. (2018) Mechanics of undulatory swimming on the surface of granular 5. Lauga E, Powers TR (2009) The hydrodynamics of swimming microorganisms. Rep matter. Available at meetings.aps.org/Meeting/DFD18/Session/L06.4 (abstr). Prog Phys 72:096601. 29. Schiebel PE, et al. (2019) Data from “SnakeScattering.” GitHub. Available at https:// 6. Holmes P, Full RJ, Koditschek D, Guckenheimer J (2006) The dynamics of legged github.com/PerrinESchiebel/SnakeScattering. Deposited January 25, 2019. locomotion: Models, analyses, and challenges. SIAM Rev 48:207–304. 30. Zhang T, Goldman DI (2014) The effectiveness of resistive force theory in granular 7. Majmudar T, Keaveny EE, Zhang J, Shelley MJ (2012) Experiments and theory of locomotion. Phys Fluids 26:101308. in a simple structured medium. J R Soc Interface 9:1809–1823. 31. Ye C, Ma S, Li B, Wang Y (2004) Turning and side motion of snake-like robot. Pro- 8. Chen L, et al. (2015) Terradynamically streamlined shapes in animals and robots ceedings of the IEEE International Conference on Robotics and Automation, (IEEE, enhance traversability through densely cluttered terrain. Bioinspir Biomim 10:046003. Piscataway, NJ), Vol 5, pp 5075–5080. 9. Sponberg S, Full RJ (2008) Neuromechanical response of musculo-skeletal structures 32. Rieser JM, et al. (2019) Dynamics of scattering in undulatory active collisions. Phys in cockroaches during rapid on rough terrain. J Exp Biol 211:433–446. Rev E 99:022606. 10. Nishikawa K, et al. (2007) Neuromechanics: An integrative approach for understand- 33. Guo ZV, Mahadevan L (2008) Limbless undulatory propulsion on land. Proc Natl Acad ing motor control. Integr Comp Biol 47:16–54. Sci USA 105:3179–3184. 11. Koditschek DE, Full RJ, Buehler M (2004) Mechanical aspects of legged locomotion 34. Tytell ED, Hsu C-Y, Williams TL, Cohen AH, Fauci LJ (2010) Interactions between inter- control. Arthropod Struct Dev 33:251–272. nal forces, body stiffness, and fluid environment in a neuromechanical model of 12. Ayali A, et al. (2015) The comparative investigation of the stick insect and cockroach lamprey swimming. Proc Natl Acad Sci USA 107:19832–19837. models in the study of insect locomotion. Curr Opin Insect Sci 12:1–10. 35. Schiebel PE, et al. (2019) Data from “MatlabSnakeModel.” GitHub. Available 13. Pearson KG, Franklin R (1984) Characteristics of leg movements and patterns of at https://github.com/PerrinESchiebel/MatlabSnakeModel. Deposited January 25, coordination in locusts on rough terrain. Int J Robotics Res 3:101–112. 2019. 14. Jayaram K, et al. (2018) Transition by head-on collision: Mechanically mediated 36. Jayne BC (1988) Muscular mechanisms of snake locomotion: An electromyographic manoeuvres in cockroaches and small robots. J R Soc Interface 15:20170664. study of lateral undulation of the Florida banded water snake (Nerodia fasciata) and 15. Liao JC (2007) A review of fish swimming mechanics and behaviour in altered flows. the yellow rat snake (Elaphe obsoleta). J Morphol 197:159–181. Philos Trans R Soc B 362:1973–1993. 37. Astley HC, Mendelson JR, Goldman DJ (2017) Side-impact collision: Obstacle 16. Moon BR, Gans C (1998) Kinematics, muscular activity and propulsion in gopher negotiation mechanics in sidewinding snakes. Integr Comp Biol 57:E196. snakes. J Exp Biol 201:2669–2684. 38. Sharpe SS, et al. (2015) Locomotor benefits of being a slender and slick sand swimmer. 17. Gans C (1975) Tetrapod limblessness: Evolution and functional corollaries. Amer Zool J Exp Biol 218:440–450. 15:455–467. 39. Griffiths DJ (1962) Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, 18. Mosauer W (1933) Locomotion and diurnal range of Sonora occipitalis, Crotalus NJ). cerastes, and Crotalus atrox as seen from their tracks. Copeia 1933:14–16. 40. Revzen S, Guckenheimer JM (2012) Finding the dimension of slow dynamics in a 19. Hu DL, Nirody J, Scott T, Shelley MJ (2009) The mechanics of slithering locomotion. rhythmic system. J R Soc Interface 9:957–971. Proc Natl Acad Sci USA 106:10081–10085. 41. McGeer T (1990) Passive dynamic walking. Int J Rob Res 9:62–82. 20. Marvi H, et al. (2014) Sidewinding with minimal slip: Snake and robot ascent of sandy 42. Saranli U, Buehler M, Koditschek DE (2001) RHex: A simple and highly mobile hexapod slopes. Science 346:224–229. robot. Int J Rob Res 20:616–631. 21. Kelley KC, Arnold SJ, Gladstone J (1997) The effects of substrate and vertebral number 43. Liljeback¨ P, Pettersen KY, Stavdahl O, Gravdahl JT (2012) A review on modelling, on locomotion in the garter snake Thamnophis elegans. Funct Ecol 11:189–198. implementation, and control of snake robots. Rob Auton Syst 60:29–40. 22. Wen Q, et al. (2012) Proprioceptive coupling within motor neurons drives C. elegans 44. Marchetti MC, et al. (2013) Hydrodynamics of soft active matter. Rev Mod Phys forward locomotion. Neuron 76:750–761. 85:1143–1189. 23. Landau LD, Lifshitz EM (1976) Course of Theoretical Physics. Mechanics (Buttersworth- 45. Drescher K, Dunkel J, Cisneros LH, Ganguly S, Goldstein RE (2011) Fluid dynamics Heinemann, Burlington, MA), Vol 16, 3rd Ed. and noise in bacterial cell–cell and cell–surface scattering. Proc Natl Acad Sci USA 24. Astley HC, et al. (2015) Modulation of orthogonal body waves enables high 108:10940–10945. maneuverability in sidewinding locomotion. Proc Natl Acad Sci USA 112:6200–6205. 46. Kantsler V, Dunkel J, Polin M, Goldstein RE (2013) Ciliary contact interactions 25. Stephens GJ, Johnson-Kerner B, Bialek W, Ryu WS (2008) Dimensionality and dynamics dominate surface scattering of swimming eukaryotes. Proc Natl Acad Sci USA in the behavior of C. elegans. PLoS Comput Biol 4:e1000028. 110:1187–1192.

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