Research Article Topology Optimization of 3-DOF

Research ArticleTopology Optimization of 3-DOF Peristaltic Structure RobotBased on Vector Continuous Mapping Matrix

GaoWang1 Dachang Zhu2 and Ning Liu1

1School of Information Science and Technology Jinan University Guangzhou Guangdong 510632 China2School of Mechanical and Electrical Engineering Guangzhou University Guangzhou Guangdong 510006 China

Correspondence should be addressed to Dachang Zhu zdc98998163com

Received 24 February 2016 Accepted 17 August 2016

Academic Editor Salvatore Strano

Copyright copy 2016 Gao Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A mechanism for topology optimization of 3-DOF parallel peristaltic structure robot with vector continuous mapping matrixusing Solid Isotropic Material with Penalization (SIMP) method is presented in this paper We focus on how to prevent thedifferential motion consistency between parallel prototype mechanisms with peristaltic structure As the conventional parallelrobot jointshinges are no longer needed after topology optimization therefore we renamed this kind of 3-DOF robot structuresas peristaltic structure In the proposed method the vector continuous mapping matrix is built as stressstrain transfer directionconditions for topology optimization of peristaltic structure and SIMP method is used for multi-inputs and multioutputs decidedby parallel prototype mechanisms Some numerical examples are presented to illustrate the validity of the proposed method

1 Introduction

Substantial researchers have been spent on the design ofmicro-nanoequipment in recent years Flexure joints aretypically manufactured monolithically therefore avoidingassembly errors In terms of operation flexure joints havelittle friction loss and do not require lubricationThey gener-ate smooth and continuous displacement without backlashSo flexure joints are usually used to compose the structureof micro-nanoequipment named as compliant mechanisms[1 2] However compliant mechanisms cannot finish spatialmultidimensional motion characteristics To overcome theshortcoming the structure of parallel mechanisms withflexure joints is investigated called compliant parallel mech-anism Yun and Li [3] presented the design and modeling ofa new 6-DOF 8-PSSSPS compliant dual redundant parallelrobot with wide-range flexure joints and the kinematicsmodel of the macro parallel mechanism system via thestiffness model and Newton-Raphson method are adaptedto build the dynamicrsquos model for the micromotion systemDong et al [4] proposed a compliant ultraprecision parallelpositioner based on the coarsefine dual architecture andflexure hinges are adopted as compliant passive joints Choi

et al [5] proposed a compliant parallel mechanism fortwo translations and applied it to 119883119884 fine motion stagedriven by piezo actuators and four flexure-based prismaticjoint chains are arranged in four sides of a target platformto implement the compliant parallel mechanism Gao andZhang [6] designed a novel three-DOF compliant parallelmechanism and its performance characterization is analyzedwhich affects the application potential Treatments of thecharacterization and design of compliant parallel mecha-nismsmay also be found in literatures [7ndash9] By using the flex-ure hinges instead of the conventional rigid joints althoughthis kind of structure composed method has spatial mul-tidimensional motion characteristics the whole stiffness isdecent obviously It is important to design a perfect structuresatisfying the two subjects high stiffness and multidimen-sional operation ability In terms of high stiffness topologyoptimization method is proposed for compliant mechanismThe advantages of topology optimization approach are thatit does not require a rigid-link mechanism configurationas a starting point and it can be used to design single-piece fully compliant mechanisms Ever since Bends Φ119890introduced the homogenization method [10] many topologyoptimization methods have been developed such as the solid

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4740586 8 pageshttpdxdoiorg10115520164740586

2 Mathematical Problems in Engineering

A

B

CP

Q

R

D

E

F

PlatformM1(1205791)

M2(1205792)

M3(1205793)

(a)

R

P QA

B

C

DE

F

c

l

a1 a2

b3

b2b1

a3

1205791

1205793

1205792

1205951 1205952

1205953

120593

F

A998400 B998400

C998400

D998400

E998400

F998400

(b)

Figure 1 3-RRR planar parallel prototype mechanism (a) 3D model of 3-RRR (b) description of 3-RRR kinematics

isotropic material with the penalization (SIMP) method [11]the ground structuremethod [12] and level set basedmethod[13] Although topology optimization has been a maturedtool for optimization design problem most of topologyoptimizationmethods are focusing on the single input-outputcondition In reality parallel compliant mechanisms withmultiple inputs and multiple outputs are widely used inthe fields of micropositioning and micromanipulation Therelationship between multiple inputs and multiple outputs isvectormappingmatrixwith certain connection Zhu et al [1415] presented a hinge-free compliant mechanism which hasone input and two outputs based on level set method Jones[16] presented a wavelet-based topology optimization formu-lation by implicitly imbeddingwavelet shrinkagemethod intooptimization formulation based on SIMP method Topologyoptimization methods for parallel compliant mechanismsmay also be found in other literatures [17ndash19] In thosemethods compared with conventional parallel prototypemechanisms themotion characteristics of designed structureare ambiguous On the other hand the mechanism has notincluded the hinge form after the topology optimizationprocess In this paper we called this structure peristalticstructure

In this paper we proposed a new method for topologyoptimization of planar 3-DOF peristaltic structure parallelrobot based on differential kinematic vector continuousmapping matrix with conventional prototype parallel mech-anism The remainder of the paper is organized as followsIn Section 2 the differential kinematic vector continuousmapping matrix with conventional prototype parallel mech-anism is built In Section 3 the optimization problems ofdesign peristaltic structure are proposed based on SIMPmethodThe shape sensitivity analysis is applied to obtain thedisplacement field for the optimization combined with thevector mapping matrix In Section 4 numerical examples arepresented to demonstrate the effectiveness of the proposedmethod Finally conclusions and a discussion for furtherwork are developed

2 Differential Kinematics with VectorContinuous Mapping Matrix

In order to keep the differential kinematic characteristicswith the prototype of parallel mechanism the differentialkinematic equations with vector continuous mapping ofmicroelastic deformation mechanism should be set Withoutloss of generality we select the 3-RRR prototype of planarparallel mechanism as an example As shown in Figure 1 thestructure of 3-RRR prototype planar parallel mechanism isestablished

Constraint equations are described as

119909119861 = 119909119860 + ℎ sdot cos120593119910119861 = 119910119860 + ℎ sdot sin120593119909119862 = 119909119860 + ℎ sdot cos (120593 + 120587)119910119862 = 119910119860 + ℎ sdot sin (120593 + 120587) (1)

Vector equation of the limb 119875119863119860may be described as997888997888rarr119874119860 = 997888997888rarr119874119875 + 997888997888rarr119875119863 + 997888997888rarr119863119860 (2)

where 119909119901 = 119910119901 = 0 Substituting (1) into (2) we obtain119909119860 = 1198861 sdot cos 1205791 + 1198871 sdot cos (1205791 + 1205951)119910119860 = 1198861 sdot sin 1205791 + 1198871 sdot sin (1205791 + 1205951) (3)

Vector equation of the limb 119861119864119876 can be described as997888997888rarr119874119861 = 997888997888rarr119875119876 + 997888997888rarr119876119864 + 997888rarr119864119861 (4)

where 119909119902 = 119888 119910119902 = 0 Substituting (1) into (4) we have119909119860 = 119888 minus 1198862 sdot cos 1205792 minus 1198872 sdot cos1205952 minus ℎ sdot cos120593119910119860 = 1198862 sdot sin 1205792 + 1198872 sdot sin1205952 minus ℎ sdot sin120593119909119861 = 119888 minus 1198862 sdot cos 1205792 minus 1198872 sdot cos1205952119910119861 = 1198862 sdot sin 1205792 + 1198872 sdot sin1205952

(5)

Mathematical Problems in Engineering 3

Vector equation of the limb 119877119865119862 can be described as

997888997888rarr119874119862 = 997888997888rarr119875119862 + 997888997888rarr119877119865 + 997888997888rarr119865119862 (6)

where 119909119877 = 1198882 119910119877 = 119897 Substituting (1) into (6) yields119909119860 = 1198882 minus 1198863 sdot cos 1205793 minus 1198873 sdot cos1205953 minus ℎ sdot cos(120593 + 1205873 )119910119860 = 119897 minus 1198863 sdot sin 1205793 minus 1198873 sdot sin1205953 minus ℎ sdot sin(120593 + 1205873 )119909119862 = 1198882 minus 1198863 sdot cos 1205793 minus 1198873 sdot cos1205953119910119862 = 119897 minus 1198863 sdot sin 1205793 minus 1198873 sdot sin1205953

(7)

We assumed that the displacements of each limb are equalto infinity and can be defined as 1205791015840119894 = 120579119894 + Δ120579119894 119894 = 1 2 3 andthen (3) (5) and (7) can be rewritten as follows with infinitedisplacements R1015840119872

R1015840119872 =

[[[[[[[[[[[[

119909101584011986011991010158401198601199091015840119861119910101584011986111990910158401198621199101015840119862

]]]]]]]]]]]]

=[[[[[[[[[[[[

1198861 sdot cos (1205791 + Δ1205791) + 1198871 sdot cos ((1205791 + 1205951) + (Δ1205791 + Δ1205951))1198861 sdot sin (1205791 + Δ1205791) + 1198871 sdot sin ((1205791 + 1205951) + (Δ1205791 + Δ1205951))119888 minus 1198862 sdot cos (1205792 + Δ1205792) minus 1198872 sdot cos (1205952 + Δ1205952)1198862 sdot sin (1205792 + Δ1205792) + 1198872 sdot sin (1205952 + Δ1205952)1198882 minus 1198863 sdot cos (1205793 + Δ1205793) minus 1198873 sdot cos (1205953 + Δ1205953)119897 minus 1198863 sdot sin (1205793 + Δ1205793) minus 1198873 sdot sin (1205953 + Δ1205953)

]]]]]]]]]]]]

(8)

Let Δ119909 = 1199091015840119860 minus 119909119860 and Δ119910 = 1199101015840119860 minus 119910119860 and thethree parameters Δ119909 Δ119910 and Δ120593 can be defined as outputdisplacement of the moving platform

The formulas of the active joints displacement and tasksdisplacement are included with (3) (5) (7) and (8) andthe vector continuous mapping matrix between the jointsdisplacement and task displacement can be expressed asfollows

Δ119909 sdot cos (1205791 + 1205951) + Δ119910 sdot sin (1205791 + 1205951) minus 1198861 sdot Δ1205791sdot sin1205951 = 0

Δ119909 sdot cos1205952 + Δ119910 sdot sin1205952 + ℎ sdot Δ120593 sdot sin (120593 + 1205952) + 1198862sdot Δ1205792 sdot sin (1205792 minus 1205952) = 0

Δ119909 sdot cos1205953 + Δy sdot sin1205953 minus ℎ sdot Δ120593 sdot sin(120593 + 1205873 minus 1205952)minus 1198863 sdot Δ1205793 sdot sin (1205793 minus 1205953) = 0

(9)

Then the formulations can be rearranged and simplifiedas follows

[[[Δ119909Δ119910Δ120593

]]]= 119869119863 sdot [[[

Δ1205791Δ1205792Δ1205793]]] (10)

where

119869119863 =[[[[[[[[[

1198861 sdot 1199041205951 sdot (119904 (120572) sdot 1199041205952 + 119904 (120574) sdot 1199041205953)minus119872 1198862 sdot 119904 (120573) sdot 119904 (120590) sdot 119904 (120572)minus119872 1198863 sdot 119904 (120573) sdot 119904 (120574) sdot 119904 (1205793 minus 1205953)minus1198721198861 sdot 1199041205951 sdot (119904 (120572) sdot 1199041205952 sdot 1198881205952 + 119904 (120574) sdot 1198881205953)119872 1198862 sdot 119888 (120573) sdot 119904 (120590) sdot 119904 (120572)119872 1198863 sdot 119888 (120573) sdot 119904 (120574) sdot 119904 (1205793 minus 1205953)minus1198721198861 sdot 1199041205951 sdot 119904 (1205953 minus 1205952)ℎ sdot 119872 1198862 sdot 119904 (1205792 minus 1205952) sdot 119904 (1205953 minus 1205791 minus 1205951)ℎ sdot 119872 1198863 sdot 119904 (1205793 minus 1205953) sdot 119904 (1205952 minus 1205791 minus 1205951)ℎ sdot 119872

]]]]]]]]]

120572 = 120593 + 1205873 minus 1205953120573 = 1205791 + 1205951120574 = 120593 + 1205952120590 = 1205792 minus 1205952119872 = 119904(120593 + 1205873 minus 1205953) sdot 119904 (1205791 + 1205951 minus 1205952) + 119904 (120593 + 1205952) sdot 119904 (1205791 + 1205951 minus 1205953)119904 (lowast) = sin (lowast) 119888 (lowast) = cos (lowast)

(11)


3 Topology Optimization of 3-DOFPeristaltic Structure

31 SIMP Model of Topology Optimization with Vector Con-tinuous Mapping Matrix The kinematic characteristics of 3-DOF peristaltic structure have three input parameters andthree output parameters respectively By using the vector con-tinuous mapping matrix the model of SIMP can be built as

min 119862 = 3sum119894=1

3sum119895=1

(119879119895119870119880119894) = sum1le119890le1198731le119894le31le119895le3

119879119890119895120588119901119890 119870119864119880119890119894st 119870119895 = 119865119895 119870119880119894 = 119865119894 119865119895 = 119869119863119865119894

119894 = 1 2 3 119895 = 1 2 3forall 997888rarr int

Ω120588119890119889Ω le 1198810 le 120588min le 120588119890 le 1 119890 = 1 2 119873

(12)

In (12) 119880119895 is the companion displacement vector119870 is thewhole postoptimality stiffness of creep structure 119880119894 is thedisplacement with actual load 119881 is postoptimality volumeof creep structure 119865119894 is 119894th actual load 119880119890119894 is the unitdisplacement vector under the 119894th actual load 119865119894 is 119895th virtualload 119890119894 is the unit displacement vector under the 119895th virtualload 119870119864 is unit stiffness 120588119890 is unit density and 119901 is penaltyfactor and satisfies 119901 ge max2(1 minus 1198810) 4(1 + V0)32 Sensitivity Analysis Sensitivity analysis in topology opti-mization includes the sensitivity of objective function andconstraint functionDifferentiating the two types of functionswith respect to design variables we can derive the directionof iterative update during topology optimization process

321 Sensitivity Analysis of Objective Function Differentiat-ing formulation (12) with respect to density 120588119890 yields

120597119862120597120588119890 =120597 (sum3119894=1sum3119895=1 (119879119895119870119880119894))120597120588119890

= 120597 (sum3119895=1 119879119895 )120597120588119890 sdot 119870 sdot 3sum119894=1

119880119894 + 3sum119895=1

119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894+ 3sum119895=1

119879119895 sdot 119870 sdot 120597 (sum3119894=1119880119894)120597120588119890= (120597 (sum3119895=1 119879119895 )120597120588119890 sdot 119870 + 3sum

119895=1

119879119895 sdot 120597119870120597120588119890) sdot 3sum119894=1

119880119894

+ 3sum119895=1

119879119895 sdot (120597119870120597120588119890 sdot3sum119894=1

119880119894 + 119870 sdot 120597 (sum3119894=1119880119894)120597120588119890 )minus 3sum119895=1

119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894(13)

From a macro perspective without loss of generality weassumed that the input and output are unconcerned withdesign variables and then differentiating formulation (12)with respect to design variables we obtain

120597119870120597120588119890 sdot3sum119895=1

119879119895 + 119870 sdot 120597 (sum3119895=1 119879119895 )120597120588119890 = 120597 (sum3119895=1 119865119895)120597120588119890 = 0120597119870120597120588119890 sdot

3sum119894=1

119880119894 + 119870 sdot 120597 (sum3119894=1119880119894)120597120588119890 = 120597 (sum3119894=1 119865119894)120597120588119890 = 0(14)

Substituting (14) into (13) we have

120597119862120597120588119890 = minus 3sum119895=1

119879119895 sdot 119870 sdot 3sum119894=1

119880119894 (15)

Substituting119870 = sum119873119890=1 120588119901119890 sdot119870119864 into (15) the computationalformula for objective function sensitivity analysis is given by

120597119862120597120588119890 =119873sum119890=1

3sum119894=1

3sum119895=1

119901119879119895 120588119901minus1119890 119870119864119880119894 (16)

322 Sensitivity Analysis of Constraint Function Differenti-ating the volume function with respect to the unit density 120588119890sensitivity of constraint function can be derived as

120597119881120597120588119890 =120597 (intΩ120588119890119889119881)120597120588119890 = 1 (17)

We find that the sensitivity of constraint function com-putational result is a constant It denotes that the direction ofiterative update of volume with design variables is ensured

33 Solution toOptimizationModel Weapply Lagrangemul-tiplier to construct the Lagrange function including objectivefunction and constraint function simultaneously By solvingthe extreme value of the Lagrange function we can constructthe steady state condition of Lagrangersquos function on designvariables using KKT conditions so the optimization value ofobjective function can be derivedThe optimization criterionalgorithm is proposed for iterative update described as

120588(119896+1)119890 = min (119898 + 1) 120588(119896)119890 1 st min (1 + 119898) sdot 120588(119896)119890 1 le (119863(119896)119890 )120577 sdot 120588(119896)119890(119863(119896)119890 )120577 sdot 120588(119896)119890 st max (1 minus 119898) sdot 120588(119896)119890 120588min lt (119863(119896)119890 )120577 sdot 120588(119896)119890 lt min (1 + 119898) sdot 120588(119896)119890 1max (1 minus 119898) 120588(119896)119890 120588min st (119863(119896)119890 )120577 sdot 120588(119896)119890 le max (1 minus 119898) sdot 120588(119896)119890 120588min

(18)


Fixed basement

Motion platform

Output

Output Output

Input Fin2

Input Fin1Input Fin3

Fout2

Fout1Fout3

(a) (b)

Figure 2 Design conditions of topology optimization for 3-DOF robot of peristaltic structure (a) and the experimental mechanism (b)

where 120588(119896)119890 is the iteration value of 119896 step 120588(119896+1)119890 is the iterationvalue of 119896 + 1 step and119898 is the moving limit constant119898 denotes the appropriate constraint to design variablesfor stabilizing the iteration process Based on literature [10]the range of parameter 119898 may be selected within 01sim03 120577is damping factor and in the range of 04sim05 and 119863(119896)119890 isexpressed as follows

119863(119896)119890 = 119901120588(119901minus1)119890 sum3119895=1 119879119895 sdot 119870119864 sdot sum3119894=1119880119894 sdot Δ119864Λ(119896)119881119890= (max (0 minus120597119862120597120588119890))Λ(119896)119881119890

(19)

where Λ(119896) is the Lagrange multiplier with volume constraintof 119896 step iteration and the update of Lagrange multiplier isadapted double convex linear programming algorithm And119881119890 is the unit volume of 119896 step iteration

34 Heaviside Filter We adapt Heaviside function to modifythe linear filter for peristaltic structure in 3-DOF robot Asthe range of minimum filter radius if 119890 gt 0 then 120588119890 = 1otherwise 119890 = 0 then 120588119890 = 0 The resulting expression oflinear filter can be given by

119894 = sum119895

120596119894119895120588119895 sum119895

120596119894119895 = 1 (20)

where 120596119894119895 is filter weight factor of the 119895th unit to the 119894th unitTo get the better discrete 01 distribution optimization

results and decent middle density unit second weight cal-culation equation and Heaviside function are adopted asfollows

120596119894119895 =

119903min minus 119889 (119894 119895)sum119896isin119873119894 (119903min minus 119889 (119894 119896)) 119895 isin 1198731198940 119895 notin 119873119894119873119894 = 119895 119903min minus 119889 (119894 119896) ge 0

(21)

where119889(119894 119895) is the displacement between 119894th unit and 119895th unitand 119903min is the minimum filter radius

We adapt again theHeaviside function to approximate theoriginal linear density filterTheHeaviside filter functionmaybe expressed as

120588119890 = 1 + exp (minus120573 sdot 119890) + 119890 sdot exp (minus120573) (22)

where 120573 is the control parameter for flat degree of Heavisidefunction If 120573 = 0 then Heaviside filter can be changed tolinear filter if 120573 = infin Heaviside filter can be changed tomaximum density filter 120588119890 is unit physical density and 119890 islinear result of unit density

Heaviside function filter can be freely switched betweenlinear density filter andmaximumdensity filter in themethodof parameters selection

4 Design Conditions ofTopology Optimization for 3-DOFPeristaltic Structure

Based on linear elastic relationship between force and dis-placement the 3 DOF of peristaltic structure have beenchanged to force output as shown in Figure 2 119865in1 119865in2 and119865in3 are the force inputs in the fixed basement and119865out1119865out2and 119865out3 are the force outputs in the motion platform center

5 Simulations and ExperimentalResults Analysis of 3-DOF PeristalticStructure Robot

Assuming that the topology optimization yield of peristalticstructure is a square profile with 14 times 14 unit size and thenumber of discrete grids is 140 times 140 the elastic modulusof solid material is 2 times 1011 Pa the elastic modulus of holesis 10minus3 Pa Poissonrsquos ratio is 035 optimum volume ratio is0278 minimum filter radius is 14 input force 119865in1 is 1200N


Figure 3 The topological structure of 3-DOF peristaltic structurerobot optimization result

20 40 60 80 100 120 140 160 1800minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Figure 4 Objective function 119862 and convergence history of 3-DOFperistaltic structure switched curve

input force 119865in2 is 1200N and input force 119865in3 is 1200NBased on the relationship between input forces and outputforces denoted as 119869119863 119865119895 = 119869119863 sdot 119865119894 the output forces can besolved as 119865out1 = 478N 119865out2 = 1570N and 119865out3 = 1240Nrespectively The optimization was run for 180 iterations andthe topology optimized result of 3-DOF peristaltic structurerobot is shown in Figure 3 and the iterative process of theobjective function 119862 optimization is shown in Figure 4

We adapt the general curve fitting for the contour of3-DOF peristaltic structure robot and the median filteringmethod to smooth the boundary of peristaltic structureoptimized result Import the contour data into SolidWorks3D software the modeling structure is shown in Figure 5

And then CAE software by ANSYS is used for staticsimulations of 3-DOF peristaltic structure robot Import the3Dmodel of the optimized result intoANSYS software divid-ing finite elementmesh and setting boundary conditions andcarry on static analysis The simulations method of 3-DOFperistaltic structure robot is shown in Figure 6 Accordingto the previous conditions set the fixed basement is in thediagonal of the simulation domain The input forces 1198651 1198652and 1198653 in acting points of 3-DOF peristaltic structure robotare transferred to the output region (the moving platform) inits elastic deformationmethod including the displacement ofdirection 119909119910 and rotational direction 119911

Figure 5 Modeling of 3-DOF peristaltic structure robot withmedian filtering method

y

x1198131

1198132

1198133

Figure 6 3-DOF peristaltic structure robot static simulationsanalysis

minus2080 minus139 minus71 minus02 66minus75

minus625

minus5

minus375

minus25

minus125

0

125

25

375

5

Valu

e

minus20800 minus139 minus771 minus02 666

(times10minus3)

008 016 024 032 04 048 056 064 072 080

Time

Figure 7 Differential displacement of 3-DOF peristaltic structurerobot with direction 119909

The differential displacement of directions 119909 and 119910 in thecenter of moving platform is shown in Figures 7 and 8 withthe maximum values minus215 nm and 180 nm respectively Thedifferential rotational displacement of direction 119911 is shown inFigure 9 with the maximum value minus415 120583rad

According to Section 2 3-DOF peristaltic structure dif-ferential kinematic analysis we set initial condition in certainsteady state in Table 1


Table 1 3-DOF peristaltic structure differential kinematic parameters

1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10

Table 2 Differential displacements of the 3-DOF peristaltic structure robot in theory and simulation

Differential displacement of 119909 Differential displacement of 119910 Differential displacement of 119911Value of theory minus215 nm 180 nm minus415 120583radValue of simulation minus615 nm 293 nm minus355 120583rad

minus144 minus75 minus05 53 132

(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time


minus44minus119 18110631minus44minus119 1118110631

008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)


Substituting these parameters shown in Table 1 into (10)we obtain

[[[minus20109 minus23827 2543221459 minus06673 0298909943 minus01023 minus06925

]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)

Substituting input parameters (Δ1205791 Δ1205792 Δ1205793) =(25 25 25) into (23) the displacement of movingplatform may be given by

[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)

Comparing the simulation results in Figures 7sim9 withtheoretical calculation results (24) in the same computationalscale the proposed 3-DOF peristaltic structure robot nolonger suffers the high stress concentration and is moresuitable for real world applications as shown in Table 2

FromTable 2 we can see that the results of orders of mag-nitude and symbols are the same in all directionsrotationsand the differential displacement of 3-DOF peristaltic struc-ture has the same characteristic with the parallel prototypemechanism The peristaltic structure robot realizes 3-DOFplanar kinematics which is displacement 119909119910 and rotationaldisplacement 119911 in micron level So simulations verify thecorrectness of vector continuous mapping topology opti-mization design method

However we note that the quantitative errors between thevalues of theory and simulation are so big and analysis resultsof error source are shown in two factors

(1) The theoretical result and simulation result adopt thestructure of 3-RRR prototype planar parallel mech-anism and the 3-DOF peristaltic structure robotrespectively The former transfers force and kinemat-ics through rigid hinges and the latter transfers forceand kinematics through its own elastic deformationDue to elastic potential energy in the latter owninside the output force 119865119895 displacement decreases

(2) We assumed the elastic deformation and displace-ment outputs approximate linearization and the stressstrain transfer in peristaltic structure is isotropic

6 Conclusions

In this paper we presented a new formulation for synthesis ofperistaltic structure with vector continuous mapping matrixbased on topology optimization method In the proposedmethod the vector continuous mapping matrix derivedwith 3-RRR parallel prototype mechanism is taken intoaccount to make the peristaltic structure to inherit the same


kinematic characteristic with parallel prototype mechanismConsequently we proposed a new topology optimizationmethod combining SIMP with vector continuous mappingmatrix to synthesise 3-DOF peristaltic structure robot Adoptthe Heaviside filter to modify optimization function depen-dence of mesh grid division We use curve fitting methodto smooth the contour for optimization result and modeling3D peristaltic structure Import the model into ANSYS andimplement static simulations analysis Numerical simulationswith vector continuous mapping matrix are presented forillustrating the validity of the presented method Someconclusions are obtained

(1) 3-DOF peristaltic structure robot moving platformrealizes 119909-displacement minus615 nm 119910-displacement293 nm and 119911-rotational displacement minus355 120583radand shows planar 3-DOF parallel mechanism differ-ential kinematics

(2) Compared with similar general parallel mechanismkinematics in approximately equal load conditionsthe results of orders of magnitude and symbolsare the same in all directionsrotations The 3-DOFperistaltic structure robot based on vector continuousmapping matrix design method delivers the samekinematic characteristic with similar general parallelmechanism which qualitatively verifies validity of themethod

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grant no 51165009 China Post-doctoral Science Foundation under Grant no 2013M541874Science and Technology Project of Guangdong Province(China) under Grant no 2013B011302002 and Science andTechnology Program of Guangzhou (China) under Grantno 2015080100252015Y2000010 The authors would like tothank Vanta Intelligent Equipment Technology Co Ltdin Guangzhou for supporting them with the experimentalsetup

References

[1] L L Howell Compliant Mechanisms John Wiley amp Sons NewYork NY USA 2013

[2] W Bejgerowski J W Gerdes S K Gupta and H A BruckldquoDesign and fabrication of miniature compliant hinges formulti-material compliant mechanismsrdquo International Journal ofAdvanced Manufacturing Technology vol 57 no 5-8 pp 437ndash452 2011

[3] Y Yun and Y Li ldquoDesign and analysis of a novel 6-DOFredundant actuated parallel robot with compliant hinges forhigh precision positioningrdquo Nonlinear Dynamics vol 61 no 4pp 829ndash845 2010

[4] W Dong Z Du L Sun and B Zheng ldquoA compliantultraminusprecision 6minusDOF parallel positioner based on thecoarsefine dual architecturerdquo in Proceedings of the 1st IEEEInternational Conference on NanoMicro Engineered and Molec-ular Systems pp 488ndash492 Zhuhai China January 2006

[5] K-B Choi J J Lee G H Kim and H J Lim ldquoA compliantparallel mechanism with flexure-based joint chains for twotranslationsrdquo International Journal of Precision Engineering andManufacturing vol 13 no 9 pp 1625ndash1632 2012

[6] Z Gao and D Zhang ldquoSimulation driven performance charac-terization of a spatial compliant parallel mechanismrdquo Interna-tional Journal of Mechanics and Materials in Design vol 10 no3 pp 227ndash246 2014

[7] Z Dachang and F Yanping ldquoStructure design of a 3minusDOFUPCtype rotational fully spatial compliant parallel manipulatorsrdquoInternational Journal of Advancements in Computing Technol-ogy vol 5 no 8 pp 70ndash81 2013

[8] D Zhang and Z Gao ldquoPerformance analysis and optimizationof a five-degrees-of-freedom compliant hybrid parallel micro-manipulatorrdquo Robotics and Computer-Integrated Manufactur-ing vol 34 pp 20ndash29 2015

[9] G Hao and H Li ldquoDesign of 3-legged XYZ compliant parallelmanipulators withminimised parasitic rotationsrdquoRobotica vol33 no 4 pp 787ndash806 2015

[10] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[11] M P Bendsoslashe andO Sigmund ldquoMaterial interpolation schemesin topology optimizationrdquoArchive of AppliedMechanics vol 69no 9-10 pp 635ndash654 1999

[12] M I Frecker G K Ananthasuresh S Nishiwaki N Kikuchiand S Kota ldquoTopological synthesis of compliant mecha-nisms using multi-criteria optimizationrdquo Journal of MechanicalDesign vol 119 no 2 pp 238ndash245 1997

[13] G Allaire F Jouve and A-M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[14] B Zhu X Zhang and N Wang ldquoTopology optimization ofhinge-free compliant mechanisms with multiple outputs usinglevel set methodrdquo Structural and Multidisciplinary Optimiza-tion vol 47 no 5 pp 659ndash672 2013

[15] B Zhu X Zhang and S Fatikow ldquoA multi-objective method ofhinge-free compliant mechanism optimizationrdquo Structural andMultidisciplinary Optimization vol 49 no 3 pp 431ndash440 2014

[16] J Jones Contact Mechanics chapter 6 Cambridge UniversityPress Cambridge UK 2000

[17] G Hao and H Li ldquoConceptual designs of multi-degree of free-dom compliant parallel manipulators composed of wire-beambased compliant mechanismsrdquo Proceedings of the Institution ofMechanical Engineers Part C Journal ofMechanical EngineeringScience vol 229 no 3 pp 538ndash555 2015

[18] G Z Lum T J Teo G Yang S H Yeo andM Sitti ldquoIntegratingmechanism synthesis and topological optimization techniquefor stiffness-oriented design of a three degrees-of-freedomflexure-based parallel mechanismrdquo Precision Engineering vol39 pp 125ndash133 2015

[19] J Beroz A J Hart and S Awtar ldquoExtensible-link kinematicmodel for characterizing and optimizing compliant mechanismmotionrdquo Journal of Mechanical Design vol 136 no 3 Article ID031008 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


A

B

CP

Q

R

D

E

F

PlatformM1(1205791)

M2(1205792)

M3(1205793)

(a)

R

P QA

B

C

DE

F

c

l

a1 a2

b3

b2b1

a3

1205791

1205793

1205792

1205951 1205952

1205953

120593

F

A998400 B998400

C998400

D998400

E998400

F998400

(b)

Figure 1 3-RRR planar parallel prototype mechanism (a) 3D model of 3-RRR (b) description of 3-RRR kinematics

isotropic material with the penalization (SIMP) method [11]the ground structuremethod [12] and level set basedmethod[13] Although topology optimization has been a maturedtool for optimization design problem most of topologyoptimizationmethods are focusing on the single input-outputcondition In reality parallel compliant mechanisms withmultiple inputs and multiple outputs are widely used inthe fields of micropositioning and micromanipulation Therelationship between multiple inputs and multiple outputs isvectormappingmatrixwith certain connection Zhu et al [1415] presented a hinge-free compliant mechanism which hasone input and two outputs based on level set method Jones[16] presented a wavelet-based topology optimization formu-lation by implicitly imbeddingwavelet shrinkagemethod intooptimization formulation based on SIMP method Topologyoptimization methods for parallel compliant mechanismsmay also be found in other literatures [17ndash19] In thosemethods compared with conventional parallel prototypemechanisms themotion characteristics of designed structureare ambiguous On the other hand the mechanism has notincluded the hinge form after the topology optimizationprocess In this paper we called this structure peristalticstructure

In this paper we proposed a new method for topologyoptimization of planar 3-DOF peristaltic structure parallelrobot based on differential kinematic vector continuousmapping matrix with conventional prototype parallel mech-anism The remainder of the paper is organized as followsIn Section 2 the differential kinematic vector continuousmapping matrix with conventional prototype parallel mech-anism is built In Section 3 the optimization problems ofdesign peristaltic structure are proposed based on SIMPmethodThe shape sensitivity analysis is applied to obtain thedisplacement field for the optimization combined with thevector mapping matrix In Section 4 numerical examples arepresented to demonstrate the effectiveness of the proposedmethod Finally conclusions and a discussion for furtherwork are developed

2 Differential Kinematics with VectorContinuous Mapping Matrix

In order to keep the differential kinematic characteristicswith the prototype of parallel mechanism the differentialkinematic equations with vector continuous mapping ofmicroelastic deformation mechanism should be set Withoutloss of generality we select the 3-RRR prototype of planarparallel mechanism as an example As shown in Figure 1 thestructure of 3-RRR prototype planar parallel mechanism isestablished

Constraint equations are described as

119909119861 = 119909119860 + ℎ sdot cos120593119910119861 = 119910119860 + ℎ sdot sin120593119909119862 = 119909119860 + ℎ sdot cos (120593 + 120587)119910119862 = 119910119860 + ℎ sdot sin (120593 + 120587) (1)

Vector equation of the limb 119875119863119860may be described as997888997888rarr119874119860 = 997888997888rarr119874119875 + 997888997888rarr119875119863 + 997888997888rarr119863119860 (2)

where 119909119901 = 119910119901 = 0 Substituting (1) into (2) we obtain119909119860 = 1198861 sdot cos 1205791 + 1198871 sdot cos (1205791 + 1205951)119910119860 = 1198861 sdot sin 1205791 + 1198871 sdot sin (1205791 + 1205951) (3)

Vector equation of the limb 119861119864119876 can be described as997888997888rarr119874119861 = 997888997888rarr119875119876 + 997888997888rarr119876119864 + 997888rarr119864119861 (4)

where 119909119902 = 119888 119910119902 = 0 Substituting (1) into (4) we have119909119860 = 119888 minus 1198862 sdot cos 1205792 minus 1198872 sdot cos1205952 minus ℎ sdot cos120593119910119860 = 1198862 sdot sin 1205792 + 1198872 sdot sin1205952 minus ℎ sdot sin120593119909119861 = 119888 minus 1198862 sdot cos 1205792 minus 1198872 sdot cos1205952119910119861 = 1198862 sdot sin 1205792 + 1198872 sdot sin1205952

(5)





(7)


R1015840119872 =

[[[[[[[[[[[[

119909101584011986011991010158401198601199091015840119861119910101584011986111990910158401198621199101015840119862

]]]]]]]]]]]]

=[[[[[[[[[[[[


]]]]]]]]]]]]

(8)






(9)


[[[Δ119909Δ119910Δ120593

]]]= 119869119863 sdot [[[

Δ1205791Δ1205792Δ1205793]]] (10)

where

119869119863 =[[[[[[[[[


]]]]]]]]]


(11)




min 119862 = 3sum119894=1

3sum119895=1


119879119890119895120588119901119890 119870119864119880119890119894st 119870119895 = 119865119895 119870119880119894 = 119865119894 119865119895 = 119869119863119865119894

119894 = 1 2 3 119895 = 1 2 3forall 997888rarr int

Ω120588119890119889Ω le 1198810 le 120588min le 120588119890 le 1 119890 = 1 2 119873

(12)



120597119862120597120588119890 =120597 (sum3119894=1sum3119895=1 (119879119895119870119880119894))120597120588119890

= 120597 (sum3119895=1 119879119895 )120597120588119890 sdot 119870 sdot 3sum119894=1

119880119894 + 3sum119895=1

119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894+ 3sum119895=1


119895=1

119879119895 sdot 120597119870120597120588119890) sdot 3sum119894=1

119880119894

+ 3sum119895=1

119879119895 sdot (120597119870120597120588119890 sdot3sum119894=1


119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894(13)


120597119870120597120588119890 sdot3sum119895=1

119879119895 + 119870 sdot 120597 (sum3119895=1 119879119895 )120597120588119890 = 120597 (sum3119895=1 119865119895)120597120588119890 = 0120597119870120597120588119890 sdot

3sum119894=1

119880119894 + 119870 sdot 120597 (sum3119894=1119880119894)120597120588119890 = 120597 (sum3119894=1 119865119894)120597120588119890 = 0(14)


120597119862120597120588119890 = minus 3sum119895=1

119879119895 sdot 119870 sdot 3sum119894=1

119880119894 (15)


120597119862120597120588119890 =119873sum119890=1

3sum119894=1

3sum119895=1

119901119879119895 120588119901minus1119890 119870119864119880119894 (16)


120597119881120597120588119890 =120597 (intΩ120588119890119889119881)120597120588119890 = 1 (17)




(18)


Fixed basement

Motion platform

Output

Output Output

Input Fin2


Fout2

Fout1Fout3

(a) (b)




(19)



119894 = sum119895

120596119894119895120588119895 sum119895

120596119894119895 = 1 (20)



120596119894119895 =


(21)












20 40 60 80 100 120 140 160 1800minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06






y

x1198131

1198132

1198133



minus625

minus5

minus375

minus25

minus125

0

125

25

375

5

Valu

e


(times10minus3)

008 016 024 032 04 048 056 064 072 080

Time






1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10




(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time



008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)




]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)


[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)






6 Conclusions






Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(7)


R1015840119872 =

[[[[[[[[[[[[

119909101584011986011991010158401198601199091015840119861119910101584011986111990910158401198621199101015840119862

]]]]]]]]]]]]

=[[[[[[[[[[[[


]]]]]]]]]]]]

(8)






(9)


[[[Δ119909Δ119910Δ120593

]]]= 119869119863 sdot [[[

Δ1205791Δ1205792Δ1205793]]] (10)

where

119869119863 =[[[[[[[[[


]]]]]]]]]


(11)




min 119862 = 3sum119894=1

3sum119895=1


119879119890119895120588119901119890 119870119864119880119890119894st 119870119895 = 119865119895 119870119880119894 = 119865119894 119865119895 = 119869119863119865119894

119894 = 1 2 3 119895 = 1 2 3forall 997888rarr int

Ω120588119890119889Ω le 1198810 le 120588min le 120588119890 le 1 119890 = 1 2 119873

(12)



120597119862120597120588119890 =120597 (sum3119894=1sum3119895=1 (119879119895119870119880119894))120597120588119890

= 120597 (sum3119895=1 119879119895 )120597120588119890 sdot 119870 sdot 3sum119894=1

119880119894 + 3sum119895=1

119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894+ 3sum119895=1


119895=1

119879119895 sdot 120597119870120597120588119890) sdot 3sum119894=1

119880119894

+ 3sum119895=1

119879119895 sdot (120597119870120597120588119890 sdot3sum119894=1


119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894(13)


120597119870120597120588119890 sdot3sum119895=1

119879119895 + 119870 sdot 120597 (sum3119895=1 119879119895 )120597120588119890 = 120597 (sum3119895=1 119865119895)120597120588119890 = 0120597119870120597120588119890 sdot

3sum119894=1

119880119894 + 119870 sdot 120597 (sum3119894=1119880119894)120597120588119890 = 120597 (sum3119894=1 119865119894)120597120588119890 = 0(14)


120597119862120597120588119890 = minus 3sum119895=1

119879119895 sdot 119870 sdot 3sum119894=1

119880119894 (15)


120597119862120597120588119890 =119873sum119890=1

3sum119894=1

3sum119895=1

119901119879119895 120588119901minus1119890 119870119864119880119894 (16)


120597119881120597120588119890 =120597 (intΩ120588119890119889119881)120597120588119890 = 1 (17)




(18)


Fixed basement

Motion platform

Output

Output Output

Input Fin2


Fout2

Fout1Fout3

(a) (b)




(19)



119894 = sum119895

120596119894119895120588119895 sum119895

120596119894119895 = 1 (20)



120596119894119895 =


(21)












20 40 60 80 100 120 140 160 1800minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06






y

x1198131

1198132

1198133



minus625

minus5

minus375

minus25

minus125

0

125

25

375

5

Valu

e


(times10minus3)

008 016 024 032 04 048 056 064 072 080

Time






1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10




(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time



008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)




]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)


[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)






6 Conclusions






Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












min 119862 = 3sum119894=1

3sum119895=1


119879119890119895120588119901119890 119870119864119880119890119894st 119870119895 = 119865119895 119870119880119894 = 119865119894 119865119895 = 119869119863119865119894

119894 = 1 2 3 119895 = 1 2 3forall 997888rarr int

Ω120588119890119889Ω le 1198810 le 120588min le 120588119890 le 1 119890 = 1 2 119873

(12)



120597119862120597120588119890 =120597 (sum3119894=1sum3119895=1 (119879119895119870119880119894))120597120588119890

= 120597 (sum3119895=1 119879119895 )120597120588119890 sdot 119870 sdot 3sum119894=1

119880119894 + 3sum119895=1

119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894+ 3sum119895=1


119895=1

119879119895 sdot 120597119870120597120588119890) sdot 3sum119894=1

119880119894

+ 3sum119895=1

119879119895 sdot (120597119870120597120588119890 sdot3sum119894=1


119879119895 sdot 120597119870120597120588119890 sdot3sum119894=1

119880119894(13)


120597119870120597120588119890 sdot3sum119895=1

119879119895 + 119870 sdot 120597 (sum3119895=1 119879119895 )120597120588119890 = 120597 (sum3119895=1 119865119895)120597120588119890 = 0120597119870120597120588119890 sdot

3sum119894=1

119880119894 + 119870 sdot 120597 (sum3119894=1119880119894)120597120588119890 = 120597 (sum3119894=1 119865119894)120597120588119890 = 0(14)


120597119862120597120588119890 = minus 3sum119895=1

119879119895 sdot 119870 sdot 3sum119894=1

119880119894 (15)


120597119862120597120588119890 =119873sum119890=1

3sum119894=1

3sum119895=1

119901119879119895 120588119901minus1119890 119870119864119880119894 (16)


120597119881120597120588119890 =120597 (intΩ120588119890119889119881)120597120588119890 = 1 (17)




(18)


Fixed basement

Motion platform

Output

Output Output

Input Fin2


Fout2

Fout1Fout3

(a) (b)




(19)



119894 = sum119895

120596119894119895120588119895 sum119895

120596119894119895 = 1 (20)



120596119894119895 =


(21)












20 40 60 80 100 120 140 160 1800minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06






y

x1198131

1198132

1198133



minus625

minus5

minus375

minus25

minus125

0

125

25

375

5

Valu

e


(times10minus3)

008 016 024 032 04 048 056 064 072 080

Time






1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10




(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time



008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)




]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)


[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)






6 Conclusions






Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Fixed basement

Motion platform

Output

Output Output

Input Fin2


Fout2

Fout1Fout3

(a) (b)




(19)



119894 = sum119895

120596119894119895120588119895 sum119895

120596119894119895 = 1 (20)



120596119894119895 =


(21)












20 40 60 80 100 120 140 160 1800minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06






y

x1198131

1198132

1198133



minus625

minus5

minus375

minus25

minus125

0

125

25

375

5

Valu

e


(times10minus3)

008 016 024 032 04 048 056 064 072 080

Time






1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10




(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time



008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)




]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)


[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)






6 Conclusions






Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











20 40 60 80 100 120 140 160 1800minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06






y

x1198131

1198132

1198133



minus625

minus5

minus375

minus25

minus125

0

125

25

375

5

Valu

e


(times10minus3)

008 016 024 032 04 048 056 064 072 080

Time






1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10




(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time



008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)




]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)


[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)






6 Conclusions






Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205791 1205792 1205793 1205951 1205952 1205953 120593 1198861 1198862 1198863 ℎ23120587 49120587 16120587 112120587 19120587 13120587 536120587 6 6 6 10




(times10minus3)

minus08

minus06

minus04

minus02

0

02

04

06

08

1

12

Valu

e

008 016 024 032 04 048 056 064 072 080

Time



008 016 024 032 04 048 056 064 072 080

Time

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

Valu

e

(times10minus3)




]]][[[Δ1205791Δ1205792Δ1205793

]]]= [[[

Δ119909Δ119910Δ120593]]] (23)


[[[Δ119909Δ119910Δ120593

]]]= (minus615 293 minus355)1015840 (24)






6 Conclusions






Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Competing Interests


Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Topology Optimization of 3-DOF