Research ArticleKinematics and Dynamics of a Tensegrity-Based Water WaveEnergy Harvester

Min Lin1 Tuanjie Li1 and Zhifei Ji2

1School of Electro-Mechanical Engineering Xidian University Xirsquoan 710071 China2College of Mechanical and Energy Engineering Jimei University Xiamen 361021 China

Correspondence should be addressed to Zhifei Ji zfji18163com

Received 5 January 2016 Accepted 22 May 2016

Academic Editor Shahram Payandeh

Copyright copy 2016 Min Lin et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A tensegrity-based water wave energy harvester is proposedThe direct and inverse kinematic problems are investigated by using ageometric method Afterwards the singularities and workspaces are discussed Then the Lagrangian method was used to developthe dynamic model considering the interaction between the harvester and water waves The results indicate that the proposedharvester allows harvesting 1359 more energy than a conventional heaving system Therefore tensegrity systems can be viewedas one alternative solution to conventional water wave energy harvesting systems

1 Introduction

Tensegrity systems are formed by a combination of rigidelements (struts) under compression and elastic elements(cables or springs) under tensionThe use of cables or springsas tensile components leads to an important reduction in theweight of the systems Due to this attractive nature tensegritysystems have been proposed to be used in many disciplinesMoreover a detailed description of the history of tensegritysystems is provided in [1 2]

The first research work that deals with tensegrity systemswas completed by Calladine [3] Since then tensegrity sys-tems have been rapidly applied as structures in the architec-tural context A tensegrity dome was proposed by Pellegrino[4] Some design methods for tensegrity domes are proposedby Fu [5] Afterwards tensegrity structures have been alsoproposed to be served as bridges [6ndash9] Moreover the use ofcables or springs in tensegrities allows them to be deployable[10 11] Due to this nature some research works are foundtowards their use as antennas [12 13] For static applicationsthe subject of form-finding of tensegrities has attracted theattention of several researchers [14 15] Moreover a review ofform-findingmethods was provided by Tibert and Pellegrino[16]The basic issues about the statics of tensegrity structureswere reviewed by Juan and Tur [17]

From an engineering point of view tensegrities are aspecial class of structures whose components may simulta-neously perform the purposes of structural force actuationsense and feedback control For such kind of structurepulleys or other kinds of actuators may stretchshorten someof the constituting components in order to substantiallychange their forms with a little variation of the structurersquosenergy Ingber [18] has demonstrated that tensegrity struc-tures are very similar to cytoskeleton structures of unicellularorganisms Afterwards the cellular tensegrity model is usedto understand the cell structure biological networks andmechanoregulation [19 20] Tensegrity structures are alsovery similar to muscle-skeleton structures of high efficiencyland animals whose speeds can reach up to 60mph Themuscle-skeleton systems of these beings are composed of onlytensional and compressional componentsThey thus have theability to run with high speed [21]

Another interesting application of tensegrities is theirdevelopment for use as mechanisms Oppenheim andWilliams [22] were the first to consider the actuation oftensegrity systems by modifying the lengths of their com-ponents in order to obtain tensegrity mechanisms After-wards several mechanisms based on tensegrity systems wereproposed such as a flight simulator [23] a space telescope

Hindawi Publishing CorporationJournal of RoboticsVolume 2016 Article ID 2190231 13 pageshttpdxdoiorg10115520162190231

2 Journal of Robotics

Lineargenerator

Spring

Water wave

Sea bed

L

Z

Y

X

O

D

B9984002B998400

3

B9984004

B9984001

B2

B3

B1

B4

Z1

Y1

O998400X1

R2

R1

P

A3

A2

A4

A1

L3

L2

L1

L4

O1

Figure 1 A tensegrity-based water wave energy harvester

[24] and a tensegrity walking robot [25ndash27] For tensegritymechanisms an interesting topic named tensegrity parallelmechanism has been proposed recently The concept oftensegrity parallel mechanism was introduced by Marshall[28] Then Shekarforoush et al [29] presented the statics ofa 3-3 tensegrity parallel mechanism Afterwards Crane III etal [30] proposed a planar tensegrity parallel mechanism andcompleted its equilibrium analysis Tensegrity systems havebeen identified as one of three main research trends in mech-anisms and robotics for the second decade of the 21st century[31] However just a few references have stated the possibilityof using tensegrity systems as water wave energy harvestersScruggs and Skelton [32]made a preliminary investigation onthe potential use of controlled tensegrity structures to harvestenergy Sunny et al [33] studied the feasibility of harvestingenergy using polyvinylidene fluoride patches mounted onvibrating prestressedmembrane Vasquez et al [34] stated thepossibility of using a planar tensegrity mechanism in oceanapplications This application is attractive since it can play animportant role in the expansion of clean energy technologiesthat help the worldrsquos sustainable development

This work presents the analysis of a tensegrity-basedwater wave energy harvester Since this is the first stage forthe development of a new application for tensegrity systemsa simplified linear model of sea waves was used to analyzethe proposed harvester The analytical solutions to the directand inverse kinematic problems are found using a geometricmethod Based on the obtained relationships between theinput and output variables the singular configurations havebeen discussed The workspaces of the proposed mechanismhave subsequently been computed Afterwards the dynamicswere investigated Finally the energy harvesting capabilitiesof the tensegrity-based harvester are compared with a con-ventional heaving system

2 Geometry of the Water WaveEnergy Harvester

A diagram of the tensegrity-based water wave energy har-vester is shown in Figure 1 It consists of a float four springsfour linear generators and one kinematic chain The lineargenerators are joining node pairs 119860

119894119861119894(119894 = 1 2 3 4) while

the springs are joining node pairs 11986011198614 11986021198611 11986031198612 and

11986041198613 The float of height 119863 is denoted by 119861

1119861211986131198614 This

harvester is obtained from a square tensegrity parallel prism[10] by connecting the top of the latter to a float

From Figure 1 it can be seen that the sides of the squaresformed by nodes 119860

1119860211986031198604 1198611119861211986131198614 and 119861

1015840

11198611015840

21198611015840

31198611015840

4

have the same length 119871 Moreover the length of the lineargenerator joining node pairs 119860

119894119861119894is denoted by 119871

119894 As

illustrated in Figure 1 the springs and the linear generatorsare connected to the float and the sea bed at nodes 119860

119894and 119861

119894

by spherical joints without frictionThe sea bed is consideredto be parallel to the horizontal plane A fixed reference frame119860 (119883 119884 119885) is located at the center of the square 119860

1119860211986031198604

with its 119883 axis parallel to the line joining nodes 1198604and 119860

1

and its 119885 axis perpendicular to the sea bed while a movingreference frame 119861 (119883

1 1198841 1198851) is located at the mass center

of the float with its 1198831axis parallel to the line joining nodes

1198614and 119861

1and its 119885

1axis perpendicular to the plane formed

by nodes 1198611 1198612 1198613 and 119861

4 Moreover the vectors specifying

the positions of nodes 119860119894and 119861

119894in the fixed reference frame

are defined as 119860a119894and 119860b

119894 respectively Also the vectors

specifying the positions of nodes 119861119894in the moving reference

frame are defined as 119861b119894

In order to obtain an appropriate kinematic model of theharvester the following hypotheses are made

(i) The springs are linear with stiffness 119870 and lengths119897119895(119895 = 1 2 3 4) and all the springs have the same free

length 1198710

Journal of Robotics 3

(ii) The water waves are traveling along the 119884 axis

In Figure 1 a passive kinematic chain denoted by 11987711198751198772

is used to connect nodes 119874 and 1198741 Nodes 119874 and 119874

1rep-

resent the centers of the squares 1198601119860211986031198604and 119861

1119861211986131198614

respectively Considering the constraints introduced by thiskinematic chain the possible movements of the float drivenby water waves are rotations about the119883 axis and translationsalong the 119884 and 119885 axes Therefore the harvester has threedegrees of freedom

The Cartesian coordinates of the mass center of the floatin the fixed reference frame are defined as (119909 119910 119911) FromFigure 1 it can be seen that119909 = 0 is always satisfiedMoreoverthe angle 120579 is used to specify the rotation of the float about119883axis Meanwhile the range of 120579 is assumed to be [minus1205872 1205872]The variables 119910 119911 and 120579 are driven by the water waves Asa consequence they are thus chosen as the inputs of thesystem Furthermore only three of the four linear generatorsrsquolengths are independent For this reason the lengths of thegenerators joining nodes 119860

11198611 11986021198612 and 119860

31198613are chosen

as the outputs of the system It follows that the harvesterrsquosoutput vector is O = [119871

1 1198712 1198713]119879 while its input vector is

I = [119910 119911 120579]119879

3 Kinematic Analysis

For the harvester the linear generators are used to convertwave motion cleanly into electricity Generally the efficiencyof electricity generation of the system is highly dependent onthemotions of linear generators To provide great insight intothe kinematics of the harvester the relationship between theinput and output vectors is developed in this section

31 Direct Kinematic Analysis The direct kinematic analysisconsists in computing the output vector O for the giveninput vector I According to [35] the most convenientapproach to set an algebraic equation system for kinematicproblem of a parallel mechanism is to use the rotation matrixparameters and the position vector of the moving platformThis approach is used in this work to deal with the kinematicproblems of the harvester The position and orientation ofthe float are described by the position vector P = OO1015840 =[0 119910 119911]

119879 and the rotationmatrix 119860R119861with respect to the fixed

reference frame FromFigure 1 it can be seen that the rotationmatrix 119860R

119861can be defined by rotating the moving reference

frame1205872 about1198851axis followed by 120579 about119884

1axis 119860R

119861thus

takes the following form

119860R119861=[[

[

0 minus1 0

cos 120579 0 sin 120579minus sin 120579 0 cos 120579

]]

]

(1)

Then the position vectors of points 119861119894(119894 = 1 2 3 4) with

respect to the fixed reference frame can be obtained

119860b119894= P +

119860R119861

119861b119894 119894 = 1 2 3 4 (2)

The vectors specifying the positions of nodes 119861119894in the

moving reference frame can be easily derived

119861b1=

[[[[[[[

[

119871

2

minus119871

2

minus119863

2

]]]]]]]

]

119861b2=

[[[[[[[

[

119871

2

119871

2

minus119863

2

]]]]]]]

]

119861b3=

[[[[[[[

[

minus119871

2

119871

2

minus119863

2

]]]]]]]

]

119861b4=

[[[[[[[

[

minus119871

2

minus119871

2

minus119863

2

]]]]]]]

]

(3)

Substituting (3) into (2) we have

119860b1=

[[[[[[

[

119871

2

119910 +119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 minus 119871

2sin 120579

]]]]]]

]

119860b2=

[[[[[[

[

minus119871

2

119910 +119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 minus 119871

2sin 120579

]]]]]]

]

119860b3=

[[[[[[

[

minus119871

2

119910 minus119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 + 119871

2sin 120579

]]]]]]

]

119860b4=

[[[[[[

[

119871

2

119910 minus119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 + 119871

2sin 120579

]]]]]]

]

(4)

4 Journal of Robotics

From Figure 1 it can also be seen that 119860a1

=

[1198712 minus1198712 0]119879 119860a2= [1198712 1198712 0]

119879 119860a3= [minus1198712 1198712 0]

119879and 119860a

4= [minus1198712 minus1198712 0]

119879 With the position vectors ofpoints 119860

119894and 119861

119894now known the vector equation of the 119894th

linear generator can be written as

L119894=119860b119894minus119860a119894 119894 = 1 2 3 4 (5)

By using (5) the solution to the direct kinematic problemis found as follows

1198711= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

(6)

1198712= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

(7)

1198713= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

(8)

Here for the latter use the length of the linear generatorjoining nodes 119860

41198614is also presented

1198714= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

+ 1198712]

12

(9)

32 Inverse Kinematic Analysis The inverse kinematic prob-lem corresponds to the computation of the input vector I forthe given output vector O The solution to this problem canbe found by solving (6)ndash(8) for the input variables 119910 119911 and120579 Subtracting the square of (7) from that of (6) yields

2119871119910 + 1198712 cos 120579 minus 119871

2minus 1198712

1+ 1198712

2minus 119863119871 sin 120579 = 0 (10)

Subtracting the square of (8) from that of (6) we obtain

2119871119910 (cos 120579 + 1) minus 119863119871 sin 120579 minus 1198712

1+ 1198712

3minus 2119871119911 sin 120579 = 0 (11)

From (10) and (11) the following expressions can bederived

119910 =1

2119871(1198712+ 1198712

1minus 1198712

2minus 1198712 cos 120579 + 119863119871 sin 120579)

119911 =1

2119871 sin 120579[1198712sin2120579 + 119863119871 sin 120579 cos 120579

+ (1198712

1minus 1198712

2) cos 120579 minus 119871

2

2+ 1198712

3]

(12)

By substituting (12) into (6) the following equation isobtained

4 (1198712minus 1198712

2) 1198712cos2120579

minus [(1198712

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

] cos 120579

minus 41198712(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

= 0

(13)

Because of the range imposed on 120579 four solutions for 120579can be arrived at by solving (13) Furthermore by substitutingthese results into (12) the solutions to the inverse kinematicproblem are found

4 Singularity Analysis

41 Jacobian Matrix The Jacobian matrix of the harvesteris defined as the relationships between a set of infinitesimalchanges of its input vectors and the corresponding infinites-imal changes of its output vectors The Jacobian matrix Jrelates 120575I to 120575O such that 120575O = J120575I J can be rewritten interms of matrices C and D such that C120575O = D120575I From (6)ndash(8) the elements of C andD can be computed and written interms of the input variables as follows

11986211= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

11986222= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

11986233= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

11986212= 11986213= 11986221= 11986223= 11986231= 11986232= 0

11986311= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 + 119871

2

11986312= 11986322= 119911 minus

119863

2cos 120579 minus 119871

2sin 120579

11986313= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

Journal of Robotics 5

11986321= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986331= 119910 minus

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986332= 119911 minus

119863

2cos 120579 + 119871

2sin 120579

11986323= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

+119863

2cos 120579)

11986333= (119911 minus

119863

2cos 120579 + 119871

2sin 120579) (119863

2sin 120579 + 119871

2cos 120579)

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

(14)

For (14) it is noted that 119862119894119895and 119863

119894119895are the elements located

on the 119894th line and 119895th column of C andD respectively

42 Singular Configurations The singular configurations ofthe harvester consist in finding the situations where therelationships between infinitesimal changes in its input andoutput variables degenerate When such a situation occursthe harvesterwill gain or lose one ormore degrees of freedomthus leading to a loss of control As a consequence suchconfigurations are usually avoided when possible Generallythe singular configurations of the harvester can be obtainedby setting det(C) = 0 det(D) = 0 or both The determinantsof C andD can be expressed as follows

det (C) = [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

sdot [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

sdot [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

det (D) = 1198712

8[(119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910]

= 0

(15)

By examining (15) it is possible to extract the expressionscorresponding to singular configurations The following isa list of these expressions as well as their descriptions withrespect to the mechanismrsquos behaviors

(i) [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

= 0

(16)

(a) The length of the linear generator joining nodes 1198601

and 1198611is equal to zero Node 119860

1is thus coincident

with node 1198611 Moreover node 119860

4is also coincident

with node 1198612

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

1and 119860

4 When this

is the case only one variable is needed to definethe system The harvester thus loses two degrees offreedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198601and 119861

4are

possible without deforming the springs and the lineargenerators

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(ii) [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

(17)

(a) The length of the linear generator joining nodes 1198603

and 1198613is equal to zero Node 119860

2is coincident with

node 1198614

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

2and 119860

3 When this

case occurs only one variable can be used to describethe rotation of the float The harvester thus loses twodegrees of freedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198602and 119861

1are

possible without deforming the springs and the lineargenerators

6 Journal of Robotics

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(iii) (119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910

= 0

(18)

(a) Actually it is impossible to extract the behaviors ofthe harvester from (18) This case corresponds tothe boundaries of the input workspace and will bemapped in Section 52 Generally speaking whenthis is the case infinitesimal movements of the inputvariables along a direction perpendicular to a certainsurface cannot be generated

From (16) and (17) it can be seen that the singular con-figuration (i) corresponds to the situation where the length ofthe linear generator119860

11198611is equal to zero while configuration

(ii) corresponds to the situation where the length of the lineargenerator 119860

31198613is equal to zero From an engineering point

of view the linear generators are generally limited to operatewithin a range of nonzero lengths However from the aspectof mechanismrsquos analysis the lengths of prismatic actuatorscan be set to be zero This case belongs to one kind of thesingular configurations of the proposed mechanism

5 Workspaces

Since the input variables 119910 119911 and 120579 are driven by waterwaves the ranges of the input variables can be used todescribe the strengths of the water waves Moreover theamount of the electricity produced by the harvester dependson the movements of the linear generators The ranges ofthe output variables can be considered as an indicator of theefficiency of energy harvesting In this section the rangesof the input vectors are referred to as the input workspacewhile the ranges of the output vectors are referred to as theoutput workspace The boundaries of the input and outputworkspaces usually correspond to singular configurationsdescribed in Section 42 From (16)ndash(18) it can be seen thatthe singular configurations are expressed in terms of the inputvariables According to these expressions the boundaries ofthe input workspace can be computed Afterwards theseboundaries will be mapped from the input domain into theoutput domain in order to generate the output workspace

51 Input Workspace The input workspace of the harvesteris a volume whose boundaries correspond to singular con-figurations discussed in Section 42 An example of such aworkspace with 119871 = 1m and119863 = 1m is shown in Figure 2

In Figure 2 the surface corresponding to the singularconfiguration (iii) is identified by surface (iii) From thisfigure it can be seen that the input workspace can be dividedinto three parts The first part is defined by minus1 le 119910 le 0 and0 le 120579 le 1205872 It is bounded by surface (iii) and the planes

151050y (m)

005

1

i

iiiiii

iii

ii

0

05

1

15

2

z (m

)

minus05minus05

minus1minus1 minus15 120579 (rad)

Figure 2 Input workspace of the harvester with 119871 = 1m and 119863 =

01m

corresponding to 119910 = minus1 120579 = 1205872 and 119911 = 2 Moreover thesecond part is defined by 0 le 119910 le 1 and 0 le 120579 le 1205872 It isbounded by the planes corresponding to 120579 = 1205872 119910 = 1 andsurface (iii) Finally the third part is defined by 0 le 119910 le 1

and minus1205872 le 120579 le 0 It is bounded by the planes correspondingto 120579 = minus1205872 119910 = minus1 and surface (iii) Furthermorefrom Figure 2 it can also be observed that curves (i) and(ii) correspond to the singular configurations (i) and (ii)respectively Since the harvester will be uncontrolled whenit reaches a singular configuration the boundaries of theinput workspace and the singular curves (i) and (ii) shouldbe avoided during the use of such a harvester

52 Output Workspace In order to obtain the outputworkspace the singular configurations detailed in Section 42should be rewritten in terms of the output variables firstlyFrom (16) and (17) it can be concluded that the singu-lar configuration (i) in the output domain corresponds to1198711= 0 while the singular configuration (ii) corresponds to

1198712= 0 Generally speaking by substituting the solutions to

the inverse kinematic problem into (18) an expression forsingular configuration (iii) in terms of the output variablescan be arrived at However this procedure is rather tediousHere Bezoutrsquos method [36] was used to derive the expressioncorresponding to singular configuration (iii) in the outputdomain due to its simplicity

Equation (13) is firstly rewritten as

1198721cos2120579 +119872

2cos 120579 +119872

3= 0 (19)

where1198721= 4 (119871

2minus 1198712

2) 1198712

1198722= minus [(119871

2

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

]

1198723= minus4119871

2(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

(20)

Moreover by substituting (12) into (18) the following equa-tion is obtained

1198731cos2120579 + 119873

2cos 120579 + 119873

3= 0 (21)

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

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International Journal of

2 Journal of Robotics

Lineargenerator

Spring

Water wave

Sea bed

L

Z

Y

X

O

D

B9984002B998400

3

B9984004

B9984001

B2

B3

B1

B4

Z1

Y1

O998400X1

R2

R1

P

A3

A2

A4

A1

L3

L2

L1

L4

O1

Figure 1 A tensegrity-based water wave energy harvester

[24] and a tensegrity walking robot [25ndash27] For tensegritymechanisms an interesting topic named tensegrity parallelmechanism has been proposed recently The concept oftensegrity parallel mechanism was introduced by Marshall[28] Then Shekarforoush et al [29] presented the statics ofa 3-3 tensegrity parallel mechanism Afterwards Crane III etal [30] proposed a planar tensegrity parallel mechanism andcompleted its equilibrium analysis Tensegrity systems havebeen identified as one of three main research trends in mech-anisms and robotics for the second decade of the 21st century[31] However just a few references have stated the possibilityof using tensegrity systems as water wave energy harvestersScruggs and Skelton [32]made a preliminary investigation onthe potential use of controlled tensegrity structures to harvestenergy Sunny et al [33] studied the feasibility of harvestingenergy using polyvinylidene fluoride patches mounted onvibrating prestressedmembrane Vasquez et al [34] stated thepossibility of using a planar tensegrity mechanism in oceanapplications This application is attractive since it can play animportant role in the expansion of clean energy technologiesthat help the worldrsquos sustainable development

This work presents the analysis of a tensegrity-basedwater wave energy harvester Since this is the first stage forthe development of a new application for tensegrity systemsa simplified linear model of sea waves was used to analyzethe proposed harvester The analytical solutions to the directand inverse kinematic problems are found using a geometricmethod Based on the obtained relationships between theinput and output variables the singular configurations havebeen discussed The workspaces of the proposed mechanismhave subsequently been computed Afterwards the dynamicswere investigated Finally the energy harvesting capabilitiesof the tensegrity-based harvester are compared with a con-ventional heaving system

2 Geometry of the Water WaveEnergy Harvester

A diagram of the tensegrity-based water wave energy har-vester is shown in Figure 1 It consists of a float four springsfour linear generators and one kinematic chain The lineargenerators are joining node pairs 119860

119894119861119894(119894 = 1 2 3 4) while

the springs are joining node pairs 11986011198614 11986021198611 11986031198612 and

11986041198613 The float of height 119863 is denoted by 119861

1119861211986131198614 This

harvester is obtained from a square tensegrity parallel prism[10] by connecting the top of the latter to a float

From Figure 1 it can be seen that the sides of the squaresformed by nodes 119860

1119860211986031198604 1198611119861211986131198614 and 119861

1015840

11198611015840

21198611015840

31198611015840

4

have the same length 119871 Moreover the length of the lineargenerator joining node pairs 119860

119894119861119894is denoted by 119871

119894 As

illustrated in Figure 1 the springs and the linear generatorsare connected to the float and the sea bed at nodes 119860

119894and 119861

119894

by spherical joints without frictionThe sea bed is consideredto be parallel to the horizontal plane A fixed reference frame119860 (119883 119884 119885) is located at the center of the square 119860

1119860211986031198604

with its 119883 axis parallel to the line joining nodes 1198604and 119860

1

and its 119885 axis perpendicular to the sea bed while a movingreference frame 119861 (119883

1 1198841 1198851) is located at the mass center

of the float with its 1198831axis parallel to the line joining nodes

1198614and 119861

1and its 119885

1axis perpendicular to the plane formed

by nodes 1198611 1198612 1198613 and 119861

4 Moreover the vectors specifying

the positions of nodes 119860119894and 119861

119894in the fixed reference frame

are defined as 119860a119894and 119860b

119894 respectively Also the vectors

specifying the positions of nodes 119861119894in the moving reference

frame are defined as 119861b119894

In order to obtain an appropriate kinematic model of theharvester the following hypotheses are made

(i) The springs are linear with stiffness 119870 and lengths119897119895(119895 = 1 2 3 4) and all the springs have the same free

length 1198710

Journal of Robotics 3

(ii) The water waves are traveling along the 119884 axis

In Figure 1 a passive kinematic chain denoted by 11987711198751198772

is used to connect nodes 119874 and 1198741 Nodes 119874 and 119874

1rep-

resent the centers of the squares 1198601119860211986031198604and 119861

1119861211986131198614

respectively Considering the constraints introduced by thiskinematic chain the possible movements of the float drivenby water waves are rotations about the119883 axis and translationsalong the 119884 and 119885 axes Therefore the harvester has threedegrees of freedom

The Cartesian coordinates of the mass center of the floatin the fixed reference frame are defined as (119909 119910 119911) FromFigure 1 it can be seen that119909 = 0 is always satisfiedMoreoverthe angle 120579 is used to specify the rotation of the float about119883axis Meanwhile the range of 120579 is assumed to be [minus1205872 1205872]The variables 119910 119911 and 120579 are driven by the water waves Asa consequence they are thus chosen as the inputs of thesystem Furthermore only three of the four linear generatorsrsquolengths are independent For this reason the lengths of thegenerators joining nodes 119860

11198611 11986021198612 and 119860

31198613are chosen

as the outputs of the system It follows that the harvesterrsquosoutput vector is O = [119871

1 1198712 1198713]119879 while its input vector is

I = [119910 119911 120579]119879

3 Kinematic Analysis

For the harvester the linear generators are used to convertwave motion cleanly into electricity Generally the efficiencyof electricity generation of the system is highly dependent onthemotions of linear generators To provide great insight intothe kinematics of the harvester the relationship between theinput and output vectors is developed in this section

31 Direct Kinematic Analysis The direct kinematic analysisconsists in computing the output vector O for the giveninput vector I According to [35] the most convenientapproach to set an algebraic equation system for kinematicproblem of a parallel mechanism is to use the rotation matrixparameters and the position vector of the moving platformThis approach is used in this work to deal with the kinematicproblems of the harvester The position and orientation ofthe float are described by the position vector P = OO1015840 =[0 119910 119911]

119879 and the rotationmatrix 119860R119861with respect to the fixed

reference frame FromFigure 1 it can be seen that the rotationmatrix 119860R

119861can be defined by rotating the moving reference

frame1205872 about1198851axis followed by 120579 about119884

1axis 119860R

119861thus

takes the following form

119860R119861=[[

[

0 minus1 0

cos 120579 0 sin 120579minus sin 120579 0 cos 120579

]]

]

(1)

Then the position vectors of points 119861119894(119894 = 1 2 3 4) with

respect to the fixed reference frame can be obtained

119860b119894= P +

119860R119861

119861b119894 119894 = 1 2 3 4 (2)

The vectors specifying the positions of nodes 119861119894in the

moving reference frame can be easily derived

119861b1=

[[[[[[[

[

119871

2

minus119871

2

minus119863

2

]]]]]]]

]

119861b2=

[[[[[[[

[

119871

2

119871

2

minus119863

2

]]]]]]]

]

119861b3=

[[[[[[[

[

minus119871

2

119871

2

minus119863

2

]]]]]]]

]

119861b4=

[[[[[[[

[

minus119871

2

minus119871

2

minus119863

2

]]]]]]]

]

(3)

Substituting (3) into (2) we have

119860b1=

[[[[[[

[

119871

2

119910 +119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 minus 119871

2sin 120579

]]]]]]

]

119860b2=

[[[[[[

[

minus119871

2

119910 +119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 minus 119871

2sin 120579

]]]]]]

]

119860b3=

[[[[[[

[

minus119871

2

119910 minus119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 + 119871

2sin 120579

]]]]]]

]

119860b4=

[[[[[[

[

119871

2

119910 minus119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 + 119871

2sin 120579

]]]]]]

]

(4)

4 Journal of Robotics

From Figure 1 it can also be seen that 119860a1

=

[1198712 minus1198712 0]119879 119860a2= [1198712 1198712 0]

119879 119860a3= [minus1198712 1198712 0]

119879and 119860a

4= [minus1198712 minus1198712 0]

119879 With the position vectors ofpoints 119860

119894and 119861

119894now known the vector equation of the 119894th

linear generator can be written as

L119894=119860b119894minus119860a119894 119894 = 1 2 3 4 (5)

By using (5) the solution to the direct kinematic problemis found as follows

1198711= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

(6)

1198712= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

(7)

1198713= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

(8)

Here for the latter use the length of the linear generatorjoining nodes 119860

41198614is also presented

1198714= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

+ 1198712]

12

(9)

32 Inverse Kinematic Analysis The inverse kinematic prob-lem corresponds to the computation of the input vector I forthe given output vector O The solution to this problem canbe found by solving (6)ndash(8) for the input variables 119910 119911 and120579 Subtracting the square of (7) from that of (6) yields

2119871119910 + 1198712 cos 120579 minus 119871

2minus 1198712

1+ 1198712

2minus 119863119871 sin 120579 = 0 (10)

Subtracting the square of (8) from that of (6) we obtain

2119871119910 (cos 120579 + 1) minus 119863119871 sin 120579 minus 1198712

1+ 1198712

3minus 2119871119911 sin 120579 = 0 (11)

From (10) and (11) the following expressions can bederived

119910 =1

2119871(1198712+ 1198712

1minus 1198712

2minus 1198712 cos 120579 + 119863119871 sin 120579)

119911 =1

2119871 sin 120579[1198712sin2120579 + 119863119871 sin 120579 cos 120579

+ (1198712

1minus 1198712

2) cos 120579 minus 119871

2

2+ 1198712

3]

(12)

By substituting (12) into (6) the following equation isobtained

4 (1198712minus 1198712

2) 1198712cos2120579

minus [(1198712

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

] cos 120579

minus 41198712(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

= 0

(13)

Because of the range imposed on 120579 four solutions for 120579can be arrived at by solving (13) Furthermore by substitutingthese results into (12) the solutions to the inverse kinematicproblem are found

4 Singularity Analysis

41 Jacobian Matrix The Jacobian matrix of the harvesteris defined as the relationships between a set of infinitesimalchanges of its input vectors and the corresponding infinites-imal changes of its output vectors The Jacobian matrix Jrelates 120575I to 120575O such that 120575O = J120575I J can be rewritten interms of matrices C and D such that C120575O = D120575I From (6)ndash(8) the elements of C andD can be computed and written interms of the input variables as follows

11986211= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

11986222= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

11986233= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

11986212= 11986213= 11986221= 11986223= 11986231= 11986232= 0

11986311= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 + 119871

2

11986312= 11986322= 119911 minus

119863

2cos 120579 minus 119871

2sin 120579

11986313= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

Journal of Robotics 5

11986321= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986331= 119910 minus

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986332= 119911 minus

119863

2cos 120579 + 119871

2sin 120579

11986323= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

+119863

2cos 120579)

11986333= (119911 minus

119863

2cos 120579 + 119871

2sin 120579) (119863

2sin 120579 + 119871

2cos 120579)

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

(14)

For (14) it is noted that 119862119894119895and 119863

119894119895are the elements located

on the 119894th line and 119895th column of C andD respectively

42 Singular Configurations The singular configurations ofthe harvester consist in finding the situations where therelationships between infinitesimal changes in its input andoutput variables degenerate When such a situation occursthe harvesterwill gain or lose one ormore degrees of freedomthus leading to a loss of control As a consequence suchconfigurations are usually avoided when possible Generallythe singular configurations of the harvester can be obtainedby setting det(C) = 0 det(D) = 0 or both The determinantsof C andD can be expressed as follows

det (C) = [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

sdot [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

sdot [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

det (D) = 1198712

8[(119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910]

= 0

(15)

By examining (15) it is possible to extract the expressionscorresponding to singular configurations The following isa list of these expressions as well as their descriptions withrespect to the mechanismrsquos behaviors

(i) [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

= 0

(16)

(a) The length of the linear generator joining nodes 1198601

and 1198611is equal to zero Node 119860

1is thus coincident

with node 1198611 Moreover node 119860

4is also coincident

with node 1198612

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

1and 119860

4 When this

is the case only one variable is needed to definethe system The harvester thus loses two degrees offreedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198601and 119861

4are

possible without deforming the springs and the lineargenerators

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(ii) [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

(17)

(a) The length of the linear generator joining nodes 1198603

and 1198613is equal to zero Node 119860

2is coincident with

node 1198614

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

2and 119860

3 When this

case occurs only one variable can be used to describethe rotation of the float The harvester thus loses twodegrees of freedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198602and 119861

1are

possible without deforming the springs and the lineargenerators

6 Journal of Robotics

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(iii) (119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910

= 0

(18)

(a) Actually it is impossible to extract the behaviors ofthe harvester from (18) This case corresponds tothe boundaries of the input workspace and will bemapped in Section 52 Generally speaking whenthis is the case infinitesimal movements of the inputvariables along a direction perpendicular to a certainsurface cannot be generated

From (16) and (17) it can be seen that the singular con-figuration (i) corresponds to the situation where the length ofthe linear generator119860

11198611is equal to zero while configuration

(ii) corresponds to the situation where the length of the lineargenerator 119860

31198613is equal to zero From an engineering point

of view the linear generators are generally limited to operatewithin a range of nonzero lengths However from the aspectof mechanismrsquos analysis the lengths of prismatic actuatorscan be set to be zero This case belongs to one kind of thesingular configurations of the proposed mechanism

5 Workspaces

Since the input variables 119910 119911 and 120579 are driven by waterwaves the ranges of the input variables can be used todescribe the strengths of the water waves Moreover theamount of the electricity produced by the harvester dependson the movements of the linear generators The ranges ofthe output variables can be considered as an indicator of theefficiency of energy harvesting In this section the rangesof the input vectors are referred to as the input workspacewhile the ranges of the output vectors are referred to as theoutput workspace The boundaries of the input and outputworkspaces usually correspond to singular configurationsdescribed in Section 42 From (16)ndash(18) it can be seen thatthe singular configurations are expressed in terms of the inputvariables According to these expressions the boundaries ofthe input workspace can be computed Afterwards theseboundaries will be mapped from the input domain into theoutput domain in order to generate the output workspace

51 Input Workspace The input workspace of the harvesteris a volume whose boundaries correspond to singular con-figurations discussed in Section 42 An example of such aworkspace with 119871 = 1m and119863 = 1m is shown in Figure 2

In Figure 2 the surface corresponding to the singularconfiguration (iii) is identified by surface (iii) From thisfigure it can be seen that the input workspace can be dividedinto three parts The first part is defined by minus1 le 119910 le 0 and0 le 120579 le 1205872 It is bounded by surface (iii) and the planes

151050y (m)

005

1

i

iiiiii

iii

ii

0

05

1

15

2

z (m

)

minus05minus05

minus1minus1 minus15 120579 (rad)

Figure 2 Input workspace of the harvester with 119871 = 1m and 119863 =

01m

corresponding to 119910 = minus1 120579 = 1205872 and 119911 = 2 Moreover thesecond part is defined by 0 le 119910 le 1 and 0 le 120579 le 1205872 It isbounded by the planes corresponding to 120579 = 1205872 119910 = 1 andsurface (iii) Finally the third part is defined by 0 le 119910 le 1

and minus1205872 le 120579 le 0 It is bounded by the planes correspondingto 120579 = minus1205872 119910 = minus1 and surface (iii) Furthermorefrom Figure 2 it can also be observed that curves (i) and(ii) correspond to the singular configurations (i) and (ii)respectively Since the harvester will be uncontrolled whenit reaches a singular configuration the boundaries of theinput workspace and the singular curves (i) and (ii) shouldbe avoided during the use of such a harvester

52 Output Workspace In order to obtain the outputworkspace the singular configurations detailed in Section 42should be rewritten in terms of the output variables firstlyFrom (16) and (17) it can be concluded that the singu-lar configuration (i) in the output domain corresponds to1198711= 0 while the singular configuration (ii) corresponds to

1198712= 0 Generally speaking by substituting the solutions to

the inverse kinematic problem into (18) an expression forsingular configuration (iii) in terms of the output variablescan be arrived at However this procedure is rather tediousHere Bezoutrsquos method [36] was used to derive the expressioncorresponding to singular configuration (iii) in the outputdomain due to its simplicity

Equation (13) is firstly rewritten as

1198721cos2120579 +119872

2cos 120579 +119872

3= 0 (19)

where1198721= 4 (119871

2minus 1198712

2) 1198712

1198722= minus [(119871

2

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

]

1198723= minus4119871

2(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

(20)

Moreover by substituting (12) into (18) the following equa-tion is obtained

1198731cos2120579 + 119873

2cos 120579 + 119873

3= 0 (21)

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

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DistributedSensor Networks

International Journal of

Journal of Robotics 3

(ii) The water waves are traveling along the 119884 axis

In Figure 1 a passive kinematic chain denoted by 11987711198751198772

is used to connect nodes 119874 and 1198741 Nodes 119874 and 119874

1rep-

resent the centers of the squares 1198601119860211986031198604and 119861

1119861211986131198614

respectively Considering the constraints introduced by thiskinematic chain the possible movements of the float drivenby water waves are rotations about the119883 axis and translationsalong the 119884 and 119885 axes Therefore the harvester has threedegrees of freedom

The Cartesian coordinates of the mass center of the floatin the fixed reference frame are defined as (119909 119910 119911) FromFigure 1 it can be seen that119909 = 0 is always satisfiedMoreoverthe angle 120579 is used to specify the rotation of the float about119883axis Meanwhile the range of 120579 is assumed to be [minus1205872 1205872]The variables 119910 119911 and 120579 are driven by the water waves Asa consequence they are thus chosen as the inputs of thesystem Furthermore only three of the four linear generatorsrsquolengths are independent For this reason the lengths of thegenerators joining nodes 119860

11198611 11986021198612 and 119860

31198613are chosen

as the outputs of the system It follows that the harvesterrsquosoutput vector is O = [119871

1 1198712 1198713]119879 while its input vector is

I = [119910 119911 120579]119879

3 Kinematic Analysis

For the harvester the linear generators are used to convertwave motion cleanly into electricity Generally the efficiencyof electricity generation of the system is highly dependent onthemotions of linear generators To provide great insight intothe kinematics of the harvester the relationship between theinput and output vectors is developed in this section

31 Direct Kinematic Analysis The direct kinematic analysisconsists in computing the output vector O for the giveninput vector I According to [35] the most convenientapproach to set an algebraic equation system for kinematicproblem of a parallel mechanism is to use the rotation matrixparameters and the position vector of the moving platformThis approach is used in this work to deal with the kinematicproblems of the harvester The position and orientation ofthe float are described by the position vector P = OO1015840 =[0 119910 119911]

119879 and the rotationmatrix 119860R119861with respect to the fixed

reference frame FromFigure 1 it can be seen that the rotationmatrix 119860R

119861can be defined by rotating the moving reference

frame1205872 about1198851axis followed by 120579 about119884

1axis 119860R

119861thus

takes the following form

119860R119861=[[

[

0 minus1 0

cos 120579 0 sin 120579minus sin 120579 0 cos 120579

]]

]

(1)

Then the position vectors of points 119861119894(119894 = 1 2 3 4) with

respect to the fixed reference frame can be obtained

119860b119894= P +

119860R119861

119861b119894 119894 = 1 2 3 4 (2)

The vectors specifying the positions of nodes 119861119894in the

moving reference frame can be easily derived

119861b1=

[[[[[[[

[

119871

2

minus119871

2

minus119863

2

]]]]]]]

]

119861b2=

[[[[[[[

[

119871

2

119871

2

minus119863

2

]]]]]]]

]

119861b3=

[[[[[[[

[

minus119871

2

119871

2

minus119863

2

]]]]]]]

]

119861b4=

[[[[[[[

[

minus119871

2

minus119871

2

minus119863

2

]]]]]]]

]

(3)

Substituting (3) into (2) we have

119860b1=

[[[[[[

[

119871

2

119910 +119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 minus 119871

2sin 120579

]]]]]]

]

119860b2=

[[[[[[

[

minus119871

2

119910 +119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 minus 119871

2sin 120579

]]]]]]

]

119860b3=

[[[[[[

[

minus119871

2

119910 minus119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 + 119871

2sin 120579

]]]]]]

]

119860b4=

[[[[[[

[

119871

2

119910 minus119871

2cos 120579 minus 119863

2sin 120579

119911 minus119863

2cos 120579 + 119871

2sin 120579

]]]]]]

]

(4)

4 Journal of Robotics

From Figure 1 it can also be seen that 119860a1

=

[1198712 minus1198712 0]119879 119860a2= [1198712 1198712 0]

119879 119860a3= [minus1198712 1198712 0]

119879and 119860a

4= [minus1198712 minus1198712 0]

119879 With the position vectors ofpoints 119860

119894and 119861

119894now known the vector equation of the 119894th

linear generator can be written as

L119894=119860b119894minus119860a119894 119894 = 1 2 3 4 (5)

By using (5) the solution to the direct kinematic problemis found as follows

1198711= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

(6)

1198712= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

(7)

1198713= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

(8)

Here for the latter use the length of the linear generatorjoining nodes 119860

41198614is also presented

1198714= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

+ 1198712]

12

(9)

32 Inverse Kinematic Analysis The inverse kinematic prob-lem corresponds to the computation of the input vector I forthe given output vector O The solution to this problem canbe found by solving (6)ndash(8) for the input variables 119910 119911 and120579 Subtracting the square of (7) from that of (6) yields

2119871119910 + 1198712 cos 120579 minus 119871

2minus 1198712

1+ 1198712

2minus 119863119871 sin 120579 = 0 (10)

Subtracting the square of (8) from that of (6) we obtain

2119871119910 (cos 120579 + 1) minus 119863119871 sin 120579 minus 1198712

1+ 1198712

3minus 2119871119911 sin 120579 = 0 (11)

From (10) and (11) the following expressions can bederived

119910 =1

2119871(1198712+ 1198712

1minus 1198712

2minus 1198712 cos 120579 + 119863119871 sin 120579)

119911 =1

2119871 sin 120579[1198712sin2120579 + 119863119871 sin 120579 cos 120579

+ (1198712

1minus 1198712

2) cos 120579 minus 119871

2

2+ 1198712

3]

(12)

By substituting (12) into (6) the following equation isobtained

4 (1198712minus 1198712

2) 1198712cos2120579

minus [(1198712

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

] cos 120579

minus 41198712(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

= 0

(13)

Because of the range imposed on 120579 four solutions for 120579can be arrived at by solving (13) Furthermore by substitutingthese results into (12) the solutions to the inverse kinematicproblem are found

4 Singularity Analysis

41 Jacobian Matrix The Jacobian matrix of the harvesteris defined as the relationships between a set of infinitesimalchanges of its input vectors and the corresponding infinites-imal changes of its output vectors The Jacobian matrix Jrelates 120575I to 120575O such that 120575O = J120575I J can be rewritten interms of matrices C and D such that C120575O = D120575I From (6)ndash(8) the elements of C andD can be computed and written interms of the input variables as follows

11986211= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

11986222= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

11986233= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

11986212= 11986213= 11986221= 11986223= 11986231= 11986232= 0

11986311= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 + 119871

2

11986312= 11986322= 119911 minus

119863

2cos 120579 minus 119871

2sin 120579

11986313= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

Journal of Robotics 5

11986321= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986331= 119910 minus

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986332= 119911 minus

119863

2cos 120579 + 119871

2sin 120579

11986323= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

+119863

2cos 120579)

11986333= (119911 minus

119863

2cos 120579 + 119871

2sin 120579) (119863

2sin 120579 + 119871

2cos 120579)

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

(14)

For (14) it is noted that 119862119894119895and 119863

119894119895are the elements located

on the 119894th line and 119895th column of C andD respectively

42 Singular Configurations The singular configurations ofthe harvester consist in finding the situations where therelationships between infinitesimal changes in its input andoutput variables degenerate When such a situation occursthe harvesterwill gain or lose one ormore degrees of freedomthus leading to a loss of control As a consequence suchconfigurations are usually avoided when possible Generallythe singular configurations of the harvester can be obtainedby setting det(C) = 0 det(D) = 0 or both The determinantsof C andD can be expressed as follows

det (C) = [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

sdot [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

sdot [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

det (D) = 1198712

8[(119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910]

= 0

(15)

By examining (15) it is possible to extract the expressionscorresponding to singular configurations The following isa list of these expressions as well as their descriptions withrespect to the mechanismrsquos behaviors

(i) [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

= 0

(16)

(a) The length of the linear generator joining nodes 1198601

and 1198611is equal to zero Node 119860

1is thus coincident

with node 1198611 Moreover node 119860

4is also coincident

with node 1198612

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

1and 119860

4 When this

is the case only one variable is needed to definethe system The harvester thus loses two degrees offreedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198601and 119861

4are

possible without deforming the springs and the lineargenerators

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(ii) [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

(17)

(a) The length of the linear generator joining nodes 1198603

and 1198613is equal to zero Node 119860

2is coincident with

node 1198614

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

2and 119860

3 When this

case occurs only one variable can be used to describethe rotation of the float The harvester thus loses twodegrees of freedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198602and 119861

1are

possible without deforming the springs and the lineargenerators

6 Journal of Robotics

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(iii) (119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910

= 0

(18)

(a) Actually it is impossible to extract the behaviors ofthe harvester from (18) This case corresponds tothe boundaries of the input workspace and will bemapped in Section 52 Generally speaking whenthis is the case infinitesimal movements of the inputvariables along a direction perpendicular to a certainsurface cannot be generated

From (16) and (17) it can be seen that the singular con-figuration (i) corresponds to the situation where the length ofthe linear generator119860

11198611is equal to zero while configuration

(ii) corresponds to the situation where the length of the lineargenerator 119860

31198613is equal to zero From an engineering point

of view the linear generators are generally limited to operatewithin a range of nonzero lengths However from the aspectof mechanismrsquos analysis the lengths of prismatic actuatorscan be set to be zero This case belongs to one kind of thesingular configurations of the proposed mechanism

5 Workspaces

Since the input variables 119910 119911 and 120579 are driven by waterwaves the ranges of the input variables can be used todescribe the strengths of the water waves Moreover theamount of the electricity produced by the harvester dependson the movements of the linear generators The ranges ofthe output variables can be considered as an indicator of theefficiency of energy harvesting In this section the rangesof the input vectors are referred to as the input workspacewhile the ranges of the output vectors are referred to as theoutput workspace The boundaries of the input and outputworkspaces usually correspond to singular configurationsdescribed in Section 42 From (16)ndash(18) it can be seen thatthe singular configurations are expressed in terms of the inputvariables According to these expressions the boundaries ofthe input workspace can be computed Afterwards theseboundaries will be mapped from the input domain into theoutput domain in order to generate the output workspace

51 Input Workspace The input workspace of the harvesteris a volume whose boundaries correspond to singular con-figurations discussed in Section 42 An example of such aworkspace with 119871 = 1m and119863 = 1m is shown in Figure 2

In Figure 2 the surface corresponding to the singularconfiguration (iii) is identified by surface (iii) From thisfigure it can be seen that the input workspace can be dividedinto three parts The first part is defined by minus1 le 119910 le 0 and0 le 120579 le 1205872 It is bounded by surface (iii) and the planes

151050y (m)

005

1

i

iiiiii

iii

ii

0

05

1

15

2

z (m

)

minus05minus05

minus1minus1 minus15 120579 (rad)

Figure 2 Input workspace of the harvester with 119871 = 1m and 119863 =

01m

corresponding to 119910 = minus1 120579 = 1205872 and 119911 = 2 Moreover thesecond part is defined by 0 le 119910 le 1 and 0 le 120579 le 1205872 It isbounded by the planes corresponding to 120579 = 1205872 119910 = 1 andsurface (iii) Finally the third part is defined by 0 le 119910 le 1

and minus1205872 le 120579 le 0 It is bounded by the planes correspondingto 120579 = minus1205872 119910 = minus1 and surface (iii) Furthermorefrom Figure 2 it can also be observed that curves (i) and(ii) correspond to the singular configurations (i) and (ii)respectively Since the harvester will be uncontrolled whenit reaches a singular configuration the boundaries of theinput workspace and the singular curves (i) and (ii) shouldbe avoided during the use of such a harvester

52 Output Workspace In order to obtain the outputworkspace the singular configurations detailed in Section 42should be rewritten in terms of the output variables firstlyFrom (16) and (17) it can be concluded that the singu-lar configuration (i) in the output domain corresponds to1198711= 0 while the singular configuration (ii) corresponds to

1198712= 0 Generally speaking by substituting the solutions to

the inverse kinematic problem into (18) an expression forsingular configuration (iii) in terms of the output variablescan be arrived at However this procedure is rather tediousHere Bezoutrsquos method [36] was used to derive the expressioncorresponding to singular configuration (iii) in the outputdomain due to its simplicity

Equation (13) is firstly rewritten as

1198721cos2120579 +119872

2cos 120579 +119872

3= 0 (19)

where1198721= 4 (119871

2minus 1198712

2) 1198712

1198722= minus [(119871

2

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

]

1198723= minus4119871

2(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

(20)

Moreover by substituting (12) into (18) the following equa-tion is obtained

1198731cos2120579 + 119873

2cos 120579 + 119873

3= 0 (21)

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Robotics

From Figure 1 it can also be seen that 119860a1

=

[1198712 minus1198712 0]119879 119860a2= [1198712 1198712 0]

119879 119860a3= [minus1198712 1198712 0]

119879and 119860a

4= [minus1198712 minus1198712 0]

119879 With the position vectors ofpoints 119860

119894and 119861

119894now known the vector equation of the 119894th

linear generator can be written as

L119894=119860b119894minus119860a119894 119894 = 1 2 3 4 (5)

By using (5) the solution to the direct kinematic problemis found as follows

1198711= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

(6)

1198712= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

(7)

1198713= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

(8)

Here for the latter use the length of the linear generatorjoining nodes 119860

41198614is also presented

1198714= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

+ 1198712]

12

(9)

32 Inverse Kinematic Analysis The inverse kinematic prob-lem corresponds to the computation of the input vector I forthe given output vector O The solution to this problem canbe found by solving (6)ndash(8) for the input variables 119910 119911 and120579 Subtracting the square of (7) from that of (6) yields

2119871119910 + 1198712 cos 120579 minus 119871

2minus 1198712

1+ 1198712

2minus 119863119871 sin 120579 = 0 (10)

Subtracting the square of (8) from that of (6) we obtain

2119871119910 (cos 120579 + 1) minus 119863119871 sin 120579 minus 1198712

1+ 1198712

3minus 2119871119911 sin 120579 = 0 (11)

From (10) and (11) the following expressions can bederived

119910 =1

2119871(1198712+ 1198712

1minus 1198712

2minus 1198712 cos 120579 + 119863119871 sin 120579)

119911 =1

2119871 sin 120579[1198712sin2120579 + 119863119871 sin 120579 cos 120579

+ (1198712

1minus 1198712

2) cos 120579 minus 119871

2

2+ 1198712

3]

(12)

By substituting (12) into (6) the following equation isobtained

4 (1198712minus 1198712

2) 1198712cos2120579

minus [(1198712

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

] cos 120579

minus 41198712(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

= 0

(13)

Because of the range imposed on 120579 four solutions for 120579can be arrived at by solving (13) Furthermore by substitutingthese results into (12) the solutions to the inverse kinematicproblem are found

4 Singularity Analysis

41 Jacobian Matrix The Jacobian matrix of the harvesteris defined as the relationships between a set of infinitesimalchanges of its input vectors and the corresponding infinites-imal changes of its output vectors The Jacobian matrix Jrelates 120575I to 120575O such that 120575O = J120575I J can be rewritten interms of matrices C and D such that C120575O = D120575I From (6)ndash(8) the elements of C andD can be computed and written interms of the input variables as follows

11986211= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

11986222= [(119911 minus

119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

11986233= [(119911 minus

119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

11986212= 11986213= 11986221= 11986223= 11986231= 11986232= 0

11986311= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 + 119871

2

11986312= 11986322= 119911 minus

119863

2cos 120579 minus 119871

2sin 120579

11986313= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

Journal of Robotics 5

11986321= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986331= 119910 minus

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986332= 119911 minus

119863

2cos 120579 + 119871

2sin 120579

11986323= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

+119863

2cos 120579)

11986333= (119911 minus

119863

2cos 120579 + 119871

2sin 120579) (119863

2sin 120579 + 119871

2cos 120579)

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

(14)

For (14) it is noted that 119862119894119895and 119863

119894119895are the elements located

on the 119894th line and 119895th column of C andD respectively

42 Singular Configurations The singular configurations ofthe harvester consist in finding the situations where therelationships between infinitesimal changes in its input andoutput variables degenerate When such a situation occursthe harvesterwill gain or lose one ormore degrees of freedomthus leading to a loss of control As a consequence suchconfigurations are usually avoided when possible Generallythe singular configurations of the harvester can be obtainedby setting det(C) = 0 det(D) = 0 or both The determinantsof C andD can be expressed as follows

det (C) = [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

sdot [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

sdot [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

det (D) = 1198712

8[(119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910]

= 0

(15)

By examining (15) it is possible to extract the expressionscorresponding to singular configurations The following isa list of these expressions as well as their descriptions withrespect to the mechanismrsquos behaviors

(i) [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

= 0

(16)

(a) The length of the linear generator joining nodes 1198601

and 1198611is equal to zero Node 119860

1is thus coincident

with node 1198611 Moreover node 119860

4is also coincident

with node 1198612

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

1and 119860

4 When this

is the case only one variable is needed to definethe system The harvester thus loses two degrees offreedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198601and 119861

4are

possible without deforming the springs and the lineargenerators

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(ii) [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

(17)

(a) The length of the linear generator joining nodes 1198603

and 1198613is equal to zero Node 119860

2is coincident with

node 1198614

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

2and 119860

3 When this

case occurs only one variable can be used to describethe rotation of the float The harvester thus loses twodegrees of freedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198602and 119861

1are

possible without deforming the springs and the lineargenerators

6 Journal of Robotics

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(iii) (119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910

= 0

(18)

(a) Actually it is impossible to extract the behaviors ofthe harvester from (18) This case corresponds tothe boundaries of the input workspace and will bemapped in Section 52 Generally speaking whenthis is the case infinitesimal movements of the inputvariables along a direction perpendicular to a certainsurface cannot be generated

From (16) and (17) it can be seen that the singular con-figuration (i) corresponds to the situation where the length ofthe linear generator119860

11198611is equal to zero while configuration

(ii) corresponds to the situation where the length of the lineargenerator 119860

31198613is equal to zero From an engineering point

of view the linear generators are generally limited to operatewithin a range of nonzero lengths However from the aspectof mechanismrsquos analysis the lengths of prismatic actuatorscan be set to be zero This case belongs to one kind of thesingular configurations of the proposed mechanism

5 Workspaces

Since the input variables 119910 119911 and 120579 are driven by waterwaves the ranges of the input variables can be used todescribe the strengths of the water waves Moreover theamount of the electricity produced by the harvester dependson the movements of the linear generators The ranges ofthe output variables can be considered as an indicator of theefficiency of energy harvesting In this section the rangesof the input vectors are referred to as the input workspacewhile the ranges of the output vectors are referred to as theoutput workspace The boundaries of the input and outputworkspaces usually correspond to singular configurationsdescribed in Section 42 From (16)ndash(18) it can be seen thatthe singular configurations are expressed in terms of the inputvariables According to these expressions the boundaries ofthe input workspace can be computed Afterwards theseboundaries will be mapped from the input domain into theoutput domain in order to generate the output workspace

51 Input Workspace The input workspace of the harvesteris a volume whose boundaries correspond to singular con-figurations discussed in Section 42 An example of such aworkspace with 119871 = 1m and119863 = 1m is shown in Figure 2

In Figure 2 the surface corresponding to the singularconfiguration (iii) is identified by surface (iii) From thisfigure it can be seen that the input workspace can be dividedinto three parts The first part is defined by minus1 le 119910 le 0 and0 le 120579 le 1205872 It is bounded by surface (iii) and the planes

151050y (m)

005

1

i

iiiiii

iii

ii

0

05

1

15

2

z (m

)

minus05minus05

minus1minus1 minus15 120579 (rad)

Figure 2 Input workspace of the harvester with 119871 = 1m and 119863 =

01m

corresponding to 119910 = minus1 120579 = 1205872 and 119911 = 2 Moreover thesecond part is defined by 0 le 119910 le 1 and 0 le 120579 le 1205872 It isbounded by the planes corresponding to 120579 = 1205872 119910 = 1 andsurface (iii) Finally the third part is defined by 0 le 119910 le 1

and minus1205872 le 120579 le 0 It is bounded by the planes correspondingto 120579 = minus1205872 119910 = minus1 and surface (iii) Furthermorefrom Figure 2 it can also be observed that curves (i) and(ii) correspond to the singular configurations (i) and (ii)respectively Since the harvester will be uncontrolled whenit reaches a singular configuration the boundaries of theinput workspace and the singular curves (i) and (ii) shouldbe avoided during the use of such a harvester

52 Output Workspace In order to obtain the outputworkspace the singular configurations detailed in Section 42should be rewritten in terms of the output variables firstlyFrom (16) and (17) it can be concluded that the singu-lar configuration (i) in the output domain corresponds to1198711= 0 while the singular configuration (ii) corresponds to

1198712= 0 Generally speaking by substituting the solutions to

the inverse kinematic problem into (18) an expression forsingular configuration (iii) in terms of the output variablescan be arrived at However this procedure is rather tediousHere Bezoutrsquos method [36] was used to derive the expressioncorresponding to singular configuration (iii) in the outputdomain due to its simplicity

Equation (13) is firstly rewritten as

1198721cos2120579 +119872

2cos 120579 +119872

3= 0 (19)

where1198721= 4 (119871

2minus 1198712

2) 1198712

1198722= minus [(119871

2

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

]

1198723= minus4119871

2(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

(20)

Moreover by substituting (12) into (18) the following equa-tion is obtained

1198731cos2120579 + 119873

2cos 120579 + 119873

3= 0 (21)

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Journal of Robotics 5

11986321= 119910 +

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986331= 119910 minus

119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2

11986332= 119911 minus

119863

2cos 120579 + 119871

2sin 120579

11986323= (119911 minus

119863

2cos 120579 minus 119871

2sin 120579) (119863

2sin 120579 minus 119871

2cos 120579)

minus (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

+119863

2cos 120579)

11986333= (119911 minus

119863

2cos 120579 + 119871

2sin 120579) (119863

2sin 120579 + 119871

2cos 120579)

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2) (

119871

2sin 120579

minus119863

2cos 120579)

(14)

For (14) it is noted that 119862119894119895and 119863

119894119895are the elements located

on the 119894th line and 119895th column of C andD respectively

42 Singular Configurations The singular configurations ofthe harvester consist in finding the situations where therelationships between infinitesimal changes in its input andoutput variables degenerate When such a situation occursthe harvesterwill gain or lose one ormore degrees of freedomthus leading to a loss of control As a consequence suchconfigurations are usually avoided when possible Generallythe singular configurations of the harvester can be obtainedby setting det(C) = 0 det(D) = 0 or both The determinantsof C andD can be expressed as follows

det (C) = [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

sdot [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

+ 1198712]

12

sdot [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

det (D) = 1198712

8[(119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910]

= 0

(15)

By examining (15) it is possible to extract the expressionscorresponding to singular configurations The following isa list of these expressions as well as their descriptions withrespect to the mechanismrsquos behaviors

(i) [(119911 minus119863

2cos 120579 minus 119871

2sin 120579)

2

+ (119910 +119871

2cos 120579 minus 119863

2sin 120579 + 119871

2)

2

]

12

= 0

(16)

(a) The length of the linear generator joining nodes 1198601

and 1198611is equal to zero Node 119860

1is thus coincident

with node 1198611 Moreover node 119860

4is also coincident

with node 1198612

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

1and 119860

4 When this

is the case only one variable is needed to definethe system The harvester thus loses two degrees offreedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198601and 119861

4are

possible without deforming the springs and the lineargenerators

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(ii) [(119911 minus119863

2cos 120579 + 119871

2sin 120579)

2

+ (119910 minus119871

2cos 120579 minus 119863

2sin 120579 minus 119871

2)

2

]

12

= 0

(17)

(a) The length of the linear generator joining nodes 1198603

and 1198613is equal to zero Node 119860

2is coincident with

node 1198614

(b) The movement of the float is reduced to a rotationabout the axis joining nodes 119860

2and 119860

3 When this

case occurs only one variable can be used to describethe rotation of the float The harvester thus loses twodegrees of freedom

(c) Infinitesimal movements of node 1198741015840 in a direction

perpendicular to the line joining nodes1198602and 119861

1are

possible without deforming the springs and the lineargenerators

6 Journal of Robotics

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(iii) (119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910

= 0

(18)

(a) Actually it is impossible to extract the behaviors ofthe harvester from (18) This case corresponds tothe boundaries of the input workspace and will bemapped in Section 52 Generally speaking whenthis is the case infinitesimal movements of the inputvariables along a direction perpendicular to a certainsurface cannot be generated

From (16) and (17) it can be seen that the singular con-figuration (i) corresponds to the situation where the length ofthe linear generator119860

11198611is equal to zero while configuration

(ii) corresponds to the situation where the length of the lineargenerator 119860

31198613is equal to zero From an engineering point

of view the linear generators are generally limited to operatewithin a range of nonzero lengths However from the aspectof mechanismrsquos analysis the lengths of prismatic actuatorscan be set to be zero This case belongs to one kind of thesingular configurations of the proposed mechanism

5 Workspaces

Since the input variables 119910 119911 and 120579 are driven by waterwaves the ranges of the input variables can be used todescribe the strengths of the water waves Moreover theamount of the electricity produced by the harvester dependson the movements of the linear generators The ranges ofthe output variables can be considered as an indicator of theefficiency of energy harvesting In this section the rangesof the input vectors are referred to as the input workspacewhile the ranges of the output vectors are referred to as theoutput workspace The boundaries of the input and outputworkspaces usually correspond to singular configurationsdescribed in Section 42 From (16)ndash(18) it can be seen thatthe singular configurations are expressed in terms of the inputvariables According to these expressions the boundaries ofthe input workspace can be computed Afterwards theseboundaries will be mapped from the input domain into theoutput domain in order to generate the output workspace

51 Input Workspace The input workspace of the harvesteris a volume whose boundaries correspond to singular con-figurations discussed in Section 42 An example of such aworkspace with 119871 = 1m and119863 = 1m is shown in Figure 2

In Figure 2 the surface corresponding to the singularconfiguration (iii) is identified by surface (iii) From thisfigure it can be seen that the input workspace can be dividedinto three parts The first part is defined by minus1 le 119910 le 0 and0 le 120579 le 1205872 It is bounded by surface (iii) and the planes

151050y (m)

005

1

i

iiiiii

iii

ii

0

05

1

15

2

z (m

)

minus05minus05

minus1minus1 minus15 120579 (rad)

Figure 2 Input workspace of the harvester with 119871 = 1m and 119863 =

01m

corresponding to 119910 = minus1 120579 = 1205872 and 119911 = 2 Moreover thesecond part is defined by 0 le 119910 le 1 and 0 le 120579 le 1205872 It isbounded by the planes corresponding to 120579 = 1205872 119910 = 1 andsurface (iii) Finally the third part is defined by 0 le 119910 le 1

and minus1205872 le 120579 le 0 It is bounded by the planes correspondingto 120579 = minus1205872 119910 = minus1 and surface (iii) Furthermorefrom Figure 2 it can also be observed that curves (i) and(ii) correspond to the singular configurations (i) and (ii)respectively Since the harvester will be uncontrolled whenit reaches a singular configuration the boundaries of theinput workspace and the singular curves (i) and (ii) shouldbe avoided during the use of such a harvester

52 Output Workspace In order to obtain the outputworkspace the singular configurations detailed in Section 42should be rewritten in terms of the output variables firstlyFrom (16) and (17) it can be concluded that the singu-lar configuration (i) in the output domain corresponds to1198711= 0 while the singular configuration (ii) corresponds to

1198712= 0 Generally speaking by substituting the solutions to

the inverse kinematic problem into (18) an expression forsingular configuration (iii) in terms of the output variablescan be arrived at However this procedure is rather tediousHere Bezoutrsquos method [36] was used to derive the expressioncorresponding to singular configuration (iii) in the outputdomain due to its simplicity

Equation (13) is firstly rewritten as

1198721cos2120579 +119872

2cos 120579 +119872

3= 0 (19)

where1198721= 4 (119871

2minus 1198712

2) 1198712

1198722= minus [(119871

2

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

]

1198723= minus4119871

2(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

(20)

Moreover by substituting (12) into (18) the following equa-tion is obtained

1198731cos2120579 + 119873

2cos 120579 + 119873

3= 0 (21)

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Robotics

(d) External forces parallel to the line 11986111198614are resisted by

the harvester

(iii) (119863119871 minus 2119863119910 minus 2119871119911) sin 2120579

+ (2119871119910 minus 2119863119911 minus 1198712) cos 2120579 + (8119911

2+ 21198632) cos 120579

+ (8119910119911 minus 4119871119911 + 2119863119871) sin 120579 + 1198712minus 6119863119911 minus 2119871119910

= 0

(18)

(a) Actually it is impossible to extract the behaviors ofthe harvester from (18) This case corresponds tothe boundaries of the input workspace and will bemapped in Section 52 Generally speaking whenthis is the case infinitesimal movements of the inputvariables along a direction perpendicular to a certainsurface cannot be generated

From (16) and (17) it can be seen that the singular con-figuration (i) corresponds to the situation where the length ofthe linear generator119860

11198611is equal to zero while configuration

(ii) corresponds to the situation where the length of the lineargenerator 119860

31198613is equal to zero From an engineering point

of view the linear generators are generally limited to operatewithin a range of nonzero lengths However from the aspectof mechanismrsquos analysis the lengths of prismatic actuatorscan be set to be zero This case belongs to one kind of thesingular configurations of the proposed mechanism

5 Workspaces

Since the input variables 119910 119911 and 120579 are driven by waterwaves the ranges of the input variables can be used todescribe the strengths of the water waves Moreover theamount of the electricity produced by the harvester dependson the movements of the linear generators The ranges ofthe output variables can be considered as an indicator of theefficiency of energy harvesting In this section the rangesof the input vectors are referred to as the input workspacewhile the ranges of the output vectors are referred to as theoutput workspace The boundaries of the input and outputworkspaces usually correspond to singular configurationsdescribed in Section 42 From (16)ndash(18) it can be seen thatthe singular configurations are expressed in terms of the inputvariables According to these expressions the boundaries ofthe input workspace can be computed Afterwards theseboundaries will be mapped from the input domain into theoutput domain in order to generate the output workspace

51 Input Workspace The input workspace of the harvesteris a volume whose boundaries correspond to singular con-figurations discussed in Section 42 An example of such aworkspace with 119871 = 1m and119863 = 1m is shown in Figure 2

In Figure 2 the surface corresponding to the singularconfiguration (iii) is identified by surface (iii) From thisfigure it can be seen that the input workspace can be dividedinto three parts The first part is defined by minus1 le 119910 le 0 and0 le 120579 le 1205872 It is bounded by surface (iii) and the planes

151050y (m)

005

1

i

iiiiii

iii

ii

0

05

1

15

2

z (m

)

minus05minus05

minus1minus1 minus15 120579 (rad)

Figure 2 Input workspace of the harvester with 119871 = 1m and 119863 =

01m

corresponding to 119910 = minus1 120579 = 1205872 and 119911 = 2 Moreover thesecond part is defined by 0 le 119910 le 1 and 0 le 120579 le 1205872 It isbounded by the planes corresponding to 120579 = 1205872 119910 = 1 andsurface (iii) Finally the third part is defined by 0 le 119910 le 1

and minus1205872 le 120579 le 0 It is bounded by the planes correspondingto 120579 = minus1205872 119910 = minus1 and surface (iii) Furthermorefrom Figure 2 it can also be observed that curves (i) and(ii) correspond to the singular configurations (i) and (ii)respectively Since the harvester will be uncontrolled whenit reaches a singular configuration the boundaries of theinput workspace and the singular curves (i) and (ii) shouldbe avoided during the use of such a harvester

52 Output Workspace In order to obtain the outputworkspace the singular configurations detailed in Section 42should be rewritten in terms of the output variables firstlyFrom (16) and (17) it can be concluded that the singu-lar configuration (i) in the output domain corresponds to1198711= 0 while the singular configuration (ii) corresponds to

1198712= 0 Generally speaking by substituting the solutions to

the inverse kinematic problem into (18) an expression forsingular configuration (iii) in terms of the output variablescan be arrived at However this procedure is rather tediousHere Bezoutrsquos method [36] was used to derive the expressioncorresponding to singular configuration (iii) in the outputdomain due to its simplicity

Equation (13) is firstly rewritten as

1198721cos2120579 +119872

2cos 120579 +119872

3= 0 (19)

where1198721= 4 (119871

2minus 1198712

2) 1198712

1198722= minus [(119871

2

1minus 1198712

2)2

+ (1198712

2minus 1198712

3)2

minus (1198712

1minus 1198712

3)2

]

1198723= minus4119871

2(1198712minus 1198712

2) minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

(20)

Moreover by substituting (12) into (18) the following equa-tion is obtained

1198731cos2120579 + 119873

2cos 120579 + 119873

3= 0 (21)

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Robotics 7

108

64

200

5

10

8

6

4

2

010

iv

v

vi

vi

vii

L1(m)

L2 (m)

L3

(m)

Figure 3Outputworkspace of thewaterwave energy harvesterwith119871 = 1m and119863 = 01m

where

1198731= (1198712

1minus 1198712

2) (1198712

2minus 1198712

3)

1198732= minus (119871

2

1minus 1198712

2)2

minus (1198712

2minus 1198712

3)2

1198733= (1198712

2minus 1198712

3) (1198712

2minus 1198712

3)

(22)

It should be noted that (21) represents singular config-uration (iii) expressed by 119871

1 1198712 1198713 and 120579 Moreover (19)

is used to compute 120579 for the given values of 1198711 1198712 and 119871

3

Generally the solutions to 120579 obtained by solving (21) shouldsatisfy (19) Furthermore both (19) and (21) can be consideredas two quadratics with respect to cos 120579 According to Bezoutrsquosmethod the condition that (19) and (21) have a comment rootfor cos 120579 is as follows

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198722

1198731

1198732

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

11987221198723

1198732

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

11987211198723

1198731

1198733

1003816100381610038161003816100381610038161003816100381610038161003816

2

= 0 (23)

Simplifying (23) yields

[(1198712

1minus 1198712

3) (1198712

1minus 21198712

2+ 1198712

3)]2

sdot (41198714minus 411987121198712

2+ 1198714

1minus 21198712

11198712

2+ 1198714

1)

sdot (41198714minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3) = 0

(24)

Equation (24) represents the surfaces corresponding tosingular configuration (iii) in the output workspace Byplotting these surfaces the output workspace of the harvestercan be obtained An example of such plots is shown inFigure 3 with 119871 = 1m and119863 = 01m

From Figure 3 it can be seen that the singular configu-ration (iii) determined by (24) corresponds to four surfaces(surfaces (iv)ndash(vii)) in the output workspace Moreoversurfaces (iv) (v) (vi) and (vii) correspond to expressions1198711minus1198713= 01198712

1minus21198712

2+1198712

3= 0 41198714minus411987121198712

2+1198714

1minus21198712

11198712

2+1198714

1= 0

and 41198714 minus 411987121198712

2+ 1198714

2minus 21198712

21198712

3+ 1198714

3= 0 respectively It can

also be observed that the output workspace of the harvester

can be divided into two parts The first part is bounded bysurface (v) surface (vi) plane 119871

1= 0 and plane 119871

3= 0while

the second part is bounded by surfaces denoted by (iv) (vi)and (vii) and planes denoted by 119871

2= 0 and 119871

1= 10 This

output workspace should be considered during the use anddesign of such a harvester

It is noted that the forward and inverse kinematics Jaco-bianmatrix andworkspaces should be consideredwhen suchharvester is being designed Moreover when the harvester isput to use the singular configurations should be avoidedThekinematics and Jacobian matrix are used to find the singularconfigurations

6 Dynamic Analysis

The efficiency of the water wave harvesting is highly depen-dent on the dynamics of the harvester Therefore it is ofutmost importance to research the dynamics of the harvesterIn this section the dynamic model of the harvester isdeveloped Furthermore in order to compare the efficiencyof a conventional heaving system with that of the proposedharvester the dynamic model of the conventional heavingsystem is firstly introduced Before introducing the dynamicmodels of the two systems it is assumed that the linear waterwaves are applied on the two systems

61 Dynamic Model of a Conventional Heaving System Adiagram of the conventional heaving wave energy harvester[37] composed of a float a bar magnet and a battery isshown in Figure 4 In order to compare the efficiency of theconventional heaving systemwith the proposed harvester thefloats of both systems are assumed to have the same sizeMoreover in this paper the weight of the bar magnet wasneglected

According to [38] themotion equation of the float drivenby linear water waves in a conventional heaving system isgiven by

(119898 + 119886119908119911)1198892119911

1198891199052+ (119887119903119911+ 119887V119911 + 119887

119901119911)119889119911

119889119905

+ (120588119892119860119908119901

+ 119873119896119904) 119911 = 119865

1199110cos (120596119905 + 120572

119911)

(25)

The coefficients in (25) are given as follows

119898 is the mass of the float119886119908119911

is the added mass119887119903119911is the damping coefficient

119887V119911 is the viscous damping coefficient119887119901119911

is the power take-off coefficient119860119908119901

is the waterplane area when the body is at rest120588 is the density of seawater119892 is the acceleration due to gravity119896119904is the spring constant of mooring lines and119873 is the

number of lines (mooring restoring force)

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Robotics

Linear wave

cH2

h

a

d

Inductance coil

Float

Bar magnet Battery

Horn

Beacon light

Figure 4 A conventional heaving wave energy harvester [37]

1198651199110is the water-induced vertical force amplitude and

120596 = 2120587119879 is the circularwave frequency (119879 is thewaveperiod)120572119911is the phase angle between the wave and force

Finally it should be noted that the computations of theabove coefficients in (25) can be found in [38]

62 Dynamic Model of the Tensegrity-Based Water WaveEnergy Harvester As stated in Section 2 the harvester hasthree degrees of freedom Therefore three generalized coor-dinates chosen as q = [1199021 119902

21199023]119879= [119910 119911 120579]

119879 are neededto develop the dynamic model

In order to derive an appropriate dynamic model of theharvester the following hypotheses are made

(i) The links of the mechanism except for the float aremassless

(ii) The springs are massless(iii) There is no friction in the harvesterrsquos revolute pris-

matic and spherical jointsThe equations of motion of the harvester are developed

using the Lagrangian approach namely119889

119889119905

120597119879

120597qminus120597119879

120597q+120597119864

120597q= Q119896 (26)

where 119879 and 119864 are the kinetic and potential energies of theharvester and Q

119896is the vector of nonconservative forces

acting on the system In [37] the translation of the float along119884 axis is defined as surge the translation of the float along119885 axis is defined as heave and the rotation of the float withrespect to 119883 axis is defined as pitch The kinetic energy dueonly to the surge heave and pitchmovements of the float canbe expressed as

T =1

2q119879Mq (27)

where

M =[[[

[

119898 + 119886119908119910

0 0

0 119898 + 119886119908119910

0

0 0 119868119910+ 119860119908

]]]

]

(28)

119868119910is the mass moment of inertia with respect to 119884 axis

and119860119908is added-massmoment of inertia due to pitchingThe

potential energies due to heaving and pitchingmotions of thetop platform are described by McCormick [37] as

119864119901119911

=1

21205881198921198601199081199011199022

2

119864119901120579

=1

21198621199022

3

(29)

where 119862 is the restoring moment constant defined for abottom-flat body in terms of the draft The total potentialenergy of the harvester becomes

119864 = 119880 + 119864119901119911+ 119864119901120579

=1

21205881198921198601199081199011199022

2+1

21198621199022

3+ 119870[radic120590

2

1+ 1205902

2minus 1198970]

2

+ 119870[radic1205902

3+ 1205902

4minus 1198970]

2

(30)

The nonconservative forces which correspond to theradiation damping force viscous damping force and waterwave induced forces can be expressed as

Q119896=[[

[

minus119887V1199101

1198651199110cos (120596119905) minus (119887

119903119911+ 119887V119911) 2

1198721205790cos (120596119905) minus 119887

1199031205793

]]

]

(31)

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Robotics 9

Substituting (27) (30) and (31) into (26) the dynamic modelof the harvester can be rewritten as

Mq + Bq + G = F (32)

where

B =[[

[

119887V119910 0 0

0 119887119903119911+ 119887V119911 + 119887

1199011199110

0 0 119887119903120579

]]

]

G = [120597119864

1205971199021

120597119864

1205971199022

120597119864

1205971199023

]

119879

F = [0 119865119911119900cos120596119905 119872

1205790sin120596119905]119879

(33)

The elements of G are detailed in the Appendix For (32)it should be noted that 119887V119910 is viscous damping coefficientcorresponding to the surge movements of the float while 119887

119903120579

is the radiation damping coefficient due to pitching motion1198721205790

is the water-induced torque amplitude (applied on thefloat) The computations of 119887V119910 119887119903120579 1198721205790 and 119862 can also befound in [37]These computations are also not repeated here

7 Energy Harvesting

In this section two energy harvesting systems are researchedrespectively One is a conventional heaving system and theother is the tensegrity-based water wave harvester Also thepowers of the two systems have been computed respectivelyThe parameters of water waves are selected as 119867 = 02m119879 = 6 s and ℎ = 100m119867 is the wave height measured fromthe trough to the crest while 119879 is the wave period ℎ denotesthe water depth Moreover the floats used in the two energyharvesting systems are supposed to have the same dimensionsas 119871 = 1m119863 = 01m and 119889 = 005m

71 ConventionalHeaving System For a conventional heavingsystem the motion of the float is expressed by (25) For thegiven water wave parameters and the dimensions of the floatthe coefficients of (25) can be calculated according to [38]The results are listed in Table 1

Solving (25) yields the position and velocity of the floatwhich are shown in Figure 5 The power of the heaving bodyis given by [37]

119875119911 (119905) = 119865

119911 (119905)119889119911 (119905)

119889119905 (34)

where 119865119911(119905) is the wave introduced heaving force on the float

The power for take-off 119875(119905) is given by the differencebetween the available power (119875

119911(119905)) and the power dissipated

due to radiation (119875119903119911(119905)) and viscous effects (119875V119911(119905))

119875 (119905) = 119875119911 (119905) minus 119875

119903119911 (119905) minus 119875V119911 (119905) (35)

The average power for take-off over one period of time isgiven by

119875ave =1

119879int119879

119875 (119905) 119889119905 (36)

Table 1 Conventional heaving float coefficients

Coefficient Value Unit119898 5150 kg119886119908119911

45419 kg119887119903119911

106550 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 m2

119873 0 mdash119896119904

0 Nm1198651199110

201210 N120572119911

0 rad120596119899119911

447 Rads119887119888119911

451860 Nsdotsm1198850

021 m

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 5 Motion of the conventional heaving system

The water wave energy and power are [37]

119864 =120588119892211986721198792119887

16120587 (37)

P =12058811989221198672119879119887

32120587i (38)

Applying (36) over two wave periods of the functionshown in Figure 6 gives an average power 119875ave = 0154 kWSince the floatrsquos breadth is 1m then we can compare thisresult with the power contained in one meter of wave frontThe maximum available power per meter of wave front is119875 = 0236 kW (computed by (38)) Therefore 6517 of thewave energy can be harvested with electrical generators

72 Tensegrity-Based Wave Energy Harvester For the har-vester considered here it has infinitesimal mechanismsinherent of many tensegrity systems This means that thereare infinitesimal deformations of the mechanism that donot require any changes in the lengths of the harvesterrsquos

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Robotics

Table 2 Tensegrity-based harvester coefficients

Coefficient Value Unit119898 5150 kg119886119908119910

202 kg119886119908119911

45419 kg119860119908

1503 kgsdotm2

119868119909

433 Kgsdotm2

119862 84958 Nsdotmrad119887119903119911

106550 Nsdotsm119887119903120579

8879 Nsdotmsdotsrad119887V119910 11491 Nsdotsm119887V119911 11491 Nsdotsm119887119901119911

0 Nsdotsm119860119908119901

1 M2

1198651199110

201210 N1198721205790

1877 Nsdotm120572119911

0 rad

t (s)120 3 6 9

3

2

1

0

25

15

05

minus05

minus1

minus15

P(t)

(kW

)

P

Pave

Figure 6 Power for take-off of the conventional heaving system

components It follows that some wave energy would notbe harvested as the mechanism could deform some degreewithout the deformation being felt by the linear generatorsHowever since the deformations are infinitesimal the effectsof infinitesimal mechanisms are negligible

Let the dimensions of the float in the tensegrity-basedwater wave harvester be the same as the conventional heavingfloat The additional constant physical parameters are 119871

0=

4m and 119870 = 10Nm Table 2 contains the values of thecoefficients (computed according to [38]) for the equation ofmotion (see (32))

The simulation is performed over two wave periods thatis 12 seconds Figures 7ndash9 show the position and velocityresponse of the float surge heave and pitch

Figure 10 shows the instantaneous power for take-offTheaverage power over two wave periods is 119875ave = 0186 kWThe

t (s)120 3 6 9

05

0

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

y(t)

(m)y(t)

(ms

)

y(t)

y(t)

Figure 7 Surge motions of the tensegrity-based water wave energyharvester

0 3 6 9 12t (s)

06

04

02

0

minus02

minus04

z(t)

(m)z(t)

(ms

)

z(t)

z(t)

Figure 8 Heave motions of the tensegrity-based water wave energyharvester

power contained in one meter of wave front is 119875 = 0236 kW(computed by (38))Therefore 7876 of the available energycould be harvested by electrical generators By comparingFigures 6 with 10 it is found that the proposed tensegrity-based harvester allows harvesting 1359more energy than aconventional heaving device under linear water wave condi-tions For the conventional heaving device the movement ofthe float is translation along the119885 axis It is proper to say thatthe conventional heaving device has one degree of freedomHowever the possible movements of the proposed harvesterare rotations about the 119883 axis and translations along the 119884and 119885 axes (see Section 2) It is thus proper to say that theproposed harvester has three degrees of freedomThat is whythe harvester can harvest more energy than a conventionaldevice

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Robotics 11

012

01

008

006

004

002

0

minus002

minus004

minus006

minus008

120579(t)

(rad

) 120579(t)

(rad

s)

120579(t)120579(t)

0 3 6 9 12t (s)

Figure 9 Pitch motions of the tensegrity-based water wave energyharvester

0 3 6 9 12

0

05

1

P

Pave

P(t)

(kW

)

minus05

t (s)

Figure 10 Tensegrity-based harvester power for take-off

8 Conclusion

A tensegrity-based water wave energy harvester was pro-posed in this work The geometry of the harvester wasdescribed The solutions to the direct and inverse kinematicproblems were found by using a geometric method TheJacobian matrix and singular configurations were subse-quently computed Then the input and output workspaceswere computed on the basis of the analysis of the obtainedsingular configurations Afterwards the dynamic analysiswas performed considering the interaction with linear waterwaves considering added mass radiation damping andviscous damping phenomena It was shown that the proposedtensegrity-based water wave energy harvester allows harvest-ing 1359 more energy than a conventional heaving system

Appendix

Elements of G

The elements of G in (32) are as follows

1198661=

120597119864

1205971199021

= 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot (1199021

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) + 2119870(119902

1+119871 cos 119902

3

2

minus119863 sin 119902

3

2minus119871

2) sdot 1

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

1198662=

120597119864

1205971199022

= 1205881198921198601199081199011199022+ 2119870(119902

2minus119863 cos 119902

3

2

+119871 sin 119902

3

2) sdot 1

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

+ 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot (1199022

minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

1198663=

120597119864

1205971199023

= 1198621199023+ 21198701

minus 1198710[(1199021minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2)

2

+ (1199022minus119863 cos 119902

3

2+119871 sin 119902

3

2)

2

]

minus12

sdot [(1199021

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Robotics

minus119871 cos 119902

3

2minus119863 sin 119902

3

2+119871

2) (

119871 sin 1199023

2

minus119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2+119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2+119871 cos 119902

3

2)] + 21198701

minus 1198710[(1199021+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2)

2

+ (1199022minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

2

]

minus12

sdot [minus (1199021

+119871 cos 119902

3

2minus119863 sin 119902

3

2minus119871

2) (

119871 sin 1199023

2

+119863 cos 119902

3

2) + (119902

2minus119863 cos 119902

3

2minus119871 sin 119902

3

2)

sdot (119863 sin 119902

3

2minus119871 cos 119902

3

2)]

(A1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (no 51375360)

References

[1] R Motro Tensegrity Structural Systems for the Future KogenPage Science Guildford UK 2003

[2] R E Skelton and M C Oliveira Tensegrity Systems SpringerNew York NY USA 2009

[3] C R Calladine ldquoBuckminster Fullerrsquos lsquoTensegrityrsquo structuresand Clerk Maxwellrsquos rules for the construction of stiff framesrdquoInternational Journal of Solids and Structures vol 14 no 2 pp161ndash172 1978

[4] S Pellegrino ldquoA class of tensegrity domesrdquo International Journalof Space Structures vol 7 no 2 pp 127ndash142 1992

[5] F Fu ldquoStructural behavior and design methods of Tensegritydomesrdquo Journal of Constructional Steel Research vol 61 no 1pp 23ndash25 2005

[6] L Rhode-Barbarigos N B Hadj Ali R Motro and I F CSmith ldquoDesigning tensegrity modules for pedestrian bridgesrdquoEngineering Structures vol 32 no 4 pp 1158ndash1167 2010

[7] L Rhode-Barbarigos N B H Ali R Motro and I F CSmith ldquoDesign aspects of a deployable tensegrity-hollow-ropefootbridgerdquo International Journal of Space Structures vol 27 no2-3 pp 81ndash95 2012

[8] S Korkmaz N B H Ali and I F C Smith ldquoConfigurationof control system for damage tolerance of a tensegrity bridgerdquoAdvanced Engineering Informatics vol 26 no 1 pp 145ndash1552012

[9] R E Skelton F Fraternali G Carpentieri and A MichelettildquoMinimum mass design of tensegrity bridges with parametricarchitecture and multiscale complexityrdquo Mechanics ResearchCommunications vol 58 pp 124ndash132 2014

[10] J Duffy J Rooney B Knight and C D Crane III ldquoReview of afamily of self-deploying tensegrity structures with elastic tiesrdquoShock and Vibration Digest vol 32 no 2 pp 100ndash106 2000

[11] A Hanaor ldquoDouble-layer tensegrity grids as deployable struc-turesrdquo International Journal of Space Structures vol 8 no 1-2pp 135ndash143 1993

[12] G Tibert Deployable tensegrity structures in space applications[PhD thesis] Royal Institute of Technology Stockholm Swe-den 2002

[13] N Fazli and A Abedian ldquoDesign of tensegrity structures forsupporting deployable mesh antennasrdquo Scientia Iranica vol 18no 5 pp 1078ndash1087 2011

[14] N Vassart and R Motro ldquoMultiparametered form findingmethod application to tensegrity systemsrdquo International Jour-nal of Space Structures vol 14 no 2 pp 147ndash154 1999

[15] K Koohestani ldquoForm-finding of tensegrity structures viagenetic algorithmrdquo International Journal of Solids and Struc-tures vol 49 no 5 pp 739ndash747 2012

[16] AG Tibert and S Pellegrino ldquoReviewof form-findingmethodsfor tensegrity structuresrdquo International Journal of Space Struc-tures vol 18 no 4 pp 209ndash223 2003

[17] S H Juan and J M M Tur ldquoTensegrity frameworks staticanalysis reviewrdquo Mechanism and Machine Theory vol 43 no7 pp 859ndash881 2008

[18] D E Ingber ldquoCellular tensegrity defining new rules of biologi-cal design that govern the cytoskeletonrdquo Journal of Cell Sciencevol 104 no 3 pp 613ndash627 1993

[19] D E Ingber ldquoThe architecture of liferdquo Scientific American vol278 no 1 pp 48ndash57 1998

[20] D E Ingber ldquoTensegrity I Cell structure and hierarchicalsystems biologyrdquo Journal of Cell Science vol 116 no 7 pp 1157ndash1173 2003

[21] SM Levin ldquoThe tensegrity-truss as amodel for spinalmechan-ics biotensegrityrdquo Journal ofMechanics inMedicine and Biologyvol 2 no 3 pp 375ndash388 2002

[22] I J Oppenheim and W O Williams ldquoTensegrity prisms asadaptive structuresrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition Dallas TexUSA November 1997

[23] C Sultan M Corless and R E Skelton ldquoTensegrity flightsimulatorrdquo Journal of Guidance Control and Dynamics vol 23no 6 pp 1055ndash1064 2000

[24] C Sultan M Corless and R E Skelton ldquoPeak-to-peak controlof an adaptive tensegrity space telescoperdquo in Smart Struc-tures and Materials 1999 Mathematics and Control in SmartStructures vol 3667 of Proceedings of SPIE pp 190ndash201 TheInternational Society for Optical Engineering Newport BeachCalif USA March 1999

[25] C Paul F J Valero-Cuevas andH Lipson ldquoDesign and controlof tensegrity robots for locomotionrdquo IEEE Transactions onRobotics vol 22 no 5 pp 944ndash957 2006

[26] A G Rovira and J M M Tur ldquoControl and simulation ofa tensegrity-based mobile robotrdquo Robotics and AutonomousSystems vol 57 no 5 pp 526ndash535 2009

[27] S Hirai and R Imuta ldquoDynamic simulation of six-struttensegrity rollingrdquo in Proceedings of the 2012 IEEE InternationalConference on Robotics and Biomimetics Guangzhou China2012

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Robotics 13

[28] M Q Marshall Analysis of tensegrity-based parallel platformdevices [MS thesis] University of Florida Gainesville FlaUSA 2003

[29] SMM ShekarforoushM Eghtesad andM Farid ldquoKinematicand static analyses of statically balanced spatial tensegritymechanism with active compliant componentsrdquo Journal ofIntelligent amp Robotic Systems vol 71 no 3-4 pp 287ndash302 2013

[30] C D Crane III J Bayat V Vikas and R Roberts ldquoKinematicanalysis of a planar tensegrity mechanism with pre-stressedspringsrdquo in Advances in Robot Kinematics Analysis and DesignJ Lenarcic and PWenger Eds pp 419ndash427 Springer Batz-sur-Mer France 2008

[31] J M McCarthy ldquo21st century kinematics synthesis compli-ance and tensegrityrdquo Journal of Mechanisms and Robotics vol3 no 2 Article ID 020201 2011

[32] J T Scruggs and R E Skelton ldquoRegenerative tensegrity struc-tures for energy harvesting applicationsrdquo in Proceedings of the45th IEEE Conference on Decision and Control (CDC rsquo06) pp2282ndash2287 IEEE San Diego Calif USA December 2006

[33] M R Sunny C Sultan and R K Kapania ldquoOptimal energyharvesting from amembrane attached to a tensegrity structurerdquoAIAA Journal vol 52 no 2 pp 307ndash319 2014

[34] R E Vasquez C D Crane III and J C Correa ldquoAnalysisof a planar tensegrity mechanism for ocean wave energyharvestingrdquo Journal of Mechanisms and Robotics vol 6 no 3Article ID 031015 2014

[35] S V Sreenivasan K J Waldron and P Nanua ldquoClosed-form direct displacement analysis of a 6-6 Stewart platformrdquoMechanism and Machine Theory vol 29 no 6 pp 855ndash8641994

[36] C D Crane III and J Duffy Kinematic Analysis of RobotManipulators Cambridge University Press New York NYUSA 1998

[37] M McCormick Ocean Wave Energy Conversion Dover Mine-ola NY USA 2007

[38] R E Vasquez Analysis of a tensegrity system for oceanwave energy harvesting [PhD thesis] University of FloridaGainesville Fla USA 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of