# Non-planar snake gaits: from Stigmatic-starts to Sidewinding

N. Charles1, R. Chelakkot2, M. Gazzola3, B. Young4, & L. Mahadevan5∗
1 Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138,
USA
2 Department of Physics, Indian Institute of Technology Bombay, Mumbai,
400076, India
3 Department of Mechanical Science and Engineering, National Center for
Supercomputing Applications,
Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-
Champaign,
Urbana, IL, USA
4 Department of Anatomy, A.T. Still University of Health Sciences,
Kirksville College of Osteopathic Medicine, Kirksville, MO 63501, USA
5 Engineering and Applied Sciences, Department of Physics,
Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA
∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
> Of the vast variety of animal gaits, one of the most striking is the non-
> planar undulating helical motion of a sidewinder. But non-planar helical
> gaits are not limited to sidewinders. Here we report a new non-planar gait
> used as an escape strategy in juvenile anacondas ($Eunectes~{}notaeus$). In
> the S(tigmatic)-start, named for its eponymous shape, transient locomotion
> arises when the snake writhes and bends out of the plane while rolling
> forward about its midsection without slippage. We present a mathematical
> model for an active non-planar filament that interacts anisotropically with
> a frictional substrate to quantify our observations and show that locomotion
> is due to a propagating localized pulse of a topological quantity, the link
> density. A phase diagram as a function of scaled body size and muscular
> torques shows that relatively light juveniles are capable of S-starts but
> heavy adults are not, consistent with our experimental observations. We
> further show theoretically that a periodic sequence of S-starts naturally
> leads to sidewinding. All together, our characterization of a novel escape
> strategy in snakes using non-planar gaits highlights the role of topology in
> locomotion, provides a phase diagram for gait feasibility as a function of
> body size, and shows that the S-start forms the fundamental kernel
> underlying sidewinding.

Snakes exhibit a wide variety of steady gaits on land associated with
rectilinear motion (?, ?), periodic undulation (?, ?, ?, ?, ?, ?, ?, ?), and
sidewinding (?, ?, ?, ?). In addition, they also employ a range of transient
gaits such as concertina locomotion (?, ?) and various striking and escape
lunges (?, ?), depending on the type of species, physical environment, and
behavioral situation. Perhaps the most dramatic of these gaits is sidewinding
(?), wherein a snake lifts part of its body out of the plane, and places it
askew; when this is done repeatedly, the snake winds itself sideways, leading
to the eponymous sidewinding gait. Sidewinding involves spatially
inhomogeneous non-planar motions that are transient. The complexity of this
gait raises a number of questions about the extent, origin and limits of this
gait from a dynamical, physiological and evolutionary perspective (?, ?).

In this letter we describe and quantify a new transient non-planar gait
observed in newborn and juvenile yellow anacondas ($Eunectes~{}notaeus$) that
gives the snake a very fast forward velocity, often used defensively to escape
a threatening situation. Inspired by a rapid aquatic gait commonly seen in
juvenile fish, known as the C-start (?, ?), we dub this terrestrial analog the
S-start, since the snake forms an S-shape consisting of three roughly co-
linear segments connected by two tightly curved regions as shown in Fig. 1a.
Motion begins when the two curved segments are lifted off the ground.
Simultaneously the outer parallel straight segments move along the ground even
as the middle straight segment remains stationary. This causes the curved
segments to travel along the snake. (see SI Movie 1). Thus, the snake
effectively moves like a ruck in a rug (?) and propels itself forward
(Fig.1b-f), as the S shape flows along its length, until one end comes off the
ground, terminating this transient mode of motility. This non-planar mode
leads to motion parallel to the snake and while reminiscent of sidewinding (?,
?) is different as it is a transient gait and does not lead to sideways
translation. However, as we will see later the S-start forms the building
block from which many non-planar gaits such as sidewinding are formed.

Figure 1:  S-start of newborn anaconda($Eunectes~{}notaeus$). (a) The starting
point of the locomotion is an ‘S’ shape including three co-linear regions,
connected by two curved regions that are elevated from the contact surface as
the middle, straight segment pushes down against the surface. (b-f) Series of
overlaid time-lapse images indicating the S-start (see SI Movie 1).

Lifting its body allows the snake to minimize frictional losses, but comes
with a cost – the snake has to overcome gravity and move out of the plane to
do so. We expect slender snakes or those with relatively stronger musculature
to be able to achieve this motion, while thicker snakes with less musculature
(and more bone) will be unable to do so. The maximum length $L$ that can be
supported out of the plane scales as $L\sim m/\rho a^{2}gh$, where $m$ is the
maximum active torque that the musculature can generate, $\rho$ the density of
the snake of cross-sectional diameter $a$, and $h$ the height above the
ground. Assuming that the maximum torque $m\sim\Gamma a^{3}f(\phi)$, where
$\Gamma$ is the maximum muscular stress, $\phi$ is the fraction of the tissue
that is muscle, and that the typical lift-off height $h\sim a$, we find that
$L\sim\Gamma f(\phi)/\rho g$. Therefore as the snake grows in girth, its
internal muscle fraction decreases (due to a concomitant increase in bone
mass) (?, ?, ?), the length it can support off the ground decreases, and the
S-start becomes physically unattainable. Indeed, our study of locomotion in
three newborn, five juvenile, and two adult anacondas with a range of lengths
and weights (SI) reveals that all the newborn and juvenile anacondas can move
using the S-start, whereas adult anacondas do not exhibit this type of
locomotion.

Figure 2:  Mathematical model of the S-start motion and comparison with
experiments. (a) A schematic of the mathematical model, with localized in-
plane torques at locations $b,d$ and a localized out-of-plane torque at $c$,
that propagate from head to tail synchronously at velocity $v_{m}$. (b) (top)
Numerical simulations capture the S-start (see SI for details, and Movie 2).
(bottom) Simulation snapshot shows the filament bending out of the plane. (c)
Kymograph of the local vertical elevation as a function of arc length, $s$ and
time, $t$, shows a minimum around $c$ where the filament is stationary
relative to ground, and is maximum at $b$ and $d$. (d) Kymograph of the local
speed with respect to the surface. The speed is nearly zero at region $c$ and
uniform for other segments.

To understand these observations quantitatively, we model the snake (Fig. 2a)
as an active elastic filament interacting frictionally with a planar
substrate. The filament is assumed to have a circular cross-section of
diameter $a$ and length $L$ with aspect ratio $L/a=50$. The axis is
parametrized by a material coordinate $s$, and the shape of the snake at a
particular time $t$ can be characterized in terms of a local position vector
$\textbf{r}(s,t)$ and an associated material orthonormal coordinate system
$\\{\textbf{d}_{1}(s,t),\textbf{d}_{2}(s,t),\textbf{d}_{3}(s,t)\\}$. We assume
that the filament has a passive bending rigidity $B$ and twisting rigidity
$C$, and we further neglect extensional and shear deformations of the
filament, a reasonable assumption for long slender elastic objects, so that
$s$ becomes the arc length and $\textbf{d}_{3}(s,t)$ is the tangent to the
center-line of the snake. Since the filament is capable of exerting muscular
couples along its body, we define an active torque vector
$\textbf{m}=\Sigma_{i=1}^{3}\textbf{m}_{i}\textbf{d}_{i}$ (Fig 2a), associated
with the two bending modes and one twisting mode of the filament at each
cross-section. Then, the motion of the filament is determined via the balance
of linear and angular momentum governed by the equations $\rho
a^{2}\partial\textbf{v}(s,t)/\partial t=\partial\textbf{f}(s,t)/\partial
s+\textbf{F}_{e}(s,t)$ and
$\textbf{I}\partial{\boldsymbol{\omega}}(s,t)/\partial
t=\partial\textbf{M}(s,t)/\partial s+\textbf{d}_{3}\times\textbf{f}$. Here
$\rho$ is the density of the filament, ${\bf v=\dot{r}}$ is the local filament
velocity, ${\bf f}$ is the vector of force resultants at a cross section,
${\boldsymbol{\omega}}$ is the local angular velocity of filament rotation,
$\bf I$ is the moment of inertia of the cross section (assumed to be
circular), ${\bf M}={\bf B}{\bf k}(s,t)+\textbf{m}(s,t)$ is sum of the passive
elastic torques (associated with the vector of bending and twisting strains
${\bf k}(s,t)$ and the matrix of bending and twisting stiffnesses
$\textbf{B}=\text{diag}\left(B,B,C\right)$) and the active muscular torques
$\textbf{m}(s,t)$ (SI), and ${\bf F}_{e}$ is the external force on the
filament associated with gravity and frictional interactions with the ground
(SI). We assume that the filament interacts with the substrate frictionally,
with a finite friction coefficient that is anisotropic. Assuming
$\mu_{f}<\mu_{b}<\mu_{s}$ to be the coefficients of kinetic friction in the
forward, backward and sideways directions respectively, we choose
$\mu_{f}:\mu_{b}:\mu_{s}=1.1:1.4:2$, based on recent experiments (?). We
discretize the filament and solve the associated discretized governing
equations using the numerical integration scheme given in references (?).

Figure 3:  (a) Snake centerline $Wr$ at one snapshot during simulated snake
motion, as a function of its body weight relative to the muscle torques ($\rho
gL^{2}/\Gamma a$). As expected, $Wr$ decreases monotonically with increasing
weight and decreasing muscle. (b) Velocity of the snake relative to its muscle
velocity ($v_{s}/v_{m}$) as a function of body centerline $Wr$. Velocity is
maximized for intermediate values of $Wr$. (c) Snake velocity as a function of
body weight relative to muscle torque, combining parts (a) and (b). Filled
symbols are simulation data and open symbols correspond to yellow anaconda
(Eunectes notaeus) assuming a value for $\Gamma\text{ [kPa]}\simeq
7.72(a\text{ [cm]})^{1.39}$ for snakes (?), measured on different surfaces,
with the error bars of one standard deviation. The S-start fails in the shaded
region, where the snake uncoils without a net translation and causes a
backward shift of snake center-of-mass and a negative $v_{s}$, and for large
weight or small muscle torque. Seen through the lens of parts (a) and (b), the
S-start requires an intermediate value of $Wr$, which is produced for
intermediate weight to muscle ratios. (d) A two dimensional phase space
indicates different locomotion regimes, characterized by two parameters, a
scaled muscular torque $\Gamma a^{3}L/B$ and the a scaled weight $\rho
a^{2}L^{3}/B$. Very small and very large values of the scaled weight are
either inertially unstable (U) or so heavy that friction dominates (F),
producing backwards motion. For extremely heavy snakes, no net motion is
observed (N). The S-start (S) is feasible only in an optimal regime of body
weight and muscle forces (see Movie S2). Note that the vertical axis over the
horizontal $(\rho a^{2}L^{3}/B)/(\Gamma a^{3}L/B)$ gives the horizontal axis
in (c). That this single ratio determines the type of locomotion explains the
straight phase boundaries in (d).

To complete the mathematical formulation of the problem, we note that the
S-start shows the snake has two localized bends associated with the in-plane
curvatures (${\boldsymbol{m}_{2}}$) at locations $(a)$ and $(c)$, and one
localized bend associated with out-of-plane curvature at the contact zone in
the central segment that is pressed downwards with an out-of-plane torque
(${\boldsymbol{m}_{1}}$) at $(b)$ (see Fig. 2a). To model this, we assume that
the three active muscular torques are localized with a magnitude $m\sim\Gamma
a^{3}\exp^{-(s-s_{0}-{v_{m}}t)/2\sigma^{2}}$, where $\Gamma$ is the maximum
magnitude of the muscular stress, $s_{0}$ is the initial location of the
torque, $\sigma$ is the axial scale over which the torque acts, and ${v_{m}}$
the speed of propagation. These three localized torques move uniformly along
the filament towards the tail with a speed ${v_{m}}$ relative to the body,
causing a pulse-like traveling wave that propels the filament parallel to
itself with a speed $v_{s}={\bf v}_{\text{cm}}\cdot\textbf{x}_{h}$
($\textbf{x}_{h}$ is the unit vector associated with the direction of the head
$s=0$, ${\bf v}_{\text{cm}}$ is the center-of-mass velocity of the snake) as
shown in Fig. 2b (see SI Movie 2). Eventually the active torques reach the
tail and the whole filament straightens out. In Fig. 2c, we plot the local
elevation of segments along the filament, showing that the traveling pulse
corresponds to a small localized region of contact to be pushed down into the
substrate while the curved regions are elevated above the surface. In Fig. 2d,
we plot the local speed of the filament $(|{\bf v}|)$ showing that the middle
section at $(c)$ never slips relative to the surface, while the other regions
move uniformly through this region.

The localized pulse that underlies the S-start has a topological
interpretation that stems from recognizing that the non-planar shape and
motion of a filament can be characterized in terms of the CFW theorem (?)
relating the topological link ($Lk$) to the twist ($Tw$) and writhe ($Wr$) via
the formula $Lk=Tw+Wr$. Here, we apply these quantities to our active filament
by treating the filament centerline $\textbf{r}(s,t)$ and first director
vector field $\textbf{d}_{1}(s,t)$ as a mathematical ribbon. While the CFW
theorem holds exactly for closed ribbons, we minimize errors for our open
system by appending a long straight edge of length $100L$ to each end of the
centerline, and compute $Lk$, $Tw$ and $Wr$ by adapting methods from (?). In
Fig. 3a we show the writhe of the snake body at a single moment during the
simulated gait; light, muscular snakes show much more $Wr$ than heavier and
less muscular snakes. In Fig. 3b, we show that the scaled forward snake speed
(averaged over a full gait cycle) is small for low values of $Wr$ because of
the effects of friction, and also low for large values because the snake tends
to flail and lift off from the ground, but is maximized for an intermediate
value of $Wr$. Combining the results of Fig. 3a-b, in Fig. 3c we plot snake
forward speed over snake weight relative to muscular strength. Seen together,
these plots reveal that maximum forward speed is achieved for the weight and
muscle that produce the optimal $Wr$ in the snake body.

To quantify the role of non-planar bending in the snake body during motion, we
employ local densities of $Lk$, $Tw$ and $Wr$, denoted by $\lambda$, $\tau$
and $\omega$, respectively, and defined by
$Tw=\int_{0}^{L}\tau(s)ds,~{}\lambda(s)=Lk(\boldsymbol{r}([s,s+\delta]))/\delta,~{}\omega=Wr(\boldsymbol{r}([s,s+\delta]))/\delta$
where $\boldsymbol{r}([s,s+\delta])$ denotes the section of the snake
centerline for which the arclength coordinate lies in the range
$[s,s+\delta]$, chosing $\delta=L/10$, determined empirically. In Fig. SI-2
and Movie S2, we show $\lambda$, $\tau$ and $\omega$ over one full gait cycle.
In the S-start, $\lambda(s,t)\neq 0$ in a small zone, and as the snake moves
forward, $\lambda(s,t)$ translates along the body without changing its
amplitude profile. In contrast, the link density for friction-dominated
locomotion $\lambda_{\text{friction}}(s,t)\neq 0$ over most of the snake body
and $|\lambda_{\text{friction}}|<|\lambda_{\text{S-start}}|$ when
$\lambda_{\text{S-start}}\neq 0$ (where $\lambda_{\text{S-start}}$ is
$\lambda$ for the S-start); during unstable motion, $\lambda(s)$ changes its
amplitude profile dramatically without cohesive translation. These
observations provide a qualitative understanding of the quantitative results
shown in Fig. 3; a successful S-start can be viewed purely topologically,
wherein the snake must have a threshold link current moving from its head to
its tail.

Given the simplicity of this escape gait, it is natural to ask when it is
feasible in terms of its morphology and musculature. We use the two parameters
associated with the scaled weight $\rho ga^{2}\ell^{3}/B$ and the scaled
muscular torque $\Gamma a^{3}\ell/B$ to classify the phase space of non-planar
S-starts. In Fig. 3d, we see that in a relatively light snake that is over-
powered, inertia dominates over weight and friction and the snake flails
without much net movement as the snake is unable to maintain sufficient
contact with the substrate (SI Movie S2: part 3). For relatively heavy snakes
that are under-powered, friction dominates inertia, and the active torques
simply uncoil the snake into a straight shape, again leading to little net
movement (SI Movie S2: part 2). However when the relative weight is
intermediate, the S-start is favorable and the filament can propel itself at a
finite velocity along the direction of its orientation. These observations
naturally yield the biomechanical limits of S-starts, and show how
developmental transitions allow a juvenile to exhibit these as a means of
rapid escape but lose that ability as they grow. In Fig. 3c, we plot data from
observations of juveniles and adults showing that this is indeed the case.

The non-planar nature of the S-start has some similarities to the sidewinding
gait, particularly in the context of how the body bends and writhes into a
helical segment even as the snake moves. However there is an important
difference: sidewinding is a periodic gait while the S-start is a transient
one. Building on previous experimental observations (?, ?, ?, ?) that have
shown the efficacy of the sidewinding gait in a variety of situations, we now
proceed to understand it mathematically. A natural premise based on our study
so far is that the S-start is the building block of a sidewinding gait. To
probe this, we periodically initiate the spatiotemporal patterns of the torque
triplets used for the S-start at the head and propagate them along the body
(Fig SI-3a). We find that as this wave propagates along the snake, it causes
the body to translate orthogonally to its tail-to-head direction, pivoting
about multiple contact points which are themselves stationary with respect to
the ground (Fig. SI-4, Movie S3). We see that sidewinding involves multiple
$\lambda$-pulses (Figs. SI-2 and SI-4), each similar to an S-start $\lambda$
profile propagating through the snake, confirming that the S-start forms the
basic building block of sidewinding.

Our study highlights an unusual transient mode of locomotion in newborn and
juvenile anacondas, emphasizing the role of out-of-plane motion in snake
locomotion. A new mathematical model for non-planar gaits allows us to
quantify the mechanism by which this transient rapid motion can be achieved
using a simple set of three localized propagating torques, and a phase diagram
shows the regime where the S-start is feasible, consistent with observations.
More generally, we show how this transient mode forms the kernel of
sidewinding, and is suggestive of how sidewinding might have evolved.

Acknowledgments. We thank NSF grants BioMatter DMR 1922321, MRSEC DMR 2011754
and EFRI 1830901, the Simons Foundation and the Henri Seydoux Fund (L.M.) for
partial financial support. All experimental data and codes used for simulation
will be deposited in a publicly accessible repository. Simulations were
carried out using a variant of the code created as part of (?).

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