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Research Article Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion Manipulator with Passive Link Hui Yang, 1 Hairong Fang , 1,2 Yuefa Fang, 1,2 and Haibo Qu 1 Robotics Research Center, School of Mechanical Electronic and Control Engineering, Beijing Jiaotong University, Beijing , China Key Laboratory of Vehicle Advanced Manufacturing, Measuring and Control Technology, Beijing Jiaotong University, Ministry of Education, Beijing, China Correspondence should be addressed to Hairong Fang; [email protected] Received 9 August 2018; Revised 3 October 2018; Accepted 21 October 2018; Published 25 November 2018 Academic Editor: Jorge Pomares Copyright © 2018 Hui Yang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In order to solve the problem of the honeycombs perfusion in the thermal protection system of the spacecraſt, this paper presents a novel parallel perfusion manipulator with one translational and two rotational (1T2R) degrees of freedom (DOFs), which can be used to construct a 5-DOF hybrid perfusion system for the perfusion of the honeycombs. e proposed 3P SS&PU parallel perfusion manipulator is mainly utilized as the main body of the hybrid perfusion system. e inverse kinematics and the Jacobian matrix of the proposed parallel manipulator are obtained. e analysis of kinematics performance for the proposed parallel manipulator including workspace, singularity, dexterity, and stiffness is conducted. Based on the virtual work principle and the link Jacobian matrix, the dynamic model of the parallel perfusion manipulator is carried out. With reference to dynamic equations, the relationship between the driving force and the mechanism parameters can be derived. In order to verify the correctness of the kinematics and dynamics model, the comparison of theoretical and simulation curves of the motion parameters related to the driving sliders is performed. Corresponding analyses illustrate that the proposed parallel perfusion possesses good kinematics performance and could satisfy the perfusion requirements of the honeycombs. e correctness of the established kinematics and dynamics models is proved, which has great significance for the experimental research of the perfusion system. 1. Introduction During the rise and reentry of the spacecraſt, the crew module of spacecraſt will suffer from a large aerodynamic heating effect. erefore, thermal protection system is needed to ensure the safety of the pilot and the normal operation of the equipment [1–3]. At present, the structure of spacecraſt ther- mal protection layer usually adopts hexagonal honeycomb structure, and then these honeycombs are spliced together and evenly spread on the surface of spherical crown. In order to achieve the effect of thermal protection, it needs to perfuse the heat-resistant material into the honeycomb structure [4]. However, due to the large size of spherical surface structure, the perfusion of heat-resistant material becomes the main problem. At present, it mainly adopts manual perfusion, which will inevitably require more manpower to complete the perfusion. Accordingly, it is necessary to introduce perfusion manipulator into the perfusion system. In the perfusion system, all the perfusion of the honey- combs is conducted based on the detection system. Due to the large size of the spherical crown surface, the perfusion object will be divided into subregion for processing. First, according to the detection system’s identification, the position of the task honeycombs can be identified. en, through the rotation of the spherical crown worktable and the motion of the parallel perfusion manipulator, the perfusion of the task honeycombs can be accomplished. When the perfusion of the identified task honeycombs is completed, the detection system will conduct the identification of the next group of honeycombs. In the article, the design and analysis of the perfusion manipulator are the focus of our research. As we all know, the traditional serial manipulator has the advantage of large workspace and the disadvantage of low stiffness, while the parallel manipulator with closed kine- matic chain has some advantages compared with the serial manipulator, such as high structural rigidity, high dynamic Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 6768947, 18 pages https://doi.org/10.1155/2018/6768947
19

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Page 1: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Research ArticleKinematics Performance and Dynamics Analysis ofa Novel Parallel Perfusion Manipulator with Passive Link

Hui Yang1 Hairong Fang 12 Yuefa Fang12 and Haibo Qu1

1Robotics ResearchCenter School ofMechanical Electronic andControl Engineering Beijing JiaotongUniversity Beijing 100044 China2Key Laboratory of Vehicle Advanced Manufacturing Measuring and Control Technology Beijing Jiaotong UniversityMinistry of Education Beijing China

Correspondence should be addressed to Hairong Fang hrfangbjtueducn

Received 9 August 2018 Revised 3 October 2018 Accepted 21 October 2018 Published 25 November 2018

Academic Editor Jorge Pomares

Copyright copy 2018 Hui Yang 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

In order to solve the problem of the honeycombs perfusion in the thermal protection system of the spacecraft this paper presentsa novel parallel perfusion manipulator with one translational and two rotational (1T2R) degrees of freedom (DOFs) which canbe used to construct a 5-DOF hybrid perfusion system for the perfusion of the honeycombs The proposed 3PSSampPU parallelperfusion manipulator is mainly utilized as the main body of the hybrid perfusion systemThe inverse kinematics and the Jacobianmatrix of the proposed parallel manipulator are obtained The analysis of kinematics performance for the proposed parallelmanipulator including workspace singularity dexterity and stiffness is conducted Based on the virtual work principle and thelink Jacobianmatrix the dynamic model of the parallel perfusion manipulator is carried out With reference to dynamic equationsthe relationship between the driving force and the mechanism parameters can be derived In order to verify the correctness ofthe kinematics and dynamics model the comparison of theoretical and simulation curves of the motion parameters related tothe driving sliders is performed Corresponding analyses illustrate that the proposed parallel perfusion possesses good kinematicsperformance and could satisfy the perfusion requirements of the honeycombs The correctness of the established kinematics anddynamics models is proved which has great significance for the experimental research of the perfusion system

1 Introduction

During the rise and reentry of the spacecraft the crewmoduleof spacecraft will suffer from a large aerodynamic heatingeffect Therefore thermal protection system is needed toensure the safety of the pilot and the normal operation of theequipment [1ndash3] At present the structure of spacecraft ther-mal protection layer usually adopts hexagonal honeycombstructure and then these honeycombs are spliced togetherand evenly spread on the surface of spherical crown In orderto achieve the effect of thermal protection it needs to perfusethe heat-resistant material into the honeycomb structure [4]However due to the large size of spherical surface structurethe perfusion of heat-resistant material becomes the mainproblem At present it mainly adopts manual perfusionwhichwill inevitably requiremoremanpower to complete theperfusion Accordingly it is necessary to introduce perfusionmanipulator into the perfusion system

In the perfusion system all the perfusion of the honey-combs is conducted based on the detection system Due tothe large size of the spherical crown surface the perfusionobject will be divided into subregion for processing Firstaccording to the detection systemrsquos identification the positionof the task honeycombs can be identified Then through therotation of the spherical crown worktable and the motion ofthe parallel perfusion manipulator the perfusion of the taskhoneycombs can be accomplished When the perfusion ofthe identified task honeycombs is completed the detectionsystem will conduct the identification of the next group ofhoneycombs In the article the design and analysis of theperfusion manipulator are the focus of our research

As we all know the traditional serial manipulator hasthe advantage of large workspace and the disadvantage oflow stiffness while the parallel manipulator with closed kine-matic chain has some advantages compared with the serialmanipulator such as high structural rigidity high dynamic

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6768947 18 pageshttpsdoiorg10115520186768947

2 Mathematical Problems in Engineering

1T2R perfusion manipulator

Arc guide way

Hexagonal honeycomb

Sphericalsurface

Perfusion head

Figure 1 3D model of the hybrid perfusion system

performance high accuracy and low moving inertia [5ndash9]For such a reason parallel manipulator has been extensivelyapplied as flight simulator [10ndash12] machine tools [13ndash15]mobile machining module [16 17] and high-speed pick-and-place robots [18ndash20] However the parallel manipulator alsohas the disadvantage of small workspace For the perfusionsystem the spherical structure is relatively large and the endof the moving platform needs to carry the heavy perfusiondevice so it requires that the perfusion manipulator shouldhave the characteristics of large workspace and high stiffnessHowever due to the low stiffness of the serial manipulatorand the small workspace of the parallel manipulator allof them could hardly satisfy the perfusion requirements ofthe perfusion manipulator Consequently in order to solvethese problems hybrid manipulator with the advantages ofserial manipulator and parallel manipulator is selected as thesuitable perfusion manipulator

Since the spherical crown surface can be seen as acomplex free-form surface the perfusion manipulator needsat least five DOFs (3T2R) in the process of the heat-resistant material perfusion Generally the 5-DOF hybridmanipulator can be achieved by adding a 2-DOF rotatinghead to the moving platform of a 3-DOF position adjustmentmanipulator which are typically represented by Tricept [2122] Trivariant [23 24] and Exechon machine tool [2526] In addition some scholars also constructed the 5-DOFhybrid manipulator by integrating a 3-DOF pose adjustmentmanipulator with two long guide rails which are particularlytaken as example as Sprint Z3 head [27 28] and the machinetool [29 30] Motivated by this idea this paper proposesa novel 1T2R parallel manipulator that can be used toconstruct 5-DOF hybrid perfusion system for the perfusionof the honeycombs in the thermal protection system of thespacecraft For the 1T2R parallel manipulator due to theintroduction of the passive link the stiffness of the proposedparallel manipulator has been improved Moreover sincethe idea that the moving platform size is larger than thefixed platform size the singularity of the parallel manipulatoris avoided effectively And the structural of the proposed

Fixed platform

PSS limb

PU limb

Moving platform

Figure 2 CADmodel of the 1T2R parallel manipulator

manipulator is simple compact symmetrical and easy tocontrol Thus because of the reasons mentioned above theproposed parallel manipulator is selected for the perfusion ofthe honeycombs in the thermal protection system

In this paper the objective of the researchmainly is aimedat the kinematics performance analysis and the establishmentof the dynamics model for the proposed 1T2R parallelperfusion manipulator The remainder of this article is givenas follows In Section 2 structure description of the proposed1T2R parallel manipulator is represented The analysis ofkinematics and performance is conducted in Sections 3 and4 respectively In Section 5 based on the principle of virtualwork dynamic model of the proposed parallel manipulator isestablishedThe simulation research is carried out by utilizingkinematics and dynamics model in Section 6 Conclusionsare given in Section 7

2 Architecture

21 ArchitectureDescription To satisfy the perfusion require-ment of spherical surface the perfusion manipulator shouldhave at least five DOFs which concludes three translationalDOFs and two rotational DOFs According to the require-ments a 5-DOF hybrid perfusion system is proposed whichis shown in Figure 1 The perfusion system consists of an arcguide way a 1T2R parallel manipulator and a honeycombworktableThe 1T2R parallel manipulator canmove along thearc guide way The honeycomb worktable can rotate aboutits own axis and the honeycomb structure is evenly spreadin spherical surface When the perfusion manipulator is ata certain position of the guide way the perfusion head canachieve the perfusion of the honeycombs within the movingplatformrsquos reachable workspace Through the rotation of theworktable and the movement of the parallel manipulatoralong the guide way all the perfusion of honeycombs can becompleted successfully

Figure 2 indicates a CAD model of the 1T2R paral-lel manipulator which consists of a fixed platform three

Mathematical Problems in Engineering 3

B

M

B1

N1

l1

M1

B4

yb1

zbxb

N2

l2

ymRm

zm xm

M2

M3

B3

N3

l3

B2

Rb

1

Figure 3 Kinematic diagram of the 1T2R parallel manipulator

actuated PSS (P denotes the actuated joint) links a movingplatform and the passive constraint link PU Every PSSlimb connects the fixed platform with perfusion platform byan active prismatic joint followed by two spherical jointsAnd every prismatic joint is driven by a servomotor Thepassive limb contains a universal joint and a prismatic jointwhich connects the perfusion platform with the fixed platesuccessivelyThe existence of the PU limbwill highly improvestiffness of the whole system

22 Parameters Description The schematic model of pro-posed manipulator in Figure 3 shows that it has two plat-forms fixed platform labeled by119861111986121198613 andmoving platformdemonstrated by119872111987221198723The three limbs are placed in 120degree intervals on base platform Links119873119894119872119894 are attached tothe moving platform by a spherical joint at119872119894 and to a sliderby a spherical joint at119873119894Themiddle limb is established fromthe fixed platform to the perfusion platform whose one endis attached to the base with a prismatic joint 1198614 and the otherend is attached to the end-effector platform by a universaljoint To facilitate the analysis the fixed reference frame119861-119909119887119910119887119911119887 is placed at the center of the base platform where 119910119887axis is along the direction of straight line 1198611198611 Similarly thecoordinate axes of moving frame are denoted by119872-119909119898119910119898119911119898in which 119910119898 axis is along the direction of straight line1198721198721Parameter 120593119894 is the angle measured from 119909119887 to 119861119861119894 and 120601119894is the angle measured from 119909119898 to 119872119872119894 The length of thelimb119873119894119872119894 is denoted by 119897119894The length of 119861119861119894 is demonstratedby 119877119887 and 119877119898 represents the length of the line 119872119872119894 Thedisplacement of the driving joint is represented by 1199041198943 Kinematics Analysis

31 Inverse Kinematics As shown in Figure 4 the rela-tionship between the coordinate systems 119861-119909119887119910119887119911119887 and119872-119909119898119910119898119911119898 can be described by the rotation matrix 119861119877119875which can be obtained by three continuous rotations of

Bi

Nii

sii

Di

xizi

lii

Mi

i

ym

zm

M

xm

xb

B

yb

zb

i

yi

Figure 4 Kinematics of the ith PSS branch

Euler angles 120572 120573 and 120574 about the fixed 119909119887 119910119887 and 119911119887 axisrespectively For the 1T2R parallel manipulator the angle 120574 iszero Thus the rotation matrix from 119872-119909119898119910119898119911119898 coordinatesystem to 119861-119909119887119910119887119911119887 coordinate system can be expressed as

119861119877119872 = Rot (119910119887 120573)Rot (119909119887 120572) = [[

[119888120573 119904120573119904120572 1199041205731198881205720 119888120572 minus119904120572

minus119904120573 119888120573119904120572 119888120573119888120572]]]

(1)

where s and c correspond to the sine and cosine functionsrespectively

To facilitate the analysis the local coordinate system119873119894-119909119894119910119894119911119894 of the ith PSS branch also has been established atthe point119873119894 and the 119911119894 axis is along the direction of straightline 119873119894119872119894 The system 119873119894-119909119894119910119894119911119894 can be considered as twocontinuous rotations of angles 120581119894 and 120595119894 about 119911119887 axis and 1199101015840119894axis (rotated 119910119887 axis) respectively Consequently the rotationmatrix can be represented as

119861119877119894 = Rot (119911119887 120581119894)Rot (1199101015840119894 120595119894)

= [[[119888120581119894119888120595119894 minus119904120581119894 119888120581119894119904120595119894119904120581119894119888120595119894 119888120581119894 119904120581119894119904120595119894minus119904120595119894 0 119888120595119894

]]]

(2)

According to Figure 4 the closed-loop vector equation ofthe ith link is given as follows

119898 + 119903119894 = 119887119894 + 119904119894119899119894 + 119886119894 + 119897119894119896119894 (3)

where 119898 is the position vector of the moving platformdescribed in 119861-119909119887119910119887119911119887 system 119903119894 = 119861119877119872119898119903119894 119903119894 and 119898119903119894represent the position vector of the straight line 119872119872119894 in119861-119909119887119910119887119911119887 and 119872-119909119898119910119898119911119898 system respectively 119887119894 is theposition vector of the straight line 119874119861119894 in 119861-119909119887119910119887119911119887 systemwhile 119899119894 and 119896119894 denote the unit vector of the straight line119861119894119863119894 and119873119894119872119894 expressed in the coordinate system 119861-119909119887119910119887119911119887

4 Mathematical Problems in Engineering

respectively 119904119894 is the displacement of the prismatic joint forthe ith PSS branch and 119897119894 is the length of the link 119873119894119872119894Therefore the vectors mentioned above can be obtained as

119898 = [[[00119911119898

]]]

119898119903119894 = [[

[1198771198981198881206011198941198771198981199041206011198940

]]]

119887119894 = [[[1198771198871198881205931198941198771198871199041205931198940

]]]

119899119894 = [[[001]]]

119886119894 = [[[119889111988812059311989411988911199041205931198940

]]]

(4)

From 2) the unit vector 119896119894 will be expressed as

119896119894 = 119861119877119894119894119896119894 = 119861119877119894 [[[001]]]= [[[119888120581119894119904120595119894119904120581119894119904120595119894119888120595119894

]]]= [[[119896119909119894119896119910119894119896119911119894

]]]

(5)

At the same time the relationship between 120581119894 and 120595119894 canbe calculated as

119888120595119894 = 119896119911119894119904120595119894 = radic1198962119909119894 + 1198962119910119894 (0 le 120595119894 le 120587) 119904120581119894 = 119896119910119894119904120595119894 119888120581119894 = 119896119909119894119904120595119894

(6)

Substituting (4) and (5) into (3) and squaring both sidesof (3) can be deduced as

1199041198861198941199042119894 + 119904119887119894119904119894 + 119904119888119894 = 0 (7)

where

119904119886119894 = 1198612119909119894 + 1198612119910119894 + 1198612119911119894119904119887119894 = 2 (119860119909119894119861119909119894 + 119860119910119894119861119910119894 + 119860119911119894119861119911119894) 119904119888119894 = 1198602119909119894 + 1198602119910119894 + 1198602119911119894 minus 1198972119894

119860119909119894 = 119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894

119861119909119894 = 0119860119910119894 = 119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119861119910119894 = 0119860119911119894 = 119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573119861119911119894 = minus1

(8)

The inverse kinematics solutions for the ith limb can bederived from (7)

119904119894 = minus119904119887119894 plusmn radic1199042119887119894 minus 41199041198861198941199041198881198942119904119886119894 (9)

From (9) there are two solutions for each driving jointthat is to say there exist eight possible solutions for a givenconfiguration However the three driving sliders are onlyallowed moving downward from point 119861119894Therefore only thenegative symbol can satisfy the motion characteristics of theparallel perfusion manipulator

32 Forward Kinematics Forward kinematics for the 3PSS-PUmechanism is to solve the pose parameters (120572 120573 119911119898) afterknowing the driving parameters (1199041 1199042 1199043) For this problemit can be obtained from (3) and (4) which can lead to thefollowing formula

119897119894119896119894 = [[[119909119897119894119910119897119894119911119897119894

]]]

= [[[119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573 minus 119904119894

]]]

(10)

According to constraint of the length of the links 119873119894119872119894the restraint equation can be obtained

1199092119897119894 + 1199102119897119894 + 1199112119897119894 = 1198972119894 (11)

To solve the forward kinematics problem 120573 and 119911119898 canbe represented by 120572 Therefore (12) can be derived from (10)and (11)

1199112119898 + (119860 119894119888120573 + 119877119894119904120573 + 119876119894) 119911119898 + (119862119894119888120573 + 119863119894119904120573 + 119864119894)= 0 (12)

where the coefficients 119860 119894 119876119894 sim 119864119894 are functions of 120572119860 119894 = 2119877119898119904120601119894119904120572119877119894 = minus2119877119898119888120601119894119876119894 = minus2s119894

Mathematical Problems in Engineering 5

C119894 = minus2119877119898119888120601119894119888120593119894 (119877119887 + 1198891) minus 2119877119898119904119894119904120572119904120601119894119863119894 = 2119877119898119904119894119888120601119894 minus 2119877119898119888120593119894119904120572119904120601119894 (119877119887 + 1198891) 119864119894 = (119877119887 + 1198891)2 + 1198772119898 + 1199042119894 minus 1198972119894

minus 2119877119898119888120572119904120601119894119904120593119894 (119877119887 + 1198891) (13)

Equation (12) also can be further simplified by subtract-ing one equation from another which yields the followingexpression

(119860 119894119895119911119898 + 119862119894119895) 119888120573 + (119877119894119895119911119898 + 119863119894119895) 119904120573 + 119876119894119895119911119898 + 119864119894119895= 0 (14)

where 119894 = 119895 and119860 119894119895 = 119860 119894 minus 119860119895119877119894119895 = 119877119894 minus 119877119895119876119894119895 = 119876119894 minus 119876119895119862119894119895 = 119862119894 minus 119862119895119863119894119895 = 119863119894 minus 119863119895119864119894119895 = 119864119894 minus 119864119895

(15)

Therefore 120573 can be obtained by 119911119898 and 120572 as

119888120573 = minus1198651199112119898 + 119866119911119898 + 1198671198681199112119898 + 119869119911119898 + 119870

119904120573 = minus1198711199112119898 + 119878119911119898 + 1198791198681199112119898 + 119869119911119898 + 119870(16)

where the coefficients 119865 sim 119870119883 are functions of 120572 which areexpressed as

119865 = 1198771211987613 minus 1198761211987713119866 = 1198631211987613 + 1198771211986413 minus 1198641211987713 minus 1198761211986313119867 = 1198631211986413 minus 1198641211986313119868 = 1198771211986013 minus 1198601211987713119869 = 1198631211986013 + 1198771211986213 minus 1198621211987713 minus 1198601211986313119870 = 1198631211986213 minus 1198621211986313119871 = 1198761211986013 minus 1198601211987613119878 = 1198641211986013 + 1198761211986213 minus 1198621211987613 minus 1198601211986413119879 = 1198641211986213 minus 1198621211986413

(17)

Since 1198882120573 + 1199042120573 = 1 substituting (16) into this formulayields

11988041199114119898 + 11988031199113119898 + 11988021199112119898 + 1198801119911119898 + 1198800 = 0 (18)

where

1198804 = 1198652 + 1198712 minus 11986821198803 = 2 (119865119866 + 119871119878 minus 119868119869) 1198802 = 1198662 + 1198782 minus 1198692 + 2119865119867 + 2119871119879 minus 21198681198701198801 = 2119866119867 + 2119878119879 minus 21198691198701198800 = 1198672 + 1198792 minus 1198702

(19)

Equation (12) consists of three independent equationsand two independent equations have been obtained as shownin (16) Substituting (16) into (12) another equation can bededuced For instance when 119894 = 1 the equation is expressedas

11988231199113119898 +11988221199112119898 +1198821119911119898 +1198820 = 0 (20)

where1198823 = 1198601119865 + 1198771119871 minus 11987611198681198822 = 1198601119866 + 1198621119865 + 1198771119878 + 1198631119871 minus 1198761119869 minus 11986411198681198821 = 1198601119867 + 1198621119866 + 1198771119879 + 1198631119878 minus 1198761119870 minus 11986411198691198820 = 1198621119867 +1198631119879 minus 1198641119870

(21)

It can be seen that (18) and (20) have a commonsolution of 119911119898 By using Bezoutrsquos method [31] the followingdeterminant should be satisfied10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198804 1198803 1198802 1198801 1198800 0 00 1198804 1198803 1198802 1198801 1198800 00 0 1198804 1198803 1198802 1198801 11988001198823 1198822 1198821 1198820 0 0 00 1198823 1198822 1198821 1198820 0 00 0 1198823 1198822 1198821 1198820 00 0 0 1198823 1198822 1198821 1198820

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (22)

Thus (22) becomes an equation about 120572 when thedisplacement 119904119894 is known Now that all the equations havebeen obtained the method to solve the forward kinematicsproblem can be given as follows (i) to calculate 120572 from (22)(ii) to calculate 119911119898 from (20) and (iii) to calculate 120573 from (16)

For step (i) the sine and cosine components of 120572 shouldbe replaced by 119905 = tan(1205722) Based on the standard trans-formation expression the equation 119904120572 = (1minus1199052)(1+1199052) 119888120572 =2119905(1+1199052) can be obtained Finally (22) becomes a polynomialalgebraic equation about the variable 119905Then according to (ii)and (iii) the forward kinematics problem can be solved

33 JacobianMatrix Taking the derivative of (3) with respectto time leads to

119904119894119899119894 + 120596119894 times 119897119894119896119894 = 119907119898 + 120596119898 times 119903119894 (23)

where 119907119898 = [0 0 119898]T and 120596119898 = [ 120573 0]T denote the linearand angular velocity vector of the hybrid perfusion platformin the fixed coordinate system 119861-119909119887119910119887119911119887 respectively

6 Mathematical Problems in Engineering

Task honeycombs

40∘

Figure 5 Structure of the spherical surface

Taking the dot product with 119896119894 on both sides of (23) thevelocity of the ith driving joint can be deduced as

119904119894 = [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

][119907119898120596119898

] = 119869119894 [119907119898120596119898

] (24)

Rewriting the velocities of the driving joints in the matrixform as

119904 = 119869minus1119904 119869119901 = 119869 (25)

where

119904 = [ 1199041 1199042 1199043]T = [119907T119898 120596T119898]T 119869119904 = diag (119896T11198991 119896T21198992 119896T31198993) 119869119901 = [ 1198961 1198962 1198963

1199031 times 1198961 1199032 times 1198962 1199033 times 1198963]T

(26)

The Jacobianmatrix between the velocity vector and thedriving joint velocity vector 119904 can be obtained as

119869 = 119869minus1119904 119869119901 = [119869T1 119869T2 119869T3 ]T (27)

4 Kinematics Performance Analysis

Following the establishment of the mechanism model andthe kinematics analysis the analysis of the kinematics perfor-mance of the 3PSS-PUparallelmanipulatorwill be conductedin this section which includes the workspace singularitydexterity and stiffness The analysis of workspace is mainlyon account of the kinematics solved in Section 31 And basedon the Jacobian matrix analyzed in Section 33 the analysesof the singularity dexterity and stiffness are all developed indetail

41 Workspace Analysis

411 Task Honeycombs Analysis As shown in Figure 5 themaximum angle between the point on the spherical crownsurface and the vertical axis is 40 degrees Due to the large

size of the spherical crown surface it adopts the method ofsubarea perfusion to accomplish the perfusion of the largeobject and the task honeycombs of subarea are shown inFigure 6 For the task honeycombs the maximum angle 120579maxof the honeycombs is 10∘ During the perfusion process andthrough the rotation of the spherical crownworktable and themovement of the parallel manipulator along the arc guide railand the motion of the moving platform the perfusion of allthe honeycombs can be completed successfully

412 Reachable Workspace Analysis In this section thereachableworkspace of the perfusionmanipulator is obtainedby the method of the geometric constraints As shown inFigure 7 the flow diagram for calculating the workspaceof the perfusion manipulator has been given in detailAccording to the flow diagram of the workspace and theparameters given in Table 1 the reachable workspace of themoving platform can be obtained

As shown in Figures 8(a) and 8(b) the 3D view andvertical view of the reachable workspace for the parallelperfusion manipulator are obtained From the 3D view ofthe workspace it can be seen that the rotation range ofthe moving platform about 119909119887 axis and 119910119887 axis remainsunchanged with the increase of the value of the variable 119911119898For the task honeycombs of the subarea the maximum angleof the task honeycombs is 10∘ and then the task workspacecan be described as the yellow region in Figure 8(b) FromFigure 8(b) it also can be easily concluded that the taskworkspace is always within the reachable workspace of theparallelmanipulatorThat is to say the proposed 1T2R parallelperfusionmanipulator can complete the perfusion of the taskhoneycombs Then by the rotation of the spherical crownworktable and the movement of the parallel manipulatoralong the arc guide rail the perfusion of all the honeycombswill be completed

42 Singularity Analysis The method based on the Jacobianmatrices is the most common method to find the singularityof a mechanism [32] In order to obtain all the singularitypositions of the parallel manipulator the determinant of theinverse Jacobian matrix 119869119904 and forward Jacobian matrix 119869119901will be conducted Based on the Jacobian matrix the singu-larity conditions of the proposed parallel manipulator canbe divided into three types which include inverse kinematicsingularity (IKS) direct kinematic singularity (DKS) andcombined singularity (CS) And the conditions for satisfyingthese types of singularity can be given as follows

IKS det (119869119904) = 0 det (119869119901119904) = 0

DKS det (119869119904) = 0 det (119869119901119904) = 0CS det (119869119904) = 0 det (119869119901119904) = 0

(28)

where 119869119901119904 is a 3 times 3 submatrix from 119869119901As for the IKS singularity it occurs when matrix 119869119904 is

not full rank ie det(119869119904) = 0 or 119896119894 sdot 119899119894 = 0 (119894 = 1 2 3)which means one or more PSS legs are perpendicular totheir corresponding sliding rails As shown in Figure 9 it

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Page 2: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

2 Mathematical Problems in Engineering

1T2R perfusion manipulator

Arc guide way

Hexagonal honeycomb

Sphericalsurface

Perfusion head

Figure 1 3D model of the hybrid perfusion system

performance high accuracy and low moving inertia [5ndash9]For such a reason parallel manipulator has been extensivelyapplied as flight simulator [10ndash12] machine tools [13ndash15]mobile machining module [16 17] and high-speed pick-and-place robots [18ndash20] However the parallel manipulator alsohas the disadvantage of small workspace For the perfusionsystem the spherical structure is relatively large and the endof the moving platform needs to carry the heavy perfusiondevice so it requires that the perfusion manipulator shouldhave the characteristics of large workspace and high stiffnessHowever due to the low stiffness of the serial manipulatorand the small workspace of the parallel manipulator allof them could hardly satisfy the perfusion requirements ofthe perfusion manipulator Consequently in order to solvethese problems hybrid manipulator with the advantages ofserial manipulator and parallel manipulator is selected as thesuitable perfusion manipulator

Since the spherical crown surface can be seen as acomplex free-form surface the perfusion manipulator needsat least five DOFs (3T2R) in the process of the heat-resistant material perfusion Generally the 5-DOF hybridmanipulator can be achieved by adding a 2-DOF rotatinghead to the moving platform of a 3-DOF position adjustmentmanipulator which are typically represented by Tricept [2122] Trivariant [23 24] and Exechon machine tool [2526] In addition some scholars also constructed the 5-DOFhybrid manipulator by integrating a 3-DOF pose adjustmentmanipulator with two long guide rails which are particularlytaken as example as Sprint Z3 head [27 28] and the machinetool [29 30] Motivated by this idea this paper proposesa novel 1T2R parallel manipulator that can be used toconstruct 5-DOF hybrid perfusion system for the perfusionof the honeycombs in the thermal protection system of thespacecraft For the 1T2R parallel manipulator due to theintroduction of the passive link the stiffness of the proposedparallel manipulator has been improved Moreover sincethe idea that the moving platform size is larger than thefixed platform size the singularity of the parallel manipulatoris avoided effectively And the structural of the proposed

Fixed platform

PSS limb

PU limb

Moving platform

Figure 2 CADmodel of the 1T2R parallel manipulator

manipulator is simple compact symmetrical and easy tocontrol Thus because of the reasons mentioned above theproposed parallel manipulator is selected for the perfusion ofthe honeycombs in the thermal protection system

In this paper the objective of the researchmainly is aimedat the kinematics performance analysis and the establishmentof the dynamics model for the proposed 1T2R parallelperfusion manipulator The remainder of this article is givenas follows In Section 2 structure description of the proposed1T2R parallel manipulator is represented The analysis ofkinematics and performance is conducted in Sections 3 and4 respectively In Section 5 based on the principle of virtualwork dynamic model of the proposed parallel manipulator isestablishedThe simulation research is carried out by utilizingkinematics and dynamics model in Section 6 Conclusionsare given in Section 7

2 Architecture

21 ArchitectureDescription To satisfy the perfusion require-ment of spherical surface the perfusion manipulator shouldhave at least five DOFs which concludes three translationalDOFs and two rotational DOFs According to the require-ments a 5-DOF hybrid perfusion system is proposed whichis shown in Figure 1 The perfusion system consists of an arcguide way a 1T2R parallel manipulator and a honeycombworktableThe 1T2R parallel manipulator canmove along thearc guide way The honeycomb worktable can rotate aboutits own axis and the honeycomb structure is evenly spreadin spherical surface When the perfusion manipulator is ata certain position of the guide way the perfusion head canachieve the perfusion of the honeycombs within the movingplatformrsquos reachable workspace Through the rotation of theworktable and the movement of the parallel manipulatoralong the guide way all the perfusion of honeycombs can becompleted successfully

Figure 2 indicates a CAD model of the 1T2R paral-lel manipulator which consists of a fixed platform three

Mathematical Problems in Engineering 3

B

M

B1

N1

l1

M1

B4

yb1

zbxb

N2

l2

ymRm

zm xm

M2

M3

B3

N3

l3

B2

Rb

1

Figure 3 Kinematic diagram of the 1T2R parallel manipulator

actuated PSS (P denotes the actuated joint) links a movingplatform and the passive constraint link PU Every PSSlimb connects the fixed platform with perfusion platform byan active prismatic joint followed by two spherical jointsAnd every prismatic joint is driven by a servomotor Thepassive limb contains a universal joint and a prismatic jointwhich connects the perfusion platform with the fixed platesuccessivelyThe existence of the PU limbwill highly improvestiffness of the whole system

22 Parameters Description The schematic model of pro-posed manipulator in Figure 3 shows that it has two plat-forms fixed platform labeled by119861111986121198613 andmoving platformdemonstrated by119872111987221198723The three limbs are placed in 120degree intervals on base platform Links119873119894119872119894 are attached tothe moving platform by a spherical joint at119872119894 and to a sliderby a spherical joint at119873119894Themiddle limb is established fromthe fixed platform to the perfusion platform whose one endis attached to the base with a prismatic joint 1198614 and the otherend is attached to the end-effector platform by a universaljoint To facilitate the analysis the fixed reference frame119861-119909119887119910119887119911119887 is placed at the center of the base platform where 119910119887axis is along the direction of straight line 1198611198611 Similarly thecoordinate axes of moving frame are denoted by119872-119909119898119910119898119911119898in which 119910119898 axis is along the direction of straight line1198721198721Parameter 120593119894 is the angle measured from 119909119887 to 119861119861119894 and 120601119894is the angle measured from 119909119898 to 119872119872119894 The length of thelimb119873119894119872119894 is denoted by 119897119894The length of 119861119861119894 is demonstratedby 119877119887 and 119877119898 represents the length of the line 119872119872119894 Thedisplacement of the driving joint is represented by 1199041198943 Kinematics Analysis

31 Inverse Kinematics As shown in Figure 4 the rela-tionship between the coordinate systems 119861-119909119887119910119887119911119887 and119872-119909119898119910119898119911119898 can be described by the rotation matrix 119861119877119875which can be obtained by three continuous rotations of

Bi

Nii

sii

Di

xizi

lii

Mi

i

ym

zm

M

xm

xb

B

yb

zb

i

yi

Figure 4 Kinematics of the ith PSS branch

Euler angles 120572 120573 and 120574 about the fixed 119909119887 119910119887 and 119911119887 axisrespectively For the 1T2R parallel manipulator the angle 120574 iszero Thus the rotation matrix from 119872-119909119898119910119898119911119898 coordinatesystem to 119861-119909119887119910119887119911119887 coordinate system can be expressed as

119861119877119872 = Rot (119910119887 120573)Rot (119909119887 120572) = [[

[119888120573 119904120573119904120572 1199041205731198881205720 119888120572 minus119904120572

minus119904120573 119888120573119904120572 119888120573119888120572]]]

(1)

where s and c correspond to the sine and cosine functionsrespectively

To facilitate the analysis the local coordinate system119873119894-119909119894119910119894119911119894 of the ith PSS branch also has been established atthe point119873119894 and the 119911119894 axis is along the direction of straightline 119873119894119872119894 The system 119873119894-119909119894119910119894119911119894 can be considered as twocontinuous rotations of angles 120581119894 and 120595119894 about 119911119887 axis and 1199101015840119894axis (rotated 119910119887 axis) respectively Consequently the rotationmatrix can be represented as

119861119877119894 = Rot (119911119887 120581119894)Rot (1199101015840119894 120595119894)

= [[[119888120581119894119888120595119894 minus119904120581119894 119888120581119894119904120595119894119904120581119894119888120595119894 119888120581119894 119904120581119894119904120595119894minus119904120595119894 0 119888120595119894

]]]

(2)

According to Figure 4 the closed-loop vector equation ofthe ith link is given as follows

119898 + 119903119894 = 119887119894 + 119904119894119899119894 + 119886119894 + 119897119894119896119894 (3)

where 119898 is the position vector of the moving platformdescribed in 119861-119909119887119910119887119911119887 system 119903119894 = 119861119877119872119898119903119894 119903119894 and 119898119903119894represent the position vector of the straight line 119872119872119894 in119861-119909119887119910119887119911119887 and 119872-119909119898119910119898119911119898 system respectively 119887119894 is theposition vector of the straight line 119874119861119894 in 119861-119909119887119910119887119911119887 systemwhile 119899119894 and 119896119894 denote the unit vector of the straight line119861119894119863119894 and119873119894119872119894 expressed in the coordinate system 119861-119909119887119910119887119911119887

4 Mathematical Problems in Engineering

respectively 119904119894 is the displacement of the prismatic joint forthe ith PSS branch and 119897119894 is the length of the link 119873119894119872119894Therefore the vectors mentioned above can be obtained as

119898 = [[[00119911119898

]]]

119898119903119894 = [[

[1198771198981198881206011198941198771198981199041206011198940

]]]

119887119894 = [[[1198771198871198881205931198941198771198871199041205931198940

]]]

119899119894 = [[[001]]]

119886119894 = [[[119889111988812059311989411988911199041205931198940

]]]

(4)

From 2) the unit vector 119896119894 will be expressed as

119896119894 = 119861119877119894119894119896119894 = 119861119877119894 [[[001]]]= [[[119888120581119894119904120595119894119904120581119894119904120595119894119888120595119894

]]]= [[[119896119909119894119896119910119894119896119911119894

]]]

(5)

At the same time the relationship between 120581119894 and 120595119894 canbe calculated as

119888120595119894 = 119896119911119894119904120595119894 = radic1198962119909119894 + 1198962119910119894 (0 le 120595119894 le 120587) 119904120581119894 = 119896119910119894119904120595119894 119888120581119894 = 119896119909119894119904120595119894

(6)

Substituting (4) and (5) into (3) and squaring both sidesof (3) can be deduced as

1199041198861198941199042119894 + 119904119887119894119904119894 + 119904119888119894 = 0 (7)

where

119904119886119894 = 1198612119909119894 + 1198612119910119894 + 1198612119911119894119904119887119894 = 2 (119860119909119894119861119909119894 + 119860119910119894119861119910119894 + 119860119911119894119861119911119894) 119904119888119894 = 1198602119909119894 + 1198602119910119894 + 1198602119911119894 minus 1198972119894

119860119909119894 = 119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894

119861119909119894 = 0119860119910119894 = 119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119861119910119894 = 0119860119911119894 = 119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573119861119911119894 = minus1

(8)

The inverse kinematics solutions for the ith limb can bederived from (7)

119904119894 = minus119904119887119894 plusmn radic1199042119887119894 minus 41199041198861198941199041198881198942119904119886119894 (9)

From (9) there are two solutions for each driving jointthat is to say there exist eight possible solutions for a givenconfiguration However the three driving sliders are onlyallowed moving downward from point 119861119894Therefore only thenegative symbol can satisfy the motion characteristics of theparallel perfusion manipulator

32 Forward Kinematics Forward kinematics for the 3PSS-PUmechanism is to solve the pose parameters (120572 120573 119911119898) afterknowing the driving parameters (1199041 1199042 1199043) For this problemit can be obtained from (3) and (4) which can lead to thefollowing formula

119897119894119896119894 = [[[119909119897119894119910119897119894119911119897119894

]]]

= [[[119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573 minus 119904119894

]]]

(10)

According to constraint of the length of the links 119873119894119872119894the restraint equation can be obtained

1199092119897119894 + 1199102119897119894 + 1199112119897119894 = 1198972119894 (11)

To solve the forward kinematics problem 120573 and 119911119898 canbe represented by 120572 Therefore (12) can be derived from (10)and (11)

1199112119898 + (119860 119894119888120573 + 119877119894119904120573 + 119876119894) 119911119898 + (119862119894119888120573 + 119863119894119904120573 + 119864119894)= 0 (12)

where the coefficients 119860 119894 119876119894 sim 119864119894 are functions of 120572119860 119894 = 2119877119898119904120601119894119904120572119877119894 = minus2119877119898119888120601119894119876119894 = minus2s119894

Mathematical Problems in Engineering 5

C119894 = minus2119877119898119888120601119894119888120593119894 (119877119887 + 1198891) minus 2119877119898119904119894119904120572119904120601119894119863119894 = 2119877119898119904119894119888120601119894 minus 2119877119898119888120593119894119904120572119904120601119894 (119877119887 + 1198891) 119864119894 = (119877119887 + 1198891)2 + 1198772119898 + 1199042119894 minus 1198972119894

minus 2119877119898119888120572119904120601119894119904120593119894 (119877119887 + 1198891) (13)

Equation (12) also can be further simplified by subtract-ing one equation from another which yields the followingexpression

(119860 119894119895119911119898 + 119862119894119895) 119888120573 + (119877119894119895119911119898 + 119863119894119895) 119904120573 + 119876119894119895119911119898 + 119864119894119895= 0 (14)

where 119894 = 119895 and119860 119894119895 = 119860 119894 minus 119860119895119877119894119895 = 119877119894 minus 119877119895119876119894119895 = 119876119894 minus 119876119895119862119894119895 = 119862119894 minus 119862119895119863119894119895 = 119863119894 minus 119863119895119864119894119895 = 119864119894 minus 119864119895

(15)

Therefore 120573 can be obtained by 119911119898 and 120572 as

119888120573 = minus1198651199112119898 + 119866119911119898 + 1198671198681199112119898 + 119869119911119898 + 119870

119904120573 = minus1198711199112119898 + 119878119911119898 + 1198791198681199112119898 + 119869119911119898 + 119870(16)

where the coefficients 119865 sim 119870119883 are functions of 120572 which areexpressed as

119865 = 1198771211987613 minus 1198761211987713119866 = 1198631211987613 + 1198771211986413 minus 1198641211987713 minus 1198761211986313119867 = 1198631211986413 minus 1198641211986313119868 = 1198771211986013 minus 1198601211987713119869 = 1198631211986013 + 1198771211986213 minus 1198621211987713 minus 1198601211986313119870 = 1198631211986213 minus 1198621211986313119871 = 1198761211986013 minus 1198601211987613119878 = 1198641211986013 + 1198761211986213 minus 1198621211987613 minus 1198601211986413119879 = 1198641211986213 minus 1198621211986413

(17)

Since 1198882120573 + 1199042120573 = 1 substituting (16) into this formulayields

11988041199114119898 + 11988031199113119898 + 11988021199112119898 + 1198801119911119898 + 1198800 = 0 (18)

where

1198804 = 1198652 + 1198712 minus 11986821198803 = 2 (119865119866 + 119871119878 minus 119868119869) 1198802 = 1198662 + 1198782 minus 1198692 + 2119865119867 + 2119871119879 minus 21198681198701198801 = 2119866119867 + 2119878119879 minus 21198691198701198800 = 1198672 + 1198792 minus 1198702

(19)

Equation (12) consists of three independent equationsand two independent equations have been obtained as shownin (16) Substituting (16) into (12) another equation can bededuced For instance when 119894 = 1 the equation is expressedas

11988231199113119898 +11988221199112119898 +1198821119911119898 +1198820 = 0 (20)

where1198823 = 1198601119865 + 1198771119871 minus 11987611198681198822 = 1198601119866 + 1198621119865 + 1198771119878 + 1198631119871 minus 1198761119869 minus 11986411198681198821 = 1198601119867 + 1198621119866 + 1198771119879 + 1198631119878 minus 1198761119870 minus 11986411198691198820 = 1198621119867 +1198631119879 minus 1198641119870

(21)

It can be seen that (18) and (20) have a commonsolution of 119911119898 By using Bezoutrsquos method [31] the followingdeterminant should be satisfied10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198804 1198803 1198802 1198801 1198800 0 00 1198804 1198803 1198802 1198801 1198800 00 0 1198804 1198803 1198802 1198801 11988001198823 1198822 1198821 1198820 0 0 00 1198823 1198822 1198821 1198820 0 00 0 1198823 1198822 1198821 1198820 00 0 0 1198823 1198822 1198821 1198820

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (22)

Thus (22) becomes an equation about 120572 when thedisplacement 119904119894 is known Now that all the equations havebeen obtained the method to solve the forward kinematicsproblem can be given as follows (i) to calculate 120572 from (22)(ii) to calculate 119911119898 from (20) and (iii) to calculate 120573 from (16)

For step (i) the sine and cosine components of 120572 shouldbe replaced by 119905 = tan(1205722) Based on the standard trans-formation expression the equation 119904120572 = (1minus1199052)(1+1199052) 119888120572 =2119905(1+1199052) can be obtained Finally (22) becomes a polynomialalgebraic equation about the variable 119905Then according to (ii)and (iii) the forward kinematics problem can be solved

33 JacobianMatrix Taking the derivative of (3) with respectto time leads to

119904119894119899119894 + 120596119894 times 119897119894119896119894 = 119907119898 + 120596119898 times 119903119894 (23)

where 119907119898 = [0 0 119898]T and 120596119898 = [ 120573 0]T denote the linearand angular velocity vector of the hybrid perfusion platformin the fixed coordinate system 119861-119909119887119910119887119911119887 respectively

6 Mathematical Problems in Engineering

Task honeycombs

40∘

Figure 5 Structure of the spherical surface

Taking the dot product with 119896119894 on both sides of (23) thevelocity of the ith driving joint can be deduced as

119904119894 = [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

][119907119898120596119898

] = 119869119894 [119907119898120596119898

] (24)

Rewriting the velocities of the driving joints in the matrixform as

119904 = 119869minus1119904 119869119901 = 119869 (25)

where

119904 = [ 1199041 1199042 1199043]T = [119907T119898 120596T119898]T 119869119904 = diag (119896T11198991 119896T21198992 119896T31198993) 119869119901 = [ 1198961 1198962 1198963

1199031 times 1198961 1199032 times 1198962 1199033 times 1198963]T

(26)

The Jacobianmatrix between the velocity vector and thedriving joint velocity vector 119904 can be obtained as

119869 = 119869minus1119904 119869119901 = [119869T1 119869T2 119869T3 ]T (27)

4 Kinematics Performance Analysis

Following the establishment of the mechanism model andthe kinematics analysis the analysis of the kinematics perfor-mance of the 3PSS-PUparallelmanipulatorwill be conductedin this section which includes the workspace singularitydexterity and stiffness The analysis of workspace is mainlyon account of the kinematics solved in Section 31 And basedon the Jacobian matrix analyzed in Section 33 the analysesof the singularity dexterity and stiffness are all developed indetail

41 Workspace Analysis

411 Task Honeycombs Analysis As shown in Figure 5 themaximum angle between the point on the spherical crownsurface and the vertical axis is 40 degrees Due to the large

size of the spherical crown surface it adopts the method ofsubarea perfusion to accomplish the perfusion of the largeobject and the task honeycombs of subarea are shown inFigure 6 For the task honeycombs the maximum angle 120579maxof the honeycombs is 10∘ During the perfusion process andthrough the rotation of the spherical crownworktable and themovement of the parallel manipulator along the arc guide railand the motion of the moving platform the perfusion of allthe honeycombs can be completed successfully

412 Reachable Workspace Analysis In this section thereachableworkspace of the perfusionmanipulator is obtainedby the method of the geometric constraints As shown inFigure 7 the flow diagram for calculating the workspaceof the perfusion manipulator has been given in detailAccording to the flow diagram of the workspace and theparameters given in Table 1 the reachable workspace of themoving platform can be obtained

As shown in Figures 8(a) and 8(b) the 3D view andvertical view of the reachable workspace for the parallelperfusion manipulator are obtained From the 3D view ofthe workspace it can be seen that the rotation range ofthe moving platform about 119909119887 axis and 119910119887 axis remainsunchanged with the increase of the value of the variable 119911119898For the task honeycombs of the subarea the maximum angleof the task honeycombs is 10∘ and then the task workspacecan be described as the yellow region in Figure 8(b) FromFigure 8(b) it also can be easily concluded that the taskworkspace is always within the reachable workspace of theparallelmanipulatorThat is to say the proposed 1T2R parallelperfusionmanipulator can complete the perfusion of the taskhoneycombs Then by the rotation of the spherical crownworktable and the movement of the parallel manipulatoralong the arc guide rail the perfusion of all the honeycombswill be completed

42 Singularity Analysis The method based on the Jacobianmatrices is the most common method to find the singularityof a mechanism [32] In order to obtain all the singularitypositions of the parallel manipulator the determinant of theinverse Jacobian matrix 119869119904 and forward Jacobian matrix 119869119901will be conducted Based on the Jacobian matrix the singu-larity conditions of the proposed parallel manipulator canbe divided into three types which include inverse kinematicsingularity (IKS) direct kinematic singularity (DKS) andcombined singularity (CS) And the conditions for satisfyingthese types of singularity can be given as follows

IKS det (119869119904) = 0 det (119869119901119904) = 0

DKS det (119869119904) = 0 det (119869119901119904) = 0CS det (119869119904) = 0 det (119869119901119904) = 0

(28)

where 119869119901119904 is a 3 times 3 submatrix from 119869119901As for the IKS singularity it occurs when matrix 119869119904 is

not full rank ie det(119869119904) = 0 or 119896119894 sdot 119899119894 = 0 (119894 = 1 2 3)which means one or more PSS legs are perpendicular totheir corresponding sliding rails As shown in Figure 9 it

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

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Page 3: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 3

B

M

B1

N1

l1

M1

B4

yb1

zbxb

N2

l2

ymRm

zm xm

M2

M3

B3

N3

l3

B2

Rb

1

Figure 3 Kinematic diagram of the 1T2R parallel manipulator

actuated PSS (P denotes the actuated joint) links a movingplatform and the passive constraint link PU Every PSSlimb connects the fixed platform with perfusion platform byan active prismatic joint followed by two spherical jointsAnd every prismatic joint is driven by a servomotor Thepassive limb contains a universal joint and a prismatic jointwhich connects the perfusion platform with the fixed platesuccessivelyThe existence of the PU limbwill highly improvestiffness of the whole system

22 Parameters Description The schematic model of pro-posed manipulator in Figure 3 shows that it has two plat-forms fixed platform labeled by119861111986121198613 andmoving platformdemonstrated by119872111987221198723The three limbs are placed in 120degree intervals on base platform Links119873119894119872119894 are attached tothe moving platform by a spherical joint at119872119894 and to a sliderby a spherical joint at119873119894Themiddle limb is established fromthe fixed platform to the perfusion platform whose one endis attached to the base with a prismatic joint 1198614 and the otherend is attached to the end-effector platform by a universaljoint To facilitate the analysis the fixed reference frame119861-119909119887119910119887119911119887 is placed at the center of the base platform where 119910119887axis is along the direction of straight line 1198611198611 Similarly thecoordinate axes of moving frame are denoted by119872-119909119898119910119898119911119898in which 119910119898 axis is along the direction of straight line1198721198721Parameter 120593119894 is the angle measured from 119909119887 to 119861119861119894 and 120601119894is the angle measured from 119909119898 to 119872119872119894 The length of thelimb119873119894119872119894 is denoted by 119897119894The length of 119861119861119894 is demonstratedby 119877119887 and 119877119898 represents the length of the line 119872119872119894 Thedisplacement of the driving joint is represented by 1199041198943 Kinematics Analysis

31 Inverse Kinematics As shown in Figure 4 the rela-tionship between the coordinate systems 119861-119909119887119910119887119911119887 and119872-119909119898119910119898119911119898 can be described by the rotation matrix 119861119877119875which can be obtained by three continuous rotations of

Bi

Nii

sii

Di

xizi

lii

Mi

i

ym

zm

M

xm

xb

B

yb

zb

i

yi

Figure 4 Kinematics of the ith PSS branch

Euler angles 120572 120573 and 120574 about the fixed 119909119887 119910119887 and 119911119887 axisrespectively For the 1T2R parallel manipulator the angle 120574 iszero Thus the rotation matrix from 119872-119909119898119910119898119911119898 coordinatesystem to 119861-119909119887119910119887119911119887 coordinate system can be expressed as

119861119877119872 = Rot (119910119887 120573)Rot (119909119887 120572) = [[

[119888120573 119904120573119904120572 1199041205731198881205720 119888120572 minus119904120572

minus119904120573 119888120573119904120572 119888120573119888120572]]]

(1)

where s and c correspond to the sine and cosine functionsrespectively

To facilitate the analysis the local coordinate system119873119894-119909119894119910119894119911119894 of the ith PSS branch also has been established atthe point119873119894 and the 119911119894 axis is along the direction of straightline 119873119894119872119894 The system 119873119894-119909119894119910119894119911119894 can be considered as twocontinuous rotations of angles 120581119894 and 120595119894 about 119911119887 axis and 1199101015840119894axis (rotated 119910119887 axis) respectively Consequently the rotationmatrix can be represented as

119861119877119894 = Rot (119911119887 120581119894)Rot (1199101015840119894 120595119894)

= [[[119888120581119894119888120595119894 minus119904120581119894 119888120581119894119904120595119894119904120581119894119888120595119894 119888120581119894 119904120581119894119904120595119894minus119904120595119894 0 119888120595119894

]]]

(2)

According to Figure 4 the closed-loop vector equation ofthe ith link is given as follows

119898 + 119903119894 = 119887119894 + 119904119894119899119894 + 119886119894 + 119897119894119896119894 (3)

where 119898 is the position vector of the moving platformdescribed in 119861-119909119887119910119887119911119887 system 119903119894 = 119861119877119872119898119903119894 119903119894 and 119898119903119894represent the position vector of the straight line 119872119872119894 in119861-119909119887119910119887119911119887 and 119872-119909119898119910119898119911119898 system respectively 119887119894 is theposition vector of the straight line 119874119861119894 in 119861-119909119887119910119887119911119887 systemwhile 119899119894 and 119896119894 denote the unit vector of the straight line119861119894119863119894 and119873119894119872119894 expressed in the coordinate system 119861-119909119887119910119887119911119887

4 Mathematical Problems in Engineering

respectively 119904119894 is the displacement of the prismatic joint forthe ith PSS branch and 119897119894 is the length of the link 119873119894119872119894Therefore the vectors mentioned above can be obtained as

119898 = [[[00119911119898

]]]

119898119903119894 = [[

[1198771198981198881206011198941198771198981199041206011198940

]]]

119887119894 = [[[1198771198871198881205931198941198771198871199041205931198940

]]]

119899119894 = [[[001]]]

119886119894 = [[[119889111988812059311989411988911199041205931198940

]]]

(4)

From 2) the unit vector 119896119894 will be expressed as

119896119894 = 119861119877119894119894119896119894 = 119861119877119894 [[[001]]]= [[[119888120581119894119904120595119894119904120581119894119904120595119894119888120595119894

]]]= [[[119896119909119894119896119910119894119896119911119894

]]]

(5)

At the same time the relationship between 120581119894 and 120595119894 canbe calculated as

119888120595119894 = 119896119911119894119904120595119894 = radic1198962119909119894 + 1198962119910119894 (0 le 120595119894 le 120587) 119904120581119894 = 119896119910119894119904120595119894 119888120581119894 = 119896119909119894119904120595119894

(6)

Substituting (4) and (5) into (3) and squaring both sidesof (3) can be deduced as

1199041198861198941199042119894 + 119904119887119894119904119894 + 119904119888119894 = 0 (7)

where

119904119886119894 = 1198612119909119894 + 1198612119910119894 + 1198612119911119894119904119887119894 = 2 (119860119909119894119861119909119894 + 119860119910119894119861119910119894 + 119860119911119894119861119911119894) 119904119888119894 = 1198602119909119894 + 1198602119910119894 + 1198602119911119894 minus 1198972119894

119860119909119894 = 119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894

119861119909119894 = 0119860119910119894 = 119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119861119910119894 = 0119860119911119894 = 119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573119861119911119894 = minus1

(8)

The inverse kinematics solutions for the ith limb can bederived from (7)

119904119894 = minus119904119887119894 plusmn radic1199042119887119894 minus 41199041198861198941199041198881198942119904119886119894 (9)

From (9) there are two solutions for each driving jointthat is to say there exist eight possible solutions for a givenconfiguration However the three driving sliders are onlyallowed moving downward from point 119861119894Therefore only thenegative symbol can satisfy the motion characteristics of theparallel perfusion manipulator

32 Forward Kinematics Forward kinematics for the 3PSS-PUmechanism is to solve the pose parameters (120572 120573 119911119898) afterknowing the driving parameters (1199041 1199042 1199043) For this problemit can be obtained from (3) and (4) which can lead to thefollowing formula

119897119894119896119894 = [[[119909119897119894119910119897119894119911119897119894

]]]

= [[[119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573 minus 119904119894

]]]

(10)

According to constraint of the length of the links 119873119894119872119894the restraint equation can be obtained

1199092119897119894 + 1199102119897119894 + 1199112119897119894 = 1198972119894 (11)

To solve the forward kinematics problem 120573 and 119911119898 canbe represented by 120572 Therefore (12) can be derived from (10)and (11)

1199112119898 + (119860 119894119888120573 + 119877119894119904120573 + 119876119894) 119911119898 + (119862119894119888120573 + 119863119894119904120573 + 119864119894)= 0 (12)

where the coefficients 119860 119894 119876119894 sim 119864119894 are functions of 120572119860 119894 = 2119877119898119904120601119894119904120572119877119894 = minus2119877119898119888120601119894119876119894 = minus2s119894

Mathematical Problems in Engineering 5

C119894 = minus2119877119898119888120601119894119888120593119894 (119877119887 + 1198891) minus 2119877119898119904119894119904120572119904120601119894119863119894 = 2119877119898119904119894119888120601119894 minus 2119877119898119888120593119894119904120572119904120601119894 (119877119887 + 1198891) 119864119894 = (119877119887 + 1198891)2 + 1198772119898 + 1199042119894 minus 1198972119894

minus 2119877119898119888120572119904120601119894119904120593119894 (119877119887 + 1198891) (13)

Equation (12) also can be further simplified by subtract-ing one equation from another which yields the followingexpression

(119860 119894119895119911119898 + 119862119894119895) 119888120573 + (119877119894119895119911119898 + 119863119894119895) 119904120573 + 119876119894119895119911119898 + 119864119894119895= 0 (14)

where 119894 = 119895 and119860 119894119895 = 119860 119894 minus 119860119895119877119894119895 = 119877119894 minus 119877119895119876119894119895 = 119876119894 minus 119876119895119862119894119895 = 119862119894 minus 119862119895119863119894119895 = 119863119894 minus 119863119895119864119894119895 = 119864119894 minus 119864119895

(15)

Therefore 120573 can be obtained by 119911119898 and 120572 as

119888120573 = minus1198651199112119898 + 119866119911119898 + 1198671198681199112119898 + 119869119911119898 + 119870

119904120573 = minus1198711199112119898 + 119878119911119898 + 1198791198681199112119898 + 119869119911119898 + 119870(16)

where the coefficients 119865 sim 119870119883 are functions of 120572 which areexpressed as

119865 = 1198771211987613 minus 1198761211987713119866 = 1198631211987613 + 1198771211986413 minus 1198641211987713 minus 1198761211986313119867 = 1198631211986413 minus 1198641211986313119868 = 1198771211986013 minus 1198601211987713119869 = 1198631211986013 + 1198771211986213 minus 1198621211987713 minus 1198601211986313119870 = 1198631211986213 minus 1198621211986313119871 = 1198761211986013 minus 1198601211987613119878 = 1198641211986013 + 1198761211986213 minus 1198621211987613 minus 1198601211986413119879 = 1198641211986213 minus 1198621211986413

(17)

Since 1198882120573 + 1199042120573 = 1 substituting (16) into this formulayields

11988041199114119898 + 11988031199113119898 + 11988021199112119898 + 1198801119911119898 + 1198800 = 0 (18)

where

1198804 = 1198652 + 1198712 minus 11986821198803 = 2 (119865119866 + 119871119878 minus 119868119869) 1198802 = 1198662 + 1198782 minus 1198692 + 2119865119867 + 2119871119879 minus 21198681198701198801 = 2119866119867 + 2119878119879 minus 21198691198701198800 = 1198672 + 1198792 minus 1198702

(19)

Equation (12) consists of three independent equationsand two independent equations have been obtained as shownin (16) Substituting (16) into (12) another equation can bededuced For instance when 119894 = 1 the equation is expressedas

11988231199113119898 +11988221199112119898 +1198821119911119898 +1198820 = 0 (20)

where1198823 = 1198601119865 + 1198771119871 minus 11987611198681198822 = 1198601119866 + 1198621119865 + 1198771119878 + 1198631119871 minus 1198761119869 minus 11986411198681198821 = 1198601119867 + 1198621119866 + 1198771119879 + 1198631119878 minus 1198761119870 minus 11986411198691198820 = 1198621119867 +1198631119879 minus 1198641119870

(21)

It can be seen that (18) and (20) have a commonsolution of 119911119898 By using Bezoutrsquos method [31] the followingdeterminant should be satisfied10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198804 1198803 1198802 1198801 1198800 0 00 1198804 1198803 1198802 1198801 1198800 00 0 1198804 1198803 1198802 1198801 11988001198823 1198822 1198821 1198820 0 0 00 1198823 1198822 1198821 1198820 0 00 0 1198823 1198822 1198821 1198820 00 0 0 1198823 1198822 1198821 1198820

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (22)

Thus (22) becomes an equation about 120572 when thedisplacement 119904119894 is known Now that all the equations havebeen obtained the method to solve the forward kinematicsproblem can be given as follows (i) to calculate 120572 from (22)(ii) to calculate 119911119898 from (20) and (iii) to calculate 120573 from (16)

For step (i) the sine and cosine components of 120572 shouldbe replaced by 119905 = tan(1205722) Based on the standard trans-formation expression the equation 119904120572 = (1minus1199052)(1+1199052) 119888120572 =2119905(1+1199052) can be obtained Finally (22) becomes a polynomialalgebraic equation about the variable 119905Then according to (ii)and (iii) the forward kinematics problem can be solved

33 JacobianMatrix Taking the derivative of (3) with respectto time leads to

119904119894119899119894 + 120596119894 times 119897119894119896119894 = 119907119898 + 120596119898 times 119903119894 (23)

where 119907119898 = [0 0 119898]T and 120596119898 = [ 120573 0]T denote the linearand angular velocity vector of the hybrid perfusion platformin the fixed coordinate system 119861-119909119887119910119887119911119887 respectively

6 Mathematical Problems in Engineering

Task honeycombs

40∘

Figure 5 Structure of the spherical surface

Taking the dot product with 119896119894 on both sides of (23) thevelocity of the ith driving joint can be deduced as

119904119894 = [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

][119907119898120596119898

] = 119869119894 [119907119898120596119898

] (24)

Rewriting the velocities of the driving joints in the matrixform as

119904 = 119869minus1119904 119869119901 = 119869 (25)

where

119904 = [ 1199041 1199042 1199043]T = [119907T119898 120596T119898]T 119869119904 = diag (119896T11198991 119896T21198992 119896T31198993) 119869119901 = [ 1198961 1198962 1198963

1199031 times 1198961 1199032 times 1198962 1199033 times 1198963]T

(26)

The Jacobianmatrix between the velocity vector and thedriving joint velocity vector 119904 can be obtained as

119869 = 119869minus1119904 119869119901 = [119869T1 119869T2 119869T3 ]T (27)

4 Kinematics Performance Analysis

Following the establishment of the mechanism model andthe kinematics analysis the analysis of the kinematics perfor-mance of the 3PSS-PUparallelmanipulatorwill be conductedin this section which includes the workspace singularitydexterity and stiffness The analysis of workspace is mainlyon account of the kinematics solved in Section 31 And basedon the Jacobian matrix analyzed in Section 33 the analysesof the singularity dexterity and stiffness are all developed indetail

41 Workspace Analysis

411 Task Honeycombs Analysis As shown in Figure 5 themaximum angle between the point on the spherical crownsurface and the vertical axis is 40 degrees Due to the large

size of the spherical crown surface it adopts the method ofsubarea perfusion to accomplish the perfusion of the largeobject and the task honeycombs of subarea are shown inFigure 6 For the task honeycombs the maximum angle 120579maxof the honeycombs is 10∘ During the perfusion process andthrough the rotation of the spherical crownworktable and themovement of the parallel manipulator along the arc guide railand the motion of the moving platform the perfusion of allthe honeycombs can be completed successfully

412 Reachable Workspace Analysis In this section thereachableworkspace of the perfusionmanipulator is obtainedby the method of the geometric constraints As shown inFigure 7 the flow diagram for calculating the workspaceof the perfusion manipulator has been given in detailAccording to the flow diagram of the workspace and theparameters given in Table 1 the reachable workspace of themoving platform can be obtained

As shown in Figures 8(a) and 8(b) the 3D view andvertical view of the reachable workspace for the parallelperfusion manipulator are obtained From the 3D view ofthe workspace it can be seen that the rotation range ofthe moving platform about 119909119887 axis and 119910119887 axis remainsunchanged with the increase of the value of the variable 119911119898For the task honeycombs of the subarea the maximum angleof the task honeycombs is 10∘ and then the task workspacecan be described as the yellow region in Figure 8(b) FromFigure 8(b) it also can be easily concluded that the taskworkspace is always within the reachable workspace of theparallelmanipulatorThat is to say the proposed 1T2R parallelperfusionmanipulator can complete the perfusion of the taskhoneycombs Then by the rotation of the spherical crownworktable and the movement of the parallel manipulatoralong the arc guide rail the perfusion of all the honeycombswill be completed

42 Singularity Analysis The method based on the Jacobianmatrices is the most common method to find the singularityof a mechanism [32] In order to obtain all the singularitypositions of the parallel manipulator the determinant of theinverse Jacobian matrix 119869119904 and forward Jacobian matrix 119869119901will be conducted Based on the Jacobian matrix the singu-larity conditions of the proposed parallel manipulator canbe divided into three types which include inverse kinematicsingularity (IKS) direct kinematic singularity (DKS) andcombined singularity (CS) And the conditions for satisfyingthese types of singularity can be given as follows

IKS det (119869119904) = 0 det (119869119901119904) = 0

DKS det (119869119904) = 0 det (119869119901119904) = 0CS det (119869119904) = 0 det (119869119901119904) = 0

(28)

where 119869119901119904 is a 3 times 3 submatrix from 119869119901As for the IKS singularity it occurs when matrix 119869119904 is

not full rank ie det(119869119904) = 0 or 119896119894 sdot 119899119894 = 0 (119894 = 1 2 3)which means one or more PSS legs are perpendicular totheir corresponding sliding rails As shown in Figure 9 it

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Page 4: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

4 Mathematical Problems in Engineering

respectively 119904119894 is the displacement of the prismatic joint forthe ith PSS branch and 119897119894 is the length of the link 119873119894119872119894Therefore the vectors mentioned above can be obtained as

119898 = [[[00119911119898

]]]

119898119903119894 = [[

[1198771198981198881206011198941198771198981199041206011198940

]]]

119887119894 = [[[1198771198871198881205931198941198771198871199041205931198940

]]]

119899119894 = [[[001]]]

119886119894 = [[[119889111988812059311989411988911199041205931198940

]]]

(4)

From 2) the unit vector 119896119894 will be expressed as

119896119894 = 119861119877119894119894119896119894 = 119861119877119894 [[[001]]]= [[[119888120581119894119904120595119894119904120581119894119904120595119894119888120595119894

]]]= [[[119896119909119894119896119910119894119896119911119894

]]]

(5)

At the same time the relationship between 120581119894 and 120595119894 canbe calculated as

119888120595119894 = 119896119911119894119904120595119894 = radic1198962119909119894 + 1198962119910119894 (0 le 120595119894 le 120587) 119904120581119894 = 119896119910119894119904120595119894 119888120581119894 = 119896119909119894119904120595119894

(6)

Substituting (4) and (5) into (3) and squaring both sidesof (3) can be deduced as

1199041198861198941199042119894 + 119904119887119894119904119894 + 119904119888119894 = 0 (7)

where

119904119886119894 = 1198612119909119894 + 1198612119910119894 + 1198612119911119894119904119887119894 = 2 (119860119909119894119861119909119894 + 119860119910119894119861119910119894 + 119860119911119894119861119911119894) 119904119888119894 = 1198602119909119894 + 1198602119910119894 + 1198602119911119894 minus 1198972119894

119860119909119894 = 119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894

119861119909119894 = 0119860119910119894 = 119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119861119910119894 = 0119860119911119894 = 119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573119861119911119894 = minus1

(8)

The inverse kinematics solutions for the ith limb can bederived from (7)

119904119894 = minus119904119887119894 plusmn radic1199042119887119894 minus 41199041198861198941199041198881198942119904119886119894 (9)

From (9) there are two solutions for each driving jointthat is to say there exist eight possible solutions for a givenconfiguration However the three driving sliders are onlyallowed moving downward from point 119861119894Therefore only thenegative symbol can satisfy the motion characteristics of theparallel perfusion manipulator

32 Forward Kinematics Forward kinematics for the 3PSS-PUmechanism is to solve the pose parameters (120572 120573 119911119898) afterknowing the driving parameters (1199041 1199042 1199043) For this problemit can be obtained from (3) and (4) which can lead to thefollowing formula

119897119894119896119894 = [[[119909119897119894119910119897119894119911119897119894

]]]

= [[[119877119898119888120601119894119888120573 + 119877119898119904120601119894119904120573119904120572 minus (119877119887 + 1198891) 119888120593119894119877119898119904120601119894119888120572 minus (119877119887 + 1198891) 119904120593119894119911119898 + 119877119898119904120601119894119888120573119904120572 minus 119877119898119888120601119894119904120573 minus 119904119894

]]]

(10)

According to constraint of the length of the links 119873119894119872119894the restraint equation can be obtained

1199092119897119894 + 1199102119897119894 + 1199112119897119894 = 1198972119894 (11)

To solve the forward kinematics problem 120573 and 119911119898 canbe represented by 120572 Therefore (12) can be derived from (10)and (11)

1199112119898 + (119860 119894119888120573 + 119877119894119904120573 + 119876119894) 119911119898 + (119862119894119888120573 + 119863119894119904120573 + 119864119894)= 0 (12)

where the coefficients 119860 119894 119876119894 sim 119864119894 are functions of 120572119860 119894 = 2119877119898119904120601119894119904120572119877119894 = minus2119877119898119888120601119894119876119894 = minus2s119894

Mathematical Problems in Engineering 5

C119894 = minus2119877119898119888120601119894119888120593119894 (119877119887 + 1198891) minus 2119877119898119904119894119904120572119904120601119894119863119894 = 2119877119898119904119894119888120601119894 minus 2119877119898119888120593119894119904120572119904120601119894 (119877119887 + 1198891) 119864119894 = (119877119887 + 1198891)2 + 1198772119898 + 1199042119894 minus 1198972119894

minus 2119877119898119888120572119904120601119894119904120593119894 (119877119887 + 1198891) (13)

Equation (12) also can be further simplified by subtract-ing one equation from another which yields the followingexpression

(119860 119894119895119911119898 + 119862119894119895) 119888120573 + (119877119894119895119911119898 + 119863119894119895) 119904120573 + 119876119894119895119911119898 + 119864119894119895= 0 (14)

where 119894 = 119895 and119860 119894119895 = 119860 119894 minus 119860119895119877119894119895 = 119877119894 minus 119877119895119876119894119895 = 119876119894 minus 119876119895119862119894119895 = 119862119894 minus 119862119895119863119894119895 = 119863119894 minus 119863119895119864119894119895 = 119864119894 minus 119864119895

(15)

Therefore 120573 can be obtained by 119911119898 and 120572 as

119888120573 = minus1198651199112119898 + 119866119911119898 + 1198671198681199112119898 + 119869119911119898 + 119870

119904120573 = minus1198711199112119898 + 119878119911119898 + 1198791198681199112119898 + 119869119911119898 + 119870(16)

where the coefficients 119865 sim 119870119883 are functions of 120572 which areexpressed as

119865 = 1198771211987613 minus 1198761211987713119866 = 1198631211987613 + 1198771211986413 minus 1198641211987713 minus 1198761211986313119867 = 1198631211986413 minus 1198641211986313119868 = 1198771211986013 minus 1198601211987713119869 = 1198631211986013 + 1198771211986213 minus 1198621211987713 minus 1198601211986313119870 = 1198631211986213 minus 1198621211986313119871 = 1198761211986013 minus 1198601211987613119878 = 1198641211986013 + 1198761211986213 minus 1198621211987613 minus 1198601211986413119879 = 1198641211986213 minus 1198621211986413

(17)

Since 1198882120573 + 1199042120573 = 1 substituting (16) into this formulayields

11988041199114119898 + 11988031199113119898 + 11988021199112119898 + 1198801119911119898 + 1198800 = 0 (18)

where

1198804 = 1198652 + 1198712 minus 11986821198803 = 2 (119865119866 + 119871119878 minus 119868119869) 1198802 = 1198662 + 1198782 minus 1198692 + 2119865119867 + 2119871119879 minus 21198681198701198801 = 2119866119867 + 2119878119879 minus 21198691198701198800 = 1198672 + 1198792 minus 1198702

(19)

Equation (12) consists of three independent equationsand two independent equations have been obtained as shownin (16) Substituting (16) into (12) another equation can bededuced For instance when 119894 = 1 the equation is expressedas

11988231199113119898 +11988221199112119898 +1198821119911119898 +1198820 = 0 (20)

where1198823 = 1198601119865 + 1198771119871 minus 11987611198681198822 = 1198601119866 + 1198621119865 + 1198771119878 + 1198631119871 minus 1198761119869 minus 11986411198681198821 = 1198601119867 + 1198621119866 + 1198771119879 + 1198631119878 minus 1198761119870 minus 11986411198691198820 = 1198621119867 +1198631119879 minus 1198641119870

(21)

It can be seen that (18) and (20) have a commonsolution of 119911119898 By using Bezoutrsquos method [31] the followingdeterminant should be satisfied10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198804 1198803 1198802 1198801 1198800 0 00 1198804 1198803 1198802 1198801 1198800 00 0 1198804 1198803 1198802 1198801 11988001198823 1198822 1198821 1198820 0 0 00 1198823 1198822 1198821 1198820 0 00 0 1198823 1198822 1198821 1198820 00 0 0 1198823 1198822 1198821 1198820

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (22)

Thus (22) becomes an equation about 120572 when thedisplacement 119904119894 is known Now that all the equations havebeen obtained the method to solve the forward kinematicsproblem can be given as follows (i) to calculate 120572 from (22)(ii) to calculate 119911119898 from (20) and (iii) to calculate 120573 from (16)

For step (i) the sine and cosine components of 120572 shouldbe replaced by 119905 = tan(1205722) Based on the standard trans-formation expression the equation 119904120572 = (1minus1199052)(1+1199052) 119888120572 =2119905(1+1199052) can be obtained Finally (22) becomes a polynomialalgebraic equation about the variable 119905Then according to (ii)and (iii) the forward kinematics problem can be solved

33 JacobianMatrix Taking the derivative of (3) with respectto time leads to

119904119894119899119894 + 120596119894 times 119897119894119896119894 = 119907119898 + 120596119898 times 119903119894 (23)

where 119907119898 = [0 0 119898]T and 120596119898 = [ 120573 0]T denote the linearand angular velocity vector of the hybrid perfusion platformin the fixed coordinate system 119861-119909119887119910119887119911119887 respectively

6 Mathematical Problems in Engineering

Task honeycombs

40∘

Figure 5 Structure of the spherical surface

Taking the dot product with 119896119894 on both sides of (23) thevelocity of the ith driving joint can be deduced as

119904119894 = [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

][119907119898120596119898

] = 119869119894 [119907119898120596119898

] (24)

Rewriting the velocities of the driving joints in the matrixform as

119904 = 119869minus1119904 119869119901 = 119869 (25)

where

119904 = [ 1199041 1199042 1199043]T = [119907T119898 120596T119898]T 119869119904 = diag (119896T11198991 119896T21198992 119896T31198993) 119869119901 = [ 1198961 1198962 1198963

1199031 times 1198961 1199032 times 1198962 1199033 times 1198963]T

(26)

The Jacobianmatrix between the velocity vector and thedriving joint velocity vector 119904 can be obtained as

119869 = 119869minus1119904 119869119901 = [119869T1 119869T2 119869T3 ]T (27)

4 Kinematics Performance Analysis

Following the establishment of the mechanism model andthe kinematics analysis the analysis of the kinematics perfor-mance of the 3PSS-PUparallelmanipulatorwill be conductedin this section which includes the workspace singularitydexterity and stiffness The analysis of workspace is mainlyon account of the kinematics solved in Section 31 And basedon the Jacobian matrix analyzed in Section 33 the analysesof the singularity dexterity and stiffness are all developed indetail

41 Workspace Analysis

411 Task Honeycombs Analysis As shown in Figure 5 themaximum angle between the point on the spherical crownsurface and the vertical axis is 40 degrees Due to the large

size of the spherical crown surface it adopts the method ofsubarea perfusion to accomplish the perfusion of the largeobject and the task honeycombs of subarea are shown inFigure 6 For the task honeycombs the maximum angle 120579maxof the honeycombs is 10∘ During the perfusion process andthrough the rotation of the spherical crownworktable and themovement of the parallel manipulator along the arc guide railand the motion of the moving platform the perfusion of allthe honeycombs can be completed successfully

412 Reachable Workspace Analysis In this section thereachableworkspace of the perfusionmanipulator is obtainedby the method of the geometric constraints As shown inFigure 7 the flow diagram for calculating the workspaceof the perfusion manipulator has been given in detailAccording to the flow diagram of the workspace and theparameters given in Table 1 the reachable workspace of themoving platform can be obtained

As shown in Figures 8(a) and 8(b) the 3D view andvertical view of the reachable workspace for the parallelperfusion manipulator are obtained From the 3D view ofthe workspace it can be seen that the rotation range ofthe moving platform about 119909119887 axis and 119910119887 axis remainsunchanged with the increase of the value of the variable 119911119898For the task honeycombs of the subarea the maximum angleof the task honeycombs is 10∘ and then the task workspacecan be described as the yellow region in Figure 8(b) FromFigure 8(b) it also can be easily concluded that the taskworkspace is always within the reachable workspace of theparallelmanipulatorThat is to say the proposed 1T2R parallelperfusionmanipulator can complete the perfusion of the taskhoneycombs Then by the rotation of the spherical crownworktable and the movement of the parallel manipulatoralong the arc guide rail the perfusion of all the honeycombswill be completed

42 Singularity Analysis The method based on the Jacobianmatrices is the most common method to find the singularityof a mechanism [32] In order to obtain all the singularitypositions of the parallel manipulator the determinant of theinverse Jacobian matrix 119869119904 and forward Jacobian matrix 119869119901will be conducted Based on the Jacobian matrix the singu-larity conditions of the proposed parallel manipulator canbe divided into three types which include inverse kinematicsingularity (IKS) direct kinematic singularity (DKS) andcombined singularity (CS) And the conditions for satisfyingthese types of singularity can be given as follows

IKS det (119869119904) = 0 det (119869119901119904) = 0

DKS det (119869119904) = 0 det (119869119901119904) = 0CS det (119869119904) = 0 det (119869119901119904) = 0

(28)

where 119869119901119904 is a 3 times 3 submatrix from 119869119901As for the IKS singularity it occurs when matrix 119869119904 is

not full rank ie det(119869119904) = 0 or 119896119894 sdot 119899119894 = 0 (119894 = 1 2 3)which means one or more PSS legs are perpendicular totheir corresponding sliding rails As shown in Figure 9 it

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Mathematical Problems in Engineering

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Page 5: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 5

C119894 = minus2119877119898119888120601119894119888120593119894 (119877119887 + 1198891) minus 2119877119898119904119894119904120572119904120601119894119863119894 = 2119877119898119904119894119888120601119894 minus 2119877119898119888120593119894119904120572119904120601119894 (119877119887 + 1198891) 119864119894 = (119877119887 + 1198891)2 + 1198772119898 + 1199042119894 minus 1198972119894

minus 2119877119898119888120572119904120601119894119904120593119894 (119877119887 + 1198891) (13)

Equation (12) also can be further simplified by subtract-ing one equation from another which yields the followingexpression

(119860 119894119895119911119898 + 119862119894119895) 119888120573 + (119877119894119895119911119898 + 119863119894119895) 119904120573 + 119876119894119895119911119898 + 119864119894119895= 0 (14)

where 119894 = 119895 and119860 119894119895 = 119860 119894 minus 119860119895119877119894119895 = 119877119894 minus 119877119895119876119894119895 = 119876119894 minus 119876119895119862119894119895 = 119862119894 minus 119862119895119863119894119895 = 119863119894 minus 119863119895119864119894119895 = 119864119894 minus 119864119895

(15)

Therefore 120573 can be obtained by 119911119898 and 120572 as

119888120573 = minus1198651199112119898 + 119866119911119898 + 1198671198681199112119898 + 119869119911119898 + 119870

119904120573 = minus1198711199112119898 + 119878119911119898 + 1198791198681199112119898 + 119869119911119898 + 119870(16)

where the coefficients 119865 sim 119870119883 are functions of 120572 which areexpressed as

119865 = 1198771211987613 minus 1198761211987713119866 = 1198631211987613 + 1198771211986413 minus 1198641211987713 minus 1198761211986313119867 = 1198631211986413 minus 1198641211986313119868 = 1198771211986013 minus 1198601211987713119869 = 1198631211986013 + 1198771211986213 minus 1198621211987713 minus 1198601211986313119870 = 1198631211986213 minus 1198621211986313119871 = 1198761211986013 minus 1198601211987613119878 = 1198641211986013 + 1198761211986213 minus 1198621211987613 minus 1198601211986413119879 = 1198641211986213 minus 1198621211986413

(17)

Since 1198882120573 + 1199042120573 = 1 substituting (16) into this formulayields

11988041199114119898 + 11988031199113119898 + 11988021199112119898 + 1198801119911119898 + 1198800 = 0 (18)

where

1198804 = 1198652 + 1198712 minus 11986821198803 = 2 (119865119866 + 119871119878 minus 119868119869) 1198802 = 1198662 + 1198782 minus 1198692 + 2119865119867 + 2119871119879 minus 21198681198701198801 = 2119866119867 + 2119878119879 minus 21198691198701198800 = 1198672 + 1198792 minus 1198702

(19)

Equation (12) consists of three independent equationsand two independent equations have been obtained as shownin (16) Substituting (16) into (12) another equation can bededuced For instance when 119894 = 1 the equation is expressedas

11988231199113119898 +11988221199112119898 +1198821119911119898 +1198820 = 0 (20)

where1198823 = 1198601119865 + 1198771119871 minus 11987611198681198822 = 1198601119866 + 1198621119865 + 1198771119878 + 1198631119871 minus 1198761119869 minus 11986411198681198821 = 1198601119867 + 1198621119866 + 1198771119879 + 1198631119878 minus 1198761119870 minus 11986411198691198820 = 1198621119867 +1198631119879 minus 1198641119870

(21)

It can be seen that (18) and (20) have a commonsolution of 119911119898 By using Bezoutrsquos method [31] the followingdeterminant should be satisfied10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198804 1198803 1198802 1198801 1198800 0 00 1198804 1198803 1198802 1198801 1198800 00 0 1198804 1198803 1198802 1198801 11988001198823 1198822 1198821 1198820 0 0 00 1198823 1198822 1198821 1198820 0 00 0 1198823 1198822 1198821 1198820 00 0 0 1198823 1198822 1198821 1198820

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (22)

Thus (22) becomes an equation about 120572 when thedisplacement 119904119894 is known Now that all the equations havebeen obtained the method to solve the forward kinematicsproblem can be given as follows (i) to calculate 120572 from (22)(ii) to calculate 119911119898 from (20) and (iii) to calculate 120573 from (16)

For step (i) the sine and cosine components of 120572 shouldbe replaced by 119905 = tan(1205722) Based on the standard trans-formation expression the equation 119904120572 = (1minus1199052)(1+1199052) 119888120572 =2119905(1+1199052) can be obtained Finally (22) becomes a polynomialalgebraic equation about the variable 119905Then according to (ii)and (iii) the forward kinematics problem can be solved

33 JacobianMatrix Taking the derivative of (3) with respectto time leads to

119904119894119899119894 + 120596119894 times 119897119894119896119894 = 119907119898 + 120596119898 times 119903119894 (23)

where 119907119898 = [0 0 119898]T and 120596119898 = [ 120573 0]T denote the linearand angular velocity vector of the hybrid perfusion platformin the fixed coordinate system 119861-119909119887119910119887119911119887 respectively

6 Mathematical Problems in Engineering

Task honeycombs

40∘

Figure 5 Structure of the spherical surface

Taking the dot product with 119896119894 on both sides of (23) thevelocity of the ith driving joint can be deduced as

119904119894 = [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

][119907119898120596119898

] = 119869119894 [119907119898120596119898

] (24)

Rewriting the velocities of the driving joints in the matrixform as

119904 = 119869minus1119904 119869119901 = 119869 (25)

where

119904 = [ 1199041 1199042 1199043]T = [119907T119898 120596T119898]T 119869119904 = diag (119896T11198991 119896T21198992 119896T31198993) 119869119901 = [ 1198961 1198962 1198963

1199031 times 1198961 1199032 times 1198962 1199033 times 1198963]T

(26)

The Jacobianmatrix between the velocity vector and thedriving joint velocity vector 119904 can be obtained as

119869 = 119869minus1119904 119869119901 = [119869T1 119869T2 119869T3 ]T (27)

4 Kinematics Performance Analysis

Following the establishment of the mechanism model andthe kinematics analysis the analysis of the kinematics perfor-mance of the 3PSS-PUparallelmanipulatorwill be conductedin this section which includes the workspace singularitydexterity and stiffness The analysis of workspace is mainlyon account of the kinematics solved in Section 31 And basedon the Jacobian matrix analyzed in Section 33 the analysesof the singularity dexterity and stiffness are all developed indetail

41 Workspace Analysis

411 Task Honeycombs Analysis As shown in Figure 5 themaximum angle between the point on the spherical crownsurface and the vertical axis is 40 degrees Due to the large

size of the spherical crown surface it adopts the method ofsubarea perfusion to accomplish the perfusion of the largeobject and the task honeycombs of subarea are shown inFigure 6 For the task honeycombs the maximum angle 120579maxof the honeycombs is 10∘ During the perfusion process andthrough the rotation of the spherical crownworktable and themovement of the parallel manipulator along the arc guide railand the motion of the moving platform the perfusion of allthe honeycombs can be completed successfully

412 Reachable Workspace Analysis In this section thereachableworkspace of the perfusionmanipulator is obtainedby the method of the geometric constraints As shown inFigure 7 the flow diagram for calculating the workspaceof the perfusion manipulator has been given in detailAccording to the flow diagram of the workspace and theparameters given in Table 1 the reachable workspace of themoving platform can be obtained

As shown in Figures 8(a) and 8(b) the 3D view andvertical view of the reachable workspace for the parallelperfusion manipulator are obtained From the 3D view ofthe workspace it can be seen that the rotation range ofthe moving platform about 119909119887 axis and 119910119887 axis remainsunchanged with the increase of the value of the variable 119911119898For the task honeycombs of the subarea the maximum angleof the task honeycombs is 10∘ and then the task workspacecan be described as the yellow region in Figure 8(b) FromFigure 8(b) it also can be easily concluded that the taskworkspace is always within the reachable workspace of theparallelmanipulatorThat is to say the proposed 1T2R parallelperfusionmanipulator can complete the perfusion of the taskhoneycombs Then by the rotation of the spherical crownworktable and the movement of the parallel manipulatoralong the arc guide rail the perfusion of all the honeycombswill be completed

42 Singularity Analysis The method based on the Jacobianmatrices is the most common method to find the singularityof a mechanism [32] In order to obtain all the singularitypositions of the parallel manipulator the determinant of theinverse Jacobian matrix 119869119904 and forward Jacobian matrix 119869119901will be conducted Based on the Jacobian matrix the singu-larity conditions of the proposed parallel manipulator canbe divided into three types which include inverse kinematicsingularity (IKS) direct kinematic singularity (DKS) andcombined singularity (CS) And the conditions for satisfyingthese types of singularity can be given as follows

IKS det (119869119904) = 0 det (119869119901119904) = 0

DKS det (119869119904) = 0 det (119869119901119904) = 0CS det (119869119904) = 0 det (119869119901119904) = 0

(28)

where 119869119901119904 is a 3 times 3 submatrix from 119869119901As for the IKS singularity it occurs when matrix 119869119904 is

not full rank ie det(119869119904) = 0 or 119896119894 sdot 119899119894 = 0 (119894 = 1 2 3)which means one or more PSS legs are perpendicular totheir corresponding sliding rails As shown in Figure 9 it

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Page 6: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

6 Mathematical Problems in Engineering

Task honeycombs

40∘

Figure 5 Structure of the spherical surface

Taking the dot product with 119896119894 on both sides of (23) thevelocity of the ith driving joint can be deduced as

119904119894 = [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

][119907119898120596119898

] = 119869119894 [119907119898120596119898

] (24)

Rewriting the velocities of the driving joints in the matrixform as

119904 = 119869minus1119904 119869119901 = 119869 (25)

where

119904 = [ 1199041 1199042 1199043]T = [119907T119898 120596T119898]T 119869119904 = diag (119896T11198991 119896T21198992 119896T31198993) 119869119901 = [ 1198961 1198962 1198963

1199031 times 1198961 1199032 times 1198962 1199033 times 1198963]T

(26)

The Jacobianmatrix between the velocity vector and thedriving joint velocity vector 119904 can be obtained as

119869 = 119869minus1119904 119869119901 = [119869T1 119869T2 119869T3 ]T (27)

4 Kinematics Performance Analysis

Following the establishment of the mechanism model andthe kinematics analysis the analysis of the kinematics perfor-mance of the 3PSS-PUparallelmanipulatorwill be conductedin this section which includes the workspace singularitydexterity and stiffness The analysis of workspace is mainlyon account of the kinematics solved in Section 31 And basedon the Jacobian matrix analyzed in Section 33 the analysesof the singularity dexterity and stiffness are all developed indetail

41 Workspace Analysis

411 Task Honeycombs Analysis As shown in Figure 5 themaximum angle between the point on the spherical crownsurface and the vertical axis is 40 degrees Due to the large

size of the spherical crown surface it adopts the method ofsubarea perfusion to accomplish the perfusion of the largeobject and the task honeycombs of subarea are shown inFigure 6 For the task honeycombs the maximum angle 120579maxof the honeycombs is 10∘ During the perfusion process andthrough the rotation of the spherical crownworktable and themovement of the parallel manipulator along the arc guide railand the motion of the moving platform the perfusion of allthe honeycombs can be completed successfully

412 Reachable Workspace Analysis In this section thereachableworkspace of the perfusionmanipulator is obtainedby the method of the geometric constraints As shown inFigure 7 the flow diagram for calculating the workspaceof the perfusion manipulator has been given in detailAccording to the flow diagram of the workspace and theparameters given in Table 1 the reachable workspace of themoving platform can be obtained

As shown in Figures 8(a) and 8(b) the 3D view andvertical view of the reachable workspace for the parallelperfusion manipulator are obtained From the 3D view ofthe workspace it can be seen that the rotation range ofthe moving platform about 119909119887 axis and 119910119887 axis remainsunchanged with the increase of the value of the variable 119911119898For the task honeycombs of the subarea the maximum angleof the task honeycombs is 10∘ and then the task workspacecan be described as the yellow region in Figure 8(b) FromFigure 8(b) it also can be easily concluded that the taskworkspace is always within the reachable workspace of theparallelmanipulatorThat is to say the proposed 1T2R parallelperfusionmanipulator can complete the perfusion of the taskhoneycombs Then by the rotation of the spherical crownworktable and the movement of the parallel manipulatoralong the arc guide rail the perfusion of all the honeycombswill be completed

42 Singularity Analysis The method based on the Jacobianmatrices is the most common method to find the singularityof a mechanism [32] In order to obtain all the singularitypositions of the parallel manipulator the determinant of theinverse Jacobian matrix 119869119904 and forward Jacobian matrix 119869119901will be conducted Based on the Jacobian matrix the singu-larity conditions of the proposed parallel manipulator canbe divided into three types which include inverse kinematicsingularity (IKS) direct kinematic singularity (DKS) andcombined singularity (CS) And the conditions for satisfyingthese types of singularity can be given as follows

IKS det (119869119904) = 0 det (119869119901119904) = 0

DKS det (119869119904) = 0 det (119869119901119904) = 0CS det (119869119904) = 0 det (119869119901119904) = 0

(28)

where 119869119901119904 is a 3 times 3 submatrix from 119869119901As for the IKS singularity it occurs when matrix 119869119904 is

not full rank ie det(119869119904) = 0 or 119896119894 sdot 119899119894 = 0 (119894 = 1 2 3)which means one or more PSS legs are perpendicular totheir corresponding sliding rails As shown in Figure 9 it

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Page 7: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 7

GR = 10∘

Figure 6 Task honeycombs of the subarea

Given the Parameters

Begin

Inverse Kinematics Calculations

No

End

Yes

Yes

Yes

Yes

No

No

No

zm = zm + Δzm

= + Δ

= + Δ

siGCH le si le siGR

zm = zmGR

= GR

= GR

RbRmlii i

= GCH = GCHzm = zmGCH

Save (zm)

Figure 7 Flow chart for the reachable workspace

Table 1 Dimensional parameters of 1T2R parallel manipulator

Parameters Values Parameters Values Parameters Values Parameters Values119877119887mm 160 1198891mm 50 120572maxrad plusmn06 119911119898maxmm 560119877119898mm 270 119904119894minmm 0 120573maxrad plusmn06 120593119894rad 1205872 + 2120587(119894 minus 1)3119897119894mm 300 119904119894maxmm 250 119911119898minmm 300 120601119894rad 1205872 + 2120587(119894 minus 1)3

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Mathematical Problems in Engineering

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Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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Page 8: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

8 Mathematical Problems in Engineering

500

400

300

z m(m

m)

(rad)

minus05

minus05

00 00

05

05

(rad)

550

500

450

400

350

300

(a) 3D view of the reachable workspace

Task workspace

(rad)minus05 00 05

(r

ad)

05

00

minus05

500

400

300

zm (mm)

(b) Vertical view of the reachable workspace

Figure 8

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 9 Manipulator positions when IKS happens

gives two cases when the first link 11987311198721 is perpendicularto the corresponding sliding rails 11986111198731 and IKS occurs withdet(119869119904) = 0 In this case the parallel manipulator loses oneDoF because an infinitesimal motion of the actuator 1198731 willcause no motion of the moving platform From Figure 9(a)it can be seen that the interference between the movingplatform and the sliding rail 11986111198731 occurs when the link11987311198721 is perpendicular to the sliding rails 11986111198731 outsideAlso according to Figure 9(b) only when the size of themoving platform is smaller than the fixed platform can theinside vertical of the link 11987311198721 and the sliding rails 11986111198731be satisfied However there is no interference between thecomponents when the moving platform moves within itsreachable workspace thus the singularity case in Figure 9(a)is not exist Also for the proposed parallel manipulator theparameter 119877119898 is larger than 119877119887 thus the singularity case inFigure 9(b) does not exist either Consequently there is noIKS existing in the proposed parallel manipulator

As for the DKS singularity it occurs when matrix 119869119901119904is not full rank eg the first PSS branch lies in the plane

119872111987221198723 with the link 11987311198721 passing through the centerpoint119872 of the moving platform which is shown in Figure 10Under this situation the moving platform obtains one moreDoF even when all the three actuators are locked and theparallel manipulator will instantaneously be out of controlie joint 1198721 can infinitesimally move along the normaldirection of the plane119872111987221198723 without actuation From thefirst case in Figure 10(a) it can be known the interferencebetween themoving platformand the sliding rail11986111198731 occurswhen the link 11987311198721 coincides with the line 1198721198721 Alsoaccording to the singularity position in Figure 10(b) the link11987311198721 is on the extension line of 1198721198721 and the size of themoving platform is smaller than the base platform Howeverfor the manipulator there is no interference between thecomponents when the moving platform moves within itsreachable workspace and the parameter 119877119898 is larger than119877119887 Consequently there is no DKS existing in the parallelmanipulator either

Similarly as for the CS singularity it occurs when bothof the matrices 119869119904 and 119869119901119904 are not full rank In this case the

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Page 9: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 9

DKS

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(a)

B1

N1

B

B2

N2

B3

N3

M1

M2

M3

M

(b)

Figure 10 Manipulator positions when DKS happens

CS

M1

B1

N1

B

B2

N2

B3

N3

M2

M3

M

Figure 11 Manipulator position when CS happens

inverse kinematic and forward kinematic singularities appearsimultaneously and the special singularity configurationis shown in Figure 11 According to Figure 11 it can beconcluded that the combined singularity occurs when therelationship 119877119898 = 119877119887 + 119897119894 exists However the equationdoes not exist for the structure parameters of the proposedparallel manipulator Therefore there is no CS happening inthe parallel manipulator either

In summary the three types of singularities mentionedabove do not happen in the 3PSS-PU parallel manipulatorbecause of the carefully designing structure parameters of themanipulator

43 Dexterity and Stiffness Analysis Dexterity and stiffnessare important kinematic performance indexes to measureparallel manipulatorrsquos working ability The dexterity mainlyreflects the ability to arbitrarily change the moving plat-formrsquos position and orientation or apply force and torquesin arbitrary directions while working And the stiffnessdirectly affects manipulatorrsquos motion accuracy Because theJacobian matrix could reflect the relationship between input

and output the condition number of Jacobian matrix isfrequently used in evaluating the dexterity of a manipulatorThe calculation method of condition number comes outby computing the eigenvalue of stiffness matrix and takingsquare root of the ratio among extreme eigenvalues which isshown in the following equation

119908119888119900119899 = 120590max120590min(29)

where119908119888119900119899 denotes the condition number of Jacobian matrixand 120590max and 120590min represent the maximum and minimumeigenvalues of stiffness matrix respectively And the stiffnessmatrix of the manipulator can be described as the followingequation [33]

120581 (119869) = 119869T119870119869119869 (30)

where119870119869 is the stiffness matrix and119870119869 = [1205871 1205872] In thispaper the actuators of the 3PSS-PU parallel manipulator canbe seen as elastic components and 120587119894 denotes the stiffness ofthe ith driving joint Here 120587119894 is set to 100KNm [7]

As shown in Figure 12 it describes the distributions of theJacobian matrix condition number of the proposed parallelmanipulator at a height of 119911119898 = 450mm From Figure 12 itcan be seen that the values of condition number within theworkspace all change from 52 to 9 smoothly The closer thevalues of condition number to 1 the better the dexterity ofthe manipulator The values of dexterity index for most areasof the workspace are all below 65 which indicate the gooddexterity performance of the proposed parallel manipulatorin the reachable workspace Also the dexterity performanceis good enough to satisfy the dexterity requirements of theperfusion

Meanwhile the distributions of stiffness at differentheight of 119911119898 are also plotted which are shown in Figures 13and 14 It can be easily observed from the figures that thestiffness of the manipulator decreases with the increasing ofthe value of 119911119898 And the distribution figures of stiffness with

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

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Page 10: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

10 Mathematical Problems in Engineering

(rad)

minus05

minus05

00

00

05

05

(rad)

9

8

7

6

wco

n

9

8

7

6

(a)

(rad)

06

04

02

00

minus02

minus04

minus06

minus06 minus04 minus02 00 02 04 06

9

8

7

6

(r

ad)

(b)

Figure 12 Distributions of condition number of Jacobian matrix at a height of 119911119898 = 450mm

(rad)

minus05

minus0500

00

05

05

(rad)

600

400

700

600

500

400

300

zm = 035m

Stiff

ness

(kN

m

)

(a)

(rad)

minus05

minus0500

00

05

05

(rad)

500

400

300

200

500400300200

zm = 040mSt

iffne

ss (k

N

m)

(b)

(rad)

minus05

00

05minus05

00

05

(rad)

300

250

200

150

300250200150

zm = 045m

Stiff

ness

(kN

m

)

(c)

(rad)

minus05

00

05minus05

00

05

(rad)

200

150

100

zm = 050m

225

200

175

150

125

100

Stiff

ness

(kN

m

)

(d)

Figure 13 Distributions of stiffness at different height of 119911119898

different value of 119911119898 have the same change trend Importantlythe maximum value of stiffness always arises at the center ofthe workspace and decreases with the angle 120572 or 120573 approach-ing to themaximum value which is in line with the structuralcharacteristics of the proposed parallel manipulator Also thevalue of the stiffness at different height of 119911119898 is mostly above150kNm and the stiffness requirement of the perfusion isabout 100kNmThus the stiffness of the parallelmanipulatoris fully capable of satisfying the perfusion of the honeycombs

5 Dynamics Model ofthe Parallel Manipulator

In this section the dynamics analysis of the perfusion mani-pulator ismainly focused on the 1T2R parallel perfusionman-ipulator Based on the principle of the virtual work [34 35]the link Jacobian matrix and the dynamics model of the per-fusion manipulator are established The applied and inertiaforces of the components are derived in the corresponding

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

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Mathematical Problems in Engineering

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Page 11: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 11

(rad)

minus05minus05

0000

05 minus05

(rad)

Stiff

ness

(kN

m

)

zm = 035m

zm = 040mzm = 045mzm = 050m

600

400

200

Figure 14 Comparison of stiffness at different height of 119911119898

coordinate system At last the expression of the driving forceis obtained

51 Link Velocity Analysis Thevelocity vector of the point119872119894in the119873119894-119909119894119910119894119911119894 coordinate system can be written as

119894119907119872119894 = 119904119894119894119899119894 + 119894120596119894 times 119897119894119894119896119894 = 119894119907119898 + 119894120596119898 times 119894119903119894 (31)

Because of 119894120596T119894119894119896119894 = 0 taking the cross product with 119894119896119894

on both sides of (31) the angular velocity of the limb119873119894119872119894 inthe119873119894-119909119894119910119894119911119894 coordinate system is derived as

119894120596119894 = 1119897119894 (119894119896119894 times 119894119907119872119894 minus 119894119896119894 times 119904119894119894119899119894)

= 1119897119894 [119879(119894119896119894) 119894119907119872119894 minus 119879 (119894119896119894) 119904119894119894119899119894](32)

where

119879(119894119896119894) = [[[[

0 minus119894119896119894119911 119894119896119894119910119894119896119894119911 0 minus119894119896119894119909minus119894119896119894119910 119894119896119894119909 0

]]]]

(33)

Substituting (24) and (31) into (32) leads to

119894120596119894 = 1119897119894 [119879(119894119896119894) 119894119877119861 minus119879(119894119896119894)119879(119894119903119894) 119894119877119861]

minus (119894119896119894 times 119894119899119894) [ 119896T119894119896T119894 119899119894

(119903119894 times 119896119894)T119896T119894 119899119894

] = 119869120596119894(34)

where

119879 (119894119903119894) = [[[[

0 minus119894119903119894119911 119894119903119894119910119894119903119894119911 0 minus119894119903119894119909minus119894119903119894119910 119894119903119894119909 0

]]]]

119894119877119861 = 119861119877minus1119894 = 119861119877T119894 (35)

With reference to (31) and (34) the linear velocity of themass center of the branch 119873119894119872119894 in the 119873119894-119909119894119910119894119911119894 coordinatesystem can be given as

119894119907119894 = 119894119907119872119894 minus 119894120596119894 times 1198971198942 119894119896119894

= [119894119877119861 minus119879 (119894119903119894) 119894119877119861] + 1198971198942119879 (119894119896119894) 119869120596119894 = 119869V119894(36)

Rewriting the velocity vector of the ith branch 119873119894119872119894 inthe matrix form yields

[ 119894119907119894119894120596119894

] = [119869V119894119869120596119894

] = 119869V120596119894 (37)

where 119869V119908119894 is the link Jacobian matrix which represents themapping between the velocity vector of the moving platformin the fixed system 119861-119909119887119910119887119911119887 and the velocity vector of the ithlink in the119873119894-119909119894119910119894119911119894 coordinate system

52 Acceleration Analysis Taking the derivative of (23) withrespect to time leads to

119904119894119899119894 + 119894 times 119897119894119896119894 + 120596119894 times (120596119894 times 119897119894k119894)= 119898 + 119898 times 119903119894 + 120596119898 times (120596119898 times 119903119894) (38)

Taking the dot product with 119896119894 on both sides of (38) andsimplifying the equation the acceleration of the ith slider canbe obtained as

119904119894 = 119869119894 + 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162](39)

Thematrix form of the accelerations of the driving sliderscan be rewritten as

119904 = 119869 + 120595 (40)

where

119904 = [ 1199041 1199042 1199043]T = [T119898 T119898]T 120595 = [1205951 1205952 1205953]T 120595119894 = 1119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898) minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162

+ 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162]

(41)

53 Link Acceleration Analysis Taking the derivative of (31)with respect to time in the119873119894-119909119894119910119894119911119894 coordinate system leadsto

119904119894119894119899119894 + 119894119894 times 119897119894119894119896119894 + 119894120596119894 times (119894120596119894 times 119897119894119894119896119894)= 119894119898 + 119894119898 times 119894119903119894 + 119894120596119898 times (119894120596119898 times 119894119903119894) (42)

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

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Page 12: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

12 Mathematical Problems in Engineering

Taking the cross product of (42) with 119894119896119894 on both sidesthe angular acceleration of the ith link119873119894119872119894 can be obtainedas

119894119894 = 1119897119894 119894119896119894 times 119894119898 + 119894119896119894 times (119894119898 times 119894119903119894) minus (119894119896119894 times 119894119899119894) 119904119894+ 119894119896119894 times [119894120596119898 times (119894120596119898 times 119894119903119894)] minus 119894119896119894times [119894120596119894 times (119894120596119894 times 119897119894119894119896119894)]

(43)

Based on the derived acceleration in (39) (43) can besimplified as

119894119894 = 119869120596119894 + 120582119894 (44)

where

120582119894 = 1119897119894 minus(119894119896119894 times 119894119899119894)119896T119894 119899119894

[(119896T119894 120596119898) (119903T119894 120596119898)minus (119896T119894 119903119894) 100381610038161003816100381612059611989810038161003816100381610038162 + 119897119894 1003816100381610038161003816119896119894 times 12059611989410038161003816100381610038162] + (119894120596T119898119894119903119894) (119894119896119894times 119894120596119898) minus 10038161003816100381610038161003816119894120596119898100381610038161003816100381610038162 (119894119896119894 times 119894119903119894)

(45)

Taking the derivative of (36) with respect to time leads to

119894119894 = 119894119872119894 minus 119894119894 times 1198971198942 119894119896119894 minus 119894120596119894 times (119894120596119894 times 1198971198942 119894119896119894)

= 119894119898 minus 119879(119894119903119894) 119894119898 + 119879(119894120596119898)119879 (119894120596119898) 119894119903119894+ 1198971198942119879(119894119896119894) 119894119894 minus 1198971198942119879(119894120596119894)119879 (119894120596119894) 119894119896119894

(46)

Substituting (44) into the above equation and simplifyingcan be derived as

119894119894 = 119869V119894 + 120578119894 (47)

where 120578119894 = 119879(119894120596119898)119879(119894120596119898)119894119903119894 + (12)119879(119894119896119894)120582119894 minus(1198971198942)119879(119894120596119894)119879(119894120596119894)11989411989611989454 Dynamic Formulation Applied and inertia forces im-posed on the center of the mass of the moving perfusionplatform can be derived as

119876119872 = [ 119891119890 + 119898119898119892 minus 119898119898119898119899119890 minus 119861119868119872119898 minus 120596119898 times (119861119868119872120596119898)] (48)

where 119891119890 and 119899119890 represent the external force and torqueimposed on the mass center of the moving perfusion plat-form 119898119898 is the mass of the moving perfusion platform and119892 = [0 0 98]T1198981199042 is the gravity 119861119868119872 = 119861119877119872119868119872119872119877119861and 119868119872 denotes the inertia matrix of the moving perfusionplatform about the mass center which is described in the119872-119909119898119910119898119911119898 coordinate system

The force system 119876119872 in (48) can be divided into foursections the acceleration term 119876119872119860 the velocity term 119876119872119881

the gravity term 119876119872119866 and the external force term 119876119872119864which can be listed as follows

119876119872 = [ minus119898119898119898minus119861119868119872119898] + [ 01times3

minus120596119898 times (119861119868119872120596119898)] + [11989811989811989201times3

]

+ [119891119890119899119890] = 119876119872119860 +119876119872119881 +119876119872119866 +119876119872119864

(49)

The force system imposed on the center of the mass of theith branch119873119894119872119894 can be deduced in the 119873119894-119909119894119910119894119911119894 coordinatesystem as

119894119876119894 = [ 119898119894119894119877119861119892 minus 119898119894119894119894

minus119894119868119894119894119894 minus 119894120596119894 times (119894119868119894119894120596119894)] (50)

where 119898119894 is the mass of the link 119873119894119872119894 and 119894119868119894 is the inertiamatrix of the ith limb119873119894119872119894 about the center of its mass whichis expressed in the119873119894-119909119894119910119894119911119894 coordinate system

Similarly the force system 119894119876119894 of the ith branch119873119894119872119894 alsocan be decomposed into three parts as

119894119876119894 = [minus119898119894119869V119894minus119894119868119894119869120596119894] + [ minus119898119894120578119894minus119894119868119894120582119894 minus 119894120596119894 times (119894119868119894119894120596119894)]

+ [11989811989411989411987711986111989201times3

] = 119894119876119860119894 + 119894119876119881119894 + 119894119876119866119894(51)

For the driving joint the force system exerted at the masscenter of the slider can be derived in the 119861-119909119887119910119887119911119887 coordinatesystem as

119865119894 = 119898119904119894119892 minus 119898119904119894 119904119894 = minus119898119904119894119899119894119869119894 minus 119898119904119894120595119894119899119894 + 119898119904119894119892= 119865119860119894 + 119865119881119894 + 119865119866119894 (52)

where119865119860119894119865119881119894 and119865119866119894 are the acceleration term the velocityterm and the gravity term of the force system119865119894 respectivelyand119898119904119894 is the mass of the ith slider

For the middle passive link the branch only can movealong 119911119887 axis Thus the force imposed on the center of themass of the middle limb can be deduced in the 119861-119909119887119910119887119911119887coordinate system as

119865119901 = minus119898119901 + 119898119901119892 = minus11989811990111989901198690 + 119898119901119892= 119865119860119901 + 119865119866119901 (53)

where 119898119901 is the mass of the passive link and 119865119860119901 and 119865119866119901are the acceleration term and the gravity term of the forcesystem 119865119901 respectively And = 11989901198690 = [0 0 119898]T1198990 = [0 0 1]T 1198690 = [0 0 1 0 0 0]

Suppose the moving perfusion platform has a virtualdisplacement in the reachable workspace of the 1T2R parallelmanipulatorThus based on the principle of the virtual work

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 13: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 13

the dynamics equation of the proposed manipulator can beobtained as follows

120575119875T119876119872 + 3sum119894=1

(119899119894120575s119894)T 119865119894 + 3sum119894=1

120575119894119909T119894 119894119876119894 + (1198990120575119911)T 119865119901+ 120575119904T119891 = 0

(54)

where 119891 = [1198911 1198912 1198913]T is the driving forces of the sliders ofthe 1T2R parallel manipulator 120575119875 and 120575s119894 denote the virtualdisplacements of the moving platform and the ith drivingslider respectively and 120575119894119909119894 is the virtual displacement of theith limb119873119894119872119894 descripted in the119873119894-119909119894119910119894119911119894 coordinate system

According to the link Jacobian matrix analyzed inSection 51 the virtual displacements mentioned above canbe represented by the virtual displacement of the movingperfusion platform 120575119875 as follows

120575119904T119894 = (119869119894120575119875)T = 120575119875T119869T119894 120575119894119909T119894 = 120575119875T119869TV120596119894120575119911T = (1198690120575119875)T = 120575119875T119869T0 120575119904119879 = 120575119875T119869T

(55)

Substituting (55) into the dynamics equation leads to

120575119875T119876119872 + 3sum119894=1

120575119875T119869T119894 119899T119894 119865119894 + 3sum119894=1

120575119875T119869TV120596119894119894119876119894+ 120575119875T119869T0119899T0119865119901 + 120575119875T119869T119891 = 0

(56)

Simplifying (56) the inverse dynamics of the 1T2R paral-lel manipulator can be derived as follows

119891 = minus119869minusT [119876119872 + 3sum119894=1

(119869T119894 119899T119894 119865119894 + 119869TV120596119894119894119876119894) + 119869T0119899T0119865119901]

= minus119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119860119894 + 119869TV120596119894119894119876119860119894) + 119869T0119899T0119865119860119901+119876119872119860] minus 119869minusT [ 3sum

119894=1

(119869T119894 119899T119894 119865119881119894 + 119869TV120596119894119894119876119881119894)

+119876119872119881] minus 119869minusT [ 3sum119894=1

(119869T119894 119899T119894 119865119866119894 + 119869TV120596119894119894119876119866119894)

+ 119869T0119899T0119865119866119901 +119876119872119866] minus 119869minusT119876119872119864

(57)

where 119869minusT are the inverse matrix of 119869T

6 Numerical Simulation

In order to verify the correctness of the kinematics anddynamics model of the parallel manipulator the comparison

Table 2 The mass parameters of the proposed 1T2R parallelmanipulator (kg)

Parameters 119898119904119894 119898119894 119898119901 119898119898Mass 20 30 30 75

of theoretical and simulation curves of themotionparametersfor the sliders based on theMathematica and Adams softwareis conducted in this sectionThree kinds of rotationalmotionsof the moving perfusion platform are selected as simulationmotion trajectory to analysis the motion and driving forceof the driving slider In the initial position the displacementof all the driving sliders is 130mm and the moving platformis parallel to the base platform The mass parameters of theparallel manipulator are given in Table 2 and the inertiamatrices used in the simulation are also given as follows

119868119872 = [[[0171 0 00 0171 00 0 0341

]]]kg sdotm2

119894119868119894 = [[[0031 0 00 0031 00 0 00003

]]]kg sdotm2

(58)

The external force and torque exerted on the movingperfusion platform can be given as follows

119891119890 = (300 minus200 250)T119899119890 = (00 minus250 00)T (59)

For the first kind of rotational motion a pure rotationabout 119909119887 axis from 0 to -03rad is given without the trans-lational motion of the moving platform and the rotationalmotion of the moving platform can be described as

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (60)

As shown in Figure 15 the simplified simulation modelin Adams is given and the curves of driving forces in Adamsare also shown Based on the simulation curves the pointsof different motion parameters can be extracted Then byutilizing theMathematica software the comparison curves ofthe displacement velocity acceleration and driving force ofthe sliders are illustrated in Figures 16 and 17 respectively Ineach figure the black solid curves are the theoretical curvesof the motion parameters of the driving sliders calculatedby Mathematica software and the red green and bluedashed curves are the simulation curves of the first thesecond and the third driving slidersrsquo motion parametersobtained from the Adams software respectively Also themotion parameters of different driving sliders are markedby characters ldquomrdquo ldquonrdquo and ldquoordquo respectively For the driving

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 14: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

14 Mathematical Problems in Engineering

Figure 15 The simplified simulation model of rotation about 119909119887 axis in Adams

Disp

lace

men

t (m

m) s2s3

s1

Time (s)

180

160

140

120

100

80

60

0 1 2 3 4

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s)

v1

v2v3

0 1 2 3 4

20

10

0

minus10

minus20

minus30

minus40

v1

v2

v3

(b)

Figure 16 The comparison curves of the displacement and the velocity for the driving sliders under the rotation about 119909119887 axis

a1a2a3

Acc

eler

atio

n (m

ms

^2)

Time (s)

0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

minus85

f1

f2

f3

(b)

Figure 17 The comparison curves of the acceleration and the driving force for the driving sliders under the rotation about 119909119887 axis

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 15

Figure 18 The simplified simulation model of rotation about 119910119887 axis in Adams

force in Figure 17 the negative sign indicates the oppositedirection along the 119911119887 axis

According to these figures it can be concluded that thetheoretical and simulation curves of the motion parametersfor the driving joints have the same change trend respectivelyAnd the curves of motion parameter change smoothly andcontinuously which indicates the parallel manipulator hasa good performance along the rotation about 119909119887 axis Forthe passive PU link one of the rotation axes of the U jointconnected to the moving platform has the same direction asthe axis 119910119887 Thus according to the symmetry of the mecha-nism when the moving platform rotates around the 119909119887 axisthe second and the third driving joints should have the samemotion trajectory Obviously the displacement velocity andacceleration curves of the second and the third driving slidersin the simulation curves are all the same respectively whichverifies the correctness of the establishment of kinematicsmodel again

For the second and third kinds of rotational motionsa pure rotation about 119910119887 axis from 0 to -03rad and asimultaneous rotation about 119909119887 axis and 119910119887 axis from 0 to -03rad are given respectivelyThen the two kinds of rotationalmotions can be described respectively as follows

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7] (61)

120572= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

120573= minus03 [35 ( 1199054)

4 minus 84 ( 1199054)5 + 70( 1199054)

6 minus 20 ( 1199054)7]

(62)

Similarly the simplified simulation models of rotationabout 119910119887 axis and rotation about 119909119887 and 119910119887 axis in Adamsare presented in Figures 18 and 21 respectively Then thecomparison curves of the displacement velocity accelera-tion and driving force of the sliders under the two kindsof rotational motions are obtained as shown in Figures 1920 22 and 23 respectively As mentioned above the black

solid curves and the red green and blue dashed curves alsorepresent the theoretical and simulation curves respectivelyAnd the motion parameters of different driving sliders arealso marked by characters ldquomrdquo ldquonrdquo and ldquoordquo respectively Italso can be concluded that the theoretical and simulationcurves of the motion parameters for the sliders all havethe same change trend respectively Moreover from thesmoothly change trend of the motion curves the good rota-tion performance of the parallel manipulator is proved again

In summary through the comparison of theoretical andsimulation curves of the displacement velocity accelerationand driving force for the sliders under the different rotationalmotions of the moving platform the correctness of thekinematics and dynamics model of the parallel manipulatorhas been verified Moreover the good rotation performanceof the proposed parallel manipulator is also proved Fromthe curves of the driving force it can be concluded that thedriving forces of the three driving sliders all vary from 68Nto 85N which indicates that all the driving joints are evenlystressed

7 Conclusions

Themain work and conclusions can be drawn as follows(1) In this paper a novel hybrid perfusion systemhas been

proposed which is constructed by a 1T2R parallel perfusionmanipulator and an arc guide way and can be used forthe perfusion of the honeycombs in the thermal protectionsystem of the spacecraft

(2) The inverse kinematics and the Jacobian matrix areanalyzed comprehensively Then based on the kinematicsanalysis the performance analysis for the parallel manipula-tor is carried outThe analysis results show that theworkspaceand the stiffness of the proposed parallel manipulator allcould satisfy the perfusion requirements and there is nosingularity position within the workspace which proved thegood kinematics performance of the parallel manipulator

(3) Through the analysis of the velocity and accelerationof the components the dynamics model has been establishedby utilizing the principle of virtual work According tothe comparison of theoretical and simulation curves of themotion parameters for the sliders the correctness of thekinematics and dynamics model of the parallel manipulatoris verified Also the verified kinematics and dynamics modelswill provide a good theoretical foundation for the optimal

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 16: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

16 Mathematical Problems in EngineeringD

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

200

180

160

140

120

100

80

60

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 19 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

30

20

10

0

minus10

minus20

minus30

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3(b)

Figure 20 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119910119887 axis

Figure 21 The simplified simulation model of rotation about 119909119887 and 119910119887 axis in Adams

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 17: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Mathematical Problems in Engineering 17D

ispla

cem

ent (

mm

)

Time (s) 0 1 2 3 4

250

200

150

100

50

s1

s2

s3

(a)

Vel

ocity

(mm

s)

Time (s) 0 1 2 3 4

60

40

20

0

minus20

minus40

v1

v2

v3

(b)

Figure 22 The comparison curves of the displacement and the velocity for the sliders under the rotation about 119909119887 and 119910119887 axis

Acc

eler

atio

n (m

ms

^2)

Time (s) 0 1 2 3 4

40

20

0

minus20

minus40

a1

a2

a3

(a)

Driv

ing

Forc

e (N

)

Time (s) 0 1 2 3 4

minus70

minus75

minus80

f1

f2

f3

(b)

Figure 23 The comparison curves of the acceleration and the driving force for the sliders under the rotation about 119909119887 and 119910119887 axis

design of the driving force and the research of the controlstrategy in the future research

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

The authors gratefully acknowledge the financial support ofthe Fundamental Research Funds for the Central Universi-ties under Grant no 2018JBZ007 and the National NaturalScience Foundation of China (NSFC) under Grants 51675037and 51505023

References

[1] P K Ackerman A L Baker and C W Newquist ldquoThermalprotection systemrdquo Patent 5322725 US 1994

[2] C Gogu S K Bapanapalli R T Haftka and B V SankarldquoComparison of materials for an integrated thermal protection

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 18: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

18 Mathematical Problems in Engineering

system for spacecraft reentryrdquo Journal of Spacecra and Rocketsvol 46 no 3 pp 501ndash513 2009

[3] L T Chauvin R B Erb and D H Greenshields ldquoApollothermal-protection system developmentrdquo Journal of Spacecraamp Rockets vol 7 pp 839ndash869 2015

[4] D Wu A Zhou L Zheng et al ldquoStudy on the thermal protec-tion performance of superalloy honeycomb panels in high-speed thermal shock environmentsrdquoeoretical Applied Mech-anics Letters vol 4 no 2 pp 19ndash26 2014

[5] V Murthy and K J Waldron ldquoPosition kinematics of thegeneralized lobster arm and its series-parallel dualrdquo Journal ofMechanical Design vol 114 no 3 pp 406ndash413 1992

[6] W Ye Y Fang and S Guo ldquoDesign and analysis of a recon-figurable parallelmechanism formultidirectional additiveman-ufacturingrdquo Mechanism and Machine eory vol 112 pp 307ndash326 2017

[7] X-J Liu Z-L Jin and F Gao ldquoOptimum design of 3-DOFspherical parallel manipulators with respect to the conditioningand stiffness indicesrdquo Mechanism and Machine eory vol 35no 9 pp 1257ndash1267 2000

[8] CWang Y Fang and SGuo ldquoDesign andAnalysis of 3R2T and3R3T Parallel Mechanisms With High Rotational CapabilityrdquoJournal of Mechanisms and Robotics vol 8 no 1 p 011004 2016

[9] Y Fang and L-W Tsai ldquoStructure synthesis of a class of 4-DoFand 5-DoF parallelmanipulatorswith identical limb structuresrdquoInternational Journal of Robotics Research vol 21 no 9 pp 799ndash810 2002

[10] WDong Z Du Y Xiao andX Chen ldquoDevelopment of a paral-lel kinematic motion simulator platformrdquoMechatronics vol 23no 1 pp 154ndash161 2013

[11] B Dasgupta and T S Mruthyunjaya ldquoThe Stewart platformmanipulator a reviewrdquoMechanism andMachineeory vol 35no 1 pp 15ndash40 2000

[12] S Guo D Li H Chen and H Qu ldquoDesign and KinematicAnalysis of a Novel Flight Simulator Mechanismrdquo in IntelligentRobotics andApplications vol 8917 ofLectureNotes in ComputerScience pp 23ndash34 Springer International Publishing Cham2014

[13] Y Xu Y Chen G Zhang and P Gu ldquoAdaptable design ofmachine tools structuresrdquo Chinese Journal of Mechanical Engi-neering vol 21 no 3 pp 7ndash15 2008

[14] Q Li L Xu Q Chen and W Ye ldquoNew Family of RPR-Equivalent Parallel Mechanisms Design and Applicationrdquo Chi-nese Journal of Mechanical Engineering vol 30 no 2 pp 217ndash221 2017

[15] JWu JWang LWang T Li and Z You ldquoStudy on the stiffnessof a 5-DOF hybrid machine tool with actuation redundancyrdquoMechanism and Machine eory vol 44 no 2 pp 289ndash3052009

[16] F Xie X Liu X Luo andMWabner ldquoMobility Singularity andKinematics Analyses of a Novel Spatial Parallel MechanismrdquoJournal ofMechanisms andRobotics vol 8 no 6 p 061022 2016

[17] F Xie X Liu J Wang and M Wabner ldquoKinematic Optimiza-tion of a Five Degrees-of-Freedom Spatial Parallel MechanismWith Large Orientational Workspacerdquo Journal of Mechanismsand Robotics vol 9 no 5 p 051005 2017

[18] J Mo Z-F Shao L Guan F Xie and X Tang ldquoDynamic per-formance analysis of the X4 high-speed pick-and-place parallelrobotrdquo Robotics and Computer-Integrated Manufacturing vol46 pp 48ndash57 2017

[19] L-X Xu and Y-G Li ldquoInvestigation of joint clearance effectson the dynamic performance of a planar 2-DOF pick-and-place parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 1 pp 62ndash73 2014

[20] G Wu S Bai and P Hjoslashrnet ldquoArchitecture optimization ofa parallel Schonflies-motion robot for pick-and-place appli-cations in a predefined workspacerdquo Mechanism and Machineeory vol 106 pp 148ndash165 2016

[21] F Caccavale B Siciliano and L Villani ldquoThe Tricept robotdynamics and impedance controlrdquo IEEEASME Transactions onMechatronics vol 8 no 2 pp 263ndash268 2003

[22] M A Hosseini H-R M Daniali and H D Taghirad ldquoDexter-ous workspace optimization of a Tricept parallel manipulatorrdquoAdvanced Robotics vol 25 no 13-14 pp 1697ndash1712 2011

[23] H Liu T Huang J Mei et al ldquoKinematic design of a 5-DOFhybrid robot with large workspacelimb-stroke ratiordquo Journalof Mechanical Design vol 129 no 5 pp 530ndash537 2007

[24] T Huang M Li X M Zhao J P Mei D G Chetwynd andS J Hu ldquoConceptual design and dimensional synthesis for a 3-DOFmodule of the TriVariant - A novel 5-DOF reconfigurablehybrid robotrdquo IEEE Transactions on Robotics vol 21 no 3 pp449ndash456 2005

[25] Z M Bi and Y Jin ldquoKinematic modeling of Exechon parallelkinematic machinerdquo Robotics and Computer-Integrated Manu-facturing vol 27 no 1 pp 186ndash193 2011

[26] Z M Bi ldquoKinetostatic modeling of Exechon parallel kinematicmachine for stiffness analysisrdquo e International Journal ofAdvanced Manufacturing Technology vol 71 no 1-4 pp 325ndash335 2014

[27] J Wahl ldquoArticulated Tool Head USrdquo 6431802 2002[28] X Chen X Liu F Xie and T Sun ldquoA Comparison Study on

MotionForce Transmissibility of Two Typical 3-DOF ParallelManipulatorsThe Sprint Z3 and A3 Tool Headsrdquo InternationalJournal of Advanced Robotic Systems vol 11 no 1 p 5 2014

[29] Y Li JWang X-J Liu and L-PWang ldquoDynamic performancecomparison and counterweight optimization of two 3-DOFpar-allel manipulators for a new hybrid machine toolrdquo Mechanismand Machine eory vol 45 no 11 pp 1668ndash1680 2010

[30] D Kanaan P Wenger and D Chablat ldquoKinematic analysis of aserialndashparallel machine tool the VERNEmachinerdquoMechanismand Machine eory vol 44 no 2 pp 487ndash498 2009

[31] M Shub and S Smale ldquoComplexity of Bezoutrsquos theorem IVprobability of success extensionsrdquo SIAM Journal on NumericalAnalysis vol 33 no 1 pp 128ndash148 1996

[32] G Yang S H Yeo and C B Pham ldquoKinematics and singularityanalysis of a planar cable-driven parallel manipulatorrdquo inProceedings of the 2004 IEEERSJ International Conference onIntelligent Robots and Systems (IROS) pp 3835ndash3840 JapanOctober 2004

[33] G Coppola D Zhang and K Liu ldquoA 6-DOF reconfigurablehybrid parallel manipulatorrdquo Robotics and Computer-IntegratedManufacturing vol 30 no 2 pp 99ndash106 2014

[34] Y Zhao and F Gao ldquoInverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual workrdquoRobotica vol 27 no 2 pp 259ndash268 2009

[35] BDanaei A ArianM TaleMasouleh andAKalhor ldquoDynam-ic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanismrdquoMultibody System Dynam-ics vol 41 no 4 pp 367ndash390 2017

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 19: Kinematics Performance and Dynamics Analysis of a Novel Parallel Perfusion …downloads.hindawi.com/journals/mpe/2018/6768947.pdf · 2019-07-30 · Kinematics Performance and Dynamics

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom