# Wheelless Soft Robotic Snake Locomotion: Study on Sidewinding and Helical
Rolling Gaits

Dimuthu D. K. Arachchige, Dulanjana M. Perera1, Sanjaya Mallikarachchi,
Iyad Kanj, Yue Chen2, and Isuru S. Godage3 School of Computing, Jarvis College
of Computing and Digital Media, DePaul University, Chicago, IL 60604, USA.
Corresponding author<EMAIL_ADDRESS>
1Department of Multidisciplinary Engineering, Texas A&M University, College
Station, TX 77843, USA.
2Department of Biomedical Engineering, Georgia Institute of Technology,
Atlanta, GA 30332, USA.
3Department of Engineering Technology & Industrial Distribution and J. Mike
Walker '66 Department of Mechanical Engineering, Texas A&M University, College
Station, TX 77843, USA.
This work is supported in part by the National Science Foundation (NSF) Grants
IIS–2008797, CMMI–2048142, and CMMI–2132994.

###### Abstract

Soft robotic snakes (SRSs) have a unique combination of continuous and
compliant properties that allow them to imitate the complex movements of
biological snakes. Despite the previous attempts to develop SRSs, many have
been limited to planar movements or use wheels to achieve locomotion, which
restricts their ability to imitate the full range of biological snake
movements. We propose a new design for the SRSs that is wheelless and powered
by pneumatics, relying solely on spatial bending to achieve its movements. We
derive a kinematic model of the proposed SRS and utilize it to achieve two
snake locomotion trajectories, namely sidewinding and helical rolling. These
movements are experimentally evaluated under different gait parameters on our
SRS prototype. The results demonstrate that the SRS can successfully mimic the
proposed spatial locomotion trajectories. This is a significant improvement
over the previous designs, which were either limited to planar movements or
relied on wheels for locomotion. The ability of the SRS to effectively mimic
the complex movements of biological snakes opens up new possibilities for its
use in various applications.

##  I Introduction

Snakes are among the few reptile species that do not require limbs to locomote
in various environments including, marshes, deserts, and dense vegetation.
Within the powerful musculature that generates movements, snakes have a
skeletal structure that protects internal organs and facilitates smooth
bending to produce unique locomotion patterns [1, 2]. Further, the small
cross-section-to-length ratio of snakes facilitates passing through confined
and narrow spaces. Thus, robotic snakes, both rigid and soft, inspired by
their biological counterparts are ideally suited for applications such as
search and rescue operations and inspection tasks[3]. Soft robotic snakes
(SRS) can generate smooth body deformation (bending) and are more adaptable to
their surroundings than rigid robotic snakes due to their inherent compliance
and continuous structures, making them the best candidate to emulate natural
snake locomotion.

Figure 1: Four-section soft robotic snake prototype.

Snakes use friction anisotropy between snakeskin and the moving surface to
generate forward propulsion necessary for planar locomotion [4]. But, typical
SRS skin cannot generate anisotropic frictional forces, unless the snakeskin
is improved by other means. Thus, some SRS prototypes, such as those reported
in [5, 6, 7, 8] utilize axially-mounted passive wheels to generate anisotropic
frictional forces and achieve locomotion. The SRSs appeared in [9, 10, 11, 12]
showed wheelless planar locomotion achieved through improved snakeskin with
anisotropic frictional properties. There are SRSs that exhibit spatial
deformation and achieve lateral undulation, sidewinding, and step-climbing
gaits [13, 14]. However, they require passive wheels attached to their bodies.
Wheels are cumbersome and can hinder locomotion. Additionally, wheels prevent
achieving other types of snake locomotion gaits. Hence, wheeled SRSs are not
suitable to study organic snake locomotion.

Note, no SRS that shows spatial rolling gaits exists. In our previous work
[15], we proposed a pneumatically actuated SRS that was capable of spatial
bending but was limited to achieving planar gaits including undulation and
rolling. The SRS only had three bending sections and thus lacked adequate
degrees of freedom (DoF) to lift the body off the ground and simultaneously
maintain sufficient ground contact forces to generate a motion. Further, the
bending sections do not contain a backbone and are thus subjected to length
change during bending which can cause undesirable reaction forces that
counteracted the forward progression.

Snakes tend to use spatial gaits when friction anisotropy does not exist
(i.e., in deserts). For example, snakes use sidewinding to spatially move the
body while minimizing concentrated ground contacts (similar to articulated
limbs of octopi). In this work, we introduce spatial bendability as an
alternative to overcome limitations associated with the anisotropic frictional
forces and wheels of SRSs. Similar to snakes’ spatial gaits, the idea is to
maintain skin-ground contact at a minimum level. We propose an SRS prototype
with four sections (Fig. 1) to circumvent the limitations of the previous SRS
design [15]. Notably, the new SRS is inextensible as the bending sections have
integrated backbones. The presence of a backbone makes the SRS design more
bioinspired because similar to snakes it has a skeletal structure to support
the locomotion. Extending [15], we, in this work, i) design and fabricate a
novel 4-section SRS, ii) present a complete kinematic model of the SRS, iii)
generate locomotion trajectories for sidewinding and helical rolling motions,
and iv) experimentally validate them on the SRS prototype. To the best of the
authors’ knowledge, this is the first demonstration of wheelless spatial
locomotion and helical rolling gaits for SRSs.

##  II Soft Robotic Snake Kinematic Modeling

###  II-A Soft Robotic Snake Prototype

Figure 2: (A) SRS prototype: serially arranged four soft bending robot
sections without its rubber skin showing spherically routed pressure supply
tubes on the backbone outer shell. The enlarged image shows the backbone
offset between adjacent sections. (B) A PMA–braided sleeve on the right side
has been removed to show the Silicone tube sealed to a push-to-connect
pneumatic union fitting. (C) One section with its PMA arrangement inside the
backbone. (D) PMA mounting plates of each section.

We adopt three, actuated DoF, hybrid soft robotic design approach proposed in
[16] to fabricate the 4-section SRS prototype shown in Fig. 1. An SRS section
(of length $240~{}mm$ and radius $20~{}mm$) is made of a flexible backbone and
three McKibben-type extending-mode pneumatic muscle actuators (PMAs) [17]. A
PMA is fabricated using a flexible Silicone tube dressed with a braided sleeve
which then is sealed by push-to-connect pneumatic union fittings at either end
(Fig. 2B). A cable carrier (Triflex R-TRL40, Igus), which is a rigid kinematic
chain, is chosen as the flexible backbone. PMAs are tri-symmetrically placed
inside the backbone around its neutral axis as shown in Fig. 2C. We mount
either side of the PMA ends to the backbone using two 3D-printed end-caps
(Fig. 2D). The thickness of the end caps is set to $5~{}mm$ to retain the
flexibility of the backbone to the fullest extent. An SRS section bends in a
constant curvature arc due to its inextensible backbone [16].

We serially connect adjacent sections with a backbone length offset
$\left(50~{}mm\right)$, creating a hollow skeletal area between each section
(Refer to the enlarged image in Fig. 2A). Accordingly, the complete SRS has
three identical length offsets between four sections along its body. These
offsets – made as extensions of the backbone length itself – facilitate
connecting pressure supply tubes to PMAs of each section while preserving the
continuum nature of the complete SRS assembly. We wrap pressure supply tubes
within the SRS body without obstructing its bendability as shown in Fig. 2A.
To obtain a uniform snakeskin (hence uniform friction), a thin rubber sleeve
is wrapped around the SRS body (Fig. 1). It covers routed pressure supply
tubes and eliminates their adversarial effects during locomotion. The SRS has
12-DoF (3-DoF in each section) in total relative to its base. Without pressure
supply tubes, the total weight and the length of the SRS prototype are
$0.94~{}kg$ and $1.11~{}m$, respectively.

The hybrid design approach adopted here enables a higher stiffness control
range with adequate structural integrity necessary for the SRS spatial
locomotion [18, 19, 16, 20, 21, 22, 23]. It should be noted, the backbone-
integrated bending units (i.e., SRS sections) are heavier than the ones
without a backbone. Hence, it is required to generate higher torques to
overcome friction and weight during spatial bending. Thus, the presence of a
backbone enables us to achieve bending at higher stiffnesses without
significant torsion in SRS sections.

###  II-B Complete Robot System Model

Figure 3:  The SRS schematics. (A) An SRS section showing its PMA arrangement.
(B) 4-section SRS showing the global, robot, and local coordinates frames and
kinematic parameters. Figure 4: Periodic discretization of the sidewinding
curve relative to the global frame, {$O$}.

In the proposed SRS, the kinematics of a single SRS section can be serially
extended to obtain the kinematics of the complete SRS. The kinematics of a
single section has been in detail discussed in [24, 16, 25]. Herein, we extend
their work to obtain the kinematics of the SRS skin. Considering any i-th
section $\left(i\in\\{1,2,3,4\\}\right)$ of the SRS, Fig. 3A illustrates the
kinematic diagram thereof. The coordinate frame $\\{O_{i}\\}$ is placed at the
center of the i-th section and the $Z$ axis is aligned with the backbone
(i.e., neutral axis). The length changes of PMAs, $l_{ij}\in\mathbb{R}$
$\left(j\in\\{1,2,3\\}\right)$, causes the section to bend in
$\theta\in[-\pi,~{}\pi]$ bending direction and $\phi\in[0,~{}\pi]$ bending
angle [24, 16]. Therefore, the total length of a PMA can be expressed as
$L_{ij}=L_{i0}+l_{ij}$, where $L_{i0}$ is the backbone length. Additionally,
due to constant curvature bending, a special kinematic constraint,
$l_{i1}+l_{i2}+l_{i3}=0$ must be satisfied [24]. Accordingly, one out of
three, kinematic DoF is redundant and we can express the jointspace variable
for the i-th section as $q_{i}=[l_{i1},l_{i2}]^{T}$. By adopting modal
kinematics proposed in [26], we can derive the homogeneous transformation
matrix (HTM), $\mathbf{T_{i}}\in\mathbb{SE}\left(3\right)$ of any point on the
surface (skin) of the i-th section as

$\displaystyle\mathbf{T}_{i}\left(\boldsymbol{q}_{i},\xi_{i}\right)$
$\displaystyle=\left[\begin{array}[]{cc}\mathbf{R}_{i}\left(\boldsymbol{q}_{i},\xi_{i}\right)&\mathbf{p}_{i}\left(\boldsymbol{q}_{i},\xi_{i}\right)\\\
\mathit{\boldsymbol{0}}&1\end{array}\right]\cdots$ (3)
$\displaystyle\qquad\left[\begin{array}[]{cc}\mathbf{R}_{z}\left(\sigma_{i}\right)&\mathcal{\mathit{\boldsymbol{0}}}\\\
\mathit{\boldsymbol{0}}&1\end{array}\right]\left[\begin{array}[]{cc}\boldsymbol{1}&\mathbf{p}_{x}\left(r_{i}\right)\\\
\mathit{\boldsymbol{0}}&1\end{array}\right]$ (8)

where $\mathbf{R}_{i}(\sigma_{i})\in\mathbb{SO}\left(3\right)$ and
$\mathbf{p}_{i}\in\mathbb{R}^{3}$ denotes the rotation matrix and position
vector of a point on the neutral axis.

A point on the skin of the i-th section is obtained by translating HTM in (8)
by $\mathbf{R}_{z}\in\mathbb{SO}\left(3\right)$ and
$\mathbf{p}_{x}(r_{i})\in\mathbb{R}^{3}$ where $\sigma_{i}\in[0,~{}2\pi]$ and
$r_{i}\in\mathbb{R}$ are the angular offset $\left(Z^{+}\right)$ and the
position offset $\left(X^{+}\right)$ of the surface point from the neutral
axis. $\xi_{i}\in\left[0,1\right]$ is a selection factor for points along the
neutral axis such that 0 gives the base point and 1 gives the tip location.
When the SRS moves, its base at $\\{O_{b}\\}$ floats on the global coordinate
frame, $\\{O\\}$. By combining a floating-base coordinate frame,
$\mathbf{T}_{b}$, we define the kinematic model of the complete SRS as

$\displaystyle\mathbf{T}\left(\boldsymbol{q}_{b},{\boldsymbol{q}_{r}},{\xi}\right)$
$\displaystyle=\mathbf{T}_{b}\left(\boldsymbol{q}_{b}\right)\prod_{i=1}^{4}\mathbf{T}_{i}\left({\boldsymbol{q}_{i}},\xi_{i}\right)$
$\displaystyle=\left[\begin{array}[]{cc}\mathbf{R}\left(\boldsymbol{q}_{b},{\boldsymbol{q}_{r}},\xi\right)&\mathbf{p}\left(\boldsymbol{q}_{b},{\boldsymbol{q}_{r}},\xi\right)\\\
\mathit{\mathbf{0}}&1\end{array}\right]$ (11)

where
$\boldsymbol{q}_{r}=[\boldsymbol{q}_{1},\boldsymbol{q}_{2},\boldsymbol{q}_{3},\boldsymbol{q}_{4}]\in\mathbb{R}^{12}$
is the vector that contains all joint variables and
${\xi}=\left[0,4\right]\in\mathbb{R}$ selects a point along the SRS (i.e., SRS
base at $\xi=0$, and SRS tip at $\xi=4$).
$\boldsymbol{q}_{b}=\left[x_{b},y_{b},z_{b},\alpha,\beta,\gamma\right]\in\mathbb{R}^{6}$
defines translation and angular offsets of $\\{O_{b}\\}$ relative to $\\{O\\}$
(Fig. 3).

##  III Locomotion Trajectory Generation

###  III-A Methodology

We generate locomotion trajectories to move the robot. Snakes show periodic
locomotion, and hence, their locomotion gaits can be modeled as cyclic
mathematical curves. First, we mathematically model the desired locomotion
gait (i.e., trajectory) on the global coordinate frame (i.e., robot
taskspace). Next, one period of the identified trajectory curve (i.e., the
mathematical curve) is discretized and the curve at discretized locations is
projected onto the robot coordinate frame; this is referred to as trajectory
sampling and coordinate transformation (from the global frame to the local
frame). Finally, the inverse kinematic solution of (11) is applied to convert
the local taskspace (i.e., local curve points – $x,y,z$) into jointspace
variables required to actuate the robot. The above procedure is applied to
obtain jointspace trajectories in Secs. III-B and III-C.

Figure 5: In sidewinding trajectory: (A) A discretized curve at $t=0$
projected onto the robot coordinate frame, $\\{O_{b}\\}$. (B) Generated
trajectory curves after projecting all discretized curve points onto {$O_{b}$}
and matched jointspace trajectories (i.e., inverse kinematic solutions).
Figure 6: (A) A trajectory curve projected onto {$O_{b}$} in planar and
helical rolling. (B) Projected total trajectory curves onto {$O_{b}$} and
generated jointspace trajectories (inverse kinematic solutions) in helical
rolling.

###  III-B Sidewinding Gait Trajectory

A time-series $\left(t\right)$ sidewinding curve is given by,

$\displaystyle\begin{split}y\left(t\right)&=A_{y}\sin\left(2\pi
f_{y}t\right)\\\ z\left(t\right)&=A_{z}\sin\left(2\pi
f_{z}t+\beta\right)\end{split}$ (12)

where $A_{y},\,A_{z}$, and $f_{y},\,f_{z}$ define amplitudes and frequencies
of $y$ and $z$ curve segments, respectively. Herein, $\beta$ denotes the phase
shift between two curve segments.

Fig. 4 shows the progression of the sidewinding curve derived with
$A_{y}=0.2~{}m,A_{z}=0.05~{}m,f_{y}=2~{}Hz,f_{z}=2~{}Hz,\beta=\frac{\pi}{3}~{}rad$,
and period, $T$. We decided on those values based on the physical dimensions
of the SRS prototype. In Fig. 4, $\\{O_{s}\\}$ and $\\{O\\}$ define the
curve’s local coordinate frame and the global coordinate frame, respectively.
First, we uniformly discretize the curve at marked locations such that those
points designate the origin of the robot coordinate frame, $\\{O_{b}\\}$ at
the discretized time instances. Next, we derive local coordinate frames at
each point relative to $\\{O\\}$. At any $n^{th}$ point, the local coordinate
frame can be defined as a tangential line to the curve (let it be local $X$)
and a line normal to that (let it be local $Y$). If $\overrightarrow{e}_{X}$
and $\overrightarrow{e}_{Y}$ define unit vectors along the local $X$ and local
$Y$, respectively, then $\overrightarrow{e}_{X}\times\overrightarrow{e}_{Y}$
gives the unit vector along the local $Z$. We encapsulate this orientation
data with the discretized position data to derive the local HTM at each point
relative to $\\{O\\}$ along the curve. Then, these local HTMs at each instance
are utilized to project the sidewinding curve onto $\\{O_{b}\\}$. Fig. 5A
shows the projection of the taskspace curve at the time instance, $t=0$ onto
$\\{O_{b}\\}$. Similar to that, all subsequent taskspace curves at each
discretized time instance within the trajectory period, $T$ are projected onto
$\\{O_{b}\\}$ as depicted by thin blue lines in Fig. 5B.

Figure 7: Optimization results, i.e., matched jointspace trajectories (length
changes) for (A) Sidewinding and (B) Helical rolling gaits within a trajectory
period, $T=1~{}s$. Note: In helical rolling, the length variables of section 1
and section 4 overlap each other due to the phase shift applied. Figure 8:
SRS experimental setup.

Finally, we obtain the SRS joint variables by mapping the trajectory curves at
$\\{O_{b}\\}$ (i.e., local taskspace curves) into the SRS jointspace using
inverse kinematics [27, 28]. However, it is impossible to obtain closed-form
inverse kinematic solutions for multi-section continuum robots [26] such as
the 4-section SRS proposed here. For that reason, we formulate the solution
for the inverse kinematics as an optimization problem between the desired
local trajectory curves and the SRS forward kinematics [29]. Therein, we apply
the kinematic model in (11) to obtain uniformly distributed points (61 points
– 15 per section) along the SRS neutral axis. Correspondingly, the cost
function is formulated as

$\displaystyle f_{cost}$
$\displaystyle=\sum_{k=1}^{61}\left\|\mathbf{p}\left({0},{q},\xi_{k}\right)-f_{gait}\left(s\right)\right\|+\sum_{j\in\left\\{1,2,3\right\\}}^{i\in\left\\{1,2,3,4\right\\}}l_{ij}^{2}$
(13)

where, $\xi_{k}=\xi$ given in (11) is discretized into 61 points along the SRS
and $f_{gait}\left(s\right)$ is the trajectory curve (with $s=61$ discretized
points) to which the SRS should fit.

The latter term of (13) ensures the stability and the smoothness of the
optimized solution. We implemented (13) using MATLAB’s global constrained
optimization routine. Fig. 5B shows the matched SRS shapes (thick multi-color
lines) with the projected taskspace curves (thin blue lines) for the
sidewinding gait.

###  III-C Helical Rolling Gait Trajectory

Helical rolling (i.e., 3D/Spatial rolling) is an extension of the planar
rolling presented in our previous work [15]. Here, first, we briefly discuss
the adopted trajectory generation procedure of the planar rolling and later,
extend it to the helical rolling. In [15], we modeled the planar rolling as
circular arc displacements on a plane. Therein, we defined a trajectory cycle
as a curve rotation about the SRS’s neutral axis (i.e., Z of $\\{O_{b}\\}$).
Due to the robot’s girth, the rotation results in a displacement on the X-Y
plane. This arc displacement within a period was discretized into several
circular arc shapes at uniformly distributed time instances (Fig. 3-b in
[15]). Next, we projected those taskspace rolling curves at each time instance
onto the robot coordinate frame, $\\{O_{b}\\}$. In the end, we applied the
optimization method proposed in (13) and obtained jointspace trajectories.
Readers are referred to [15] for more details on deriving planar rolling
trajectories.

Figure 9: The SRS progression for (A) Sidewinding gait at $4~{}bar-1.00~{}Hz$,
and (B) Helical rolling gait at $4~{}bar-0.50~{}Hz$.

We generate the helical rolling trajectories by applying the same procedure of
the planar rolling. However, in this case, we actuate adjacent SRS sections
with an angular phase shift, $\vartheta=\frac{\pi}{3}$. Accordingly, the
cumulative phase shift of each adjacent section (sections 2, 3, and 4) becomes
$\frac{\pi}{3},\frac{2\pi}{3},$ and $\pi$, respectively, relative to the first
section. Because of that, the SRS spatially moves out of its current plane
along the X, Y, and Z directions relative to its base at $\\{O_{b}\\}$, which
does not otherwise occur in planar rolling as illustrated in Fig. 6A. Note
that, no angular phase shift is applied to the first SRS section relative to
its base. Due to applied phase shifts, the SRS is expected to maintain skin-
ground contacts at three points. We mathematically derive the taskspace curve
of the helical rolling on the global framework, $\\{O\\}$ by applying
$\vartheta=\frac{\pi}{3}$ between adjacent sections of the planar rolling
curve. Similar to Sec. III-B, the derived curve is discretized and the curves
at discretized locations are projected onto $\\{O_{b}\\}$ and matched for
jointspace variables as shown in Fig. 6B. Figs. 7A and 7B show optimization
results, i.e., jointspace trajectories (or length changes) for sidewinding and
helical rolling gaits during one cycle. Here, the SRS actuation rate (i.e.,
gait frequency) and the SRS bending can be controlled by adjusting the gait
period and the jointspace amplitude, respectively.

##  IV Results and Discussion

Figure 10: Locomotion tracking: (A) Sidewinding, (B) Helical rolling.

###  IV-A Experimental Setup

The experimental setup employed in validation studies is shown in Fig. 8. The
air compressor provides steady $8~{}bar$ pressure to 12 digital proportional
pressure regulators (SMC ITV3050) that are then connected to individual PMAs
(12 PMAs in 4 sections) of the SRS. The MATLAB Simulink Real-Time model
computes the length changes and generates $0-10~{}V$ voltage signals via a
data acquisition card (National Instruments PCI-6221) to control the pressure
regulators in real-time at $20~{}Hz$. During each test, the SRS prototype was
actuated for $12~{}s$ on a carpeted floor with uniform friction. Note that,
the jointspace trajectories (i.e., length changes of PMAs) obtained in Fig. 7
should be converted into actuation pressure trajectories and input to PMAs via
the experimental setup (Fig. 8) to obtain the SRS locomotion. We adopted the
jointspace – pressure mapping reported in [15] to generate corresponding
pressure inputs.

###  IV-B Validation of Sidewinding Gait

We tested the SRS’s ability to achieve sidewinding locomotion by applying
different pressure amplitude and frequency combinations. We initiated the
testing at pressure ceiling, $p=3~{}bar$, and actuation frequency,
$f=0.25~{}Hz$. We chose those values because we found that low pressures
$\left(p<3~{}bar\right)$ onto PMAs could not provide an adequate bending in
SRS sections and low frequencies $\left(f<0.25~{}Hz\right)$ could not provide
an adequate forward momentum for moving, hence the robot was unable to show
meaningful locomotion. Thus, the SRS testing was repeated by increasing the
pressure ceiling and frequency by $0.25~{}bar$ and $0.05~{}Hz$ steps,
respectively. We observed that the SRS can achieve a fairly stable sidewinding
gait at $4~{}bar$ – $1.00~{}Hz$. It was further revealed that high frequencies
$\left(f>1.25~{}Hz\right)$ result in incomplete sidewinding trajectories
because those exceed the operational bandwidth of PMAs. On the other hand, a
high-pressure ceiling $\left(p>4~{}bar\right)$ overbent sections result in
twisting hence the distorted trajectories. The SRS movements were video
recorded using a fixed camera station. Fig. 9A shows the progression of the
SRS during its sidewinding movement at $4~{}bar-1.00~{}Hz$ pressure ceiling –
frequency combination. Complete videos of the experiments are given in our
supplementary file.

###  IV-C Validation of Helical Rolling Gait

For testing helical rolling gait, we adopted the same procedure applied in the
sidewinding testing. Accordingly, we initiated the testing at $p=3~{}bar$
pressure and $f=0.25~{}Hz$ frequency combination, and increased values on a
trial-and-error basis. It must be noted that the choice for the pressure
ceiling depends on the properties of custom-made PMAs and overall SRS assembly
including the length of pressure supply tubes. Similar to the sidewinding
testing, the SRS recorded its best helical rolling trajectory replication at
$4~{}bar$ pressure amplitude. However, unlike the sidewinding trajectory, we
observed that the SRS was capable of achieving helical rolling throughout the
applied frequency range $\left(0.25~{}Hz<f<1~{}Hz\right)$ at all times. When
the frequency was gradually increased from $0.25~{}Hz$ to $1~{}Hz$, the out-
of-plane bending amplitude of the rolling trajectory (i.e., the displacement
along the $Z$ axis of $\\{O\\}$) decreased. This was expected since low-
frequency actuation allows PMAs to realize the desired bending profile – i.e.,
at a low actuation rate, air pressure reaches to PMAs in due time through long
pressure supply tubes. Fig. 9B shows the progression of the SRS during its
helical rolling at $4~{}bar-0.5~{}Hz$, pressure – frequency combination in
which the SRS showed the best gait replication. Please refer to the
accompanying supplementary video file that shows the helical rolling gait at
low-medium-high frequencies and different amplitudes.

###  IV-D Gait Analysis

We employed the image processing method (perspective image projection)
reported in [15] to estimate the robot displacement on the actuated plane (on
the X-Y plane) using video feedback and geometric blocks on the carpeted floor
as illustrated in Fig. 10. Correspondingly, we computed the SRS velocities
shown in Table I. It shows that the SRS replicated the sidewinding locomotion
faster than the helical rolling. This is obvious since the SRS replicated
sidewinding trajectory only at higher frequencies
$\left(\approx~{}1~{}Hz\right)$. On the contrary, the SRS replicated its best
helical rolling trajectory at mid-range frequencies $\left(\approx
0.5~{}Hz\right)$. As expected, the SRS maintained skin-ground contacts at
three points during helical rolling (Fig. 10B). The SRS was expected to
maintain skin-ground contacts at two points during sidewinding locomotion
(Fig. 10A). In reality (as witnessed in experimental videos), even though the
SRS touches the ground at more points, the skin-ground contact has been
dramatically reduced.

TABLE I: Locomotion Performance in Each Gait Gait Type | Travelling velocity
---|---
$[cms^{-1}]$
$V_{x}$ | $V_{y}$
Sidewinding | 13.38 | 14.12
Helical rolling | 04.56 | 07.27

##  V Conclusions

We proposed a wheelless soft robotic snake that utilizes spatial deformation
to achieve snake locomotion gaits, namely sidewinding and helical rolling,
without friction anisotropy. We reported the four-bending section (12 DoF)
robot construction and developed a complete, floating-base kinematic model. We
derived the parametric mathematical models for the aforementioned locomotion
gaits and discretized the backbone curves thereof for one cycle. Those curves
were then projected to the SRS coordinate frame and jointspace curves for each
were derived using an optimization-based inverse kinematic approach. The joint
space trajectories were then mapped to pressure trajectories and applied to
the prototype SRS to experimentally validate the wheelless locomotion gaits
under various parametric values (i.e., cyclic frequencies and amplitudes). The
SRS deformation matched well with the mathematical models and demonstrated the
feasibility and efficacy of spatial deformation in achieving wheelless SRS
locomotion. Our future research will focus on dynamic modeling and validation
of spatial locomotion of SRSs.

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