# The Effect of Internal Damping on Locomotion in Frictional Environments

Brian Van Stratum  Jonathan Clark  Kourosh Shoele<EMAIL_ADDRESS>Department of Mechanical Engineering, FAMU-FSU College of Engineering,
Tallahassee, FL, 32310, USA

###### Abstract

The gaits of undulating animals arise from a complex interaction of their
central nervous system, muscle, connective tissue, bone, and environment. As a
simplifying assumption, many previous studies have often assumed that
sufficient internal force is available to produce observed kinematics, thus
not focusing on quantifying the interconnection between muscle effort, body
shape, and external reaction forces. This interplay, however, is critical to
locomotion performance in crawling animals, especially when accompanied by
body viscoelasticity. Moreover, in bio-inspired robotic applications, the
body’s internal damping is indeed a parameter that the designer can tune.
Still, the effect of internal damping is not well understood. This study
explores how internal damping affects the locomotion performance of a crawler
with a continuous, visco-elastic, nonlinear beam model. Crawler muscle
actuation is modeled as a traveling wave of bending moment propagating
posteriorly along the body. Consistent with the friction properties of the
scales of snakes and limbless lizards, environmental forces are modeled using
anisotropic Coulomb friction. It is found that by varying the crawler body’s
internal damping, the crawler’s performance can be altered, and distinct gaits
could be achieved, including changing the net locomotion direction from
forward to back. We will discuss this forward and backward control and
identify the optimal internal damping for peak crawling speed.

##  I Introduction

The locomotion of long slender crawlers arises from the interaction of muscles
and ligaments and external forces such as Coulomb friction and viscous drag
from fluids. The mechanism of movement for long thin organisms has been the
subject of many previous investigations [e.g. 1, 2, 3]. Long slender organisms
across length scales and various media have adopted a variety of locomotion
gaits. Terrestrial snake locomotion, for example, can be classified into one
of four main locomotion modes: lateral undulation, concertina, sidewinding,
and rectilinear [1]. It is known that snakes switch between these modes
subject to their environment as snakes are motile in diverse media such as
granular, aquatic, and aerial domains [4, 5, 6]. From a robotics perspective,
the hyper-redundant body morphology of snakes, which provides mastery of such
varied terrain and stability, makes them excellent subjects for biologically
inspired designs of snake robots, surgical devices, and targeted drug delivery
mechanisms.

Figure 1: Modeling a Snake-like organism as a visco-elastic beam

Many animals use lateral undulation to move in all sorts of environments.
Examples include crawling snakes in grass, eels in water and sandfish in
sedimentary environments such as sand [7]. Further, some animals combine
lateral undulation with other motility techniques, such as fracture and crack
propagation[8], burrowing by sediment removal, and forms of two anchor
crawling utilizing peristaltic waves[9]. In addition to the utility of lateral
undulation in various environments, it also occurs across a wide range of
length scales, from reticulated pythons all the way down to nematode worms,
such as beet eelworms, which propagate waves of curvature from head to tail
down their body to move in sand and soft mud[10]. Since lateral undulation
relies on a sliding interaction with the ground, friction is both essential
and destructive to the crawler. Without friction, there can be no net forces
driving locomotion, yet the friction results in energy dissipation at the
crawler’s skin and affects their cost of transport. Yet, the transport cost of
snake-like animals moving by sliding across the ground is comparable to birds
and limbed creatures. Because of the applicability of lateral undulation in
various environments, across length scales and for various body morphologies,
it forms the focus of this study. [11, 12].

Lateral undulation has also been employed in a new class of snake robots that
are soft and compliant in nature[13, 14, 15]. These soft robots hold out the
promise of highly customized robots whose body motion and dynamics can be
programmed into the material of the body. Recent soft robotic snakes can mimic
their animal prototypes and use curvature waves for locomotion [16]. These
robots can be used for complex search operations, inspection, and maintenance
in congested and chaotic environments.

Figure 2: Free body diagram of the local coordinate system, environmental
force density, and internal torque density.

Different techniques have been proposed to model the kinematics and dynamics
of crawling robots [17]. These models represent the interface between the
crawler and its environment as either no-slip or frictional slip conditions.
For the no-slip condition, the ground interaction is modeled as a non-
holonomic constraint at every point on the body’s surface. This constraint
treats a snake belly as a flexible ice skate. In the slipping condition, on
the other hand, the environmental forces are represented with specific
frictional or resistive models, and the body is allowed to move laterally
relative to the environment. Such a model has been frequently used for
studying the slithering snake motion, notably [18, 19]. Slithering snakes
distribute their weight away from the points of maximum curvature along their
bodies in a locomotion strategy known as sinus lifting. Hu et al. [18]
advanced the understanding of slithering locomotion by introducing a model
explaining the critical role of weight distribution in snake locomotion. This
sinus-lifting action has also been leveraged to improve the functionality of
snake robots without wheels [20, 21].

Researchers have used frictional models to study the optimal locomotion
problem, studying the interaction of a crawler’s body with a frictional
environment. [18, 22, 23]. Alben studied a planar crawler subject to sliding
friction in the forward, transverse, and backward directions and used a quasi-
Newton method to find the time and space curvature trajectory that optimizes
locomotion efficiency [22]. Among other things, Alben discovered that the
optimal direction of locomotion changes as a function of environmental
friction. Later, it was shown that the optimal gait is mainly a function of
the transverse and forward friction coefficient ratio, and optimal crawling
performance is in the direction of the curvature wave propagation when
transverse friction is low and opposite that when the transverse friction is
large [23]. Alben referred to these as “direct” and “retrograde” curvature
waves. Guo et al. [19] investigated the coupled effect of internal and
external parameters, including the viscous and elastic moment of a visco-
elastic crawler with the non-holonomic, no-slip velocity constraint. They
found that parameters in the external environment mainly determine a crawler’s
body shape, and the speed of the crawler is primarily determined by internal
parameters (exogenous and endogenous parameters, respectively.) McMillen found
that damping plays a critical role in the swimming of Lamprey [24]. McMillen
showed that muscle contraction in slender swimmers happens slightly after
muscle activation, and this delay increases progressively from head to tail.
McMillen related this observation to the viscoelastic property of the tissue,
the geometry of the swimmer, and the muscle dynamics, more so than the effect
of fluid forces. This work indicates that damping is important in lateral
undulation in general and significant for designing embodied intelligence into
robotic systems.

The role of body viscoelastic damping is especially important for
pneumatically actuated soft robots. Damping is inherent to these robots, which
are composed of materials with very low Young’s moduli and include fluidic
channels for actuation [25, 26]. By tuning the damping behavior of soft robot
actuators, it is also possible to improve their performances [27]. However,
most of the prior research focused on the importance of internal compliance
within the body of a crawler [28, 29]. It has been demonstrated experimentally
that by leveraging the robot’s flexibility, it is possible to increase the
snake robot’s ability to traverse two and three-dimensional obstacles[30, 31].
However, little is known about how internal damping affects the locomotion of
long slender organisms and robots relative to their stiffness. The notable
exception is the work done by Guo, which investigated the role of internal
viscous damping on a crawler with the non-holonomic constraint[19]. However,
this non-holonomic constraint does not apply to the general locomotion case
and particularly does not apply to soft bio-inspired robots where substantial
lateral slipping motion is seen [32]. In this study, we explore the effects of
internal damping and its connection with the lateral friction coefficient on
the crawling performance of a modeled slender system that can serve as a guide
for making decisions about the effect of damping in soft robot designs and
also provide insight into bio locomotion.

##  II Model

We study the locomotion of long slender organisms using a three-dimensional
nonlinear Euler–Bernoulli beam formulation originally proposed by Tjavaras et
al.[33]. This model has been employed before to study the dynamics of animal
locomotion from swimming fish to climbing lamprey [34, 35, 36]. In the model,
an Euler-Lagrangian dual coordinate system is employed to formulate the
motion, namely the global coordinate system($x$,$y$,$z$) and a local reference
system with unit vectors $\bm{\tau}$, n, and b in the tangential, normal, and
bi-normal directions of the centerline of long cylindrical locomotor,
respectively. The two coordinate systems are related by

$\left[\begin{gathered}\bm{\tau}\hfill\\\ \bm{n}\hfill\\\ \bm{b}\hfill\\\
\end{gathered}\right]=\mathbf{C}\cdot\left[\begin{gathered}\bm{i}\hfill\\\
\bm{j}\hfill\\\ \bm{k}\hfill\\\ \end{gathered}\right],$ (1)

where $\bm{i}$, $\bm{j}$, and $\bm{k}$ are unit vectors in $x$, $y$, and $z$
directions, respectively. C is an orthogonal rotation tensor representing the
change of orientation from the global to the local reference frames. C is
constructed by using the singularity-free method of Euler parameters
${\beta_{0}}$, ${\beta_{1}}$, ${\beta_{2}}$, and ${\beta_{3}}$ to describe the
rotation based on a single angle $\vartheta$ and a principle direction
$\bm{\ell}$ as:

$C=\left[\begin{gathered}\beta_{0}^{2}+\beta_{1}^{2}-\beta_{2}^{2}-\beta_{3}^{2}\quad\;2\left({{\beta_{1}}{\beta_{2}}+{\beta_{0}}{\beta_{3}}}\right)\quad
2\left({{\beta_{1}}{\beta_{3}}-{\beta_{0}}{\beta_{2}}}\right)\hfill\\\
\;2\left({{\beta_{1}}{\beta_{2}}-{\beta_{0}}{\beta_{3}}}\right)\quad\beta_{0}^{2}-\beta_{1}^{2}+\beta_{2}^{2}-\beta_{3}^{2}\quad
2\left({{\beta_{2}}{\beta_{3}}+{\beta_{0}}{\beta_{1}}}\right)\hfill\\\
\;2\left({{\beta_{1}}{\beta_{3}}+{\beta_{0}}{\beta_{2}}}\right)\quad
2\left({{\beta_{2}}{\beta_{3}}-{\beta_{0}}{\beta_{1}}}\right)\quad\beta_{0}^{2}-\beta_{1}^{2}-\beta_{2}^{2}+\beta_{3}^{2}\hfill\\\
\end{gathered}\right]$ (2)

where

$\left[\begin{gathered}{\beta_{0}}\hfill\\\ {\beta_{1}}\hfill\\\
{\beta_{2}}\hfill\\\ {\beta_{3}}\hfill\\\
\end{gathered}\right]=\left[\begin{gathered}\cos\left({\vartheta/2}\right)\hfill\\\
{\ell_{x}}\sin\left({\vartheta/2}\right)\hfill\\\
{\ell_{y}}\sin\left({\vartheta/2}\right)\hfill\\\
{\ell_{z}}\sin\left({\vartheta/2}\right)\hfill\\\ \end{gathered}\right],$ (3)

Dynamic equations are derived by imposing the conservation of translational
and angular momenta of an infinitesimal segment of the body. We have

$m\left({\frac{{\partial{\bm{V}}}}{{\partial
t}}+{\bm{\omega}}\times{\bm{V}}}\right)=\frac{{\partial{\bm{T}}}}{{\partial
s}}+{\bm{\Omega}}\times{\bm{T}}+\left({1+\varepsilon}\right){{{\bm{F}}_{e}}}$
(4)

and

$\frac{{\partial{\bm{M}}}}{{\partial
s}}+{\bm{\Omega}}\times{\bm{M}}+\left({1+{\varepsilon}}\right)^{3}\bm{\tau}\times{\bm{T}}+\left({1+{\varepsilon}}\right)^{2}\bm{Q}=0,$
(5)

where $m$ is the mass per unit length. $s$ is the distance from this point to
the tail end along the unstretched crawler.
${\bm{V}}(s,t)={V_{\tau}}\bm{\tau}+{V_{n}}{\bm{n}}+{V_{b}}{\bm{b}}$ and
${\bm{\omega}}(s,t)={\omega_{\tau}}\bm{\tau}+{\omega_{n}}{\bm{n}}+{\omega_{b}}{\bm{b}}$
are the translational and angular velocities, respectively.
${\bm{T}}(s,t)={T_{\tau}}\bm{\tau}+{T_{n}}{\bm{n}}+{T_{b}}{\bm{b}}$ is the
internal force and
${\bm{M}}(s,t)={M_{\tau}}\bm{\tau}+{M_{n}}{\bm{n}}+{M_{b}}{\bm{b}}$ is the
internal moment. $\varepsilon(s,t)$ is the axial strain and
${\bm{\Omega}}(s,t)={\Omega_{\tau}}\bm{\tau}+{\Omega_{n}}{\bm{n}}+{\Omega_{b}}{\bm{b}}$
is the Darboux vector measuring the material torsion and curvatures of the
body centerline. ${{\bm{F}}_{e}}$ is the frictional force density, and
$\bm{Q}$ is the internal moment density generated by the body, both to be
defined later.

One compatibility relation is required to ensure a continuous body
configuration in space and time,

$\frac{{\partial\varepsilon}}{{\partial
t}}{{\bm{\tau}}}+\left({1+\varepsilon}\right){\bm{\omega}}\times{{\bm{\tau}}}=\frac{{\partial{\bm{V}}}}{{\partial
s}}+{\bm{\Omega}}\times{\bm{V}},$ (6)

The internal forces and moments are related to the strain $\varepsilon$ and
the Darboux vector ${\bm{\Omega}}$ through the constitutive relations
${T_{\tau}}=EA\varepsilon$, ${M_{\tau}}=GJ{\Omega_{\tau}}$,
${M_{n}}=EI{\Omega_{n}}$, ${M_{b}}=EI{\Omega_{b}}$. By definition, $A$ is the
cross-sectional area. EI and GJ are the bending and torsional stiffness,
respectively. Aggregate elastic Young’s modulus $E_{el}$ and visco-elastic
coefficient $D=\alpha E_{el}$ represents the combined contribution from all
the passive elements during bending. This, in real animals, includes a
viscoelastic spine, skin, white muscle, and the inactive part of red muscles.
Mathematically, linear viscoelasticity is introduced by defining
$E=E_{el}(1+\alpha\,\partial/\partial t)$. The final system of equations is

$\frac{{\partial{\bm{Y}}}}{{\partial
s}}+{\mathbf{H}}\frac{{\partial{\bm{Y}}}}{{\partial t}}+{\bm{P}}=0,$ (7)

where
${\bm{Y}}={\left[{\varepsilon\;{T_{n}}\;{T_{b}}\;{V_{\tau}}\;{V_{n}}\;{V_{b}}\;{\beta_{0}}\;{\beta_{1}}\;{\beta_{2}}\;{\beta_{3}}\;{\Omega_{\tau}}\;{\Omega_{n}}\;{\Omega_{b}}}\right]^{T}}$.
A detailed description of the matrices H and $\bm{P}$ is provided by Tjavaras
et al. (1998).

Overall, thirteen boundary conditions are required at the two ends of body. At
the tail end, we apply seven boundary conditions: ${\beta_{0}}=0$; $\bm{T}=0$
and $\bm{M}=0$. Six additional boundary conditions are applied at the head:
$\sum_{i}\beta_{i}^{2}=1$; $\bm{T}=0$; and ${M_{n}=0}$ and ${M_{b}=0}$. To
solve Eq. (7), we divide the body into ${N_{p}}-1$ segments, each with length
$\Delta s$, by uniform distribution of points ${s_{k}}$ ($k=1,\cdots,{N_{p}}$)
along the unstretched length of the body. An implicit box method is then
applied to integrate the system of equations from the time step ${t_{i-1}}$ to
${t_{i}}={t_{i-1}}+\Delta t$ [33]. The method is second-order accurate in
space and first-order accurate in time. A lower-order time integration scheme
is used for better numerical stability. Following the convergence study,
$N_{p}=50$ is found to be sufficient for the current study.

###  II.1 Environmental Friction

Snakes and other crawling animals possess body features such as scales that
give their bodies higher coefficient of friction when sliding sideways
relative to when they are sliding forward or backward [18, 37, 22]. To
replicate this property in the current mathematical model, an anisotropic
Coulomb friction model is used. In this model, the friction force is
calculated in the body attached Lagrangian coordinate system as a function of
the body velocity, $\bm{V}$, as,

$\displaystyle\bm{F}_{e}$
$\displaystyle=-m\,g(\mu_{\tau}\,\hat{V}_{\tau}\bm{\tau}\,+\,\mu_{n}\,\hat{V}_{n}\bm{n})$
(8)

where $\hat{V}_{\tau}$ and $\hat{V}_{n}$ are the regularized direction of
velocity vectors along the body in the tangential and normal directions,
respectively. They are defined as,

$\displaystyle(\hat{V}_{\tau},\hat{V}_{n})$
$\displaystyle=\frac{\left(V_{\tau},V_{n}\right)}{\sqrt{V^{2}_{\tau}+V^{2}_{n}+\epsilon^{2}}}$
(9)

where $\epsilon$ is a small regularized parameter chosen to ensure the
friction forces change continuously with the body’s velocity [23]. This
parameter is chosen from sensitivity analysis such that the results do not
change more than $1\%$.

###  II.2 Body actuation

It is known that body actuation for snakes and limbless lizards that employ
lateral undulation for locomotion is caused by muscle actuation that is
propagated along its body from head to tail [3]. This internal body actuation
is modeled as a traveling wave of muscle torque given by,

$\bm{Q}(s,t)=Q_{\text{amp}}\cos\left[\frac{2\pi\,n_{w}}{L}(s-c\,t)\right]\bm{b}$
(10)

where $L$ is the body length, $n_{w}$ is number of waves along the body and
$c$ is the wave propagation speed. For this study, $n_{w}$ is set to unity
based on the previous finding that a curvature wavelength close to unity
results in an optimized planar crawling locomotion [22]

Figure 3: For this work, shape ratio is the ratio of the tail-to-head lateral
excursion

###  II.3 Problem Setup

Equations 7 and 8 are scaled by the body length $L$, body mass per unit length
$m$, and the wave propagation speed of the muscle torque. The following
nondimensional parameters quantify the effects of viscoelastic damping, body
stiffness, and muscle torque amplitude,

$\tilde{\alpha}=\frac{\alpha\,c}{L},\hskip
14.22636pt\tilde{EI}=\frac{EI}{m\,L^{2}\,c^{2}},\hskip
14.22636pt\tilde{Q}=\frac{Q_{\text{amp}}}{m\,c^{2}}$ (11)

and the frictional environmental force is quantified with two nondimensional
parameters below.

$\displaystyle\kappa_{1}=\frac{\mu_{\tau}}{\mu_{n}},\hskip
14.22636pt\kappa_{2}=\frac{g\,L\,\mu_{n}}{c^{2}}$ (12)

As a nominal value for our study, we select a material stiffness
$\tilde{EI}=13$ and Torque amplitude $\tilde{Q}=66$. These numbers are chosen
to produce locomotion (for low damping case) with a traveling waveform with an
amplitude to wavelength ratio of $\approx 0.2$ consistent with organisms that
“swim” in granular media as well as efficient organisms at low Reynolds number
[7, 38]. We note that the produced body curvature of this “swimming” is
$\approx 2$, which is lower than what is observed for terrestrial snakes whose
body curvature could reach a maximum of $\approx 7$ [39]. We explore the role
of $\kappa_{1}$ while fixing the value of $\kappa_{2}=7.8$ consistent with the
friction values of snake scales traveling at normal speeds based on measured
values from previous studies [19, 18].

It is well-known that various taxa, including terrestrial snakes that utilize
undulation for locomotion, propagate waves of curvature from head to tail down
their bodies. As in anguilliform locomotion, these bodywaves often increase in
amplitude from the head to the tail. We quantify this increased amplitude with
a dimensionless ratio $d_{\text{tail}}/d_{\text{head}}$ shown in figure 3.
Here $d_{head}$ and $d_{tail}$ are the difference between the maximum and
minimum lateral excursions of the head and tail, respectively.

At any given moment in time, a part of the friction density contributes to
thrust, and the other part contributes to drag. To separate these, we defined
two operators of $\mathcal{H}^{+}$ and $\mathcal{H}^{-}$ as,

$\displaystyle\mathcal{H}^{+}(f)=\frac{f+|f|}{2},\hskip
14.22636pt\mathcal{H}^{-}(f)=\frac{f-|f|}{2}$ (13)

and define thrust and drag forces acting on the swimmer along the locomotion
direction as,

$\displaystyle F_{\text{thrust}}(t)$
$\displaystyle=\int_{0}^{L}\mathcal{H}^{+}({\bm{F}_{e}}(s,t)\cdot\mathbf{t}_{\bar{\bm{v}}})ds$
(14) $\displaystyle F_{\text{drag}}(t)$
$\displaystyle=\int_{0}^{L}\mathcal{H}^{-}({\bm{F}_{e}}(s,t)\cdot\bm{t}_{\bar{\bm{v}}})ds$
(15)

where $\bm{t}_{\bar{\bm{v}}}$ it the mean travel direction.

We quantify the efficiency of the crawler by comparing the work done by the
thrust force to the total work done by the muscles. We address the time
variance of the work and thrust by identifying the locomotion period and using
the phase average of thrust work and total work. We call the distance traveled
in one period of body shape variables a stride length $\lambda$. Further,
having established instantaneous thrust and drag functions, we can compute
power and efficiency using the following equations:

$\displaystyle P_{\text{muscle}}(t)$
$\displaystyle=\int_{0}^{L}\mathcal{H}^{+}\left(\bm{\omega}(s,t)\cdot\bm{Q}(s,t)\right)\,ds$
$\displaystyle\eta$
$\displaystyle=\frac{\overline{F_{\text{thrust}}}|\bar{\bm{v}}|}{\overline{P_{\text{muscle}}}}$

It is assumed that the swimmer cannot harvest power from the environment.
Hence, the contribution to the power expenditure through the length is always
non-negative.

##  III Results

Figure 4 shows computed steady locomotion results for different internal
damping $\tilde{\alpha}$ values. Here the environmental friction is normal
direction dominant with $\kappa_{1}=0$. We find that for low internal damping,
each cycle produces motion opposite the muscle moment wave propagation
direction. At the value of $\tilde{\alpha}\approx 1$, the crawler undulates in
place with no net locomotion. Damping values beyond unity produce first
increasing stride length in the direction of torque propagation and then
quickly drop off to zero. The crawler achieves the maximum efficiency at the
critical value of $\tilde{\alpha}\approx 10$.

Figure 4: Locomotion performance dependence on damping in a normal force
dominated friction ($\kappa_{1}=\mu_{\tau}/\mu_{n}=0$). Vertical axis is
stride length $\lambda$, efficiency $\eta$ and shape ratio
$d_{tail}/d_{head}$.  Figure 5: Absolute value of stride length (distance
traveled per cycle) as a function of damping and environmental friction
Figure 6: Lateral Excursion ratio as a function of environmental friction and
damping. We quantify that effect with the shape ratio or lateral excursion
ratio ($d_{tail}/d_{head}$ as defined in figure 3).

In Figure 4, the shape ratio, $d_{tail}/d_{head}$, is around 0.8 for cases
with low internal damping. The Shape ratio increases to a maximum of $\approx
4$ near the critical damping condition before returning to approximate unity
as $\tilde{\alpha}$ increases. In other words, for sufficiently low or high
damping cases, the head deviates approximately as much from the direction of
travel as the tail. However, at the critical damping point, the tail deviates
approximately four times as much as the tail.

Expanding our study of the environmental effects and damping to a larger
$40\,\times\,40$ computational grid, Figure 5 shows how the stride length
modifies over a range of $\mu_{\tau}/\mu_{n}$. On closer inspection, we find
that there is a minimum in the parameter space at
${\mu_{\tau}}/{\mu_{n}}\approx 0.5$, and a minimum for $\tilde{\alpha}\approx
1$. These local minima divide the parameter space into four quadrants. The
global maximum of the stride length is obtained in the bottom right quadrant
with higher damping and normal-dominant friction condition. The second optimal
stride length is obtained in the bottom left quadrant with low damping and
normal-dominant friction condition. The top left quadrant has a much smaller
local maximum for stride length that corresponds to high tangential friction
and low damping. High damping and tangential-direction-dominant friction
produce a global minimum in the top right quadrant.

Figure 6 shows how the shape ratio $d_{tail}/d_{head}$ changes with respect to
$\tilde{\alpha}$ and $\mu_{\tau}/\mu_{n}$. Taking fig. 5 and 6 together, we
see that a high shape ratio is associated with regimes with small stride
lengths and often accompanied by no net locomotion. Conversely, a low shape
ratio over the whole parameter space corresponds to longer stride lengths,
except for the unique case of high tangent friction and high damping.

Further summarizing the results, figure 7 labels areas of interest from
Figures 5 and 6 in the friction, damping parametric space and Figure 9
presents the crawlers motion over one stride cycle in each region. Locomotion
is slow in the tangentially dominated regions shown in fig. 9 (a) and (b); a
salient difference is that the crawler body exhibits more lateral motion for
the lower damping cases of (a). In case (b), in contrast, the crawler
undergoes approximate rigid body pitching rotation.

Figure 7: Overview of behavior in various friction and damping regimes.

Figure 9 (c), and (d) illustrate the gait changes that cause the crawler to
change direction as a function of damping. In subfigure (c), the crawler moves
from left to right (red to blue), opposite the moment propagation direction.
The different direction of motion is observed in subfigure (d), where it is
shown that the crawler is moving from right to left in the same direction as
the moment wave. A gait transition happens between left-to-right and right-to-
left motion in the restructuring area. We note that in this region, the tail
has a significant deviation outside the path the head follows. Figure 9 (d)
offers the highest stride length gait. Here the amplitude of the crawler is
significantly attenuated and characterized by a lower curvature amplitude and
reduced bending wavelength.

The curvature ($\Omega_{b}$) contour plots in figure 9 show that in both the
highly damped cases, i.e., (b) and (d), isoclines are increasing from left to
right, showing that curvature waves are propagating from tail to head.
However, we see the opposite isocline slope in (a) and (c).

The thrust distribution plots,
${\bm{F}_{e}}(s,t)\cdot\mathbf{t}_{\bar{\bm{v}}}$ in figure 9 show that region
(a) is uniquely dysfunctional in that very high thrust is being generated but
is being dissipated entirely by corresponding negative forces. Moreover, we
can see that even though the stride of (d) is longer than (c), the locomotion
generates lower forces. This is explained by the reduced gait waveform
amplitude, which exposes the crawler to less drag force.

###  III.1 Sensitivity Analysis

The previous section showed the dependence of a crawler’s gait on two
parameters identified to be most influential for locomotion, the damping
parameter $\tilde{\alpha}$ and the environmental friction ratio
$\mu_{\tau}/\mu_{n}$. Figure 8 shows the dependence of stride length on the
remaining dimensionless groups: $\mu_{n}$, $\tilde{Q}$, and $\tilde{EI}$. This
sensitivity analysis reveals that the fastest forward gait shown in fig. 9(c)
is not sensitive to variation in environmental friction, internal stiffness,
or internal actuation. Likewise, the reverse gait shown in fig. 9(d) is not
sensitive to either internal torque or material stiffness. However, this
fastest gait is sensitive to the normal friction coefficient, such that low
normal friction can cause the crawler to come to a complete stop in this
regime, and similarly, a 50% increase can cause the stride length to double.
For the transverse dominated friction regimes shown in 7 (a) and (c), an
increase of bending moment or normal friction, or reduction of stiffness
causes an increase in the stride length. This is the same trend that agrees
with the effect of the $\mu_{\tau}/\mu_{n}$ ratio in figure 5.

Figure 8:  The sensitivity of different crawling gates in Figure 7. The
sensitivity of the stride length of a crawler to variation of friction is
shown with yellow, the internal distributed torque with blue, and the body
stiffness with red.  Figure 9: Crawler trajectories in various regions of
interest with labels (a) (b) (c) and (d) corresponding to Figure 7. The first
row shows crawler trajectories in a global$(x,y)$ reference frame. The second
row shows the slithering body shapes for the regions of interest in a
reference frame $(x^{\prime},y^{\prime})$ that moves with the body’s average
velocity. The third row shows how the curvature $\Omega_{b}$ develops along
the body in space and time. The last row shows how the force distribution
${\bm{F}_{e}}(s,t)\cdot\mathbf{t}_{\bar{\bm{v}}}$ develops along the body in
space and time. The high tangential friction of (a) and (c) is not conducive
to locomotion by lateral undulation. Subfigures (c) and (d) show how damping
changes the direction of locomotion for transverse-dominated friction regimes,
such as wheeled snake robots.

##  IV Discussion

This work using the visco-elastic beam equations shows that lateral undulation
is effective when normal or lateral friction is very high relative to
tangential friction. This finding is consistent with the prevalence of wheels
on snake robots that move using mainly lateral undulation since wheels provide
a low value for $\mu_{\tau}/\mu_{n}$. This is also consistent with the
findings of a numerical study by Hu et al. [39] and the observation of Bruce
Jayne that terrestrial snakes in isotropic friction environments tend to favor
other gaits such as concertina over lateral undulation [2].

Comparing the effect of damping in figure 9, especially subfigures (c) and
(d), we see that internal damping has the effect of stretching out the body of
an undulating crawler. From a locomotion standpoint, this has the benefit of
increasing stride length. It also changes the crawler’s shape waveform by
changing the crawling direction from opposite the torque wave propagation to
moving in the same direction as the torque wave. This phenomenon could be used
to control the direction of a robotic crawler by changing the internal damping
during runtime, perhaps through a modulated aperture inside the crawler’s
body. Many mechanisms and fluidic circuitry can now be built into the material
of a soft robot through the use of specialized manufacturing processes [40].
Here the external force is primarily in the normal direction to the surface,
and the tangential friction is approaching zero. This friction regime is
comparable to a wheeled snake robot.

Considering the motion on the bottom of the parameter space in figure 5 and 6,
we showed that when the forward resistance is sufficiently low, there is a
benefit in using higher internal damping. The fact that locomotion is very
slow in the tangentially dominated friction regions of figure 9 (a) and (b) is
not surprising. Alben has demonstrated that a different curvature wave is
optimal for locomotion in this friction regime. He also identified that
efficient waves in this regime are still traveling waves but do not have a
uniform amplitude along their spatial dimension see [22] figure 10 (compare
Alben’s $\mu_{t}=0.1$ and $\mu_{t}=6$).

Figure 9 (c) and (d) shows that the low internal damping gait is such that the
crawler’s tail follows the trajectory of its head with a very slight deviation
of the tail outside the path, and the whole body follows a traveling wave
pattern. Thus, the shape ratio is almost a predictor of locomotion efficiency,
such that shape ratios close to unity are often efficient crawlers. The sole
exception is region (b), where motion is practically entirely damped out.

When eel transition from swimming to crawling on land, the amplitude of the
curvature of their gait increases. [41]. Gillis suggested that eels change
their body actuation to produce these changes. Figure 9 (c) and (d) shows that
damping reduces the amplitude of body curvature waves. This numerical result
suggests that, similarly, the eel’s body curvature amplitude reduction may be
partly explained by the presence of damping (external in this case.) Thus,
changes in muscle actuation may not be necessary to effect this curvature
change.

Considering these results in light of biological organisms, the stride length
minima in the parameter space at $\mu_{\tau}/\mu_{n}\approx 0.5$ is roughly on
the edge of the friction regimes measured for the scales of unconscious snakes
($\mu_{\tau}/\mu_{n}\approx 0.6$) [39]. In this friction regime, we find that
a traveling internal torque wave results in a traveling curvature wave of
constant amplitude along the body, similar to figure 9 (c), but with a more
lateral excursion. Alben et al. found an optimal gait in this comparable
friction regime see figure 10 of [22] with $\mu_{\tau}=6$. Figure 5 shows that
crawlers in this natural regime do not move faster with increased damping
value and, thus, do not benefit from increasing internal damping. However,
damping can still steer forward and backward motion in this regime. More work
is needed here since the effects of scale angle activation and weight
distribution are known to be significant for terrestrial snakes.

Normal or transverse-dominated friction is also comparable to organisms moving
in sedimentary media. Some organisms on various length scales, such as razor
clams and polychaete worms, can move forward and backward in sedimentary
media[42]. Recently biologically inspired robots have been developed that
mimic this locomotion[43]. Using internal damping as a control strategy to
change the gait of a crawler in lateral force-dominated locomotion regimes
might be useful in these types of soft robots to alter their burrowing
direction. Considering the non-dimensional parameters of equation 11, we can
see that damping is a more significant factor at smaller length scales. Thus,
this locomotion modification strategy is especially relevant for small robots.

##  V Conclusion

The effect of viscoelastic properties on the locomotion properties of a
flexible crawler is studied here. A traveling wave is employed as the body
actuation, and the swimmer’s response at different environmental friction
conditions is quantified. The study reveals an optimal damping value for the
crawlers with dominant normal friction forces. This condition can be leveraged
to improve the performance of certain bio-inspired robots, such as wheeled
snake robots, and systems that can adjust the normal friction coefficient
through surface interface manipulation. The results reveal that the internal
damping reduces the amplitude of a crawler’s undulatory waveform and changes
the direction of curvature propagation with an accompanied change in
locomotion direction. The unique functionality of the internal damping in
switching the direction of the locomotion can be used in future robotic design
to enable a seamless transition between forward to backward motion by
controlling internal damping.

## Acknowledgments

This work was funded by the National Science Foundation Expanding Frontiers in
Research and Innovation Program, grant number 1935278.

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