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Research Article
Kinematics and Dynamics of a Tensegrity-Based Water Wave
Energy Harvester
Min Lin,1Tuanjie Li,1and Zhifei Ji2
1School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China
2College of Mechanical and Energy Engineering, Jimei University, Xiamen 361021, China
Correspondence should be addressed to Zhifei Ji; zi@.com
Received January ; Accepted May
Academic Editor: Shahram Payandeh
Copyright © Min Lin et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A tensegrity-based water wave energy harvester is proposed. e direct and inverse kinematic problems are investigated by using a
geometric method. Aerwards, the singularities and workspaces are discussed. en, the Lagrangian method was used to develop
the dynamic model considering the interaction between the harvester and water waves. e results indicate that the proposed
harvester allows harvesting .% more energy than a conventional heaving system. erefore, tensegrity systems can be viewed
as one alternative solution to conventional water wave energy harvesting systems.
1. Introduction
Tensegrity systems are formed by a combination of rigid
elements (struts) under compression and elastic elements
(cables or springs) under tension. e use of cables or springs
as tensile components leads to an important reduction in the
weight of the systems. Due to this attractive nature, tensegrity
systems have been proposed to be used in many disciplines.
Moreover, a detailed description of the history of tensegrity
systems is provided in [, ].
e rst research work that deals with tensegrity systems
was completed by Calladine []. Since then, tensegrity sys-
tems have been rapidly applied as structures in the architec-
tural context. A tensegrity dome was proposed by Pellegrino
[]. Some design methods for tensegrity domes are proposed
by Fu []. Aerwards, tensegrity structures have been also
proposed to be served as bridges [–]. Moreover, the use of
cables or springs in tensegrities allows them to be deployable
[, ]. Due to this nature, some research works are found
towards their use as antennas [, ]. For static applications,
the subject of form-nding of tensegrities has attracted the
attention of several researchers [, ]. Moreover, a review of
form-nding methods was provided by Tibert and Pellegrino
[]. e basic issues about the statics of tensegrity structures
were reviewed by Juan and Tur [].
From an engineering point of view, tensegrities are a
special class of structures whose components may simulta-
neously perform the purposes of structural force, actuation,
sense,andfeedbackcontrol.Forsuchkindofstructure,
pulleys or other kinds of actuators may stretch/shorten some
of the constituting components in order to substantially
change their forms with a little variation of the structure’s
energy. Ingber [] has demonstrated that tensegrity struc-
tures are very similar to cytoskeleton structures of unicellular
organisms. Aerwards, the cellular tensegrity model is used
to understand the cell structure, biological networks, and
mechanoregulation [, ]. Tensegrity structures are also
very similar to muscle-skeleton structures of high eciency
land animals whose speeds can reach up to mph. e
muscle-skeleton systems of these beings are composed of only
tensional and compressional components. ey thus have the
ability to run with high speed [].
Another interesting application of tensegrities is their
development for use as mechanisms. Oppenheim and
Williams [] were the rst to consider the actuation of
tensegrity systems by modifying the lengths of their com-
ponents in order to obtain tensegrity mechanisms. Aer-
wards, several mechanisms based on tensegrity systems were
proposed,suchasaightsimulator[],aspacetelescope
Hindawi Publishing Corporation
Journal of Robotics
Volume 2016, Article ID 2190231, 13 pages
http://dx.doi.org/10.1155/2016/2190231
Journal of Robotics
Linear
generator
Spring
Wat er wave
Sea bed
L
Z
Y
X
O
D
B
2
B
3
B
4
B
1
B2
B3
B1
B4
Z1
Y1
OX1
R2
R1
P
A3
A2
A4
A1
L3
L2
L1
L4
O1
F : A tensegrity-based water wave energy harvester.
[],andatensegritywalkingrobot[–].Fortensegrity
mechanisms, an interesting topic named tensegrity parallel
mechanism has been proposed recently. e concept of
tensegrity parallel mechanism was introduced by Marshall
[]. en, Shekarforoush et al. [] presented the statics of
a - tensegrity parallel mechanism. Aerwards, Crane III et
al. [] proposed a planar tensegrity parallel mechanism and
completed its equilibrium analysis. Tensegrity systems have
been identied as one of three main research trends in mech-
anismsandroboticsfortheseconddecadeofthestcentury
[]. However, just a few references have stated the possibility
of using tensegrity systems as water wave energy harvesters.
Scruggs and Skelton [] made a preliminary investigation on
the potential use of controlled tensegrity structures to harvest
energy. Sunny et al. [] studied the feasibility of harvesting
energy using polyvinylidene uoride patches mounted on
vibrating prestressed membrane. Vasquez et al. [] stated the
possibilityofusingaplanartensegritymechanisminocean
applications. is application is attractive since it can play an
importantroleintheexpansionofcleanenergytechnologies
that help the world’s sustainable development.
is work presents the analysis of a tensegrity-based
water wave energy harvester. Since this is the rst stage for
the development of a new application for tensegrity systems,
a simplied linear model of sea waves was used to analyze
theproposedharvester.eanalyticalsolutionstothedirect
and inverse kinematic problems are found using a geometric
method. Based on the obtained relationships between the
input and output variables, the singular congurations have
been discussed. e workspaces of the proposed mechanism
have subsequently been computed. Aerwards, the dynamics
were investigated. Finally, the energy harvesting capabilities
of the tensegrity-based harvester are compared with a con-
ventional heaving system.
2. Geometry of the Water Wave
Energy Harvester
A diagram of the tensegrity-based water wave energy har-
vester is shown in Figure . It consists of a oat, four springs,
four linear generators, and one kinematic chain. e linear
generators are joining node pairs 𝑖𝑖(= 1,2,3,4)while
thespringsarejoiningnodepairs14,21,32,and
43.eoatofheightis denoted by 1234.is
harvester is obtained from a square tensegrity parallel prism
[] by connecting the top of the latter to a oat.
From Figure , it can be seen that the sides of the squares
formed by nodes 1234,1234,and
1
2
3
4
havethesamelength.Moreover,thelengthofthelinear
generator joining node pairs 𝑖𝑖is denoted by 𝑖.As
illustrated in Figure , the springs and the linear generators
areconnectedtotheoatandtheseabedatnodes𝑖and 𝑖
by spherical joints without friction. e sea bed is considered
to be parallel to the horizontal plane. A xed reference frame
(,,)is located at the center of the square 1234
with its axis parallel to the line joining nodes 4and 1
and its axis perpendicular to the sea bed, while a moving
reference frame (1,1,1)is located at the mass center
of the oat with its 1axis parallel to the line joining nodes
4and 1and its 1axis perpendicular to the plane formed
by nodes 1,2,3,and4. Moreover, the vectors specifying
the positions of nodes 𝑖and 𝑖in the xed reference frame
are dened as 𝐴a𝑖and 𝐴b𝑖,respectively.Also,thevectors
specifying the positions of nodes 𝑖in the moving reference
frame are dened as 𝐵b𝑖.
In order to obtain an appropriate kinematic model of the
harvester, the following hypotheses are made:
(i) e springs are linear with stiness and lengths
𝑗(=1,2,3,4)andallthespringshavethesamefree
length 0.
Journal of Robotics
(ii) e water waves are traveling along the axis.
In Figure , a passive kinematic chain denoted by 12
is used to connect nodes and 1. Nodes and 1rep-
resent the centers of the squares 1234and 1234,
respectively. Considering the constraints introduced by this
kinematic chain, the possible movements of the oat driven
by water waves are rotations about the axis and translations
along the and axes. erefore, the harvester has three
degrees of freedom.
e Cartesian coordinates of the mass center of the oat
in the xed reference frame are dened as (,,). From
Figure , it can be seen that =0is always satised. Moreover,
the angle is used to specify the rotation of the oat about
axis. Meanwhile, the range of is assumed to be −/2 /2.
e variables ,,andare driven by the water waves. As
a consequence, they are thus chosen as the inputs of the
system. Furthermore, only three of the four linear generators’
lengths are independent. For this reason, the lengths of the
generators joining nodes 11,22,and33are chosen
as the outputs of the system. It follows that the harvester’s
output vector is O=[
1,2,3]𝑇while its input vector is
I=[,,]𝑇.
3. Kinematic Analysis
For the harvester, the linear generators are used to convert
wave motion cleanly into electricity. Generally, the eciency
of electricity generation of the system is highly dependent on
the motions of linear generators. To provide great insight into
the kinematics of the harvester, the relationship between the
input and output vectors is developed in this section.
3.1. Direct Kinematic Analysis. e direct kinematic analysis
consists in computing the output vector Ofor the given
input vector I. According to [], the most convenient
approach to set an algebraic equation system for kinematic
problem of a parallel mechanism is to use the rotation matrix
parameters and the position vector of the moving platform.
isapproachisusedinthisworktodealwiththekinematic
problems of the harvester. e position and orientation of
the oat are described by the position vector P=OO=
[0,,]𝑇and the rotation matrix 𝐴R𝐵with respect to the xed
reference frame. From Figure , it can be seen that the rotation
matrix 𝐴R𝐵can be dened by rotating the moving reference
frame /2about 1axis followed by about 1axis. 𝐴R𝐵thus
takes the following form:
𝐴R𝐵=
0−10
cos 0sin
−sin 0cos
.()
en, the position vectors of points 𝑖(=1,2,3,4)with
respect to the xed reference frame can be obtained:
𝐴b𝑖=P+𝐴R𝐵𝐵b𝑖, =1,2,3,4. ()
e vectors specifying the positions of nodes 𝑖in the
moving reference frame can be easily derived:
𝐵b1=
2
−
2
−
2
,
𝐵b2=
2
2
−
2
,
𝐵b3=
−
2
2
−
2
,
𝐵b4=
−
2
−
2
−
2
.
()
Substituting () into (), we have
𝐴b1=
2
+
2cos −
2sin
−
2cos −
2sin
,
𝐴b2=
−
2
+
2cos −
2sin
−
2cos −
2sin
,
𝐴b3=
−
2
−
2cos −
2sin
−
2cos +
2sin
,
𝐴b4=
2
−
2cos −
2sin
−
2cos +
2sin
.
()
Journal of Robotics
FromFigure,itcanalsobeseenthat𝐴a1=
[/2,−/2,0]𝑇,𝐴a2= [/2,/2,0]𝑇,𝐴a3= [−/2,/2,0]𝑇,
and 𝐴a4= [−/2,−/2,0]𝑇. With the position vectors of
points 𝑖and 𝑖now known, the vector equation of the th
linear generator can be written as
L𝑖=𝐴b𝑖−𝐴a𝑖, =1,2,3,4. ()
By using (), the solution to the direct kinematic problem
is found as follows:
1=−
2cos −
2sin 2
++
2cos −
2sin +
221/2 ,()
2=−
2cos −
2sin 2
++
2cos −
2sin −
22+21/2 ,()
3=−
2cos +
2sin 2
+−
2cos −
2sin −
221/2 .()
Here, for the latter use, the length of the linear generator
joining nodes 44is also presented:
4=−
2cos +
2sin 2
+−
2cos −
2sin +
22+21/2 .()
3.2. Inverse Kinematic Analysis. e inverse kinematic prob-
lem corresponds to the computation of the input vector Ifor
the given output vector O.esolutiontothisproblemcan
be found by solving ()–() for the input variables ,,and
. Subtracting the square of () from that of () yields
2+2cos −2−2
1+2
2−sin =0. ()
Subtracting the square of () from that of (), we obtain
2(cos +1)−sin −2
1+2
3−2sin =0. ()
From () and (), the following expressions can be
derived:
= 1
22+2
1−2
2−2cos +sin ,
= 1
2sin 2sin2+sin cos
+2
1−2
2cos −2
2+2
3.
()
By substituting () into (), the following equation is
obtained:
42−2
22cos2
−2
1−2
22+2
2−2
32−2
1−2
32cos
−422−2
2−2
1−2
22−2
2−2
32=0.
()
Becauseoftherangeimposedon,foursolutionsfor
can be arrived at by solving (). Furthermore, by substituting
these results into (), the solutions to the inverse kinematic
problem are found.
4. Singularity Analysis
4.1. Jacobian Matrix. e Jacobian matrix of the harvester
is dened as the relationships between a set of innitesimal
changes of its input vectors and the corresponding innites-
imal changes of its output vectors. e Jacobian matrix, J,
relates Ito Osuch that O=JI.Jcanberewrittenin
terms of matrices Cand Dsuch that CO=DI.From()–
(), the elements of Cand Dcan be computed and written in
terms of the input variables as follows:
11 =−
2cos −
2sin 2
++
2cos −
2sin +
221/2 ,
22 =−
2cos −
2sin 2
++
2cos −
2sin −
22+21/2 ,
33 =−
2cos +
2sin 2
+−
2cos −
2sin −
221/2 ,
12 =13 =21 =23 =31 =32 =0,
11 =+
2cos −
2sin +
2,
12 =22 =−
2cos −
2sin ,
13 =−
2cos −
2sin
2sin −
2cos
−+
2cos −
2sin +
2
2sin
−
2cos ,
Journal of Robotics
21 =+
2cos −
2sin −
2,
31 =−
2cos −
2sin −
2,
32 =−
2cos +
2sin ,
23 =−
2cos −
2sin
2sin −
2cos
−+
2cos −
2sin −
2
2sin
+
2cos ,
33 =−
2cos +
2sin
2sin +
2cos
+−
2cos −
2sin −
2
2sin
−
2cos .
()
For (), it is noted that 𝑖𝑗 and 𝑖𝑗 are the elements located
on the th line and th column of Cand D,respectively.
4.2. Singular Congurations. e singular congurations of
the harvester consist in nding the situations where the
relationships between innitesimal changes in its input and
output variables degenerate. When such a situation occurs,
the harvester will gain or lose one or more degrees of freedom,
thus leading to a loss of control. As a consequence, such
congurations are usually avoided when possible. Generally,
the singular congurations of the harvester can be obtained
by setting det(C)=0, det(D)=0, or both. e determinants
of Cand Dcan be expressed as follows:
det (C)=−
2cos −
2sin 2
++
2cos −
2sin +
221/2
⋅−
2cos −
2sin 2
++
2cos −
2sin −
22+21/2
⋅−
2cos +
2sin 2
+−
2cos −
2sin −
221/2 =0,
det (D)=2
8−2−2sin 2
+2−2−2cos 2+82+22cos
+8−4+2sin +2−6−2
=0. ()
By examining (), it is possible to extract the expressions
corresponding to singular congurations. e following is
a list of these expressions as well as their descriptions with
respect to the mechanism’s behaviors:
(i)−
2cos −
2sin 2
++
2cos −
2sin +
221/2 =0. ()
(a) e length of the linear generator joining nodes 1
and 1is equal to zero. Node 1is thus coincident
with node 1.Moreover,node4is also coincident
with node 2.
(b) e movement of the oat is reduced to a rotation
about the axis joining nodes 1and 4.Whenthis
isthecase,onlyonevariableisneededtodene
the system. e harvester thus loses two degrees of
freedom.
(c) Innitesimal movements of node in a direction
perpendicular to the line joining nodes 1and 4are
possible without deforming the springs and the linear
generators.
(d) External forces parallel to the line 14areresistedby
the harvester.
(ii)−
2cos +
2sin 2
+−
2cos −
2sin −
221/2 =0. ()
(a) e length of the linear generator joining nodes 3
and 3is equal to zero. Node 2is coincident with
node 4.
(b) e movement of the oat is reduced to a rotation
about the axis joining nodes 2and 3.Whenthis
case occurs, only one variable can be used to describe
the rotation of the oat. e harvester thus loses two
degrees of freedom.
(c) Innitesimal movements of node in a direction
perpendicular to the line joining nodes 2and 1are
possible without deforming the springs and the linear
generators.
Journal of Robotics
(d) External forces parallel to the line 14are resisted by
the harvester.
(iii)−2−2sin 2
+2−2−2cos 2+82+22cos
+8−4+2sin +2−6−2
=0.
()
(a)Actually,itisimpossibletoextractthebehaviorsof
the harvester from (). is case corresponds to
the boundaries of the input workspace and will be
mapped in Section .. Generally speaking, when
this is the case, innitesimal movements of the input
variables along a direction perpendicular to a certain
surface cannot be generated.
From()and(),itcanbeseenthatthesingularcon-
guration (i) corresponds to the situation where the length of
the linear generator 11is equal to zero while conguration
(ii) corresponds to the situation where the length of the linear
generator 33is equal to zero. From an engineering point
of view, the linear generators are generally limited to operate
within a range of nonzero lengths. However, from the aspect
of mechanism’s analysis, the lengths of prismatic actuators
can be set to be zero. is case belongs to one kind of the
singular congurations of the proposed mechanism.
5. Workspaces
Since the input variables ,,andare driven by water
waves, the ranges of the input variables can be used to
describe the strengths of the water waves. Moreover, the
amount of the electricity produced by the harvester depends
on the movements of the linear generators. e ranges of
the output variables can be considered as an indicator of the
eciency of energy harvesting. In this section, the ranges
of the input vectors are referred to as the input workspace
whiletherangesoftheoutputvectorsarereferredtoasthe
output workspace. e boundaries of the input and output
workspaces usually correspond to singular congurations
described in Section .. From ()–(), it can be seen that
the singular congurations are expressed in terms of the input
variables. According to these expressions, the boundaries of
theinputworkspacecanbecomputed.Aerwards,these
boundaries will be mapped from the input domain into the
output domain in order to generate the output workspace.
5.1. Input Workspace. e input workspace of the harvester
is a volume whose boundaries correspond to singular con-
gurations discussed in Section .. An example of such a
workspace with =1mand=1misshowninFigure.
In Figure , the surface corresponding to the singular
conguration (iii) is identied by surface (iii). From this
gure,itcanbeseenthattheinputworkspacecanbedivided
intothreeparts.erstpartisdenedby−1 ≤ ≤ 0 and
0≤≤/2. It is bounded by surface (iii) and the planes
1.5
1
0.5
0
y (m)
0
0.5
1
i
iii
iii
iii
ii
0
0.5
1
1.5
2
z (m)
−0.5
−0.5
−1
−1 −1.5
𝜃(rad)
F : Input workspace of the harvester with =1mand=
0.1m.
corresponding to =−1,=/2,and=2.Moreover,the
second part is dened by 0≤≤1and 0≤≤/2.Itis
bounded by the planes corresponding to =/2,=1,and
surface (iii). Finally, the third part is dened by 0≤≤1
and −/2≤≤0. It is bounded by the planes corresponding
to =−/2,=−1, and surface (iii). Furthermore,
fromFigure,itcanalsobeobservedthatcurves(i)and
(ii) correspond to the singular congurations (i) and (ii),
respectively. Since the harvester will be uncontrolled when
it reaches a singular conguration, the boundaries of the
input workspace and the singular curves (i) and (ii) should
beavoidedduringtheuseofsuchaharvester.
5.2. Output Workspace. In order to obtain the output
workspace, the singular congurations detailed in Section .
shouldberewrittenintermsoftheoutputvariablesrstly.
From () and (), it can be concluded that the singu-
lar conguration (i) in the output domain corresponds to
1=0while the singular conguration (ii) corresponds to
2=0. Generally speaking, by substituting the solutions to
the inverse kinematic problem into (), an expression for
singular conguration (iii) in terms of the output variables
can be arrived at. However, this procedure is rather tedious.
Here, Bezout’s method [] was used to derive the expression
corresponding to singular conguration (iii) in the output
domain due to its simplicity.
Equation () is rstly rewritten as
1cos2+2cos +3=0, ()
where
1=42−2
22,
2=−2
1−2
22+2
2−2
32−2
1−2
32,
3=−422−2
2−2
1−2
22−2
2−2
32.
()
Moreover,bysubstituting()into(),thefollowingequa-
tion is obtained:
1cos2+2cos +3=0, ()
Journal of Robotics
10
8
6
4
2
00
5
10
8
6
4
2
0
10
iv
v
vi
vi
vii
L1(m)
L
2
(m)
L3(m)
F : Output workspace of the water wave energy harvester with
=1mand =0.1m.
where
1=2
1−2
22
2−2
3,
2=−2
1−2
22−2
2−2
32,
3=2
2−2
32
2−2
3.
()
Itshouldbenotedthat()representssingularcong-
uration (iii) expressed by 1,2,3,and.Moreover,()
is used to compute for the given values of 1,2,and3.
Generally, the solutions to obtained by solving () should
satisfy (). Furthermore, both () and () can be considered
as two quadratics with respect to cos . According to Bezout’s
method, the condition that () and () have a comment root
for cos is as follows:
12
12
23
23
−13
13
2=0. ()
Simplifying () yields
2
1−2
32
1−22
2+2
32
⋅44−422
2+4
1−22
12
2+4
1
⋅44−422
2+4
2−22
22
3+4
3=0.
()
Equation () represents the surfaces corresponding to
singular conguration (iii) in the output workspace. By
plotting these surfaces, the output workspace of the harvester
canbeobtained.Anexampleofsuchplotsisshownin
Figure with =1mand=0.1m.
FromFigure,itcanbeseenthatthesingularcongu-
ration (iii) determined by () corresponds to four surfaces
(surfaces (iv)–(vii)) in the output workspace. Moreover,
surfaces (iv), (v), (vi), and (vii) correspond to expressions
1−3=0,2
1−22
2+2
3=0,44−422
2+4
1−22
12
2+4
1=0,
and 44−422
2+4
2−22
22
3+4
3=0,respectively.Itcan
also be observed that the output workspace of the harvester
can be divided into two parts. e rst part is bounded by
surface (v), surface (vi), plane 1=0,andplane3=0while
the second part is bounded by surfaces denoted by (iv), (vi),
and (vii) and planes denoted by 2=0and 1=10.is
outputworkspaceshouldbeconsideredduringtheuseand
design of such a harvester.
It is noted that the forward and inverse kinematics, Jaco-
bian matrix, and workspaces should be considered when such
harvester is being designed. Moreover, when the harvester is
put to use, the singular congurations should be avoided. e
kinematics and Jacobian matrix are used to nd the singular
congurations.
6. Dynamic Analysis
e eciency of the water wave harvesting is highly depen-
dent on the dynamics of the harvester. erefore, it is of
utmost importance to research the dynamics of the harvester.
In this section, the dynamic model of the harvester is
developed. Furthermore, in order to compare the eciency
of a conventional heaving system with that of the proposed
harvester, the dynamic model of the conventional heaving
system is rstly introduced. Before introducing the dynamic
models of the two systems, it is assumed that the linear water
wavesareappliedonthetwosystems.
6.1. Dynamic Model of a Conventional Heaving System. A
diagram of the conventional heaving wave energy harvester
[] composed of a oat, a bar magnet, and a battery is
shown in Figure . In order to compare the eciency of the
conventional heaving system with the proposed harvester, the
oatsofbothsystemsareassumedtohavethesamesize.
Moreover, in this paper, the weight of the bar magnet was
neglected.
According to [], the motion equation of the oat, driven
by linear water waves, in a conventional heaving system is
given by
+𝑤𝑧2
2+𝑟𝑧 +V𝑧+𝑝𝑧
+𝑤𝑝 +𝑠=𝑧0 cos +𝑧. ()
e coecients in () are given as follows:
is the mass of the oat.
𝑤𝑧 is the added mass.
𝑟𝑧 is the damping coecient.
V𝑧is the viscous damping coecient.
𝑝𝑧 is the power take-o coecient.
𝑤𝑝 isthewaterplaneareawhenthebodyisatrest.
is the density of seawater.
istheaccelerationduetogravity.
𝑠isthespringconstantofmooringlinesandis the
number of lines (mooring restoring force).
Journal of Robotics
Linear wave
cH/2
h
a
d
Inductance coil
Float
Bar magnet Battery
Horn
Beacon light
F : A conventional heaving wave energy harvester [].
𝑧0 is the water-induced vertical force amplitude and
=2/isthecircularwavefrequency(is the wave
period).
𝑧is the phase angle between the wave and force.
Finally, it should be noted that the computations of the
abovecoecientsin()canbefoundin[].
6.2. Dynamic Model of the Tensegrity-Based Water Wave
Energy Harvester. As stated in Section , the harvester has
three degrees of freedom. erefore, three generalized coor-
dinates, chosen as q=123𝑇=
𝑇, are needed
to develop the dynamic model.
In order to derive an appropriate dynamic model of the
harvester, the following hypotheses are made:
(i)elinksofthemechanism,exceptfortheoat,are
massless.
(ii) e springs are massless.
(iii) ere is no friction in the harvester’s revolute, pris-
matic, and spherical joints.
eequationsofmotionoftheharvesteraredeveloped
using the Lagrangian approach; namely,
q−
q+
q=Q𝑘,()
where and are the kinetic and potential energies of the
harvester and Q𝑘is the vector of nonconservative forces
acting on the system. In [], the translation of the oat along
axis is dened as surge, the translation of the oat along
axis is dened as heave, and the rotation of the oat with
respect to axis is dened as pitch. e kinetic energy, due
onlytothesurge,heave,andpitchmovementsoftheoat,can
be expressed as
T=1
2
q𝑇M
q,()
where
M=
+𝑤𝑦 00
0+
𝑤𝑦 0
00
𝑦+𝑤
.()
𝑦is the mass moment of inertia with respect to axis
and 𝑤is added-mass moment of inertia due to pitching. e
potential energies due to heaving and pitching motions of the
top platform are described by McCormick [] as
𝑝𝑧 =1
2𝑤𝑝 2
2,
𝑝𝜃 =1
22
3,()
where is the restoring moment constant, dened for a
bottom-at body in terms of the dra. e total potential
energy of the harvester becomes
=+𝑝𝑧 +𝑝𝜃
=1
2𝑤𝑝 2
2+1
22
3+2
1+2
2−02
+2
3+2
4−02.
()
e nonconservative forces, which correspond to the
radiation damping force, viscous damping force, and water
wave induced forces, can be expressed as
Q𝑘=
−V𝑦
1
𝑧0 cos ()−𝑟𝑧 +V𝑧
2
𝜃0cos ()−𝑟𝜃
3
.()
Journal of Robotics
Substituting (), (), and () into (), the dynamic model
oftheharvestercanberewrittenas
M
q+B
q+G=F,()
where
B=
V𝑦00
0
𝑟𝑧 +V𝑧+𝑝𝑧 0
00
𝑟𝜃
,
G=
1
2
3𝑇,
F=0
𝑧𝑜 cos 𝜃0 sin 𝑇.
()
e elements of Gare detailed in the Appendix. For (),
it should be noted that V𝑦is viscous damping coecient
corresponding to the surge movements of the oat while 𝑟𝜃
is the radiation damping coecient due to pitching motion.
𝜃0 is the water-induced torque amplitude (applied on the
oat). e computations of V𝑦,𝑟𝜃,𝜃0,andcan also be
foundin[].esecomputationsarealsonotrepeatedhere.
7. Energy Harvesting
In this section, two energy harvesting systems are researched,
respectively. One is a conventional heaving system and the
other is the tensegrity-based water wave harvester. Also, the
powers of the two systems have been computed, respectively.
e parameters of water waves are selected as = 0.2m,
=6s, and =100m. is the wave height measured from
the trough to the crest while isthewaveperiod.denotes
the water depth. Moreover, the oats used in the two energy
harvesting systems are supposed to have the same dimensions
as =1m, =0.1m, and =0.05m.
7.1. Conventional Heaving System. For a conventional heaving
system, the motion of the oat is expressed by (). For the
given water wave parameters and the dimensions of the oat,
the coecients of () can be calculated according to [].
e results are listed in Table .
Solving () yields the position and velocity of the oat
which are shown in Figure . e power of the heaving body
is given by []
𝑧()=
𝑧()()
,()
where 𝑧()is the wave introduced heaving force on the oat.
e power for take-o, (), is given by the dierence
between the available power (𝑧())and the power dissipated
duetoradiation(𝑟𝑧())and viscous eects (V𝑧()):
()=
𝑧()−𝑟𝑧 ()−V𝑧().()
e average power for take-o over one period of time is
given by
ave =1
𝑇(). ()
T : Conventional heaving oat coecients.
Coecient Value Unit
. kg
𝑤𝑧 . kg
𝑟𝑧 . N⋅s/m
V𝑧. N⋅s/m
𝑝𝑧 N⋅s/m
𝑤𝑝 m
2
—
𝑠N/m
𝑧0 . N
𝑧rad
𝑛𝑧 . Rad/s
𝑐𝑧 . N⋅s/m
0. m
036912
t(s)
0.6
0.4
0.2
0
−0.2
−0.4
z(t) (m), ̇
z(t) (m/s)
z(t)
̇
z(t)
F : Motion of the conventional heaving system.
e water wave energy and power are []
=222
16 ,()
P=22
32 i.()
Applying () over two wave periods of the function
showninFiguregivesanaveragepowerave = 0.154kW.
Since the oat’s breadth is m, then we can compare this
result with the power contained in one meter of wave front.
e maximum available power per meter of wave front is
= 0.236kW (computed by ()). erefore, .% of the
wave energy can be harvested with electrical generators.
7.2. Tensegrity-Based Wave Energy Harvester. For the har-
vester considered here, it has innitesimal mechanisms
inherent of many tensegrity systems. is means that there
are innitesimal deformations of the mechanism that do
not require any changes in the lengths of the harvester’s
Journal of Robotics
T : Tensegrity-based harvester coecients.
Coecient Value Unit
. kg
𝑤𝑦 . kg
𝑤𝑧 . kg
𝑤. kg⋅m2
𝑥. Kg⋅m2
. N⋅m/rad
𝑟𝑧 . N⋅s/m
𝑟𝜃 . N⋅m⋅s/rad
V𝑦. N⋅s/m
V𝑧. N⋅s/m
𝑝𝑧 N⋅s/m
𝑤𝑝 M
2
𝑧0 . N
𝜃0 . N⋅m
𝑧rad
t (s)
120369
3
2
1
0
2.5
1.5
0.5
−0.5
−1
−1.5
P(t) (kW)
P
Pave
F : Power for take-o of the conventional heaving system.
components. It follows that some wave energy would not
be harvested as the mechanism could deform some degree
without the deformation being felt by the linear generators.
However, since the deformations are innitesimal, the eects
of innitesimal mechanisms are negligible.
Let the dimensions of the oat in the tensegrity-based
water wave harvester be the same as the conventional heaving
oat. e additional constant physical parameters are 0=
4mand=10N/m. Table contains the values of the
coecients (computed according to []) for the equation of
motion (see ()).
e simulation is performed over two wave periods, that
is,seconds.Figures–showthepositionandvelocity
response of the oat: surge, heave, and pitch.
Figure shows the instantaneous power for take-o. e
average power over two wave periods is ave =0.186kW. e
t (s)
120 3 6 9
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
y(t) (m), ̇
y(t) (m/s)
y(t)
̇
y(t)
F : Surge motions of the tensegrity-based water wave energy
harvester.
036912
t(s)
0.6
0.4
0.2
0
−0.2
−0.4
z(t) (m), ̇
z(t) (m/s)
z(t)
̇
z(t)
F : Heave motions of the tensegrity-based water wave energy
harvester.
power contained in one meter of wave front is =0.236kW
(computed by ()). erefore, .% of the available energy
could be harvested by electrical generators. By comparing
Figures with , it is found that the proposed tensegrity-
based harvester allows harvesting .% more energy than a
conventional heaving device under linear water wave condi-
tions. For the conventional heaving device, the movement of
the oat is translation along the axis. It is proper to say that
the conventional heaving device has one degree of freedom.
However,thepossiblemovementsoftheproposedharvester
are rotations about the axis and translations along the
and axes (see Section ). It is thus proper to say that the
proposed harvester has three degrees of freedom. at is why
the harvester can harvest more energy than a conventional
device.
Journal of Robotics
0.12
0.1
0.08
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
𝜃(t) (rad), ̇
𝜃(t) (rad/s)
𝜃(t)
̇
𝜃(t)
036912
t(s)
F : Pitch motions of the tensegrity-based water wave energy
harvester.
0 3 6 912
0
0.5
1
P
Pave
P(t) (kW)
−0.5
t(s)
F : Tensegrity-based harvester: power for take-o.
8. Conclusion
A tensegrity-based water wave energy harvester was pro-
posed in this work. e geometry of the harvester was
described. e solutions to the direct and inverse kinematic
problems were found by using a geometric method. e
Jacobian matrix and singular congurations were subse-
quently computed. en, the input and output workspaces
were computed on the basis of the analysis of the obtained
singular congurations. Aerwards, the dynamic analysis
was performed considering the interaction with linear water
waves, considering added mass, radiation damping, and
viscous damping phenomena. It was shown that the proposed
tensegrity-based water wave energy harvester allows harvest-
ing .% more energy than a conventional heaving system.
Appendix
Elements of G
e elements of Gin () are as follows:
1=
1=21
−01−cos 3
2−sin 3
2+
22
+2−cos 3
2+sin 3
22−1/2⋅1
−cos 3
2−sin 3
2+
2+21+cos 3
2
−sin 3
2−
2⋅1
−01+cos 3
2−sin 3
2−
22
+2−cos 3
2−sin 3
22−1/2,
2=
2=𝑤𝑝2+22−cos 3
2
+sin 3
2⋅1
−01−cos 3
2−sin 3
2+
22
+2−cos 3
2+sin 3
22−1/2+21
−01+cos 3
2−sin 3
2−
22
+2−cos 3
2−sin 3
22−1/2⋅2
−cos 3
2−sin 3
2,
3=
3=3+21
−01−cos 3
2−sin 3
2+
22
+2−cos 3
2+sin 3
22−1/2⋅1
Journal of Robotics
−cos 3
2−sin 3
2+
2sin 3
2
−cos 3
2+2−cos 3
2+sin 3
2
⋅sin 3
2+cos 3
2+21
−01+cos 3
2−sin 3
2−
22
+2−cos 3
2−sin 3
22−1/2⋅−1
+cos 3
2−sin 3
2−
2sin 3
2
+cos 3
2+2−cos 3
2−sin 3
2
⋅sin 3
2−cos 3
2.
(A.)
Competing Interests
e authors declare that they have no competing interests.
Acknowledgments
isresearchissupportedbytheNationalNaturalScience
Foundation of China (no. ).
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