Author response: Mitigating memory effects during undulatory locomotion on hysteretic materials

Article Figures and data Abstract Introduction Results and discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract While terrestrial locomotors often contend with permanently deformable substrates like sand, soil, and mud, principles of motion on such materials are lacking. We study the desert-specialist shovel-nosed snake traversing a model sand and find body inertia is negligible despite rapid transit and speed dependent granular reaction forces. New surface resistive force theory (RFT) calculation reveals how wave shape in these snakes minimizes material memory effects and optimizes escape performance given physiological power limitations. RFT explains the morphology and waveform-dependent performance of a diversity of non-sand-specialist snakes but overestimates the capability of those snakes which suffer high lateral slipping of the body. Robophysical experiments recapitulate aspects of these failure-prone snakes and elucidate how re-encountering previously deformed material hinders performance. This study reveals how memory effects stymied the locomotion of a diversity of snakes in our previous studies (Marvi et al., 2014) and indicates avenues to improve all-terrain robots. Introduction Movement is crucial to the survival of most organisms and a necessary ability in robots used in fields like medicine (Taylor, 2006), search and rescue (Murphy et al., 2008), and extraterrestrial exploration (Lindemann and Voorhees, 2005; Shrivastava et al., 2020). Successful locomotion depends on the execution of body-shape-changes which generate appropriate reaction forces from the terrain, a relationship that can be further complicated if the body motion permanently changes the state of the substrate. Much of our knowledge of terrestrial locomotion is in the regime of rigid materials where the terrain is not affected by passage of the animal (Koditschek and Full, 1999; Sponberg and Full, 2008; Kelly et al., 1997; Farley et al., 1993), an understanding which led to the development of robots which are effective on hard ground (Saranli et al., 2001; Liljebäck et al., 2012; Blickhan et al., 2007). Little is known about locomotion in non-rigid materials which are plastically deformed by the movement of animals or robots, leaving tracks or footprints. Deformable terrains represent a spectrum from materials like water which continuously flow toward the undisturbed, zero shear state (Lautrup, 2011) to those which are remodeled by the interaction, like sand. Previous work has mainly focused on motion in fluids, where disturbances dissipate (Ijspeert, 2008; Sfakiotakis et al., 1999; Liao, 2007); the impact of soft material hysteresis on locomotion is not well-understood (Mazouchova et al., 2013; Zhang and Goldman, 2014). At one end of the spectrum, where deformations are short-lived (Boyle, 2010), small fluid swimmers like the nematode Caenorhabditis elegans (Wen et al., 2012), spermatazoa (Gray, 1953), and bacteria in water (Rodenborn et al., 2013) are propelled by the viscous force of the material resisting the animal's body shape changes. In these low-Reynolds-number systems (Re, the ratio of inertial to viscous forces), the inertia of the microscopic animal is negligible compared to the fluid viscosity such that if the animal stops self-deforming it very rapidly stops translating and/or rotating. Such resistive-force-dominated swimming is also found at the macroscale in frictional-fluid swimmers like the sandfish lizard Scincus scinus and the shovel-nosed snake Chionactis occipitalis moving subsurface through granular matter (GM) (Maladen et al., 2009; Sharpe et al., 2015). The motion of these animals is similarly dominated by resistive forces, in this case the frictional interactions between grains. Notably, in both viscous and frictional fluids, the material surrounding the submerged swimmers continuously re-flows around the body of the animal such that the animal is always surrounded by material. At the other end of the spectrum, animals move in materials whose state depends on the history of interaction. Swimmers in fluids at high-Re, where inertia dominates fluid viscosity, can experience time-dependent flow (Smits, 2019; Oza et al., 2019). Less well-understood is how animals manage material memory (Keim et al., 2019) when traversing terrestrial substrates like soil, mud, or sand. Such materials will yield in response to forces applied by a locomotor, similar to a fluid. However, unlike fluids, soft materials can typically bear internal stress. At the free surface, gravity is often insufficient to return mounds of material created by a locomotor to the undisturbed state, exemplified by the tracks left behind by a passing animal. Our previous work (Mazouchova et al., 2013; McInroe et al., 2016) revealed that performance of limbed animals and robots is affected by the ability to avoid interactions with previously-disturbed material. Using limbless lateral undulation on deformable substrates to generate appropriate propulsive forces is a non-trivial task, as demonstrated by the failure of both a snake-like robot (Maladen et al., 2011a) and a number of snake species (Marvi et al., 2014) attempting to traverse the surface of GM. However, it is unclear how such animals (or robot models) might manage environmental interactions to move effectively when, unlike limbed locomotors, they cannot increase step size to avoid their own tracks. In this paper we gain understanding of movement through complex terrestrial materials by combining several approaches, each of which provided insight into the number of interrelated elements governing the system. We begin with a study of a variety of non-sand-specialist snake species moving on the surface of GM as well as a desert-specialist snake. We quantify the kinematics of the snakes, which move with varying degrees of success using a range of waveforms. To understand the connection between waveform and performance, as mediated by the GM, we measure the granular response to surface drag. We develop a model for speed dependence that indicates that the granular physics is unchanged over the range of speeds observed in the animal. We then find a characteristic drag anisotropy curve that is largely independent of drag depth and speed. Based on the granular drag experiments and our animal observations we hypothesize that, despite the fast movement and complex granular flows, body inertia is negligible. Thus, we introduce a surface granular resistive force theory (RFT). Using RFT, we identify a trade off between actuation speed and torque from the GM that explains the stereotyped waveform used by the sand-specialist as that which maximizes movement speed under an anatomical power constraint. We show that performance of the non-sand-specialist snakes depends on both morphology and waveform, although RFT only accurately predicts performance when lateral slipping of the body is less than a threshold value. In these cases we often observe the animals re-interacting with material remodeled in previous undulation cycles. We study the interplay of waveform and material remodeling using a robophysical model. RFT calculation of robot locomotion, as in the high-slip snakes, is inaccurate. Particle image velocimetry (PIV) measurements indicate RFT over-estimates performance when the robot body is pushing material lateral to the direction of motion. RFT does, however, provide insight into the regime of locomotor failure, where we find the desert-specialist operates far from failure due to a combination of its waveform, slender body, low-friction scales, and lifting of segments that only produce drag. We close with a summary of our results and a brief discussion of the implications. Better understanding of the connection between body self-deformations, substrate remodeling and force generation, and the resulting locomotor performance will elucidate principles for effective motion on hysteretic materials. This can be applied to the next generation of all-terrain robots. Further, future studies can leverage such knowledge of the benefits and limitations of different body–terrain interaction modes to tease apart neuromechanical control strategies for contending with natural terrains (Schiebel et al., 2019). Results and discussion Performance of limbless locomotors on granular material Snakes occupy a variety of habitats and display a wide range of morphologies. We took advantage of this natural diversity to explore how different body shapes and patterns of self-deformation fare on GM (granular matter, see Table 1 for a list of symbols and abbreviations). We studied 22 species of snakes representing five families from the collection at Zoo Atlanta. Snake body plans ranged from short and stout to long and slender (Figure 1(a,b); Table 2), and their natural habitats encompassed a broad range from wetlands and swamps, to wet and dry forests and rainforest canopies, to rocky deserts and mountains. Figure 1 Download asset Open asset Body shape, waveform, and ability to progress across GM varies among snake species. (a–c) Snapshots of snakes moving on the surface of GM. Dashed lines roughly indicate the area of material disturbed by the motion of the animal. (a) The generalist pygmy rattlesnake Sistrurus miliarius attempting to move on natural sand collected from Yuma, Arizona, USA. The animal has completed several undulations, sweeping GM lateral to the midline of the body. This snake failed to progress further than pictured. (b) The generalist eastern indigo snake Drymarchon couperi. On the same GM as (a) (c) The sand-specialist shovel-nosed snake Chionactis occipitalis in the lab on 300 μm glass particles. (d–i) Digitized midlines of animals. Color indicates time from beginning to end of the trial. All scaled to the 10 cm scale bar shown. (d–g) are on Yuma sand and (h,i) are on glass particles. (d) Tiger rattlesnake Crotalus tigris, a rocky habitat specialist. The animal was unable to progress on the GM. Total length of the trial tt⁢o⁢t= 25.7 s, time between plotted midlines δ⁢t=100 ms (e) Rock rattlesnake Crotalus lepidus, a rocky habitat specialist. tt⁢o⁢t= 9.4 s, δ⁢t=33 ms (f) Egg-eating snake Dasypeltis scabra inhabits a wide range of habitats in Africa. tt⁢o⁢t= 1.6 s, δ⁢t=33 ms (g) Nerodia sipedon, a water snake inhabiting most of the Eastern US extending into Canada. tt⁢o⁢t= 6.0 s, δ⁢t=33 ms (h) C. occipitalis 130 and (i) 128. These trials represent the inter-individual variation in kinematics. Some of the animals moved approximately 'in a tube' where all segments of the body followed in the path of their rostral neighbors (e.g. (h), tt⁢o⁢t= 1.25 s, δ⁢t=12 ms) while others used this strategy only on the anterior portion of the body, appearing to drag the posterior segments in a more or less straight line behind themselves (e.g. (i), tt⁢o⁢t= 1.01 s, δ⁢t=12 ms). Table 1 List of frequently-used symbols and abbreviations SymbolDefinitionGMGranular MaterialRFTResistive Force TheoryReReynolds number. Ratio of inertial to viscous forces in a system.θangle between body segment tangent and average direction of motionθmaverage maximum θ, attack angleξspatial frequency, number of waves on the bodyvC⁢o⁢Mcenter-of-mass velocityvs⁢e⁢gsegment velocity (as if riding on the trunk)ωC⁢o⁢Mangular velocity about the center-of-massβdgranular drag anglevdplate drag velocityzdepth intrusion into GM, measured from the free surfaceρdensity of granular materialβssnake body slip angleσncomponent of granular stress normal to area elementσtcomponent of granular stress tangential to area elementσn/σtanisotropy factor. Ratio of normal to tangential stressμscale-GM friction coefficientLtotal body lengthL/wAspect Ratio, snake body length divided by width at the widest pointggravitational constant = 9.81 ms-2sarclength along snake body midline measured from the head Table 2 Anatomical information for the non-sand-specialist species. SpeciesSubfamilyFamilyLength (cm)Max width (cm)Mass (g)Acranthophis dumeriliBoidae1837.95620Agkistrodon bilineatusCrotalinaeViperidae75.43.3306.7Agkistrodon contortrixCrotalinaeViperidae803.1359.5Agkistrodon piscivorusCrotalinaeViperidae924.5569.5Agkistrodon piscivorusCrotalinaeViperidae874.5Agkistrodon piscivorusCrotalinaeViperidae572.7161Aspidites ramsayiPythonidae1.33.4749Bothriechis schlegeliiCrotalinaeViperidae66.71.7104Crotalus lepidusCrotalinaeViperidae432.244Crotalus molossusCrotalinaeViperidae82.14370.7Crotalus tigrisCrotalinaeViperidae71.13.1331.6Crotalus willardiCrotalinaeViperidae473134.5Dasypeltis scabraColubridae711.151Drymarchon couperiColubridae1624.21018Epicrates subflavusBoidae1533738Lampropeltis getulaColubrinaeColubridae1213680Lichanura trivirgataBoidae712.6243.5Loxocemus bicolorLocoxemidae1113.3607.5Nerodia sipedonNatricinaeColubridae793.5453.5Senticolus triaspisColubrinaeColubridae1011.8198Sistrurus catenatusCrotalinaeViperidae522.8174.5Sistrurus miliariusCrotalinaeViperidae472.8146 As a counterpoint to the variety of snakes, which were either terrain generalists or specialized to habitats which did not have an omnipresent granular substrate, we also studied the shovel-nosed snake Chionactis occipitalis (Figure 1(c); Table 3). This species is specialized to bury within (Kavanau and Kavanau, 1966; Sharpe et al., 2015) and move across (Mosauer, 1933) the dry sand of their desert habitat. We used nine individuals collected from the desert near Yuma, Arizona, USA (see Materials and methods for details). C. occipitalis uses a stereotyped waveform (Schiebel et al., 2019) to quickly move across the sand with little slipping of the body (Mosauer, 1933). Table 3 Anatomical information for the individual sand-specialist snakes. The species Chionactis occipitalis is in the family Colubridae. Average and standard dev. of lengths 38.0±1.3 cm. IndividualLength (cm)Max width (cm)Mass (g)12036.41.12012239.21.02112340.11.12012438.11.12012536.61.018128381.01612939.31.22413037.11.02013237.11.018 Kinematics and ability vary among species The non-desert specialist snakes from the collection at Zoo Atlanta were tested in a 2 × 1 m2 air-fluidized bed filled with sand collected in Yuma County, Arizona, USA. The snakes' movement was captured at 30 frames per second (fps) using an overhead camera (data collected for Marvi et al., 2014). The desert-specialists were tested in an air-fluidized trackway of area 152 × 53 cm2 filled with 297 ± 40 μm particles (Potters Industries spherical glass Ballotini beads). C. occipitalis moved quickly relative to the non-sand-specialists so its kinematics were captured at 250 fps. In all experiments, a blower attached to the bottom cavity of the fluidized bed was turned on and air flow increased until the GM was in a fluid-like state. The air flow was then decreased until the GM settled in a loose packed state (as described in Maladen et al., 2009). Air flow was always off during experiments and the penetration depth of the snakes (O(mm)) was much less than the total depth of the GM (O(cm)). We used custom MATLAB code to digitize the midlines of the snakes; image processing functions identified the non-sand-specialist and the method described in Sharpe et al., 2015 tracked the sand-specialist's banded black markings (Schiebel et al., 2020). A cubic spline fit to the tracked data evenly divided the snake midlines into 100 measurements over the entire body (non-sand-specialists) or from neck to vent (C. occipitalis). The use of alternating left and right bends, typical of lateral undulation in snakes, on the body was ubiquitous, although specifics of the waveform such as amplitude and number of bends on the body varied (Figure 1). Among the non-specialists, some snakes used uniform, periodically repeating bends (e.g. Figure 1 (b,e,f)) like C. occipitalis (Figure 1 (c,h,i)). Others used bends of varying amplitude and wavelength along the body (Figure 1(g)). We used the tangent angle, θ, the angle between the local body tangent vector and the average direction of motion of the animal (Figure 2 (a,left)), to represent body posture at each instant in time. Consistent with previous work (Schiebel et al., 2019), we found C. occipitalis kinematics were well-described using a serpenoid curve (Figure 2—figure supplement 1; Hirose, 1993), (1) θ⁢(s,t)=θm⁢sin⁢(2⁢π⁢ξL⁢(s+vs⁢e⁢g⁢t)). Figure 2 with 2 supplements see all Download asset Open asset Snake waveform parameters measured in experiment. (a, left) θ=acos⁢(x^⋅t^) for local tangent angle, t^ and average direction of motion x^. Example body posture shown, colored by θ. (a, right) Example serpenoid curves (Equation 1) of different ξ on a body of fixed length and θm=48 deg. (b) C. occipitalis experimental measurements plotted in the (θm,ξ) parameter space. Color indicates animal number. Markers are the mean and range of each individual taken over all trials. N = 9 individuals, n = 30 trials. The gray region in the upper right corner are waves which are inaccessible given the flexibility of the snake (Sharpe et al., 2015). θm was comparable to values measured from images of tracks taken in the field (Figure 2—figure supplement 2). (inset) A close-up of the data with mean and range from each trial plotted individually. Color consistent with the main plot (Linear fit with 95% confidence interval to inset data: slope −0.016 (-0.020,–0.011) intercept 2.66 (2.42, 2.90), R-square = (c) as in black is the C. occipitalis are the mean and range of values measured on a in the different species by N = 22 individuals, n = trials. The speed that the wave the body, how quickly the shape changes in At the shape of the snake as a of along the body, s, is by the also to as the attack angle, and frequency, or number of waves on the body, ξ (Figure an curve with these parameters are In the and width of the body both how quickly the midline can and how the body can although far from these θm and ξ While be not which these in this work we to measure their values and study the connection to performance as mediated by the GM. study explore the neuromechanical of the shapes and its connection to the observed between different species. by this we snake using θm and We attack angle and at each in a trial (see Materials and methods for and took the mean over all frames to θm and ξ of the average snake the results for each of the nine C. occipitalis in the ξ θm parameter revealed that the animals used a of the shapes they were of (Figure was a in the relationship between ξ and θm when both each individual and each trial (Figure the relationship between ξ and we all individual measurements and the impact of the average C. occipitalis taken as the mean and range of the (Figure black to C. and consistent with (Figure was variety among the wave parameters measured on the non-sand-specialist species (Figure However, as in C. the snakes did not the of The of and performance the forces on the snakes at in on the shape while the of to the of internal and/or We observed deformations in the granular substrate by the motion of the animals (Figure This that the shape changes by the animals the a which in reaction forces on the animal from the GM. We the connection between body segment motion and the forces by granular drag experiments using a submerged drag measurements governing subsurface swimming in GM were by granular drag measurements (Maladen et al., 2009). These experiments revealed that during subsurface swimming in GM the material as a frictional fluid. a in a viscous propulsive forces from the of the GM to the animal's shape changes than viscous forces the with each other normal and frictional Maladen et al., 2009; Zhang and Goldman, 2014). We observed the of granular as the snakes on the granular surface (Figure 3 Figure 1). This that, consistent with subsurface swimming (Maladen et al., and surface (Mazouchova et al., 2013; McInroe et al., animals themselves using the granular forces from the body the material not frictional anisotropy of the et al., 2009). The interaction between the body and the GM can be by the of relative to the substrate, where t^ and are the local velocity and tangent of a body segment (Figure 3 (a) Sharpe et al., 2015). from the between the granular stress and the animal's self-deformation Figure 3 with 1 supplement see all Download asset Open asset granular forces by snakes by stress on a (a) of C. occipitalis moving on the Snake is moving from left to The surface was the of sand by the line were by the motion of the snake. the snake body segment with t^ of the midline moving in direction (inset) of a snake. is the angle between the local tangent and velocity (b) 3 × × The was at a constant from the undisturbed free surface of the GM to the bottom of the (c) of the moving in a direction at angle and of the total (d) drag data collected at and as a of drag of the The upper black curve is the gray is Force data collected at plotted by a of 10 to We measured the granular stress on a model for a snake body to move at drag angle between the drag velocity vector and the tangent in the (Figure 3 attached the to a force that into those normal and to the (Figure 3 This was on the end of a robot which The robot the to then submerged it to depth measured from the free surface to the bottom (Figure 3 next it the to the surface for cm at constant and and A bed the same 297 ± 40 μm glass particles used in the C. occipitalis experiments the material to an state using by a The was the same as in the snake trials. subsurface drag which to the state (Maladen et al., at the surface stress increased over several (Figure This is due to the free surface flow of the a of sand the surface, like those created by the snakes, at the of the at the of drag and increased in until a between the and those around the of the Previous studies of drag at the surface measured a similar dependent force and observed a region of GM, beginning at the bottom of the and extending the surface, whose were and gravity et al., et al., stress captured by inertia model The snake speeds were from to Figure supplement 1) and the intrusion depth of the snakes' into the GM ranged from at the of the wave which the snake off of the to (Figure supplement 2). Previous studies that granular drag stress depends on both speed et al., 2011) and depth et al., 1999; Marvi et al., 2014). with those we fixed and and found normal stress increased as increased from 1 to of the robot Figure for and normal stress increased with from the depth where force be to 40 where the was submerged with the 10 the surface (Figure Figure with 2 supplements see all Download asset Open asset drag stress as a of speed and (a) normal to the as a of at a constant depth and As in the at the upper for all trials such that the total stress was to Markers are mean and of trials. bar is not is than the curve is the model for and curve for (inset) force measured in and as by the model Color is consistent with the main is the average force measured in one all trials shown. (b) normal to the as a of at All experimental data in this are from 10 to cm drag this the force state and the robot was moving at the snakes moving on body inertia was small compared to the frictional forces between the and the surface et al., 2009). While C. occipitalis was moving we in line with (Gray, that motion when the animals the This is observed in swimmers at where the resistive forces of the The animal that friction between the body and the GM and interactions within the GM were the forces in the system. the drag was in the state, we did not observe time-dependent in the GM. Thus, we used the that stress on the be as stress and material inertia The in the is the for pushing the granular material of the and the stress is the stress to of the is the of the which we using the of and the by motion of the The stress on the be using of the grains. In fluids, is given by given gravity and local depth In granular materials is a similar depth however, the can internal the and where can be an of than in a fluid et al., In our the from the free surface to depth so We is to that at and at which are similar to the measured can be more for theory et al., or theory et al., However, for the of this we were in understanding the stress dependence on drag speed and the for animal Thus, to with the the curves plotted in Figure were using from the experimental was by from measured at the speeds collected at a given and the Using the the

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