SS symmetry
ArticleInvestigation of Snake Robot Locomotion
Possibilities in a Pipe
Ivan Virgala 1, * , Michal Kelemen 1 , Pavol Božek 2 , Zdenko Bobovský 3 , Martin Hagara 1 ,Erik Prada 1 , Petr Oščádal 3 and Martin Varga 1
1

2
3

*

Faculty of Mechanical Engineering, Technical University of Košice, 04200 Košice, Slovakia;
michal.kelemen@tuke.sk (M.K.); martin.hagara@tuke.sk (M.H.); erik.prada@tuke.sk (E.P.);
martin.varga.2@tuke.sk (M.V.)
Faculty of Material Science and Technology in Trnava, Slovak University of Technology in Bratislava,
917 24 Trnava, Slovakia; pavol.bozek@stuba.sk
Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic;
zdenko.bobovsky@vsb.cz (Z.B.); petr.oscadal@vsb.cz (P.O.)
Correspondence: ivan.virgala@tuke.sk; Tel.: +421-905-723-153

Received: 13 March 2020; Accepted: 7 May 2020; Published: 3 June 2020


Abstract: This paper analyzed the locomotion of a snake robot in narrow spaces such as a pipeor channel. We developed a unique experimental snake robot with one revolute and one linear
joint on each module, with the ability to perform planar motion. The designed locomotion pattern
was simulated in MATLAB R2015b and subsequently verified by the experimental snake robot.
The locomotion of the developed snake robot was also experimentally analyzed on dry and viscoussurfaces. The paper further describes the investigation of locomotion stability by three symmetricalcurves used to anchor static modules between the walls of the pipe. The stability was experimentally
analyzed by digital image correlation using a Q-450 Dantec Dynamics high-speed correlation system.
The paper presents some input symmetrical elements of locomotion and describes their influence on
the results of locomotion. The results of simulations and experiments show possibilities of snake
robot locomotion in a pipe.
Keywords: concertina; friction; locomotion; pipe; snake robot

1. Introduction
Snake robots are a special kind of kinematically redundant mechanism that have been a focus ofresearchers for several decades. Their kinematic structure allows them to be very flexible and adaptive
to rough terrain. Conventional robots based on a wheeled or tank-belted undercarriage are suitable
for many applications, and they are simple from the perspective of control [1]. There are also manypipe mechanisms based on biological mechanisms [2]. However, there are many situations where theycannot be used, for instance in terrain with large height differences or very narrow spaces such as
pipelines or channels. Climbing on rods or trees is another very uncomfortable environment for any
kind of robot. Snake robots can potentially be used in underwater applications, military exploration,
medical operations, searching for survivors after disasters, and many more scenarios. Because snake
robots are kinematically redundant, they can also perform motion optimization tasks within theircontrol [3].
Our goal was to implement locomotion of an experimental snake robot in a pipe and investigate
its possibilities while changing some of the input parameters. Our research was based on simulations
as well as experiments, which can be divided into three areas. The block diagram of our research
can be also seen in Figure 1. The paper is divided as follows: The second section introduces theSymmetry 2020, 12, 939; doi:10.3390/sym12060939

www.mdpi.com/journal/symmetry

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current state of snake robots designed for locomotion in a pipe or channel. Next, the locomotion
pattern is introduced and subsequently simulated in MATLAB R2015b. The designed pattern is also
Symmetry 2020, 12, x FOR PEER REVIEW
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experimentally
verified. Further experiments present the investigation of locomotion on dry,2 oily,
and lubricated surfaces. The last part of the paper presents an experimental analysis of the possibilities
and lubricated surfaces. The last part of the paper presents an experimental analysis of theof anchoring modules between the pipe walls by symmetrical curves. The analysis assumes a circularpossibilities of anchoring modules between the pipe walls by symmetrical curves. The analysis
pipe as well as a rectangular cross-section. The conclusion discusses the possibilities for snake robot
assumes a circular pipe as well as a rectangular cross‐section. The conclusion discusses thelocomotion in a pipe.
possibilities for snake robot locomotion in a pipe.

Figure 1. Block diagram of investigation of snake robot locomotion possibilities.
Figure 1. Block diagram of investigation of snake robot locomotion possibilities.

2. Review of Snake Robot Applications in a Pipe
2. Review of Snake Robot Applications in a Pipe
Research on snake robots started in the 1970s with the development of the first snake robot
Research
on snakerobotsthen,startedin the1970s with snake
the development
the first
snake robot
by Shigeo
Hirose[4]. Sincemanyexperimental
robots haveofbeen
developed,
withbya
ShigeoHirose[Error!
Referencesourcenotfound.].
Since
then,
many
experimental
snake
robotsfocus on applications such as climbing, swimming, inspecting pipes, and moving through obstacles,
have beendeveloped,
witha focuson applications
such as onclimbing,
among
others.
Althoughmost
previous
works concentrated
a lateralswimming,
undulationinspecting
pattern of pipes,
snake
and
movingthrough
obstacles,
among
others.
Althoughmost
previous
works
concentrated
on arobots with or without active/passive wheels, few works dealt with pipe inspections.
lateralundulation
patternofsnake
robotswithorwithout
active/passive
wheels,
few
works
dealt
Wakimoto et al. [5] presented the development of a micro robot that moves through pipes
with pipe
inspections.
withdifferent
diameters using a traveling wave pattern. The maximum velocity of the robot was
Wakimoto
et al.able[Error!
Referencesourcefound.]frompresented
of a micro
36 mm/s and it was
to movein pipes
withnot
diameters
18 mmtheupdevelopment
to 100 mm. Greenfield
robot
that
moves
throughpipes
withdifferent
diameters
usingatraveling
wave
pattern.
et al. described the locomotion pattern of a snake robot moving in a pipe [6]. They developedTheanmaximum
velocity
of
the
robot
was
36
mm/s
and
it
was
able
to
move
in
pipes
with
diametersfromalgorithm for climbing, and the robot stabilized itself against gravity by pressing outward to induce
18 mm up
100 Brunete
mm. Greenfield
et al. describedlocomotion
patterna snakemovinginfriction.
Into
2006,
et al., developed
a microthesnake
robot with
two ofdegreesofrobot
freedom(DOF)
a pipe[Error![7].
Referencesourcefound.].
They developed
climbing,
and theInrobot
permoduleThe robot
wasnotable
to perform
locomotionaninalgorithm
different forpipes
and elbows.
[8],
stabilized
itselfagainstgravitybypressing
outwardtoinducefriction.
In
2006,
Bruneteetal.
Barazandeh proposed a mathematical model of a climbing snake robot using a concertina pattern.
developed
a
micro
snake
robot
with
two
degreesoffreedom(DOF)
permodule[Error!
Reference
By optimizing the proposed model, the results showed that self-locking could occur during thesource notfound.]. The
robot wasableactuator
to perform
locomotion
in different
pipesthat
andthere
elbows.
In [Error!
courseof climbing,
resultingin zerotorque.
The authors
claimedshouldbe aReferencesourcenotfound.],
Barazandeh
proposedamathematical
model
of
a
climbing
snake
robot
minimum number of contact points to activate self-locking in a climbing robot. This number depends
using
concertina
pattern.
By optimizing
thefriction
proposedmodel, theresults
showedthat self‐locking
ontheaweightof therobot modules
and thecoefficient.
Wrightet al.
developeda modular
could
occur
duringthecourseofclimbing,
resultinginzeroactuator
torque.
Theauthors
claimedthatsnake robot consisting of 16 modules with a modified hobby servomotor [9]. This robot was able
tothereshouldbeaminimum
numberofcontact
pointstoactivateself‐locking
inaclimbing
robot.
This
climb inside and outside of a pipe. Another study [10] presented a snake robot with the ability tonumberdepends
on thethe robotthreemodulesthe friction
coefficient.
Wrightet al.
travelthroughpipes.
Theweight
authorsofproposed
controland
algorithms:
force
control, andadjustingdevelopedamodularsnake
robot
consistingof
16
moduleswithamodifiedhobby
servomotor[Error!
controls to changing pipe diameter and curved pipe. This team realized traveling in several kinds ofReferencebysource
not found.].
This
robot
was13able
to climb
inside andoutsideof a pipe.
Anotherpipelines
an experimental
snake
robot
using
intelligent
actuators.
In [11],
Andruskaand Peterson
study [Error! Reference
source anot
found.] presented
a snakerobot
the ability
to travel
throughdemonstrated
how to control
hyper-redundant
snake
robot
in awith
compliantelastic
environment
pipes.
The
authors
proposed
threecontrolalgorithms:
force
control,andadjustingcontrols
tosuch that it was able to navigate paths with unknown curvature radius. Akbarzadeh et al. [12]changing pipe diameter and curved pipe. This team realized traveling in several kinds of pipelines
by an experimental snake robot using 13 intelligent actuators. In [Error! Reference source notfound.], Andruska and Peterson demonstrated how to control a hyper‐redundant snake robot in acompliant elastic environment such that it was able to navigate paths with unknown curvatureradius. Akbarzadeh et al. [Error! Reference source not found.] presented a novel kinematic
modelling approach for snake robot motion with a concertina pattern. Their approach provides

Symmetry 2020, 12, 939

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presented a novel kinematic modelling approach for snake robot motion with a concertina pattern.
Their approach provides flexibility for natural locomotion. The paper also presented a dynamic curve
whose shape can be easily modified by a single parameter. Their approach was verified by simulationsin Webots. In 2012, Marvi and Hu performed a series of experiments to measure the friction of cornsnakes while climbing through a channel [13]. They proposed a theoretical model to predict thesnake’s traverse force. Their investigations showed that the snake moved in a channel by a seriesof extensions and contractions in which portions of its body extended forward while the rest wasanchored. Other researchers [14] modelled snake locomotion on the basis of autonomous decentralized
control. By simulations, they demonstrated that the locomotion pattern smoothly changed fromsinus-lifting to concertina. Our research team also contributed to this area [15]. Our paper dealt with asnake robot moving in a pipe using a traveling wave pattern. Locomotion in straight and curved pipes
was demonstrated by an experimental snake robot consisting of 11 DOF. An analysis of a snake robot
moving by rectilinear motion in a pipe is presented in [16]. A novel gait based on cylinders as well asarboreal concertina locomotion for a snake robot was proposed in [17]. The authors also introduced
a kinematic and dynamic analysis of gait, and the developed approach was verified by simulations
in V-REP. Within the simulations, the snake robot climbed on a rod using the concertina principle.
Studies on the low-cost application of a snake robot at a large-scale nuclear power facility to sense
pipe components was presented in [18]. Within this study, the locomotion method was developedand executed in a narrow space. The developed algorithm was based on angular changes of various
modules of the snake robot.
By summarizing previous works focused on snake robot applications in pipes or channels, we cansay that the three main locomotion patterns are concertina, rectilinear, and traveling wave. Looking
at the whole research area of snake robots, a small percentage of previous works were devoted to
locomotion in confined spaces such as pipes or channels. We would like to contribute to this area withthis study, including the design of a new locomotion pattern, simulations, the design of a new uniqueexperimental snake robot, and an experimental analysis of locomotion.
3. Design and Analysis of Locomotion Pattern in a Pipe
Snake robots have great potential to be applied in various kinds of environments, such as
narrow spaces, rough terrains, smooth surfaces, underwater, tall heights, and many more. In allthese environments, a biological snake would perform different kinds of natural locomotion patterns.
For example, to move underwater it could use a lateral undulation pattern, for heights it could use aclimbing pattern, and in a pipe it could use a concertina pattern.
Our analysis deals with applications in narrow spaces. From previous studies on biological snakes,
it is clear that snakes use concertina locomotion in narrow spaces [19]. Concertina locomotion consistsof several phases. First, the snake pushes against the constrained environment with the rear part ofits body. This activity we consider to be an anchoring of snake modules. Subsequently, the snake
straightens its body along the direction of motion, while the rear part is anchored. The anchoredpart can be considered static. The next phase is based on pushing the front of the body against theconstrained environment (the front part is now static, or anchored), while it pulls its rear parts to thefront. By repeating these phases, the snake is able to achieve forward locomotion. In special cases of
constrained environments such as very narrow spaces, a snake is also able to use rectilinear locomotion.
During rectilinear locomotion, the rear parts of the body are static, while the front parts move forward
by muscular activity. After this phase, by the weight of the static front parts, the rear parts can be
drawn to the front along the direction of motion. A snake uses this kind of locomotion especially when
it is in a very narrow space and is unable to perform a different kind of locomotion.
These two biological snake patterns, rectilinear and concertina, are the basis of this study. Let us
consider a snake robot as a kinematically redundant mechanism consisting of many modules imitatinga biological snake body. In order to analyze snake robot locomotion in narrow spaces, we consider asnake robot module having two DOFs: the first one considers revolute joints, the second considers

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linear joints. By connecting several modules together, each module is able to perform revolute andconsiders linear joints. By connecting several modules together, each module is able to perform
linear motion, according to the requirements of the desired application.
revolute
linearmotion,according toseveral
the requirementsof the desiredconsidersand
linearjoints.
By connecting
modules together,each application.
module is able to perform
Figure 2 shows a video sequence of biological corn snake locomotion in a pipe [13]. The snake
Figure2 shows
a video sequencecorn snake
locomotion
in a pipe [Error! Reference
revolute
andlinear motion,accordingof
tobiological
the requirementsof thedesired application.
moves according to pattern described above.
sourcenotfound.].
The
snake
moves
accordingtopattern
described
above.
Figure 2 shows a video sequence of biological corn snake locomotion in a pipe [Error! Reference
source not found.]. The snake moves according to pattern described above.

Figure 2. Corn snake
concertina
locomotion[Error!
Reference source not found.].
Figureconcertina
2. Corn snake
concertina
locomotion
locomotion[Error!
Reference[13]source not found.].
Figure 2. Corn snake

Referring to Figure
Figure 2, we
we designed aa locomotion
locomotion pattern forfor a snake robot
robot with two
two DOFs perper
Referring
Referring toto Figure 2,
2, we designed
designed a locomotion patternpattern for aa snake
snake robot with
with two DOFs
DOFs permodule.
As
can
be
seen
from
Figure3,
there
wassymmetricalbehavior
of
the
snake
robot
body.
module.
As can be seen from Figure 3, there was symmetrical behavior
behavior of the snake robot body.
module. As
Except forfor thethe first
first andand second
second phases,
phases, all
all otherother final
final statesstates ofof individual
individual phases
phases were symmetrical,
symmetrical,
ExceptExcept for the first and second phases, all other final states of individual phases were
were symmetrical,
consideringthat
thatthetheverticalverticalaxis
axisofofsymmetry
symmetrywas
wasin
inthe
themiddle
middlejoint
jointofofthe
thesnake
snakerobot.
robot.
considering
considering that the vertical axis of symmetry was in the middle joint of the snake robot.

Figure 3. Locomotion pattern in a pipe.
FigureFigure 3.
3. Locomotion pattern in a pipe.

Figure 3 assumes a triangular shape for the initial position of the locomotion cycle. The
Figurea triangular
shape
for theof the locomotion
cycle. The cycle.
amplitude
Figure 33assumes
assumesa triangularshapeforinitial
the position
initial positionof the locomotionTheamplitude of the triangular shape is primarily formed by one robot module. However, if pipeofthetriangularshapeisprimarily
formedbyone
robot
module.
However,if
pipe
diameterd(orDamplitude of the triangular shape is primarily formed by one robot module. However, if pipe
diameter d (or 𝐷 for a theoretical, link‐based robot) is greater than module length L, the amplitude
foratheoretical,
link-based
robot)
isgreaterthanmodulelengthL,theamplitudeofthetriangular
diameter d (or 𝐷 for a theoretical, link‐based robot) is greater than module length L, the amplitudeof the triangular shape can be formed by two or more modules. By considering pipe diameter D,shape
be formedby twomodules.
pipe diameter
D, moduleL, andof the cantriangularshape
can orbemoreformedby twoByorconsidering
more modules.
By considering
pipelength
diameterD,
module length L, and maximal protrusion of linear joint l, the traveled distance of one locomotion
maximalprotrusionoflinearjoint
l,thetraveled
distanceofone
locomotion
cycle
can
be
expressed
asmodule length L, and maximal protrusion of linear joint l, the traveled distance of one locomotion
cycle can be expressed as
cycle can be expressed as


 
D
+ 2l.
(1)
2L 1 − cos𝛿δ = 2𝐿
𝑐𝑜𝑠 asin
𝑎𝑠𝑖𝑛
2𝑙.
(1)
𝛿 2𝐿 1 𝑐𝑜𝑠 𝑎𝑠𝑖𝑛L2𝑙.
(1)
Equation (1)(1) considers exactly
exactly the patternfrom Figure
Figure 3, so
so it means
means that thethe robot used
used three
Equation
Equation (1) considers
considers exactly thethe patternpattern fromfrom Figure 3,
3, so itit means thatthat the robot
robot used three
three
modulestoanchor
its
body
and
two
modulesforlinearmotion.
Thepatternfrom
Figure3canbe also
also
modulesmodules toto anchoranchor its
its body
body andand two
two modules
modules forfor linear
linear motion.
motion. The
The patternpattern fromfrom Figure
Figure 33 cancan bebe alsodescribed byby a general form:
form:
described
described by aa generalgeneral form:𝑛 𝐿 𝑙
𝛿 𝐿 𝑐𝑜𝑠 𝑎𝑠𝑖𝑛 D  2𝑓 𝑚
(2)𝛿δ = 𝐿L𝑐𝑜𝑠
𝑎𝑠𝑖𝑛
(2)
(22𝑓
) + n(𝑛L 𝐿+ l)𝑙
cos asinf − m𝑚(2)
L
where f is the number of anchoring modules, m is the number of snake robot modules, and n is thewhere f is the number of anchoring modules, m is the number of snake robot modules, and n is thenumbermodules
betweenthe anchoredrear
modules;see robot
the third
phase andin Figure3.
where
f isofthe
numberof anchoring
modules,m andis thefront
numberof snake
modules,n is thenumber of modules between the anchored rear and front modules; see the third phase in Figure 3.
From
Figure3,dcanbeexpressedasnumber of modules between the anchored rear and front modules; see the third phase in Figure 3.
From Figure 3, d can be expressed as
From Figure 3, d can be expressed as
(3)
𝑑 𝐷 𝑤
(3)
𝐷+ w
𝑤
d𝑑= D
(3)
where 𝑤 represents the width of a snake robot module. Considering the link‐based model, we canwhere 𝑤represents
representsthe
the widthofofaasnake
snake robotmodule.
module. Considering thethe link-basedlink‐based model,
model, we can
can
where
assumewonly
parameterwidth𝐷, but for a realrobot
snake robot itConsidering
is necessary to consider
the widthweof theassumeonlyparameter
𝐷,
but
for
a
real
snake
robot
it
is
necessarytoconsider
the
width
of
the
assumefor diametera real snake
it is necessary
to considerl theof themodule.
module.only
Theparameter
influenceD,ofbutpipe
D robot
and linear
joint protrusion
on widthtraveled
distanceismodule. The influence of pipe diameter D and linear joint protrusion l on traveled distance isexpressed in Figure 4.
expressed in Figure 4.

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The
influence
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Figure 4.

Figurebased onon pipepipe diameter
diameter andand linear
linear joint
joint protrusion.
protrusion.
Figure 4.
4. Traveled
Traveled distancedistance ofof snake
snake robot
robot based
Figure 4. Traveled distance of snake robot based on pipe diameter and linear joint protrusion.

ofof
modulelengthL.
The real application
application ofof aa snake
snakerobot,
robot,ofofcourse,
course,considers
considersthe
thefixed
fixedvalue
value
modulelengthThe
realapplicationof traveleda snakerobot,of course,
considersthetofixedofthe
modulelengthL.
Figure5 shows
how
the thetraveled
distancechangesin relation
to changes
ofvalue
theof
pipe
diameter,while
L.
Figure5 showshow
distancechangesin relation
changes
pipe diameter,
Figurelinear
5 shows
thetraveled
distancechangesinrelation
to changes
ofdescribed
the pipe
diameter,while
linearjoint
protrusion
is neglected(shownin blue).
The
locomotion
patternin Figure
3 can
while
jointhowprotrusion
is neglected(shownin blue).
The locomotion
pattern
described
in Figure3
linearjoint
protrusionisneglected(showninblue).
The
locomotion
pattern
described
inFigure3 canassumea versionwithor without
linearjoint
protrusion.
Assuming
thetheprotrusion
is is𝑙 li =0.05𝑚,canassumea versionwithor without
linearjoint
protrusion.
Assuming
protrusion
0.05
m,assumea versionwithlinearcycle
jointwould
protrusion.
is 𝑙
0.05
𝑚,
distanceof or
onewithout
locomotion
be
mm
(see
Figureinthetraveled
locomotion
cycle
would
be 100100Assuming
mm moremorethe(seeprotrusion
Figure 55 (shown(shownin red)).
red)).
thetraveled
distanceofone
locomotion
cycle
would
be
100
mm
more
(see
Figure5(showninred)).
addinga alinear
linearjoint
to module,
the module,
the traveled
distance
per locomotion
cycle significantly
By addingjoint
to thethe traveled
distance
per locomotion
cycle significantly
increases.
By adding a linear joint to the module, the traveled distance per locomotion cycle significantly
increases.
increases.

Traveled distance based on change of pipe diameter.
Figure 5. Traveled
diameter.
Figure 5. Traveled distance based on change of pipe diameter.

4.
Simulation ofof Locomotion
Locomotion
4. Simulation
4. Simulation of LocomotionThis
a simulationof theofintroduced
locomotion
pattern. pattern.
The aim oftheaimsimulation
This section
sectionpresents
presents
a simulation
the introduced
locomotion
Theof theistoperform
locomotion
in
a
pipe,
and
also
to
verify
it
on
the
basis
of
the
mathematical
model
from
This
sectionpresentsasimulationoftheintroduced
locomotion
pattern.
The
aimof thethesimulation is to perform locomotion in a pipe, and also to verify it on the basis of the mathematicalprevious
section.
The simulationareasfollows:
theof the
snake
robot
is
simulation
isthe
to previous
perform
locomotion
a pipe,andalso to verify
it on
the
basis
of
the
mathematical
model
from
section. parameters
Theinsimulation
parameters
arelengthasfollows:
thelengthof module
the snake
◦
L
=
0.13
m,
the
maximum
linearprotrusion
is
l
=
0.05
m,
the
maximum
revolute
joint
motion
is
±90
model
from
the
previous
section.
The
simulation
parametersareasfollows:
thelength
of
the
snake
i
robot module is 𝐿 0.13 𝑚, the maximum linear protrusion is 𝑙0.05 𝑚, the maximum revolute,andthenumberofmodules
N linear= robot
8. Thesnake robotin𝑚,a snakepipe
witha rectangular
robot
module𝐿 snake0.13robot
𝑚,numberthemaximum
protrusionis0.05
the maximum
revolute
joint
motionisis±90°,
andtheof issnake
modulesis 𝑙𝑁moves
8. The
robot
moves
in a
cross-section
withdiameter
d
=
0.12
m.
joint
motion
is
±90°,
and
the
number
of
snake
robot
modules
is
𝑁
8.
The
snake
robot
moves
in a
pipe with a rectangular cross‐section with diameter 𝑑 0.12 𝑚.
simulations
were
done
in
MATLAB
R2015b.
Joint
anglesq
wereassembledinvectorpipeThewitharectangularcross‐section
withdiameter𝑑0.12
𝑚.
i assembled in vector 𝐪
The
Joint angles 𝑞 wereh simulations were
iT done in MATLAB R2015b.

p q =The2(N−1) , MATLAB
simulations
were
done
in
R2015b.
Joint
angles𝑞
wereassembled
in
𝐪
(
)
q
,
.
.
.
,
q
∈
R
where
i
∈
1,
.
.
.
,
2
N
−
1
and
p
is
the
numberof vector
locomotion
2∈(N−1
𝑞 , … , 𝑞1
ℝ ), where 𝑖 ∈ 1, … , 2 𝑁 1 and 𝑝 is the numberof locomotion
phases,hiT.
.
.
𝑞 ,…,𝑞∈ℝ, where 𝑖 ∈ 1, … , 2 𝑁 p 1 and 𝑝 is the number 2oflocomotion
phases,(N−1) as the vector
phases,. , 5}.defined
We alsodefined∈ Rof𝑞 , … , q𝑞 = q1 , .∈. .ℝ, q2(N−1as
vectorof joint velocities𝑝 ∈ 1, …p, 5∈ .{1,We. .also
vector𝐪 vector) thehi𝑞 ,…,𝑞∈p ℝ
velocities𝑝 ∈ 1, … , 5 . We also defined vector 𝐪.. as the.. vectorTof joint..
for a velocities
particularfor
locomotionphaseand vector,…,𝑞 q = q ∈ℝas the∈ vector
joint
a particularlocomotion
phase𝐪and 𝑞vector
R2(N−1)ofasjoint
the
1 , . . . , q2(N−1)
for a particularlocomotion
phase and vector𝐪𝑞 , … , 𝑞all phases∈ ℝof locomotion,
as the vectorof joint
accelerations
forparticularlocomotion
phase.
Consideringvectorofvector of joint accelerations for particular locomotion phase. Considering all phases ofthe
locomotion,
accelerations
forparticularlocomotion
phase.
Consideringallphasesoflocomotion,
thevectorofjoint angles 𝐪 is based on locomotion properties such as module length L and pipe diameter D.
joint
angles
𝐪
is
based
on
locomotion
properties
suchasmodulelengthL
andpipe
diameterD.
Vectors 𝐪 and 𝐪 are then determined on the basis of the fifth‐order polynomial of particular joint
Vectors 𝐪
determined
the fifth‐orderjoint
angles𝑞 and𝑡 𝐪𝑎 are𝑎then𝑡 𝑎𝑡𝑎 𝑡 on𝑎 the𝑡 basis𝑎 𝑡 of, where𝑖 ∈ 1, … , polynomial
2 𝑁 1 , 𝑝ofisparticular
the numberofangles
𝑞
𝑡
𝑎
𝑎
𝑡
𝑎
𝑡
𝑎
𝑡
𝑎
𝑡
𝑎
𝑡
,
where
𝑖
∈
1,
…
,
2
𝑁
1
,
𝑝
is
the
numberofactive phases of locomotion, and 𝑡 is the determined time for a particular phase. By using the fifth‐
active phases of locomotion, and 𝑡 is the determined time for a particular phase. By using the fifth‐Symmetry 2020, 12, 939

6 of 25the vector of joint angles q is based on locomotion properties such as module length L and pipe.
..
diameter D. Vectors q and q are then determined on the basis of the fifth-order polynomial of
particular joint angles p q i (t) = a0 + a1 t + a2 t2 + a3 t3 + a4 t4 + a5 t5 , where i ∈ 1, . . . , 2(N − 1) , p is
Symmetry 2020, 12, x FOR PEER REVIEW
6 of 26
the numberof active phases of locomotion, and t is the determined time for a particularphase.
Symmetry 2020, 12, x FOR PEER REVIEW
6 of 26
By using
thefifth-orderpolynomial
insteadofthethird-orderpolynomial,
we
can
also
controlorder polynomial instead of the third‐order polynomial, we can also control the acceleration of joints theaccelerationof jointsinstead[20]. The
third-order
allowsuscontrol
only topositionorder polynomialofthird‐orderpolynomial,
can also
theaccelerationof joints[Error!
Referencesourcenotthefound.].
The.polynomial
third‐order wepolynomial
allowsuscontrol
only tothe
control
the andpq
p q third‐order
2 +allows3+
4 , and[Error!
Referencesourcenotfound.].
The
polynomial
us
only
to
controlvelocity[21].
By
derivation
of
we
get
=
a
+
2a
t
+
3a
t
4a
t
5a
t
subsequently
position and velocity [Error! Reference
source𝑎 the
2 By derivation
3
5 get 𝑞4of 𝑞 weii not1found.].
.
..
p
p
2
3
positionandvelocity[Error!
Referencesourcenotfound.].
By
derivation
of
𝑞
we
get
𝑞
by derivation+ 20a5 t .ofOn𝑞the
4𝑎𝑡 get
5𝑎 𝑡q i, =and2asubsequently
by4 t derivation
webasisget of𝑞 these2𝑎 terms
6𝑎 𝑎𝑡 and by
2𝑎 𝑡 3𝑎 𝑡of q
2 + 6a3 t + 12a
i we
2𝑎
𝑡
3𝑎
𝑡
4𝑎
𝑡
5𝑎
𝑡
,
and
subsequently
byderivation
of
𝑞
we
get
𝑞
2𝑎
6𝑎 𝑡
12𝑎 𝑡
20𝑎 𝑡 .andOn thebasis
of these terms
and by into
substituting
initial andconditions
substituting
initialfinal
conditionsof motionthese terms,
we final
get thevector ofofmotioncoefficients.motion
..
T
12𝑎
𝑡
20𝑎
𝑡
.
On
the
basis
of
these
terms
and
by
substituting
initialand
final
conditions
of
6
p
p
we
get
vectorofcoefficients vectors
𝐚𝑎q,, …q,, 𝑎 p q ∈for all[a0 , . . .these
ai =into, a5 ] ∈ terms,
R for a certain
phase theof locomotion.
By computing
into
these
terms,
wegetthevectorofcoefficients
𝐚
𝑎
,
…
,
𝑎
ℝ offora certain pphase. By computing
vectorsrobot
𝐪 , for𝐪one, 𝐪cyclefor canall phasesof∈phaseslocomotion∈ {1, of. . . locomotion, 5}, the locomotionof the snake
be determined.
ℝforacertainphaseoflocomotion.
By
computing
vectors
𝐪
,
𝐪
,
𝐪
for
all
phasesoflocomotion𝑝 ∈ results
1, … , 5 of, thelocomotionof therobot
snakelocomotion
robot for one
canarebe determined.
The graphicalsimulatedsnake
incyclea pipeshown in Figure 6.
locomotion𝑝 ∈ 1,results… , 5 , thelocomotionof thesnake
robot for one
canshown
be determined.
The graphicalof simulated
snake
robot
locomotion
in a cycle
pipe arein Figure 6.
The graphical results of simulated snake robot locomotion in a pipe are shown in Figure 6.

Figure 6.
Snake robot locomotion in rectangular pipe using concertina and rectilinear
Figure 6. Snake robot locomotion in rectangular pipe using concertina and rectilinear locomotion
locomotion
patterns.
Figure 6. Snake robot locomotion in rectangular pipe using concertina and rectilinear locomotion
patterns.
patterns.

Figure7 shows timecourses ofofsnake
robot joints
joints duringconcertina
concertina locomotion.
The individual
FigureFigure77shows
showstime
timecourses
courses ofsnake
snakerobot
robot jointsduring
during concertinalocomotion.
locomotion.The
Theindividual
individual
phasesof locomotion
arearedenotedlines.
phasesof locomotion
denotedby
bydotted
dotted red
red lines.
phases of locomotion are denoted by dotted red lines.

Figure 7. Time courses of snake robot joints during locomotion.

Figure7. 7.
Timerobotjoints
jointsduring
duringlocomotion.
FigureTimecourses
coursesofof snake
snake robot
locomotion.

Symmetry 2020, 12, 939

7 of 25

From the initial position of the snake robot, it extends most of its modules along the directionof motion. By pushing the last three modules against the walls of the pipe, the anchor effect occurs.
Symmetry
2020,anchor,
12, x FOR the
PEERfront
REVIEW
7 ofmotion.
26
Becauseof thismodules are able to move forward without any backward
Symmetry 2020, 12, x FOR PEER REVIEW
7 of 26
The linear motion of the two modules is shown in Figure 6 in red. They move as much as the range ofFrom the initial position of the snake robot, it extends most of its modules along the direction of
linearactuators
enables
them.
After
this
phase,
thetofirst
three
modulesnow
serve
as effect
anchors
in orderBecauseofanchor,thelastfront
modulesareagainst
able
move
forwardany
backward
motion.
motion.
Bythispushingthethreemodulesthewallsof thewithout
pipe,
the
anchoroccurs.
Thelinearof thetwofront
modulesshown
in in
Figure6 inforward
red.theThey
asmodules
much
as needthemotion.
range
to pulltherearmodulesto thefront.
It isisclear
phase,
lastmovethree
to release
Becauseofmotion
this anchor,
themodulesarethatable
tothismove
withoutany
backwardoflinearactuators
enables
them.
After
this
phase,
the
first
three
modules
now
serve
as
anchorsinthe pressure
against
the
walls
of
the
pipe.
Figure6represents
onlyone
locomotion
cycle.
The linear motion of the two modules is shown in Figure 6 in red. They move as much as the range
Tlast three
m
order
to
pull
the
rear
modulestothefront.
It
is
clear
that
in
this
phase,
themodulesneed[xh yh ] nowof linear
actuators
them. Afteristhis
phase, the firstas anchorsThe
vectorof the enables
head coordinates
represented
by xthree∈ R serve, wherem = 2 indue tohead =modules
toorder
releasethe pressure
against theof theFigure6 represents
only
locomotion
cycle.
to pullthe rear modules
to walls
theof
front.
It pipe.
is clearthat inthis phase,the onelastlocomotion
three
modulesneed
the planar
task.
Simulated
coordinates
the headmoduleduringconcertina
areshown in
The vectorof the head
coordinatesby 𝐱 6 represents
𝑥 𝑦 only∈ℝ𝑚 cycle.
2 due
to14,
releasethewithpressureagainstthedata.
wallsisofrepresented
the pipe. Figure
one, where
locomotion
Figurealongexperimental
to theTheplanartask.
Simulated
coordinatesof the headbymodule
duringlocomotion
vectorof thehead coordinates
is represented𝐱𝑥 𝑦 concertina∈ ℝ , where𝑚 2 are
due
shown
in
Figure
14,
along
with
experimental
data.
to the planaroftask.
Simulated coordinatesof the head module during concertina locomotion are
5. Development
Experimental
Snake Robotshown in Figure 14, along with experimental data.
5.
Developmentof Experimental
Snake Robot analysis of snake robot locomotion. Our team fromThis
section deals
with the experimental

Cognitics
Labsection
at Technical
University
ofRobot
Košice,Slovakia,
developed
a 14 DOF
robot for5. DevelopmentofExperimental
Snake
This
deals
with
the experimental
analysisof snakerobot locomotion.
Oursnaketeam fromexperimental
Cognitics
Lab
at Technicalof Košice,Slovakia,
developed
a 14 DOF snake
robotfromforThis purposes.
sectiondeals withUniversity
the experimental
analysisof snakerobot locomotion.
Our team
experimental
purposes.
Cognitics Lab
at Technical University of Košice, Slovakia, developed a 14 DOF snake robot forexperimental purposes.
5.1. Mechanical Design of Snake Robot

5.1. Mechanical Design of Snake Robot

Our developed snake robot had a significant structure. Considering the joints of the robot module,
Mechanical
Designsnakeof Snake
Robot
Ourexperimental
developed
hadhavea significant
structure.
Considering
the jointsourof therobot
while5.1.
mostsnakerobot
robotsone or two
revolute
joints per module,robot
had onemodule,while
mostexperimental
snake
robots
have
one
or
two
revolute
jointspermodule,ourrobot
Ouronedevelopedsnake
had aThesignificant
structure.
Considering
jointsinofpipes.
the robot
revolute andlinear joint
perrobotmodule.
robot wasdesignedfor planar the
motionModuleshad
one revoluteandexperimental
one linear joint
per module.
Theonerobot
designedfor planar
motion ourin pipes.
module,
while
most
snake
robots haveor wastwoHS-645MG
revolute joints
per module,
robot
consistedof onerevolute
and one linear
servomotor.
The Hitec
revolute
servomotor
has theModulesconsistedofone
revolute
andone
linearservomotor.
The
Hitec
HS‐645MG
revolute
hadone
revolute
andone
linearjoint
permodule.
The
robot
was
designedforplanarmotioninpipes.
following parameters: maximum catalogue torque 0.939 Nm (6 V DC), weight 55.2 g, and maximum
servomotor
has the following
parameters:
torque(6 V DC), revoluteweightModulesconsistedof one revoluteand maximum
one linear catalogue
servomotor.
The0.939
HitecNm
HS‐645MG
speed
60◦g,/sec(6VDC). Forlinearmotion,a Firgelli
L12 linear
servomotor
was
used,
with the following
55.2
andmaximum
speed
60°/sec
(6V
DC).
For
linearmotion,aFirgelli
L12
linearservomotor
was
servomotor has the following parameters: maximum catalogue torque 0.939 Nm (6 V DC), weightparameters:
maximumforce
42 N, maximum
speed
13 N,
mm/s,
gear speed
ratio 100:1,
andgearweight
40 g.
used,
with
the
followingparameters:
maximumforce
42
maximum
13
mm/s,
ratio
55.2 g, and maximum speed 60°/sec (6V DC). For linear◦motion, a Firgelli L12 linear servomotor was
The rangeofrevolute
motionforeach
modulewas±90(seeFigure8a).
100:1,
andweight
40
g.
The
range
of
revolute
motionforeach
module
was
±90°
(see
Figure8a).
used, with the following parameters: maximum force 42 N, maximum speed 13 mm/s, gear ratio
100:1, and weight 40 g. The range of revolute motion for each module was ±90° (see Figure 8a).

(a)
(a)

(b)
(b)

Figure8. Snakerobot
modulesa)a) Revolute motion
ofofmoduleb) b)Linearmotionof module.
Figure8. Snakerobot
modulesmoduleLinearmotionof module.
Figure8. Snakerobot
modules a)Revolute
Revolute motion
motion ofmoduleb) Linear
motionof module.

range
of linear
motionformodule
each module
50 mm(see8b).
FigureThe kinematic
The Therange
of linear
motionfor each
was 50wasmm
Figure
The 8b).
kinematic
configuration
The rangeof linear
motionfor each modulewas (see
50 mm(see Figure8b). The kinematicof therobot
modulecan beseenin Figure9. Theinsideof thesnakeisrobot
moduleisof theconfiguration
robot
module
can
be
seen
in
Figure9.
Theinside
of
the
snake
robot
moduleshown
in Figureconfiguration of the robot module can be seen in Figure 9. The inside of the snake robot moduleis 10.
shown in Figure 10. The experimental snake robot consisted of eight identical modules, hence theshown in Figure
10. Theexperimental
robot
consistedof eight identicalmodules,
The experimental
snake
robot
consisted snakeof eight
identical
modules,hence thewhole hence
robotthehad 14whole robot had 14 DOF. Such kinematic redundancy represents great potential for the robot to be
robot had 14
DOF. Such kinematic
redundancy
represents
great
potential
the robot
to be or
DOF.whole
Such kinematic
redundancy
represents
great potential
for therobot
to beforappliedin narrowapplied in narrow or geometrically complicated environments. The final model of the robot is shown
applied incomplicatednarrow or geometricallycomplicatedenvironments.
The
final ismodelof therobot is 11.
shown
geometricallyenvironments.
The final
model of the
robot
shown
in Figure
The topin Figure 11. The top image in Figure 11 shows the final CAD model, and the bottom image showsin
Figure
11.
The
top
image
in
Figure
11
shows
the
final
CAD
model,
andthebottom
image
shows
image
in
Figure
11
shows
the
final
CAD
model,
andthebottom
image
shows
the
real
snake
robot.
the real snake robot.
the real snake robot.

Figure9.9.Kinematic
snake
robot
modules.
FigureKinematic configuration
configuration ofofsnake
robot
modules.

Symmetry 2020, 12, x FOR PEER REVIEW
Symmetry 2020, 12, x FOR PEER REVIEW

Symmetry 2020, 12, 939

Figure 9. Kinematic configuration of snake robot modules.

8 of 26
8 of 26

8 of 25

Figure 9. Kinematic configuration of snake robot modules.

Figure10. Inside
Inside of
snake
robot
module.
Figureof snake
snakerobot
robotmodule.
module.
Figure 10.
10. Inside ofAllcomponents
for snakerobot modules
wereprintedby
3D
printerfromABSmaterial.
EachAll
components
were printed
printedby
byaaa3D
3Dprinter
printerfrom
fromABSmaterial.
EachAll
componentsfor
forsnake
snakerobot
robot modules
modules wereABSmaterial.
Eachmoduleweighed
271
g
and
had
dimensions
of
130
×
80
×
47
mm.
moduleweighedof 130130 ××80
80××4747mm.
mm.
moduleweighed271
271 ggandandhadhaddimensions
dimensions of

Figure 11. CAD model (top) and experimental snake robot (bottom).
Figure11. CAD model (top) and experimental snake robot (bottom).
Figure 11. CAD model (top) and experimental snake robot (bottom).

ElectricalandandControl
ControlSystem
SystemofofSnake
Snake Robot
Robot
5.2.5.2.
Electrical5.2. Electrical and Control System of Snake Robot
Eachmoduleof snakethe snakerobot
had a separate
circuitboard
on anmicrocontroller
ATmega16
Eachmoduleof therobot hada separate
circuit board
based
on anbased
ATmega16
microcontroller
workingatfrequency
𝑓
16
MHz.
From
Figure
12
it
can
be
seen
that
the
robot
Each
module
of
the
snake
robot
had
a
separate
circuit
board
based
on
an
ATmega16
working at frequency fosc = 16 MHz. From Figure 12 it can be seen that the robot modules communicatedmodules
communicated
with
each
other
by
RS422.
The
main
controlsystemran
in
MATLAB
R2015b,microcontroller
workingatfrequency
𝑓
16
MHz.
From
Figure
12
it
can
be
seen
that
the
robot
with each other by RS422. The main control system ran in MATLAB R2015b, which communicatedwhich
communicated
with
the
snake
robot.
Microcontrollers
processed
received
signalswithmodules
communicatedwith each other
by RS422.
The maincontrolsystemran revolutionin MATLAB
R2015b,withthe snakerobot. Microcontrollers
processed
received
signalswithrequiredandlinearrequired
revolution andlinearposition,andsubsequently
controlled
the servomotor
positions.
which
communicated
with
the
snake
robot.
Microcontrollers
processed
received
signalswith
position, and subsequently controlled the servomotor positions.
required revolution and linear position, and subsequently controlled the servomotor positions.

Symmetry 2020, 12, 939
Symmetry 2020, 12, x FOR PEER REVIEW

9 of 25

9 of 26

Figure 12. Snake robot module data flow.
Figure 12. Snake robot module data flow.

The snake robot had on its head module an optical distance sensor, with the opportunity to add a
The snake robot had on its head module an optical distance sensor, with the opportunity to add
camera. The robot was supplied by external voltage.
a camera. The robot was supplied by external voltage.
6. Experiments with Snake Robot in a Pipe
6. Experiments with Snake Robot in a Pipe
This section describes three types of experiments. The purpose of the first experiment wasThis section describes three types of experiments. The purpose of the first experiment was toto verify the designed pattern locomotion. The second experiment investigated the influence ofverify the designed pattern locomotion. The second experiment investigated the influence of different
different kinds of pipe surfaces on locomotion. The last experiment analyzed the anchoring ofkinds of pipe surfaces on locomotion. The last experiment analyzed the anchoring of modules
modules between pipe walls by means of three symmetrical curves applied to the anchored modules.
between pipe walls by means of three symmetrical curves applied to the anchored modules. The
The
analysiswasperformed
the point
of of
view
of stabilityof therobot onmovingon ansurface
inclined
analysiswasperformed
fromfrom
the pointof viewstabilityof the robot
movingan inclined
surfaceduringlocomotion.
during locomotion.
6.1.6.1.
Locomotion
LocomotionPattern
PatternininaaPipe
Pipe
The
first
experimentintroducedininprevious
previoussections.
sections.
The
snake
The
first
experimenttested
testedthe
thelocomotion
locomotion patternpattern introduced
The
snake
robot
was
tested
in
a
pipe
with
a
rectangularcross-section
withdiameter
d
=
0.12
m.
The
lengthrobot was tested in a pipe with a rectangular cross‐section with diameter 𝑑 0.12 𝑚. The length of of
each
snake
robot
modulewasL=m andthe thenumberof modules
waswas
N =𝑁8. The
robot
used
eight
each
snake
robot
modulewas
𝐿 0.130.13
𝑚 and
numberof modules
8. Therobot
used
modules
for locomotion,
performing
revolute
as wellasaswelllinearmotion.
The maincontrolsystemran ineight modules
for locomotion,
performing
revolute
as linear
motion.
The maincontrolsystem
MATLAB
R2015b. R2015b.
ran in MATLAB
One
locomotion
Consideringthe
thelocomotion
locomotionpatternpattern
introducedOne
locomotioncycle
cycleisisshown
showninin Figure
Figure 13. Considering
introduced
in in
FigureFigureshowsthethesymmetricalsymmetrical behavior
behavior as well.
a locomotion
cycle
Figure3, 3,Figure1313shows
well. The
Thekeykeypoint
pointduring
duringa locomotion
cycle
was
to ensuresufficientpressure
betweenthe anchoredmodulestheofwallsof theBy
wasto ensuresufficientpressure
betweenthe anchoredmodulesand theandwalls
the pipe.
By pipe.
comparingcomparing
simulationand experimental
results,
we canthat realdistancetraveledwas
distancelower
thansimulation
andexperimental
results, we cansee thatrealseetraveled
lowerwasthantheoreticaltheoreticaltraveled
distance(see
Figure
14).
As
can
be
seen
from
Figure
14,
due
to
the
linearmotiontraveled distance (see Figure 14). As can be seen from Figure 14, due to the linear motion of theof the servomotors,
robot performeda small backward
motion (shown
red).
The wasreasonwas
servomotors,
the robotthe
performed
a small backward
motion (shownin red). inThe
reasonimperfect
imperfect
anchoringofmodules
betweenthepipewalls.
Other
reasonsforthevariation
intraveledanchoring of modules between the pipe walls. Other reasons for the variation in traveled distance weredistance
wereimpreciselinear servo
protrusion,
idealizingof pipesnake
robot, and among
pipe dimensions,Symmetry
2020,servo
12, x FOR
PEER REVIEW
10 of 26
impreciselinearprotrusion,
idealizingof snakerobot, anddimensions,others.
among others.

Figureinaapipe.
pipe.
Figure13.
13.Locomotion
Locomotion in

Because of the high gear ratio in the linear servomotor, the slowest part of locomotion was linear
motion, which took around 4–5 s. In general, snake robots are not very fast; their superiority andexcellence are in other properties. Nevertheless, the experiments showed the ability of a snake robot
to perform locomotion in a narrow space. The proposed mathematical background also provided anSymmetry 2020, 12, 939

10 of 25

Figure 13. Locomotion in a pipe.

BecauseBecauseofofthe
thehighhighgeargearratio
ratioininthe
thelinearlinearservomotor,
servomotor,thetheslowest
slowestpart
partofoflocomotion
locomotionwas
waslinearlinearmotion,which
tookaround4–5s.
In
general,snake
robots
are
not
very
fast;
their
superiority
motion, which took around 4–5 s. In general, snake robots are not very fast; their superiorityand
andexcellence
Nevertheless,thetheexperiments
showedthetheabilityof aofsnake
robot
toexcellenceare
areininother
otherproperties.
properties.
Nevertheless,experiments
showedabilitya snakerobot
perform
locomotioninanarrow
space.
The
proposedmathematicalbackgroundalso
providedanto perform locomotion in a narrow space. The proposed mathematical background also provided an
approximate
approximateprediction
predictionofoflocomotion.
locomotion.

FigureFigure14.
14.Traveled
Traveleddistancedistanceofofhead
headmodule
moduleduring
duringconcertina
concertinalocomotion.
locomotion.

6.2. Locomotion on Different Surfaces
6.2. Locomotion on Different Surfaces
Let us consider a real application of a snake robot moving in a pipe. For a real application,
Let us consider a real application of a snake robot moving in a pipe. For a real application, we
we should also consider that the surface of the pipe would not be ideal. In a real application, the surface
should also consider that the surface of the pipe would not be ideal. In a real application, the surface
could be dirty and contain different kinds of sediment with dry or viscous properties, and the roughness
could be dirty and contain different kinds of sediment with dry or viscous properties, and theof the surface could affect locomotion [22]. In many technical applications, high-precision motion
roughness of the surface could affect locomotion [Error! Reference source not found.]. In many
control is required. However, their performance is limited by various nonlinear factors such as friction.
technical applications, high‐precision motion control is required. However, their performance is
For this reason, friction needs to be compensated many times [23].
limited by various nonlinear factors such as friction. For this reason, friction needs to be compensated
The following section discusses basic friction models.
many times [Error! Reference source not found.].
following
section discusses basic friction models.
6.2.1. The
Friction
ModelsOneof theModels
basic friction models is called stiction. Stiction occurs in the case of zero mass velocity.
6.2.1.
Friction
The force required to overcome static friction is called breakaway force, which can be set according to
One of the basic friction models is called stiction. Stiction occurs in the case of zero mass velocity.
Equation (4):The force required to overcome static(friction is called breakaway force, which can be set according
Fe , v = 0 ∧ |Fe < Fs |Ff =(4)
to Equation (4):Fs sgn(Fe ), v = 0 ∧ |Fe | ≥ Fs
where Fs is static friction force and Fe is external force affecting the mass [24]. Once Fe overcomes Fs ,the mass starts to move. From the point of view of numerical simulation, one of the main disadvantagesof this friction model is its discontinuity at zero velocity of mass. This problem is usually overcome bysome approximation, for example, by arctan function with suitable slope. On the other hand, the slopeof arctan function can cause failure during numerical simulation due to very short integration time.
The next basic model is Coulomb friction mode. This model depends on mass velocity direction(not magnitude) sgn(v). Considering the same materials of the surface, Coulomb friction coefficientµc is usually lower than static friction coefficient µs . Coulomb friction force, also known as kinetic
friction, can be expressed by Equation (5) [25]:Fc = µc FN sgn(v)(5)

with suitable slope. On the other hand, the slope of arctan function can cause failure during numericalsimulation due to very short integration time.
The next basic model is Coulomb friction mode. This model depends on mass velocity direction(not magnitude) 𝑠𝑔𝑛 𝑣 . Considering the same materials of the surface, Coulomb friction coefficient
𝜇 is usually lower than static friction coefficient 𝜇 . Coulomb friction force, also known as kinetic
Symmetry 2020, 12, 939
11 of 25
friction, can be expressed by Equation (5) [Error! Reference source not found.]:𝐹
𝜇 𝐹 𝑠𝑔𝑛 𝑣
where FN is a normal load between two contact surfaces. Function sgn(v) can be stated aswhere 𝐹 is a normal load between two contactsurfaces. Function 𝑠𝑔𝑛 𝑣 can be stated as
+1 v > 0


𝑣 0
 1
(
)
sgn
v
=
0
v = 𝑣0 0


𝑠𝑔𝑛 𝑣
0

 −1 v < 0.
1
𝑣 0.

(5)

(6)
(6)

Thenext
next model,
model,thethe so-calledso‐calledviscousviscousfriction
frictionmodel,
model,occurs
occursin
inthe
thepresence
presenceof
ofoil.
oil. OilOil oror lubricant
lubricantThedecreasesthefrictioncoefficient
betweentwo
surfaces.
The
viscousfrictionmodel
isdescribed
bydecreases the friction coefficient between two surfaces. The viscous friction model is described by
Equation(7)[Error!
Referencesourcenotfound.]:
Equation (7) [26]:F𝐹v = µ𝜇v F𝐹N 𝑣
v

(7)
(7)

where
where µ𝜇v isisthe
theviscousviscousfriction
frictioncoefficient.
coefficient. This
This kind
kind ofof modelmodel isis different
different fromfrom thethe two
two previous
previous
modelsbecauseofitslinearity.
models because of its linearity.
The
model
is theStribeck
frictionmodel.
This model
describes
surfacessurfaces
with the presence
Thelastlastbasic
basic
model
is theStribeck
frictionmodel.
This model
describes
with theofliquid
or
solid
oil.
Stribeck
frictioncanbeexpressedbyEquation(8):
presence of liquid or solid oil. Stribeck friction can be expressed by Equation (8):v k
F f = Fc + (Fs − Fc )e−| vs | sgn(v) + µv v
𝐹
𝐹
𝐹 𝐹 𝑒
𝑠𝑔𝑛 𝑣
𝜇 𝑣

(8)
(8)

where v is sliding speed, vs is Stribeck velocity, and k is an exponent. All mentioned friction models arewhere 𝑣 is sliding speed, 𝑣 is Stribeck velocity, and k is an exponent. All mentioned friction models
shown in Figure 15.
are shown in Figure 15.

Figure 15. Friction models: (a) Coulomb (without static friction), (b) Coulomb + viscous friction
Figure 15.
Frictionmodels:(a) Coulomb(without
staticfriction),(b) Coulomb + viscous friction(without
staticfriction),(c) stiction+ Coulomb+ viscous
friction,(d) Stribeck.
(without static friction), (c) stiction + Coulomb + viscous friction, (d) Stribeck.

6.2.2. Testing of Locomotion on Different Kinds of Surfaces
In this section, snake robot locomotion on three kinds of surfaces is investigated: a clean surface,and surfaces with oil and lubricant.
Only four snake robot modules were used for the experiment, following our previous analysis [16].
For measurements, we used a LARM MSL50 incremental linear sensor with measuring step of 5 µm.
During the experiments, only the head module moved, while the other modules were static. The resultsof the first surface measurement are shown in Figure 16.

6.2.2.
Testingof12,
Locomotion
on Different Kinds of Surfaces
Symmetry
2020,
x FOR PEER REVIEW
12 of 26
In this section, snake robot locomotion on three kinds of surfaces is investigated: a clean surface,In this section, snake robot locomotion on three kinds of surfaces is investigated: a clean surface,and6.2.2.
surfaceswithoil and lubricant.
Testingof Locomotion
on Different Kinds of Surfaces
and surfaces with oil and lubricant.
Only four snake robot modules were used for the experiment, following our previous analysis
In fourthis section,
snake modules
robot locomotion
on threekinds
of surfacesfollowing
is investigated:a clean surface,
Onlysnake robot
were used
for theexperiment,
our previous
analysis[Error! Reference source not found.]. For measurements, we used a LARM MSL50 incremental linear
andsurfaces
with
oil
and
lubricant.
[Error!
Referencesource not found.]. For measurements, we used a LARM MSL50 incremental linear
Symmetry
2020,
12,
939
12
of 25
sensor withmeasuring
step of 5 μm. wereDuringthe experiments,
only the headour
modulemoved,
while
Onlymeasuring
four snake step
robotofmodules
usedthe experiment,
previousanalysissensor with
5 μm. During
theforexperiments,
onlyfollowing
the head module
moved,
while
the[16].
othermodules
were static.
The results
the first
surface measurement
aremeasuring
shown in step
FigureFormeasurements,
we usedLARM ofincremental
linear sensor withof 516.
the othermodules
were static.
The aresultsofMSL50
the first
surface measurement
are shownin Figure16.
μm. During the experiments, only the head module moved, while the other modules were static. The
results of the first surface measurement are shown in Figure 16.

Figure 16.
16. CleanClean
surfaceof pipe.
pipe.
Cleansurface
surfaceof
FigureFigure 16. Clean surface of pipe.

Figure 16
16 represents
represents measured
measured friction
friction force
based onlinear module
module
The graphgraph onon thethe leftleft inin Figureforce basedon linear
The
The
graph
on
the
left
in
Figure16
represents
measured
friction
force
based
on
linearmodule
The
graph
on
the
right
representsasampleofamovingmodulebycertainlinearvelocity.
velocity.
The
graph
on
the
right
representsasampleofamovingmodulebycertainlinearvelocity.
velocity. The graph on the right represents a sample of a moving module by certain linear velocity.
velocity. The
graphon therepresents
a sampleof a moving
modulebycan
certainlinear velocity.
Considering
these
results
andright
basic
friction models
modelsshown
in Figure
Figure15, wewe
recognize
Coulomb
Considering
these
results
andbasic
friction
shown
in
15,
can recognize
aa Coulomb
Considering
these
results
andbasic
frictionmodelsshown
inFigure15,
wecanrecognizeaCoulomb
Becausethe
theresults
resultsin
inbothbothdirections
directions ofof motion
motion were
were almost
almost thethe same,same, thethe final
friction model.
model. Because
Becausetheresultsin
both
directionsofmotionwerealmost
the
same,
the
final
frictioninboth directionsof motion were almost the same, the final
friction model. Because the results
hadasymmetricalpattern.
measured
frictioncharacteristic
measured friction characteristic had a symmetrical pattern.
measured friction characteristic had a symmetrical pattern.
experiment was
was performed
performed on
on a surface
surface withoil. TheThe results
results are
Next, thethe experimentexperimentwasperformedonwith a layer
layer ofof oil.
The
resultsare shownshown inin
Next,
Next, theexperiment was
performed onaa surface
surface with aa layerof oil.
The results
are shownin
Figure17.
Figure 17.
Figure 17.

Figure17.
Surfacewithoil.
oil.
Figure17.
Surfacewithoil.
Figure17.
17.Surface
Surface with
with
Figureoil.

Withincreasing
snake
robot
modulevelocity,
friction
also also
increased,
indicating
the viscous
frictionWith
increasing
snake
robot
modulevelocity,
friction
indicating
thetheviscousWith
increasing
snake
robot
modulevelocity,
friction
alsoincreased,
increased,
indicating
theviscousWith
increasing
snake
robot
modulevelocity,
friction
also
increased,
indicating
viscous
model+
Coulomb
frictionmodel.
The
resultantfrictioncharacteristic
wasroughlysymmetricalagain.
friction
model+Coulomb
Coulombfriction
friction model.
model. The
The resultant
resultantroughlyfrictionmodel
frictionmodel.
The
resultant friction
friction characteristic
characteristicwas
wasroughlyfrictionmodel++ Coulomb
frictioncharacteristic
wasroughly
For
the
next
experiment,
there
was
a
layer
of
lubricant
on
the
pipe
surface.
The
results
are
shown
in
symmetricalagain.
For
thenextexperiment,
experiment,there
there waswas aaa layerthepipe
surface.
TheThesymmetricalagain.
For
thenextexperiment,
there
was
layerofoflubricant
lubricantonon
on
the
pipe
surface.
Thesymmetricalagain.
For
the
next
layer
of
lubricantthepipe
surface.
Figure18.
results
shownin
Figureresults
areareshown
in Figure
Figure18.18.
results
are
shown
in
18.

Figure 18. Surface with lubricant.

Figure 18.
18. Surface
Surface with
with lubricant.
lubricant.
FigureFigure18. Surface
with lubricant.

As before, the resultant friction characteristic was roughly symmetrical. We can conclude theseexperiments by stating that the resultant friction characteristics of all measurements offered an almost
symmetrical shape.
Figure 19 presents the combination of all measurements of friction.

Symmetry
Symmetry 2020,
2020, 12,
12, xx FOR
FOR PEER
PEER REVIEWREVIEW

13
13 of
of 26
26

As
As before,
before, thethe resultant
resultant friction
friction characteristic
characteristic was
was roughlyroughly symmetrical.
symmetrical. We
We cancan conclude
conclude these
theseexperimentsbystatingthattheresultantfrictioncharacteristics
ofallmeasurements
offeredexperiments by stating that the resultant friction characteristics of all measurements offered anan almostalmostSymmetry 2020, 12, 939
13 of 25
symmetricalsymmetrical shape.
shape.
FigureFigure 19
19 presents
presents thethe combination
combination ofof allall measurements
measurements ofof friction.
friction.

Figure19.
Allmeasured
Figure19.
19. All
All measured
measured surfaces.
surfaces.
Figuresurfaces.

Measured
frictionforceswerealmost
the
for
both
of
The
Measured
frictionforceswerealmostthe samefor bothof module
motion.motion.
The following
Measured
frictionforceswerealmostthe samesame
fordirections
both directions
directionsof module
modulemotion.
Thefollowing
experimentsshowthetraveled
distanceofone
modulewiththedifferentkindsofsurfaces.
experiments
show the traveled
distanceof onemodulewiththe differentof surfaces.
results
following experiments
show thetraveled
distanceof onemodulewith thekinds
different
kinds of The
surfaces.
resultsare20.
It
obviousthattheresults
were
with
only
The
resultsare shown
shown
inisFigure
Figure20. that
It is
is theobviousthatthe very
results
were very
very
similar,withdeviations.
only small
small
areTheshown
in Figure
20. Itin
obviousresultsweresimilar,withsimilar,
only small
deviations.
deviations.

Figure20. Traveled
distance ofsnake robot
module ondifferent
different kindsofofsurfaces.
surfaces.
FigureFigure 20.
20. Traveled
Traveled distancedistance ofof snake
snake robot
robot module
module onon different kinds
kinds of surfaces.

Figure21 shows
the resultsof measured
electrical current
current consumptionduring
during linear
motion of
FigureFigure 21
21 shows
shows thethe results
results ofof measured
measured electricalelectrical current consumption
consumption during linear
linear motion
motion ofofthethehead
module.
On
the
basis
of
these
results,
there
were
no
significant
differences
betweendry, oil,the headhead module.
module. On
On thethe basis
basis ofof these
these results,
results, there
there were
were nono significant
significant differences
differences between
between dry,
dry, oil,oil,andlubricant surfaces.
Figure 22shows thesnake robot
robot moduleduring
during itsmotion
motion onthe
the lubricant
andand lubricant
lubricant surfaces.
surfaces. Figure
Figure 22
22 shows
shows thethe snakesnake robot module
module during its
its motion on
on the lubricant
lubricantsurface.
Acertain
layer
of
the
lubricantcausedfasterlinearmotionincomparison
withdrysurface.
Acertain
layer
of
the
lubricantcausedlinearmotionincomparison
withdrySymmetry
12, x layer
FOR PEER
REVIEW
14 friction.
of 26
surface.
A2020,
certainof thelubricant caused faster linear motion in comparison with dry friction.
friction.

FigureElectricalcurrent
currentconsumption
consumption during
of of
head
module.
Figure21.21.
Electricalduringmotion
motionhead
module.

Symmetry 2020, 12, 939

14 of 25

Figure 21. Electrical current consumption during motion of head module.

Figure 22. Layer of lubricant on the surface.
Figure22. Layer of lubricant on the surface.

The results of these experiments can be summarized as follows. With a dry surface under the
The results of these experiments can be summarized as follows. With a dry surface under thesnake robot module, the resultant friction was according to the Coulomb friction model. A layer ofsnake robot module, the resultant friction was according to the Coulomb friction model. A layer ofoil or lubricant resulted in the Coulomb friction + viscous friction model or, rather, Stribeck friction
oil or lubricant resulted in the Coulomb friction + viscous friction model or, rather, Stribeck friction
model. With respect to the measurement of friction in both directions, the results had symmetricalmodel. With respect to the measurement of friction in both directions, the results had symmetricalcharacteristics. From the point of view of time duration of linear motion, the slowest motion occurredcharacteristics. From the point of view of time duration of linear motion, the slowest motion occurred
for the dry surface and the fastest motion occurred for the lubricant surface. With regard to electricalfor the dry surface and the fastest motion occurred for the lubricant surface. With regard to electricalcurrent consumption, the three cases were almost the same.
current consumption, the three cases were almost the same.
Symmetry
2020,
x FOR PEERrectilinear
REVIEW
15 of 26
Next,
we12,performed
locomotion in a pipe (see Figure 23).
Next, we performed rectilinear locomotion in a pipe (see Figure 23).

FigureFigure 23.
23. Rectilinear
Rectilinear locomotion
locomotion ofof snake
snake robot.
robot.

During rectilinear locomotion, only one module moved, while others served as support mass.
During rectilinear locomotion, only one module moved, while others served as support mass.
Experimental results from measuring the motion of the head module on different kinds of surfaces
Experimental results from measuring the motion of the head module on different kinds of surfaces
were presented in a previous section. The following results present rectilinear locomotion during threewere presented in a previous section. The following results present rectilinear locomotion during
cycles. The robot moved on dry surface, surface with oil, and surface with a layer of lubricant.
three cycles. The robot moved on dry surface, surface with oil, and surface with a layer of lubricant.
Figure 24 presents the traveled distance of the head module during three locomotion cycles.
Figure 24 presents the traveled distance of the head module during three locomotion cycles.
During locomotion linear protrusion li = 40 mm was used for all linear servomotors. The experimental
During locomotion linear protrusion 𝑙
40 𝑚𝑚 was used for all linear servomotors. The
results were quite interesting. With dry friction, the snake robot traveled the maximum possible
experimental results were quite interesting. With dry friction, the snake robot traveled the maximum
distance. However, with oil and lubricant, we can see some decrease in traveled distance. In both cases,
possible distance. However, with oil and lubricant, we can see some decrease in traveled distance. In
after the head module traveled its path, during the subsequent motion of other modules, the snake
both cases, after the head module traveled its path, during the subsequent motion of other modules,
robot started to slide backward under the influence of forces between the modules. This resulted in
the snake robot started to slide backward under the influence of forces between the modules. Thisless traveled distance. That is, real applications need to consider the influence of the environment,
resulted in less traveled distance. That is, real applications need to consider the influence of thewhich, as we can see, could have an impact on locomotion properties.
environment, which, as we can see, could have an impact on locomotion properties.

experimental results were quite interesting. With dry friction, the snake robot traveled the maximum
possible distance. However, with oil and lubricant, we can see some decrease in traveled distance. In
both cases, after the head module traveled its path, during the subsequent motion of other modules,
the snake robot started to slide backward under the influence of forces between the modules. This
resulted in less traveled distance. That is, real applications need to consider the influence of theSymmetry 2020, 12, 939
15 of 25
environment, which, as we can see, could have an impact on locomotion properties.

FigureFigure24.
24.Rectilinear
Rectilinearlocomotion
locomotionon
ondifferent
differentsurfaces.
surfaces.

6.3. Experimental Analysis of Stability
6.3. Experimental Analysis of Stability
Section 3 presented the locomotion pattern of the snake robot moving in a pipe. The locomotion
Sectionpresentedcertain
the locomotion
pattern ofthe
thewalls
snake
movinginothera pipe.
The locomotion
was basedon 3anchoring
modules between
ofrobot
the pipe,
while
modules
moved
was
based
on
anchoringcertainmodules
between
the
walls
of
the
pipe,
while
other
modules
moved
freely in the direction of motion. The introduced locomotion pattern considered anchored modules
tofreelyinthedirectionofmotion.
The
introduced
locomotion
pattern
consideredanchoredmoduleshave a triangular curved shape (see Figure 3). Of course, anchoring of modules between pipe walls
to havetriangularcurvedshape
Figure assumed3). Of course,
anchoring
modulespipe
could
be ausedin several
ways.
Our(seeapproachdifferent
curvesofthat
couldbetween
be formed
bywalls
the
could
be
used
in
severalways.
Our
approachassumeddifferent
curves
that
could
be
formedbythebody of the snake robot. It was clear that the more modules the snake robot had, the more precisely
the body could form the chosen curve.
The aim of this section is to investigate how the different curves formed by the discrete body ofthe snake robot affected concertina locomotion stability. In other words, the snake robot used different
kinds of anchoring during concertina locomotion, and we investigated if these ways of anchoring weresuitable for locomotion. That is, the following section discusses three symmetrical curves that wereapplied for anchoring of modules during locomotion and were experimentaly analyzed by the DIC
method. The snake robot moved through the rectangular and circular pipe while the slope of the pipe
was changed. Our focus was on analyzing whether these anchoring curves were able to hold the snake
robot in the pipe.
6.3.1. Mathematical Background of Symmetrical Curves
The concept of symmetrical curves relates in our case to the rear modules of the robot, whichwere used to anchor it between the pipe walls. We chose the following curves due to their roughapplicability to our snake robot having eight modules [27].
Triangular Curve
The triangular curve can be described by Equation (9):σ(t) =



8
1
1
(
)
(
)
(
)
sin
ωt
−
sin
3ωt
+
sin
5ωt
−
.
.
.
9
25
π2

(9)

where ω is angular frequency. Figure 25 shows three triangular curves with different frequencies.
The amplitude of these curves was based on the pipe diameter D. As can be seen, decreasing frequency
caused expansion of the curve. The magnitude of frequency depended on the length of the snake robot
module. The longer the module, the lower the frequency of the curve.

𝜎 𝑡

𝜋

𝑠𝑖𝑛 𝜔𝑡9

𝑠𝑖𝑛 3𝜔𝑡

25

𝑠𝑖𝑛 5𝜔𝑡...

(9)

where ω is angular frequency. Figure 25 shows three triangular curves with different frequencies.
The amplitude of these curves was based on the pipe diameter D. As can be seen, decreasing
frequency
causedof25theSymmetry
2020, 12,939 expansion of the curve. The magnitude of frequency depended on the length
16 ofsnake robot module. The longer the module, the lower the frequency of the curve.

Figure 25. Triangular curves.
Figure 25. Triangular curves.

Cycloid
Curve
CycloidCurve
The
cycloidcurve
candescribed
Equation(10):
The cycloid
curve
can
bebe
described
byby
Equation(10):
r−y
𝑟 𝑎𝑟𝑐𝑐𝑜𝑠σ( y𝜎) 𝑦= r arccos−
r

q(2r2𝑟− y)𝑦
y𝑦

(10)
(10)

where
is theradiusof thecurve.
Figureshows
cycloidcurve
describedbybyEquation
Equation(10).
(10).
where
r is𝑟 theradiusof thecurve.
Figure2626
showsthethecycloidcurve
described
From
Equation(10),
obviousthat
thiscurve
curvecould
couldbebeinfluenced
influencedbybyitsitsradius
radiusr. 𝑟.
From
this
reason,From
Equation
(10),
it it
is is
obvious
that
this
From
this
reason,Symmetry 2020, 12, x FOR PEER REVIEW
17 of 26
radius𝑟 wasaccordingto pipediameterradiusr was
set setaccordingto pipediameter
D. D.

Figure26.26.Cycloid
FigureCycloidcurve.
curve.

However, there was a way to shorten or lengthen the cycloid curve by using the following
However, there was a way to shorten or lengthen the cycloid curve by using the following
parametric expression:
parametric expression:x = rt − d sin(t)
(11)
(t) 𝑡
y𝑥= r𝑟𝑡− d cos𝑑 𝑠𝑖𝑛(11)
where d denotes shortening or expanding of thecurve.
𝑦 cycloid𝑟 𝑑 𝑐𝑜𝑠𝑡 If d < r, the curve was shortened, and if
d > r, the curve was expanded.
where 𝑑 denotes shortening or expanding of the cycloid curve. If 𝑑 𝑟, the curve was shortened,
Gaussian
and if 𝑑 curve𝑟, the curve was expanded.
Gaussiancurve curve is expressed by Equation (12):The Gaussian
The Gaussian curve is expressed by Equation (12):−(x−b)2σ(x) = ae 2c2
𝜎 𝑥
𝑎𝑒

(12)
(12)

where 𝑎, 𝑏, 𝑐 𝜖 ℝ; 𝑎 represents pipe diameter D, 𝑏 represents the coordinate system origin shift,and 𝑐 represents the change of angle between two robot modules in contact with the wall of the pipe.
In other words, parameter 𝑐 shows how “sharp” the Gaussian curve will be (see Figure 27).

where 𝑑 denotes shortening or expanding of the cycloid curve. If 𝑑
and if 𝑑 𝑟, the curve was expanded.
Gaussian curve
The Gaussian curve is expressed by Equation (12):𝑟, the curve was shortened,

Symmetry 2020, 12, 939

17 of 25

𝜎 𝑥

𝑎𝑒

(12)

where a, b, c  R; a represents pipe diameter D, b represents the coordinate system origin shift, and cwhere 𝑎,the𝑏, 𝑐change
𝜖 ℝ; 𝑎of represents
pipetwodiameterD, 𝑏 represents
coordinate
originshift,
represents
angle between
robot modules
in contactthewiththe wall ofsystem
the pipe.
In otherand 𝑐parameter
representsc thechangeangle between
two robot
modulesin contact
words,
shows
howof“sharp”
the Gaussian
curve
will be (seeFigurewith27). the wall of the pipe.
In other words, parameter 𝑐 shows how “sharp” the Gaussian curve will be (see Figure 27).

FigureGaussiancurves.
curves.
Figure27.27.Gaussian

Thesebody (see(see Figure
Figure28).
28).InIngeneral,
general,
Thesesymmetricalsymmetricalcurves
curveswere
wereapplied
applied toto the snake robot bodybybyconsidering
considering
the
fact
that
a hyper-redundantmechanism
consistedmany
same
modules,
the
fact
that
a hyper‐redundantmechanism
consisted
of of
many
of of
thethesame
modules,
we
we
could
applyrobot
robotmodules
modulestotothese
thesecurves.
curves. However,
However, we
we used aa snake
could
apply
snakerobot
robotconsisting
consistingofofeight
eight
modulesfourforforanchoring),
anchoring),and
andhence
henceititwas
wasnotnotpossible
possibletotoprecisely
preciselycopy
copythese
thesecurves,
curves,modules(with(withfourwhich
wasnotnoteven
the goal.
We could
use onlyof these curves.
us consider
which
was
even
the goal.
We could
useroughonly approximations
rough approximationsof theseLet
curves.
Let usFigure28.
In
thisfigure,anchoredmodulesaredenoted
in
gray,
and
dashedverticallines
representtheconsider Figure 28. In this figure, anchored modules are denoted in gray, and dashed vertical lines
Symmetry
2020,
12,
x
FOR
PEER
REVIEW
18
of
26
axis
of symmetry
for ofdiscreteapplicationof symmetricalcurvesfor rear anchored
represent
the axis
symmetryfor discrete
applicationof symmetricalcurvesmodules.
for rear anchoredmodules.

Figure 28. Anchoring of snake robot rear by three symmetrical curves.
Figure 28. Anchoring of snake robot rear by three symmetrical curves.

6.3.2. Digital Image Correlation Method
6.3.2. Digital Image Correlation Method
Digital image correlation allowed us to perform measurements commonly using a single or stereo
Digitalimage
correlation
allowedwas
us toperform
commonlyusing
a singleorcamerasystem.
Whenthe measurement
donewith a measurements
single camera (seeFigure 29a),
it waspossiblestereocamerasystem.
Whenthemeasurement
was
done
with
a
singlecamera(see
Figure29a),
itwas
to perform only two-dimensional (2D) displacement/deformation analysis. However, it was necessary
possible
perform
only two‐dimensional(2D)object
displacement/deformation
However,wastoensuretothat
the surfaceof the investigatedwould be parallel toanalysis.
the camera
image itplanethat the surfaceof theUsing
investigatedobjectsystem
would beparallelto the camera
image
necessary
ensureduringthetowholemeasurement
process.
a correlation
withstereoscopicallyarrangedplaneduringthewholemeasurement
process.
Usingacorrelation
systemwithstereoscopicallycameras (Figure 29b), this condition was not valid and the results were in the form of three-dimensionalarranged cameras (Figure 29b),
this condition was not valid and the results were in the form of three‐displacement/deformation
fields.
dimensional displacement/deformation fields.

stereo camera system. When the measurement was done with a single camera (see Figure 29a), it waspossible to perform only two‐dimensional (2D) displacement/deformation analysis. However, it wasnecessary to ensure that the surface of the investigated object would be parallel to the camera image
plane during the whole measurement process. Using a correlation system with stereoscopically
Symmetry
2020,
12, 939 (Figure 29b), this condition was not valid and the results were in the form of18
of 25
arrangedcamerasthree‐
dimensional displacement/deformation fields.

(a)

(b)

FigureFigure 29. Scheme
Scheme ofof (a)(a) single‐camera
single-camera (2D)(2D) andand (b)(b) stereo
stereo camera (3D) correlation system.

We carriedmeasurements
witha commercial
Q-450
DantecDynamics
high-speed
carriedout
outallalldescribed
described
measurements
witha commercial
Q‐450
DantecDynamics
high‐digitaldigital
image image
correlation
system,system,
the technicalspecificationsof which
listedTablein1.TableThe system
speedcorrelation
the technicalspecificationsof arewhich
areinlisted
1. Theconsistsconsistsof two SpeedSense
9070 cameras
CMOS
sensors
resolution
systemof two SpeedSense
9070 withcameraswith
CMOSallowing
sensors maximal
allowingimage
maximalimageof 1 Mpx (1080× 800(1080px). Optimal
light
conditions
assuredby theuse ofspecialresolutionof 1 Mpx
× 800 px).
Optimallight wereconditions
wereassuredbya theuse high-powerof a special
halogen lampwith naturallightnaturaland frequency
offrequency
70 kHz. of 70 kHz.
high‐power
halogenlamp withlight and
Table 1.
Technical specifications
specifications ofof Q‐450
Q-450 Dantec
Dantec Dynamics
Dynamics high‐speed
high-speed cameras
cameras used
used forTable
1. Technicalexperimental
analysis.
experimental analysis.

Field
Field ofof viewview

2 to several2 m2Adjustablefrom several
Adjustablefrom several
mm2mm
to severalm

Typical size from 70 × 50 mm2 to 400 × 300 mm2

Typical size from 70 × 50 mm2 to 400 × 300 mm2

Measuring range

Displacement: ≈10–5 of field of view (100 mm field corresponds to 1 µm)

Exposure time

Adjustable from 300 ns

Resolution

Max. 1280 × 800 px up to sampling frequency of 3140 fps

Results form

3D contour of analyzed object, 3D displacement

Control electronics

Laptop with Windows 7 and Istra4D control software, 16-bit AD/DA converter(eight channels with voltage range from ±0.05 V to ±10 V)

Illumination

High-power halogen lamp with natural light

The principle of the Dantec Dynamics correlation system is based on the correlation of images
along small elements called facets. The standard size of facets is from 15 × 15 px up to 30 × 30 px. As ameasuring point, the center of the facet is considered.
To obtain the transform coordinates of each object surface point, the digital image correlation
system used a modern approach based on the pseudo-affine transform. If a0 . . . a7 denotes the transformparameters of potential translation, stretch, shear, and distortion of the facet, then using the algorithm,
the transform coordinates x∗ and y∗ can be expressed by Equation (13):x∗ (a0 , a1 , a2 , a3 , x, y) = a0 + a1 x + a2 y + a3 xy,y∗ (a4 , a5 , a6 , a7 , x, y) = a4 + a5 x + a6 y + a7 xy.

(13)

The unknown parameters a0 . . . a7 can be determined by minimizing the difference between thegray value intensity of the points on deformed image g(x∗ , y∗ ) and reference image f (x, y) via

To obtain the transform coordinates of each object surface point, the digital image correlation
system used a modern approach based on the pseudo‐affine transform. If 𝑎 … 𝑎 denotes thetransform parameters of potential translation, stretch, shear, and distortion of the facet, then using
the algorithm, the transform coordinates 𝑥 ∗ and 𝑦 ∗ can be expressed by Equation (13):𝑥 ∗ 𝑎 , 𝑎 , 𝑎 , 𝑎 , 𝑥, 𝑦𝑎𝑎 𝑥 𝑎 𝑦 𝑎 𝑥𝑦,
Symmetry 2020, 12, 939
19 of 25
(13)∗𝑦 𝑎 , 𝑎 , 𝑎 , 𝑎 , 𝑥, 𝑦𝑎𝑎 𝑥 𝑎 𝑦 𝑎 𝑥𝑦.
The unknown parameters 𝑎 … 𝑎 can be determined by minimizing the difference between theX∗ ∗gray value intensity of the points, 𝑦y), andimage 𝑓 𝑥, 𝑦 via(14)f (x, y) − (imagei + i g𝑔(x𝑥∗ (x,y∗ (x,referencey)))min on deformed
a0 ,...,a7 ,i0 ,i1

0

1

x,y

𝑖 𝑔 𝑥 ∗ 𝑥, 𝑦 , 𝑦 ∗ 𝑥, 𝑦𝑖𝑚𝑖𝑛 ∑ , 𝑓 𝑥, 𝑦(14)
, ,where i0 and i1 are the illumination parameters. If all object point coordinates are determined, then thewhere 𝑖 and 𝑖 are the illumination parameters. If all object point coordinates are determined, then3D coordinates of the surface are obtained.
the 3D coordinates of the surface are obtained.
Reference images with a resolution of 1280 × 448 px (density equal to 6 px/mm) captured by the
Reference images with a resolution of 1280 × 448 px (density equal to 6 px/mm) captured by the
Q-450 Dantec Dynamics 3D correlation system are depicted in Figure 30. The cameras were calibrated
Q‐450 Dantec Dynamics 3D correlation system are depicted in Figure 30. The cameras were calibrated
by a calibration target with 9 × 9 black and white fields with a field size of 8 × 8 mm. The facet sizeby a calibration target with 9 × 9 black and white fields with a field size of 8 × 8 mm. The facet sizeused for evaluating the measurement was 17 × 17 px and the facets overlaid themselves in 5 px.
used for evaluating the measurement was 17 × 17 px and the facets overlaid themselves in 5 px.
,...,

Figure30.
30. Typical
Typicalreference
referenceimages
imagescaptured
capturedby
byQ-450
Q‐450Dantec
DantecDynamics
Dynamicshigh-speed
high‐speeddigital
digitalimage
image
Figurecorrelation
correlationsystem.
system.

6.3.3.
6.3.3. Measurement
Measurementof
ofStability
Stabilityby
byDIC
DICMethod
MethodThe
Theintroduced
introducedsymmetricalsymmetricalcurves
curveswere
wereexperimentallyexperimentallyanalyzed.
analyzed. The
Thesnake
snakerobot
robotperformed
performed
forwardmotionbylinearjointsandsubsequently
thestabilityofanchoredmoduleswastestedforward motion by linear joints and subsequently the stability of anchored modules was testedusing
using
triangular,
cycloid,and Gaussian
Gaussian curves.
curves.For
For
experimental
analysis,theDIC
methodwasused.
triangular,
cycloid,andexperimental
analysis,theDIC
method
was
used.
The
Symmetry 2020, 12, x FOR PEER REVIEW
20 of 26
The
measuring
methodis shown
in Figuremeasuring
methodis shown
in Figure31. 31.

Figure 31. Measuring anchoring possibilities of snake robot during its motion.
Figure 31. Measuring anchoring possibilities of snake robot during its motion.

As Figure 31 shows, a measuring pattern was stuck to one static (anchored) snake robot module.
As Figureshows,thea measuring
patternwasstuck toone static(anchored)snake
robot module.
The aimwas to31
observedisplacementof thismodulewhile
free modules
moved
forward.
The aimto observe
theexperiment
displacementof thismoduleWe
while
freethemodules
moved forward.
The wasconditionsof thewere
as follows.
testedthree mentioned
symmetricalTheof theexperimentfollows.
Wemeasurements
tested the three
mentioned
symmetricalcurvesonconditions
surfaces withslopesof 0◦ , 10◦were, 20◦ ,asand30◦ . The
weredone for two
kinds ofof 0°, 10°, 20°,
measurements
wereFordone
formeasurement,
two kinds ofcurvesonwithsurfaceswith slopes
pipe,
one
a rectangularcross-section,
and andone 30°.
withThea circularcross-section.
eachpipe, one
withofa 4V
rectangularcross‐section,
andOfonewithwitha circularcross‐section.
Forsupplyvoltage
DC and 8V
DC was used.
course,a supply
voltage of 8V
DCeachformeasurement, supply voltage of 4V DC and 8V DC was used. Of course, with a supply voltage of 8V
DC for servomotors, the results would have better characteristics. The reason for using 4V DC wasthat lower electrical current consumption would be required during locomotion. This solution would
be suitable for real applications, such as long‐time inspection of pipes without an available external
voltage supply, whereas the robot would be dependent on batteries.

As Figure 31 shows, a measuring pattern was stuck to one static (anchored) snake robot module.
The aim was to observe the displacement of this module while free modules moved forward.
The conditions of the experiment were as follows. We tested the three mentioned symmetricalcurveson2020,surfaces
Symmetry
12, 939 with slopes of 0°, 10°, 20°, and 30°. The measurements were done for two kinds
20 ofof25
pipe, one with a rectangular cross‐section, and one with a circular cross‐section. For eachmeasurement, supply voltage of 4V DC and 8V DC was used. Of course, with a supply voltage of 8V
servomotors,
the results
bettercharacteristics.
The reason
forreason
using for4V using
DC wasDCfor servomotors,
the would
resultshave
would
have
better characteristics.
The
4VthatDClowerwas
electricalcurrent
consumption
wouldberequired
duringlocomotion.
Thissolutionwouldbesuitablethat lower electrical current consumption would be required during locomotion. This solution would
forsuitablereal applications,such as long-timeinspection
ofinspection
pipes without
an available
voltage
supply,befor real applications,such aslong‐timeof pipes
withoutexternalan available
external
whereas
the
robot
would
be
dependent
onbatteries.
voltage supply, whereas the robot would be dependent on batteries.
First,wewepresent
present thethe results
results ofinFirst,of thethe experimentexperimentwith
withpipes
pipeswithwithaarectangular
rectangularcross-section,
cross‐section,shown
shown
Figures
32–35.
inFigures
32–35.

(a)(b)◦

Figure32.
Figure 12,
32. Pipe
Pipewithwithrectangular
rectangularcross‐section
cross-sectionand
andslope
slope0°:0 :(a)(a)supply
supplyvoltage
voltage4V
4VDC,
DC,(b)(b)supply
supply 21 of 26
Symmetry
Symmetry2020,
2020, 12,xxFORFORPEERPEER REVIEWREVIEW
21 of 26
voltagevoltage8V
8VDC.
DC.

(a) (a)(b)(b)

Figure 33. Pipe with rectangular cross‐section and slope 10°: (a) supply voltage 4V DC, (b) supply◦ : (a)Figure33.33.Pipe
slope 1010°:(a)supply
supplyvoltage
voltage4V
4VDC,
DC,
supplyFigurePipewithwithrectangular
rectangularcross-section
cross-section andand slope(b)(b)
supplyvoltage 8V DC.
voltage
8V
DC.
voltage 8V DC.

(a)

(a)

(b)

(b)

◦ : (a) voltage
FigureFigure
34. Pipecross‐section
and slope(a) 20supplyDC, (b)34.withPiperectangularwith rectangularcross-section
and20°:
slope
supply 4V
voltage
4VsupplyDC, (b)voltage
supply8VDC.
voltage
DC.rectangular cross-section and slope 20°: (a) supply voltage 4V DC, (b) supply voltage
Figure 34.
Pipe8Vwith

8V DC.

34.
Pipe with rectangular cross‐section and slope 20°: (a) supply voltage 4V DC, (b) supply 21 of 25
SymmetryFigure
2020, 12,939
voltage 8V DC.

◦ : triangle
FigurePipewithrectangularcross-section
cross‐sectionand
andslope
slope 30
30°:
Figure35.35.
Pipewithrectangulartrianglefixation
fixationcurve,
curve,supply
supplyvoltage
voltage
8V
DC.
8V DC.

◦ , bothConsidering
theslope
slopeofof the surface
0°,0bothgraphsshowedsatisfactory
displacement.
The
Consideringthesurfacetotobebe
graphsshowedsatisfactory
displacement.
◦
snake
robot
would
be
able
to
perform
locomotion.
In
the
case
of
surface
slope
of
10°
and
supplyThe snake robot would be able to perform locomotion. In the case of surface slope of 10 and supply
able
voltageDC,applying
applying cycloid
cycloid andthethesnake
robot
slid down
and wasvoltageof of
4V4VDC,
andGaussian
Gaussiancurves,
curves,snake
robot
slid down
andnot
was
not
to to
perform
forwardlocomotion(see (seeFigure33a). 33a).
Only thetriangularcurve wassatisfactory.
From
able
perform
forwardlocomotion
FigureOnly
the triangularcurve
was satisfactory.
Figure33b,33b,
it wasthat theGaussiancurve failed.
resultstheof cycloid
Gaussian
From
Figureit obvious
was obvious
that
the Gaussian
curveBecause
failed. theBecauseresults andof cycloid
andcurves
using
4V
DC
were
unsatisfactory,
only
thetriangular
curve
with
4V
DC
was
measured.
In theGaussian curves using 4V DC were unsatisfactory, only the triangular curve with 4V DC was measured.
case of surface slope of 20° and◦supply voltage of 4V DC, the triangular curve also failed (see Figure
In thecase of surface slope of 20 and supply voltage of 4V DC, the triangular curve also failed (seeSymmetry
2020, 12, x FOR
PEERonly
REVIEW
22 of 2634a). Assuming
8V DC,
the triangular curve was stable (see Figure 34b). In the case of surface
Figure 34a). Assuming 8V DC, only the triangular curve was stable (see Figure 34b). In the case of
Symmetry
2020,
12,
x
FOR
PEER
REVIEW22
of 26
slope of 30° and ◦supply voltage of 8V DC, the triangular curve also failed (see Figure 35).
surfaceFrom
slopeallofmeasurements
30 and supply
voltageof rectangular
8V DC, the cross‐section,
triangular curve
also
failed(see
Figure 35).of
of
pipe
with
it
was
clear
that
for
anchoring
From all measurements of pipe with rectangular cross‐section, it was clear that for anchoring ofFromsnakeall measurements
of of
pipe
with
rectangularcross-section,
was
clear
for anchoringmeasurements
rectangular
it itwasclear
thatthatfor anchoringof of ofcertainrobot
modules,best
solution
was
use
curve
with
supplycertainFrom
snakeallrobot
modules,thethepipebestwithsolution
wasto
tocross‐section,useaatriangulartriangular
curve
with
supplyvoltage
voltageofcertainsnake
robot
modules,thebest
solution
was
to
use
a
triangular
curve
with
supplyvoltageofcertainsnake
robot
modules,thebest
solution
was
to
use
a
triangular
curve
with
supplyvoltage
of
8V
DC.
8V
8VDC.
DC.
8VThe
DC.
The
next
measurements
relate
to
pipes
with
a
circularcross-section.
The
results
areshown
next measurements relate to pipes with a circular cross‐section. The results are shown in inThenext measurements relate to pipes with a circular cross‐section. The results are shown in
Figures
36–39.
Figures
36–39.

Figures 36–39.

(a)
(a)

(b)
(b)

Figure36.36.
Pipewithcircularcross‐section
and
slope
0°:
voltage
4VDC,
DC,
(b)
supplyvoltage
FigurePipewithcircularcross‐section
andandslope
0°: (a)(a)supplyvoltage
4V(b)supplyvoltage
8V8V
Figure
36.
Pipe
with
circularcross-section
slope
0◦supply: (a) supply
voltage4VDC,(b) supply
voltage
DC.8V
DC.DC.

(a)
(a)

(b)
(b)

◦ :(a)
Figure37. Pipewithcircularcross‐sectionand
andslope
slope 10
10°:
voltage
4V4V
DC,DC,(b) supply
voltage
Figure37. Pipewithcircularcross-section(a)supply
supplyvoltage(b) supply
voltage
Figure 37. Pipe with circular cross‐section and slope 10°: (a) supply voltage 4V DC, (b) supply voltage
8V DC.
8V DC.
8V DC.

(a)(b)

Figure 37. Pipe with circular cross-section and slope 10°: (a) supply voltage 4V DC, (b) supply voltage
22 of 25
8V DC.

Symmetry 2020, 12, 939

(a)

(b)

Figure38. Pipe
circularcross-section
and andslopeslope
20°: 20(a)◦ :supply
voltage
4V DC,(b) supply
voltage
8V
Figure38. with
Pipe withcircularcross-section(a) supply
voltage
4V DC,(b) supply
voltage
Symmetry 2020, 12, x FOR PEER REVIEW
23 of 26
DC. 8V DC.

(a)

(b)

◦ , slope:Figure39. Pipewith withcircularcross-section
and andsupplyvoltage
8V DC
withwith(a) slope30°,30(b)
40°.40◦ .
Figure39. Pipe
circularcross-section
supplyvoltage
8V DC(a) slope(b) slope:◦ , both graphs showed stable results for locomotion
Considering
Considering thethe slopeslope ofof thethe surface
surfaceto
tobe
be00°,
both graphs showed stable results for locomotion◦(see(seeFigure
Figure36).
36).ByBy increasing
increasingthe
theslope
slopetoto10
10°andand usingusing aa supply
supply voltage
voltage ofof 4V
4V DC,
DC, only
only thethe triangulartriangularcurve
curve gave
gave stablestable results.
results. One
One difference
difference between
between pipes
pipes with
with circular
circular andand rectangular
rectangular cross-sections
cross-sections◦ slope and 8V DC. Although the Gaussian curve failed in the rectangularwas
was ininthe
thecasecaseof
ofthe
the10
10°
slope and 8V DC. Although the Gaussian curve failed in the rectangularpipe,
pipe,ininthe
thecircular
circularpipe
pipeititwas
wasaastablestablesolution
solution(see(seeFigure
Figure37b).
37b). Another
Another difference
difference waswasthethecasecaseofof◦
slope
slope 2020°andand 4V
4V DC.
DC. InIn thethecircular
circularpipe,
pipe,thethetriangulartriangular curve
curve waswasquitequitesuccessful,
successful, whereas
whereas inin thethe◦ slope and 8V DC, the cycloid curve was quite
rectangular
rectangularititfailed
failed(see(seeFigure
Figure38a).
38a). InIn thethe casecase ofof 20
20°
slope and 8V DC, the cycloid curve was quite◦ and 8V DC,
stablestableinincomparison
comparisonwith
withthetherectangular
rectangularpipe
pipe(see(seeFigure
Figure38b).
38b). Assuming
Assuming aa slope
slope of
of 30
30°
and
◦.
the
(Figure39a)andforforthisthisreasonwe wealsoalso
measured
for aforslope
of 40ofthetriangulartriangularcurve
curvewaswasstill
stillstablestable(Figure39a)andreasonmeasureda slope
However,in thisthe triangularcurvecurvealso failed(see Figure39b). 39b).
40°. However,incase,this case,the triangularalso failed(see FigureThewith symmetricalsymmetricalcurves
curvesused
usedforforanchoringanchoring
snake
robot
modules
between
The experiments with
snake
robot
modules
between
the
the
walls
a pipebe evaluatedas follows.
the perspectiveof stability,
thesolution
best solution
walls
of aofpipe
can canbe evaluatedas follows.
FromFrom
the perspectiveof stability,
the best
was towasuse a triangularcurve.
Thesolution
worst solution
the Gaussian
curve. Consideringofuse atotriangularcurve. Theworst
was the was
Gaussiancurve. Considering
the use ofthe
theuse
snake
the
snake
robot
in real applications
with the requirementof lowenergyitintensity,
it be
would
not be arobot
in realapplications
with the requirementof low energy
intensity,
would nota satisfactorysatisfactory
solution.
The robot
would
be able toperform locomotion
only on a horizontal
plane.
With
solution. The
robot would
be ableto perform
locomotion
only on a horizontal
plane. With
increased
increasedsurfaceslope, significantly
stability significantly
decreased.
Comparing
locomotion
in rectangularandsurface slope,stability
decreased.
Comparing
locomotion
in rectangularand circular
circularpipes,
the snakeshowedsignificantly
betterstability
circularpipe.
pipe. Whereas
Whereas in thepipes, thesnakerobot robot
showedsignificantly
betterstability
in inthethecirculartherectangular
rectangularpipe
pipesome
somecurves
curvesfailed,failed,in
inthe
thecircular
circularpipe
pipethey
theywere
werestill
stillstable.
stable.

7.
6. Conclusions
Conclusion
This
This paper
paperanalyzed
analyzed snake
snake robot
robotlocomotion
locomotionin
inaapipe.
pipe. First,
First, wewe designed
designed aalocomotion
locomotionpatternpatternusing
revolute
andlinearjoints
in
the
snake
robot
modules.
Subsequently,
weverifiedthepatternusing revolute and linear joints in the snake robot modules. Subsequently, we verified the pattern bybysimulationin MATLAB
R2015b.
Then
we introduced
our newly
developed
experimental
simulationin MATLAB
R2015b.
Then
we introduced
our newly
developed
experimental
snakesnake
robot.
robot.
Its
uniqueness
lay
in
its
combinationofrevolute
andlinearjointsinone
module.
Thissnake
Its uniqueness lay in its combination of revolute and linear joints in one module. This snake robot
robot
performed
a seriesof experiments
described
in thispaper.FirstFirstofofall,
all,wewe verified
verified theperformed
a seriesof experiments
described
in thispaper.
the designed
designed

locomotion pattern by moving the snake robot in a pipe with a rectangular cross-section. Considering
real applications of the snake robot in pipes, in which there could be sludge on the pipe surface, we
tested its rectilinear locomotion on different kinds of surfaces: dry, with oil, and with lubricant. Next,
the experiment was concerned with locomotion of the snake robot in pipes with circular andrectangular cross-sections. Three symmetrical curves were used for anchoring certain modules

Symmetry 2020, 12, 939

23 of 25

locomotion pattern by moving the snake robot in a pipe with a rectangular cross-section. Considering
real applications of the snake robot in pipes, in which there could be sludge on the pipe surface,
we tested its rectilinear locomotion on different kinds of surfaces: dry, with oil, and with lubricant.
Next, the experiment was concerned with locomotion of the snake robot in pipes with circular andrectangular cross-sections. Three symmetrical curves were used for anchoring certain modules between
the walls of the pipe. We experimentally analyzed the individual symmetrical curves in terms of
their stability under the influence of changed slope of the pipe. The DIC method was used for theseexperiments. The contributions of the paper are as follows:
•
•
•
•

•

Design of concertina locomotion based on revolute and linear joints of snake robot module. Designof mathematical model and analysis of the influence of input parameters.
Development of a unique experimental snake robot consisting of eight modules, each with one
revolute and one linear joint.
Simulation and experimental analysis of proposed concertina locomotion.
Experimental analysis of snake robot locomotion on different types of surfaces: dry, with a layerof oil, and with a layer of lubricant. Measured friction in both directions for all cases led toalmost symmetrical characteristics. The results showed that electrical current consumption for allcases was almost the same. The robot traveled its maximum possible distance with dry friction.
However, with oil or lubricant on the surface, some modules slid backward, resulting in lesstraveled distance per three locomotion cycles.
A new approach based on anchoring of snake robot modules during locomotion in a pipe bymeans of symmetrical curves. Experimental analysis of their stability with changing slope of thepipe by modern DIC methodology using Q-450 Dantec Dynamics high-speed correlation system.
The best solution was the use of a triangular curve. The circular pipe offered better stability forsnake robot locomotion in the pipe.

The contributions of the paper are unique. This study fills a gap in the area of snake robot
locomotion in a confined space such as a pipe or channel.
Author Contributions: Analysis, programming, simulations, and experiments: I.V. and E.P.; experiments:
M.K. and M.V.; DIC experiments: M.H.; mechanical design of snake robot: Z.B. and P.O.; supervision andconceptualization: P.B. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by Slovak Grant Agency VEGA 1/0389/18 “Research on kinematicallyredundant mechanisms”. This work was also supported by the European Regional Development Fund in the
Research Centre of Advanced Mechatronic Systems project, project number CZ.02.1.01/0.0/0.0/16_019/0000867,
within the Operational Programme Research, Development and Education. The work was also funded by research
project VEGA 1/0019/20 and is sponsored by project 013TUKE-4/2019: Modern educational tools and methods forforming creativity and increasing practical skills.
Acknowledgments: The authors would like to thank the Slovak Grant Agency VEGA 1/0389/18 “Researchon kinematically redundant mechanisms”. This work was also supported by the European Regional
Development Fund in the Research Centre of Advanced Mechatronic Systems project, project number
CZ.02.1.01/0.0/0.0/16_019/0000867 within the Operational Programme Research, Development and Education.
The work was also funded by research project VEGA 1/0019/20 and was sponsored by project 013TUKE-4/2019:
Modern educational tools and methods for forming creativity and increasing practical skills.
Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of thestudy; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to
publish the results.

References
1.
2.

Lipták, T.; Virgala, I.; Frankovský, P.; Šarga, P.; Gmiterko, A.; Baločková, L. A geometric approach to modelingof four- and five-link planar snake-like robot. Int. J. Adv. Robot. Syst. 2016, 13. [CrossRef]
Ostertag, O.; Ostertagová, E.; Kelemen, M.; Kelemenová, T.; Virgala, I.; Buša, J. Miniature Mobile Bristled
In-Pipe Machine. Int. J. Adv. Robot. Syst. 2014, 11, 189. [CrossRef]

Symmetry 2020, 12, 939

3.
4.
5.

6.

7.

8.

9.

10.

11.
12.
13.
14.

15.
16.

17.
18.
19.
20.
21.
22.
23.
24.

24 of 25

Kelemen, M.; Virgala, I.; Lipták, T.; L’ubica, M.; Filakovský, F.; Bulej, V. A Novel Approach for a Inverse
Kinematics Solution of a Redundant Manipulator. Appl. Sci. 2018, 8, 2229. [CrossRef]
Hirose, S. Biologically Inspired Robots: Snake-Like Locomotors and Manipulators; Oxford University Press: Oxford,
UK, 1993.
Wakimoto, S.; Nakajima, J.; Takata, M.; Kanda, T.; Suzumori, K. A micro snake-like robot for small pipe
inspection. In Proceedings of 2003 International Symposium on Micromechatronics and Human Science (IEEE Cat.
No.03TH8717); Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 2004.
Greenfield, A.; Rizzi, A.A.; Choset, H. Dynamic Ambiguities in Frictional Rigid-body Systems with
Application to Climbing via Bracing. In Proceedings of the 2005 IEEE International Conference on Robotics andAutomation; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 2006; pp. 1947–1952.
Brunete, A.; Gambao, E.; Torres, J.; Hernando, M. A 2 DoF Servomotor-based Module for Pipe Inspection
Modular Micro-robots. In Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots andSystems; Institute of Electrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 2006; pp. 1329–1334.
Barazandeh, F.; Bahr, B.; Moradi, A. How self-locking reduces actuators torque in climbing snake robots. In
Proceedings of the 2007 IEEE/ASME International Conference on Advanced Intelligent Mechatronics; Institute ofElectrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 2007; pp. 1–6.
Wright, C.; Johnson, A.M.; Peck, A.; Mccord, Z.; Naaktgeboren, A.; Gianfortoni, P.; Gonzalez-Rivero, M.;
Hatton, R.; Choset, H. Design of a modular snake robot. In Proceedings of the 2007 IEEE/RSJ International
Conference on Intelligent Robots and Systems; Institute of Electrical and Electronics Engineers (IEEE): Piscataway,
NJ, USA, 2007; pp. 2609–2614.
Kuwada, A.; Wakimoto, S.; Suzumori, K.; Adomi, Y. Automatic pipe negotiation control for snake-like robot.
In Proceedings of the 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics; Institute ofElectrical and Electronics Engineers (IEEE): Piscataway, NJ, USA, 2008; pp. 558–563.
Andruska, A.M.; Peterson, K.S. Control of a Snake-Like Robot in an Elastically Deformable Channel.
IEEE/ASME Trans. Mechatron. 2008, 13, 219–227. [CrossRef]
Akbarzadeh-T, M.-R.; Safehian, J.; Safehian, J.; Kalani, H. Generating Snake Robot Concertina Locomotion
Using a New Dynamic Curve. Int. J. Model. Optim. 2011, 1, 134–140. [CrossRef]
Marvi, H.; Hu, D.L. Friction enhancement in concertina locomotion of snakes. J. R. Soc. Interface 2012, 9,3067–3080. [CrossRef] [PubMed]
Kano, T.; Satake, F.; Date, H.; Inoue, K.; Ishiguro, A. Doing Well in Narrow Aisle! Decentralized Control
Mechanism Underlying Adaptive Concertina Locomotion of Snakes. Int. Symp. Nonlinear Theory Appl. 2014,
32–35. [CrossRef]
Trebuňa, F.; Virgala, I.; Pástor, M.; Lipták, T.; L’ubica, M. An inspection of pipe by snake robot. Int. J. Adv.
Robot. Syst. 2016, 13. [CrossRef]
Virgala, I.; Kelemen, M.; Prada, E.; Surovec, R. Motion analysis of snake robot segment. In Proceedings of the2013 IEEE 11th International Symposium on Applied Machine Intelligence and Informatics (SAMI), Herl’any,
Slovakia, 31 January–2 February 2013; pp. 145–148. [CrossRef]
Tang, C.; Shu, X.; Meng, D.-Y.; Zhou, G. Arboreal concertina locomotion of snake robots on cylinders. Int. J.
Adv. Robot. Syst. 2017, 14. [CrossRef]
Huang, C.-W.; Huang, C.-H.; Hung, Y.-H.; Chang, C.-Y. Sensing pipes of a nuclear power mechanism using
low-cost snake robot. Adv. Mech. Eng. 2018, 10. [CrossRef]
Gray, J. The mechanism of locomotion in snakes. J. Exp. Boil. 1946, 23, 101–120.
Virgala, I.; Kelemen, M.; Varga, M.; Kuryło, P. Analyzing, Modeling and Simulation of Humanoid Robot
Hand Motion. Procedia Eng. 2014, 96, 489–499. [CrossRef]
Bozek, P.; Ivandić, Ž.; Lozhkin, A.; Lyalin, V.; Tarasov, V. Solutions to the characteristic equation for industrial
robot’s elliptic trajectories. Teh. Vjesn.-Tech. Gaz. 2016, 23, 1017–1023. [CrossRef]
Krolczyk, G.M.; Legutko, S. Experimental Analysis by Measurement of Surface Roughness Variations in
Turning Process of Duplex Stainless Steel. Metrol. Meas. Syst. 2014, 21, 759. [CrossRef]
Huang, S.; Liang, W.; Tan, K.K. Intelligent Friction Compensation: A Review. IEEE/ASME Trans. Mechatron.
2019, 24, 1763–1774. [CrossRef]
Virgala, I.; Frankovský, P.; Frankovský, M.K.P.; Kenderová, M. Friction Effect Analysis of a DC Motor. Am. J.
Mech. Eng. 2013, 1, 1–5. [CrossRef]

Symmetry 2020, 12, 939

25.
26.
27.

25 of 25

Liu, L.; Xu, W.; Yue, X.-L.; Huang, D. Stochastic Bifurcation of a Strongly Non-Linear Vibro-Impact Systemwith Coulomb Friction under Real Noise. Symmetry 2018, 11, 4. [CrossRef]
Yamada, S. Nanotribology of Symmetric and Asymmetric Liquid Lubricants. Symmetry 2010, 2, 320–345.
[CrossRef]
Prada, E.; Valášek, M.; Virgala, I.; Gmiterko, A.; Kelemen, M.; Hagara, M.; Lipták, T.; Erik, P. New approachof fixation possibilities investigation for snake robot in the pipe. In Proceedings of the 2015 IEEE International
Conference on Mechatronics and Automation (ICMA); Institute of Electrical and Electronics Engineers (IEEE):
Piscataway, NJ, USA, 2015; pp. 1204–1210.
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