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Tribology Letters

, 66:119 | Cite as

Numerical Model of the Slithering Snake Locomotion Based on the Friction Anisotropy of the Ventral Skin

  • A. E. Filippov
  • G. Westhoff
  • A. Kovalev
  • S. N. Gorb
Original Paper

Abstract

Snakes are able to dynamically change their frictional interactions with a surface by at least three different methods: (1) adjusting the attitude of their scales, (2) redistributing their weight at various points of contact with the substrate, and (3) changing of their winding angles. In the present study, we have observed that snakes change their winding angles, when either friction anisotropy is suppressed by particular roughness of the substrate, or when the external force displacing snake overcomes friction resistance during their locomotion on inclines. In order to understand this behaviour and may be even to predict the specific way of the snake locomotion depending on the interactions between the ventral surface of the snake skin and the substrate, numerical modelling was undertaken. Adaptation of the winding curvature considered in the present study has something to do with an enhancement of friction anisotropy in critical behavioural situations, such as on low-friction substrates or while moving up and down a slope.

Keywords

Biotribology Friction Anisotropy Snake Locomotion Surface 

1 Introduction

In order to facilitate slithering or serpentine locomotion, snakes keep the ventral body surface almost in continuous contact with the substrate [2]. The friction forces, generated in contact, have a crucial importance for propulsion generation. Due to the presence of the specific surface microstructure [1, 7, 11, 12, 13, 15, 17, 19, 20, 21, 24], the ventral scales of the snake generate lower friction, supporting sliding in the forward direction, and simultaneously produce higher friction in lateral direction, enabling propulsive force generation during lateral winding [4, 5, 22]. Since friction depends on the surface energy, material properties and surface roughness of both bodies in contact [6, 23 23], the slithering behaviour of the snake should change on substrates with different surface properties.

The friction anisotropy, required for propulsion generation [14, 18], may be rather sensitive to the roughness of the substrate, on which the snake moves [9]. At some specific relationship between dimensions of the snake skin microstructure and substrate asperities, friction anisotropy may be very low. Previously, we numerically studied interactions between the microstructure of the ventral surface of the snake skin in contact with various sizes of substrate asperities [9]. Our model showed that friction anisotropy appears on the snake skin only on the substrates with a characteristic range of roughness, which is less or comparable with dimensions of the skin microstructure. This has an important tribological consequence for the snake locomotion: at some substrate roughness, friction anisotropy may not support propulsion generation at normal slithering. In this case, a snake tries to adapt its body shape in the particular manner, to generate propulsion. In the present paper, we aimed at modelling behavioural adaptation of the snake, to maintain friction anisotropy effect during locomotion on substrates with low friction or on inclines. In some previous snake locomotion models [3, 25], the locomotion minimizing energy expenses functional was analysed. This approach certainly helped to understand the snake locomotion in general and it suits better for robotic applications. Our approach is based on the analysis of a particular locomotion pattern of a real snake with friction parameters within narrow interval characteristic for the ventral snake skin.

2 Observations and Experiments

We performed an experiment, in which we video recorded snakes (Schokari Sand Racer, Psammophis schokari) moving on the substrates with different roughness. For a control experiment, a horizontally oriented epoxy resin replica of P150 polishing paper (3M Deutschland GmbH, Neuss, Germany), Ra=17 µm, was taken, which was considered as a rough substrate, similar to one used in one of the previous studies [14]. The snakes moved forward easily on the control substrate demonstrating a close to sine-function body shape (Supplementary Fig. 1). As a low-friction substrate, epoxy resin replica of a fine polishing paper (Buehler, Lake Bluff, IL, USA) with 300 nm asperity size (Ra=0.35 µm) was taken, which is known to decrease friction anisotropy [9]. The maximum incline (6°) was chosen in the experiments, at which the snakes did not slipped down. Specific configuration of the body shape of the snake during slithering locomotion on the low-friction substrate is shown in Fig. 1. It is quite striking to observe that the snake tries to compensate low-friction anisotropy with stronger curvatures of its windings (Supplementary movie 1).

Fig. 1

Specific configuration of the body shape of the snake Psammophis schokari during slithering locomotion on the inclined substrate with low friction. See also supplementary movie 1. Please note the typical shape of the snake body, while it is moving on the low-friction surface. It is moving almost without (or with minimal) forward propulsion. This can be well recognized due to the snake’s position relative to the markings on the substrate. Additionally, typical soliton-like wave (arrow) propagating along the body against direction of the motion, as well as a generation of new soliton at the cranial part of the body and its annihilation of the old one at the caudal end of the body, is clearly seen

In order to understand this behaviour and qualitatively predict the specific way of the snake locomotion, which depends on the interactions between the ventral surface of the snake skin and the substrate, numerical modelling was undertaken. We aimed at modelling of the body shape change of a real snake during slithering locomotion and considered the role of the friction anisotropy. Previously, in another numerical experiment, we showed the general effect of stiffness of the surface structures on friction anisotropy [8], whereas in the present work we concentrated on the role of the combination of friction anisotropy and winding behaviour in snake locomotion. Especially, we attempted at modelling behavioural adaptations aimed at overcoming locomotion problems on substrates with fine roughness, where friction anisotropy is suppressed, or/and on inclines.

3 Numerical Model

To better understand which friction parameters are important in the slithering snake locomotion, in this section, we constructed and investigated a numerical model of the snake-like motion. Our goal here is to build a simple model of the locomotion that is able to account for the anisotropic friction properties of the snake body within narrow interval of values reported for natural systems. If compared with other snake locomotion models, where the snake body shape was presented as a sinusoidal or triangular wave [25] or some arbitrary smooth function minimizing energy expenses functional [3], we aimed at modelling the shape of a real snake during locomotion at the substrates with low friction. Our snake model is organized as follows.

The body of animal is represented by an array of elastically connected segments, which are allowed to move in two \(\{ x,y\}\) directions. Each segment is numbered as \(j \equiv \{ 1:N\}\), where \(N\) is the total number of the segments and defined by two end points \(\{ {x_j},{y_j}\}\) and \(\{ {x_{j+1}},{y_{j+1}}\}\). Considering that that the length \(L\) of the studied animal is equal to \(L=105\,{\text{cm}}\), and taking into account that numerical simulation should not be too time consuming, it is convenient to use \(N=36\) points of the numerical chain. In this case, for the straight line \(\{ {x_j}(t=0),{y_j}(t=0)=0\}\) applied further as the initial condition and equidistant \(N - 1=35\)segments \({\text{d}}{x_j}={x_{j+1}} - {x_j}\), each corresponds to 3 cm long part of the real body:
$${\text{d}}{x_j}(t=0) \equiv {\text{d}}{l_0}=3\,{\text{cm}}.$$
(1)
At arbitrary time moment \(t\), the coordinates \({y_j}\) of the moving segment certainly are not equal to zero. However, as in real body, the length of each segment must be conserved:
$${\text{d}}{l_j}=\sqrt {{\text{d}}x_{j}^{2}+{\text{d}}y_{j}^{2}} ={\text{const}} \equiv {\text{d}}{l_0}.$$
(2)
This condition is quite tricky to maintain exactly in the simple model, based on the elastic force only. One way to satisfy this condition is to use strong two valley interaction between two neighbouring points in both directions \(\{ {x_{j - 1}},{y_{j - 1}}\}\),\(\{ {x_j},{y_j}\}\) and \(\{ {x_{j+1}},{y_{j+1}}\}\), produced by isotopic Higgs-like potential \(U_{j}^{{{\text{eff}}}}={{{U_o}\,{\text{d}}l_{j}^{2}\,({\text{d}}l_{0}^{2} - {\text{d}}l_{j}^{2}/2)} \mathord{\left/ {\vphantom {{{U_o}\,{\text{d}}l_{j}^{2}\,({\text{d}}l_{0}^{2} - {\text{d}}l_{j}^{2}/2)} 2}} \right. \kern-0pt} 2}\). This potential leads to \(x\) and \(y\)components of corresponding force
$$F_{{j;x}}^{{{\text{eff}}}}={U_o}\,{\text{d}}x_{j}^{{}}\,({\text{d}}l_{0}^{2} - {\text{d}}l_{j}^{2}),\quad F_{{j;y}}^{{{\text{eff}}}}={U_o}{\text{d}}y_{j}^{{}}\,({\text{d}}l_{0}^{2} - {\text{d}}l_{j}^{2}).$$
(3)
At a strong interaction constant \({U_o}\), such forces keep (isotropically) all the values \({\text{d}}l_{j}^{{}}\) as close as possible to the trial length of the segments \({\text{d}}l_{j}^{{}} \simeq {\text{d}}l_{0}^{{}}\). In addition, we suppose that every point of the array \(\{ {x_j},{y_j}\}\) undergoes an elastic force \(f_{j}^{{{\text{elastic}}}}\), which returns the body of our numerical snake to its unperturbed straight form:
$$F_{{j,x}}^{{{\text{elastic}}}}=K({x_j}(t=0) - {x_j}),\quad F_{{j,y}}^{{{\text{elastic}}}}=K({y_j}(t=0) - {y_j}).$$
(4)

Here \(K\) is corresponding elastic constant, which controls a transversal rigidity of the snake’s body. For given initial conditions, an unperturbed configuration corresponds to the straight line along \(x\) coordinate \({x_j}={x_j}(t=0) \ne 0,{y_j}(t=0)=0\).

All the perturbations in the frame of the model come from the excitations which are in the real biological system produced by the snake muscles and in the wave-form regularly driven along the snake body. These perturbations alternatively displace \(y\) coordinate from unperturbed configuration \(\{ {y_j}\} =0\) in both directions \(y>0\) and \(y<0\). Our numerical model is expected to reproduce observed real configurations of the body as naturally as possible.

As it is seen directly from the videos of real snake and Fig. 1, the typical shape of the snake body, while it is moving on the low-friction surface, has a form of almost planar waves in positive \(y>0\) and negative \(y<0\) directions propagating along the body against direction of the motion, separated by regions, which are quite sharp and perpendicular to the motion direction. Each elementary wave can be treated as a smoothed step-function with well defined positive and negative plateaus and regulated width of the transition of transition between them. It is convenient to generate such wave by application of so-called Fermi–Dirac functions:
$${f_k}(x;t)={{ \pm \sqrt {{F_k}} } \mathord{\left/ {\vphantom {{ \pm \sqrt {{F_k}} } {\left\{ {1+\exp \left[ { - \left( {x - {X_k}(t))/\Delta } \right)} \right]} \right\}}}} \right. \kern-0pt} {\left\{ {1+\exp \left[ { - \left( {x - {X_k}(t))/\Delta } \right)} \right]} \right\}}}.$$
(5)
It can be easily proven that function \({F_k}(x;t)\)tends to a constant situated far from the bending point \({X_k}(t)\). The interval of variation of the function Eq. 5 is determined by the width \(\Delta\), and its amplitude can be regulated by the factor\(\sqrt {{F_k}}\). If two such functions, having different signs of the exponentials and shifted to the values \(\pm \frac{{\Delta X}}{2}\) in different directions from the position\({X_k}(t)\), are multiplied as
$${F_k}(x;t)={{{{ \pm {F_k}} \mathord{\left/ {\vphantom {{ \pm {F_k}} {\left\{ {1+\exp \left[ { - \left( {x - {X_k}(t)+\frac{{\Delta X}}{2}} \right)} \right]} \right\}}}} \right. \kern-0pt} {\left\{ {1+\exp \left[ { - \left( {x - {X_k}(t)+\frac{{\Delta X}}{2}} \right)} \right]} \right\}}}} \mathord{\left/ {\vphantom {{{{ \pm {F_k}} \mathord{\left/ {\vphantom {{ \pm {F_k}} {\left\{ {1+\exp \left[ { - \left( {x - {X_k}(t)+\frac{{\Delta X}}{2}} \right)} \right]} \right\}}}} \right. \kern-0pt} {\left\{ {1+\exp \left[ { - \left( {x - {X_k}(t)+\frac{{\Delta X}}{2}} \right)} \right]} \right\}}}} {\left\{ {1+\exp \left[ {{{\left( {x - {X_k}(t) - \frac{{\Delta X}}{2}} \right)} \mathord{\left/ {\vphantom {{\left( {x - {X_k}(t) - \frac{{\Delta X}}{2}} \right)} \Delta }} \right. \kern-0pt} \Delta }} \right]} \right\}}}} \right. \kern-0pt} {\left\{ {1+\exp \left[ {{{\left( {x - {X_k}(t) - \frac{{\Delta X}}{2}} \right)} \mathord{\left/ {\vphantom {{\left( {x - {X_k}(t) - \frac{{\Delta X}}{2}} \right)} \Delta }} \right. \kern-0pt} \Delta }} \right]} \right\}}},$$
(6)

we will get localized (positive or negative) plateau, which has total width \(\Delta X\) and moves together with \({X_k}(t)\), Fig. 2a. Below we use \({X_k}(t)=Vt\)with constant dimensionless velocity\(V=1\), which normalizes other units of the problem.

Fig. 2

Conceptual structure of the model. The snake segments are represented by black points. The “head” is marked by the open circle. Some important shape parameters are shown in the subplot (a). Three different shapes of the snake corresponding to \(\Psi =8.9\), \(\Psi =3.7\) and \(\Psi =1.9\) are presented in the subplots (bd), respectively

Now, it is ready to be applied as a wave of force dynamically producing deformations of the snake’s body. We will take sequential waves acting alternatively in both \(y>0\) and \(y<0\) directions and the factors \({F_k}\) and width \(\Delta X\) must be adjusted ‘a posteriori’ in the course of the numerical experiment, to create realistic shapes of the snake, corresponding to different regimes of its motion.

In particular, let us note that by watching locomotion of real snakes, we observed as a rule two pairs \(\{ k\} =1,...4\) of alternating positive \({F_1}={F_3}=+F\), and negative waves \({F_2}={F_4}=+F\), moving along the body. This fact and the observed relationship between the length of each wave and the length of the region where it the line turns were originally applied by us to get a good zero approximation for the widths \(\Delta\) and \(\Delta X\). For the length \(L=105\,{\text{cm}}\) and \(\{ k\} =1, \cdots 4,\) the widths are approximately equal to \(\Delta \approx 2.5\,{\text{cm}}\), \(\Delta X \approx 47.5\,{\text{cm}}\), and for typical relations \(\Psi ={L_x}/{L_y}\) between maximum and minimum in \(x\) and \(y\) directions the constant \(3<\Psi <9.\) The factor \(F\) is varied in interval \(5.8\,{\text{cm}}<F<17.5\,{\text{cm}}\).

The remaining forces of the problem are longitudinal and transversal components of the friction, which are crucially important to generate its locomotion caused by waves that produced by the muscular forces \({F_k}(x;t)\). According to the main goal of this study, we suppose both (1) the existence of the anisotropic friction \(F_{{{\text{friction}}}}^{\parallel }\) along the longitudinal body axis and (2) another component of the force \(F_{{{\text{friction}}}}^{ \bot }\) which is perpendicular to the body axis (lateral direction) and symmetric (the same friction coefficient for sinistral and dextral directions). According to this, we define the longitudinal component of the force as
$$F_{{{\text{friction}}}}^{\parallel }=\left\{ \begin{gathered} F_{{v>0}}^{\parallel },\quad {\text{sign}}({v_\parallel })=+1; \hfill \\ F_{{v<0}}^{\parallel },\quad {\text{sign}}({v_\parallel })= - 1 \hfill \\ \end{gathered} \right.,$$
(7)
where \(F_{{v>0}}^{\parallel }\) and \(F_{{v<0}}^{\parallel }\) are two different specific friction forces, corresponding to different friction force values per segment, when the projection of the velocity along given segment is parallel \({\text{sign}}({v_\parallel })=+1\) (forward motion of the stretched snake, caudad motion of the substrate) or antiparallel \({\text{sign}}({v_\parallel })= - 1\) (backward motion of the stretched snake, craniad motion of the substrate) to the body. According to the observations, these forces must be connected by the inequality \(F_{{v>0}}^{\parallel } \leqslant F_{{v<0}}^{\parallel }\). In the frame of the model, it is convenient to normalize both forces onto smaller one \(F_{{v>0}}^{\parallel }\). Below we will vary the relation between the longitudinal forces in the relatively wide interval \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel }=[1, \cdots ,3.5]\) in order to explore and finally determine an optimal value. The same is correct for the force orthogonal to the body. Our estimation for the particular snake is \(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }=1.75\). But, for generality of the model description we will vary it in the interval \(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }=[0, \cdots ,5]\).
It is supposed in this paper that sum of the travelling waves caused by muscular forces \(\sum\limits_{{k=1}}^{4} {{F_k}({x_j},t)}\), which deform snake’s body in \(y\)direction, is the only source of the locomotion. Counting on this complete system of the equations of motion can be written as follows:
$$\left\{ \begin{gathered} \gamma \frac{{\partial {x_j}}}{{\partial t}}=F_{{j,x}}^{{{\text{eff}}}}+F_{{j,x}}^{{{\text{elastic}}}} - F_{{{\text{friction}}}}^{{ \bot ,x}} - F_{{{\text{friction}}}}^{{\parallel ,x}}; \hfill \\ \gamma \frac{{\partial {y_j}}}{{\partial t}}=F_{{j,y}}^{{{\text{eff}}}}+F_{{j,y}}^{{{\text{elastic}}}}+\sum\limits_{{k=1}}^{4} {{F_k}({x_j},t)} - F_{{{\text{friction}}}}^{{ \bot ,y}} - F_{{{\text{friction}}}}^{{\parallel ,y}}; \hfill \\ \end{gathered} \right.,$$
(8)
where \(F_{{{\text{friction}}}}^{{ \bot ,x}}\),\(F_{{{\text{friction}}}}^{{\parallel ,x}}\), \(F_{{{\text{friction}}}}^{{ \bot ,y}}\)and \(F_{{{\text{friction}}}}^{{\parallel ,x}}\)denote \(x\) and \(y\) projections of the perpendicular and longitudinal friction forces, and \(\gamma\) is damping constant which defines time scale of the problem. In the system of Eq. [8], the motion is treated as over-damped one, so all inertial terms are ignored in accordance with [14].

The friction term here corresponds to the so-called “dry dynamic friction”. It means that these forces are calculated as the vectors of constant length, which do not depend on the velocity and are directed opposite to the instantaneous direction of motion of every segment. At nonzero time step, \(\Delta t\) of the numerical procedure must be controlled also, so that formally calculated components of the friction force do not exceed those values, which completely stop the motion of the segment (because the friction force cannot cause motion in other direction by itself).

Figure 2b–d presents conceptual structure of the model for three different configurations of the snake. The segments of the body are marked by the black points. At fixed total length of \(L=105\,{\text{cm}}\), the shapes differ by the relation \(\Psi ={L_x}/{L_y}\) between maximum and minimum in \(x\) and \(y\) directions \({L_{x,y}}\). As it was already mentioned, the specific values of the form factor \(\left| {{F_k}} \right|=F\,\,\) must be adjusted to match different \(\Psi\) values of the real snake. Three representative variants, corresponding to \(\Psi =8.9\), \(\Psi =3.7\) and \(\Psi =1.9,\) are depicted in the subplots Fig. 2a–c, respectively.

Further procedure was as follows. We fixed one of the form factors \(\Psi\), and varied two other parameters (longitudinal \(F_{{{\text{friction}}}}^{\parallel }\) and transversal \(F_{{{\text{friction}}}}^{ \bot }\) friction forces). Each time we get different dynamic scenarios of the motion. Due to travelling waves, sent in one direction along the body by the perpendicular forces \(\left| {{F_k}} \right|\), and in the presence of the anisotropy of friction, the “head” of the snake permanently shifts in the direction, which is opposite to the motion of the waves. Kinetic behaviour of the model, presented in supplementary movie 2, looks quite similar to the retrograde wave propagation in a real snake presented in “Observations and experiments” section. The soliton-like waves used in our model considerably differ from the sinusoidal (Supplementary Fig. 1) or triangular waves used in the model of, for example, Wang et al. [25]. When the ratio \(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }\) is small, the travelling wave motion is also observed in the model of Alben. However, the latter model just coarsely addresses far from simple motions, when the ratio \(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }\) is neither large nor small [3]. Some of the locomotion patterns (e.g. bending/unbending or curling at the ends) are predicted by the Alben model. This behaviour minimizes the energy expenses functional, but is not related to the locomotion of real snakes.

Since the frictional coefficient is ranging from 0.1 to 0.4, the energy expenses for overcoming friction are rather low. That is why we think the locomotion speed is biologically more important parameter than the energy lost due to surface interactions. The simplest way to characterize the speed is to calculate a distance \({x_1}(t)\), which the “head” covers during fixed time \({t_{\hbox{max} }}\) of the run: \(\Delta x={x_1}({t_{\hbox{max} }};F_{{{\text{friction}}}}^{ \bot },F_{{{\text{friction}}}}^{\parallel }) - {x_1}(0;F_{{{\text{friction}}}}^{ \bot },F_{{{\text{friction}}}}^{\parallel })\). The speed of the “head” depends on \(\Psi\), \(F_{{{\text{friction}}}}^{\parallel }\) and \(F_{{{\text{friction}}}}^{ \bot }\). At fixed \(\Psi ={\text{const}}\) and one of the parameters \(F_{{{\text{friction}}}}^{ \bot }={\text{const}}\), this routine produce a family of the trajectories \({x_1}(t)\). One of such family is presented as an example for \(\Psi =8.9\) in the subplot Fig. 3a.

Fig. 3

The dependences of time-depending \({x_1}(t)\) and final \({x_{{\text{final}}}}={x_1}(t={t_{\hbox{max} }};F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel },F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel })\) positions of the snake “head” (subplots (a) and (b), respectively) shown as the functions \(\Delta x={x_{{\text{final}}}} - {x_1}(0;F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel },F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel })\) of the friction parameters \(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }\)and \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel }\). Corresponding ratio increase is marked by an arrow. Dash-dotted line presents final time moment \(t={t_{\hbox{max} }}\) which terminates the runs

It is possible to collect the final points of different runs, obtained at \(t={t_{\hbox{max} }}\), and plot them as a function of \(F_{{{\text{friction}}}}^{\parallel }\). If the same routine is repeated at varied value of \(F_{{{\text{friction}}}}^{ \bot }\), the procedure gives complete family of the final points depending on both parameters \(\Delta x={x_1}({t_{\hbox{max} }};F_{{{\text{friction}}}}^{ \bot },F_{{{\text{friction}}}}^{\parallel }) - {x_1}(0;F_{{{\text{friction}}}}^{ \bot },F_{{{\text{friction}}}}^{\parallel })\). It is shown in the second subplot (b) of the same Fig. 3.

A lines density at large transversal friction \(F_{{{\text{friction}}}}^{ \bot }\) is clearly seen in the subplot (b) of Fig. 3. It means that at given parameters the further increase of \(F_{{{\text{friction}}}}^{ \bot }\) does not lead to the increase of the speed, whereas the amplitude of triangular waves in the snake model of Wang et al. [25] and locomotion cost both monotonously decrease by increasing \(F_{{{\text{friction}}}}^{ \bot }\). To elucidate reason of such a behaviour, let us present the same result as a gray scale map of \(\Delta x\)in the parameters space \(\left\{ {F_{{{\text{friction}}}}^{\parallel },\,F_{{{\text{friction}}}}^{ \bot }} \right\}\). The map is shown in Fig. 4a. The interval of colour gradations corresponds exactly to the limits of variation \(\Delta x\) directly seen in Fig. 4b. Dark and light colours correspond to smaller and larger values of \(\Delta x\), respectively. Contour lines are added also to enhance visibility of the relief. It is seen directly from the map that starting from some region (its lower boundary is marked by tilted dashed grey line), transversal component of friction almost does not influence the distance \(\Delta x\) covered by the snake to the moment of time \(t={t_{\hbox{max} }}\).

Fig. 4

Gray scale maps: a for the distance \(\Delta x={x_{{\text{final}}}} - {x_1}(0;F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel },F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel })\) and b for the difference \(\left| {\delta {F^ \bot }} \right|=\left| {F_{{{\text{friction}}}}^{ \bot } - \gamma {v^ \bot }} \right|\) between friction force and \(F_{{{\text{friction}}}}^{ \bot }\) and transversal projection \({v^ \bot }\) of the vector \(\gamma \overrightarrow v\). Grey line in both cases marks a region, where increase of transversal friction does not enhance locomotion

Let us also calculate the absolute value of the difference between the transversal friction force \(F_{{{\text{friction}}}}^{ \bot }\) and the transversal projection \(\gamma {v^ \bot }\) of the vector \(\gamma \overrightarrow v =\left\{ {\gamma \frac{{\partial x}}{{\partial t}},\gamma \frac{{\partial y}}{{\partial t}}} \right\}\):
$$\left| {\delta {F^ \bot }} \right|=\left| {F_{{{\text{friction}}}}^{ \bot } - \gamma {v^ \bot }} \right|.$$
(9)

The gray scale map for this combination is shown in the same parameters space \(\left\{ {F_{{{\text{friction}}}}^{\parallel },\,F_{{{\text{friction}}}}^{ \bot }} \right\}\) (Fig. 4b) for the comparison with previously plotted map of \(\Delta x\).

One can check directly that combination \(\left| {\delta {F^ \bot }} \right|=\left| {F_{{{\text{friction}}}}^{ \bot } - \gamma {v^ \bot }} \right|\) practically turns to zero near the same line (\(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }\) is around 3), above which \(F_{{{\text{friction}}}}^{ \bot }\) almost does not influence the speed. In turn, from the system of equations of motion (Eq. 8) we see that in the region, where \(\left| {F_{{{\text{friction}}}}^{ \bot } - \gamma {v^ \bot }} \right| \simeq 0\) all other forces of the problem become mutually compensated and force \(F_{{{\text{friction}}}}^{ \bot }\)almost completely determines motion of segments perpendicular to the body axis. From the biological point of view, it means that above this line there is no advantage in spending more energy for bending of the body, because it does not enhance propulsion during locomotion. This regime is similar to the locomotion in the snake model on a critical slope described in Wang et al. [25].

The same calculations have been repeated by us for different values of \(\Psi\). To compare the results at different \(\Psi\), it is useful to integrate both values \(\Delta x\) and \(\left| {\delta {F^ \bot }} \right|\) over vertical coordinate of Fig. 4, normalize obtained results on their maximums and represent them in one plot. As an example, Fig. 5 contains such information for already presented \(\Psi =8.9\), as well as for two additional values:\(\Psi =5.9\), \(\Psi =3.7\). The curves for integrated values \(<\Delta x>\) and \(<\left| {\delta {F^ \bot }} \right|>\) are shown by the bold and thin lines, respectively. The increase of the ratio \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel }\) above 2.5 does not lead to further increase of \(<\Delta x>\). Similarly, the increase of the ratio \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel }\) above 3.5 does not lead to further decay of \(<\left| {\delta {F^ \bot }} \right|>\). The tendencies of the values \(<\Delta x>\) and \(<\left| {\delta {F^ \bot }} \right|>\) at increasing \(\Psi\) are marked by arrows (Fig. 5). The increase of the form factor \(\Psi\) attenuates both the speed (\(<\Delta x>\)) increase and tangential friction (\(<\left| {\delta {F^ \bot }} \right|>\)) decay, when \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel }\) ratio increases (Fig. 5). This effect is not possible to observe in the models where typical locomotion on high friction substrates was studied [3, 25].

Fig. 5

Values of \(\Delta x\)and \(\left| {\delta {F^ \bot }} \right|\) integrated over parallel friction coordinate \(F_{{{\text{friction}}}}^{\parallel }\)and normalized to their maximums, calculated for the parameters \(\Psi =8.9\),\(\Psi =5.9\)and \(\Psi =3.7\). The curves for \(<\Delta x>\)and \(<\left| {\delta {F^ \bot }} \right|>\)are shown by the bold and thin lines, respectively. The tendencies at increasing the parameter \(\Psi\)are marked by the arrows

4 Biological Interpretation of Numerical Results

Snakes are likely able to dynamically change their frictional interactions with a substrate by at least three different methods: (1) adjusting the attitude of their scales, (2) redistributing their weight at various points of contact with the substrate [14, 18], and (3) changing of their winding angles [3; and this paper]. It has been previously shown that at high speeds, snakes lift the curved parts of their bodies off of the ground as they travel in lateral undulation and in sidewinding [10, 16]. By side winding locomotion, an animal pushes into the direction with the highest friction coefficient. Recently, theoretical modelling has predicted that snakes might be able to use the weight redistribution concentrating its weight on specific points of contact [14]. These points of contact correspond approximately to points of zero body curvature. Also, snakes are likely able to dynamically change their frictional interactions with a surface by adjusting the attitude of their scales [18].

In the present study, we have observed that snakes change their winding angles, (1) when either friction anisotropy is suppressed by particular roughness of the substrate or (2) when the external force displacing snake backwards overcomes friction resistance during their locomotion on inclines. Adaptation of the winding curvature considered in the present study has presumably something to do with an enhancement of friction anisotropy in critical behavioural situations, such as low-friction substrate or moving up and down a slope.

Friction properties of the ventral skin have been previously tested in different snake species. Its friction anisotropy was clearly demonstrated: friction coefficient obtained in the cranial direction was always significantly lower than that in the lateral direction [5]. Some enhancement of friction anisotropy was found in our previous experiments for cushioned (soft underlying layer) skin of the snake Lampropeltis getula in comparison to the uncushioned one (rigid underlying layer) in contact with rough rigid substrate [4]. The comparison of friction experiments with anesthetized snakes on relatively smooth and rough surfaces demonstrated friction anisotropy, which almost completely disappeared on the smooth surface [14]. Previous experimental and numerical data show that two hierarchical levels of structures of ventral skin (scales and microdenticles) are adapted to enhance friction anisotropy on different dimensions of roughness [9]. This is the reason why snakes experience significantly less locomotory ability on smooth substrates or substrates with fine roughness.

The undulating way of the snake locomotion is modelled here by generating of four solitary waves (two on each side), which correspond to the original action of lost extremities. These waves allow an effective use of both kinds of frictional anisotropies (longitudinal and transversal ones). Longitudinal friction anisotropy in snakes is limited due to the particular geometry of skin microstructure to maximum \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel } \leqslant 1.75\). This limitation requires an additional use of transversal anisotropy. In order to get the advantage of such an effect, the snake changes its form factor [3]. However, according to our model, the growth of transversal friction, which intuitively must strongly enhance propulsion during undulating locomotion, enhances propulsion to some limited extent only. We showed that the increase of the ratio \(F_{{{\text{friction}}}}^{ \bot }/F_{{v>0}}^{\parallel }\) above around three leads to rather weak advantage in locomotion, but has stronger energetic costs. These costs are related to the fact that the snake must generate solitary waves in caudal direction. That is why the wave amplitude increase made the snake speed in our model less dependent on \(F_{{v<0}}^{\parallel }/F_{{v>0}}^{\parallel }\) ratio, but also had higher energy costs (similarly to the model of Wang et al. [25]), and even did not enhance the propulsion. The wave propagation pattern strongly differs from the motion of typical snake-inspired robots using wheels, because they do not send the waves, but rather generate undulation by changing the trajectory by turning of the head side.

Notes

Acknowledgements

This work was partly supported by the Georg Forster Research Award (Alexander von Humboldt Foundation, Germany) to A.E.F. The preparation of this paper was partly supported by the Leverhulme Trust (project CARBTRIB ‘Nanophenomena and functionality of modern carbon-based tribo-coatings’).

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References

  1. 1.
    Abdel-Aal, H.A., Vargiolu, R., Zahouani, H., El Mansori, M.: Preliminary investigation of the frictional response of reptilian shed skin. Wear. 290–291, 51–60 (2012)CrossRefGoogle Scholar
  2. 2.
    Abdel-Aal, H.A.: Surface structure and tribology of legless squamate reptiles. J Mech Behav Biomed Mat. 79, 354–398 (2018).  https://doi.org/10.1016/j.jmbbm.2017.11.008 CrossRefGoogle Scholar
  3. 3.
    Alben, S.: Optimizing snake locomotion in the plane. Proc R Soc A. 469, 20130236 (2013).  https://doi.org/10.1098/rspa.2013.0236 CrossRefGoogle Scholar
  4. 4.
    Baum, M.J., Kovalev, A.K., Michels, J., Gorb, S.N.: Anisotropic friction of the ventral scales in the snake Lampropeltis getula californiae. Tribol Lett. 54(2), 139–150 (2014).  https://doi.org/10.1007/s11249-014-0319-y CrossRefGoogle Scholar
  5. 5.
    Berthé, R.A., Westhoff, G., Bleckmann, H., Gorb, S.N.: Surface structure and frictional properties of the Amazon tree boa Corallus hortulanus (Squamata, Boidae). J Comp Physiol A. 195, 311–318 (2009)CrossRefGoogle Scholar
  6. 6.
    Bowden, F.P., Tabor, D.: The friction and lubrication of solids. Clarendon Press, Wotton-under-Edge (1986)Google Scholar
  7. 7.
    Chiasson, R.B., Lowe, C.H.: Ultrastructural scale patterns in Nerodia and Thamnophis. J Herpetol. 23, 109–118 (1989)CrossRefGoogle Scholar
  8. 8.
    Filippov, A., Gorb, S.N.: Frictional-anisotropy-based systems in biology: structural diversity and numerical model. Sci Rep. 3, 1240 (2013).  https://doi.org/10.1038/srep01240 CrossRefGoogle Scholar
  9. 9.
    Filippov, A.E., Gorb, S.N.: Modelling of the frictional behaviour of the snake skin covered by anisotropic surface nanostructures. Sci Rep. 6, 23539 (2016).  https://doi.org/10.1038/srep23539 CrossRefGoogle Scholar
  10. 10.
    Gans, C.: Slide-pushing: a transitional locomotor method of elongate squamates. Symp Zool Soc Lond. 52, 12–26 (1984)Google Scholar
  11. 11.
    Gower, D.J.: Scale microornamentation of uropeltid snakes. J Morphol. 258, 249–268 (2003)CrossRefGoogle Scholar
  12. 12.
    Hazel, J., Stone, M., Grace, M.S., Tsukruk, V.V.: Nanoscale design of snake skin for reptation locomotions via friction anisotropy. J. Biomech. 32, 477–484 (1999)CrossRefGoogle Scholar
  13. 13.
    Hoge, A.R., Santos, P.S.: Submicroscopic structure of “stratum corneum” of snakes. Science. 118, 410–411 (1953)CrossRefGoogle Scholar
  14. 14.
    Hu, L.D., Nirody, J., Scott, T., Shelley, M.J.: The mechanics of slithering locomotion. Proc Natl Acad Sci USA. 106, 10081–10085 (2009)CrossRefGoogle Scholar
  15. 15.
    Irish, F.J., Williams, E.E., Seling, E.: Scanning electron microscopy of changes in epidermal structure occurring during the shedding cycle in squamate reptiles. J Morphol. 197, 105–126 (1988)CrossRefGoogle Scholar
  16. 16.
    Jayne, B.C.: Kinematics of terrestrial snake locomotion. Copeia. 22, 915–927 (1986)CrossRefGoogle Scholar
  17. 17.
    Maderson, P.F.A.: When? why? and how? Some speculations on the evolution of the vertebrate integument. Am Zool. 12, 159–171 (1972)CrossRefGoogle Scholar
  18. 18.
    Marvi, H., Hu, D.L.: Friction enhancement in concertina locomotion of snakes. J R Soc Interface. 9–76, 3067–3080 (2012)CrossRefGoogle Scholar
  19. 19.
    Picado, C.: Epidermal microornaments of the Crotalinae. Bull Antivenin Inst Am. 4, 104–105 (1931)Google Scholar
  20. 20.
    Price, R.M.: Dorsal snake scale microdermatoglyphics: ecological indicator or taxonomic tool? J Herpetol. 16, 294–306 (1982)CrossRefGoogle Scholar
  21. 21.
    Price, R.M., Kelly, P.: Microdermatoglyphics: basal patterns and transition zones. J Herpetol 23, 244–261 (1989)CrossRefGoogle Scholar
  22. 22.
    Renous, S., Gasc, J.P., Diop, A.: Microstructure of the tegumentary surface of the Squamata (Reptilia) in relation to their spatial position and their locomotion. Fortschr Zool. 30, 487–489 (1985)Google Scholar
  23. 23.
    Scherge, M., Gorb, S.N.: Biological micro- and nanotribology. Springer, New York (2001)CrossRefGoogle Scholar
  24. 24.
    Schmidt, C.V., Gorb, S.N.: Snake scale microstructure: phylogenetic significance and functional adaptations. Schweizerbart Science Publisher, Stuttgart (2012)Google Scholar
  25. 25.
    Wang, X., Osborne, M.T., Alben, S.: Optimizing snake locomotion on an inclined plane. Phys Rev E. 89, 012717 (2014)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Functional Morphology and Biomechanics, Zoological InstituteKiel UniversityKielGermany
  2. 2.Donetsk Institute for Physics and EngineeringNational Academy of Sciences of UkraineDonetskUkraine
  3. 3.Tierpark Hagenbeck GGmbH, Tropen-AquariumHamburgGermany

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