Availableonlineatwww.sciencedirect.com
Journal of Applied Research and Technology
www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology14(2016)9–37
Original
An integrated study of the workspace and singularity for a Schönflies parallel manipulator
J. Jesús Cervantes-Sánchez
∗, José María Rico-Martínez, Víctor Hugo Pérez-Mu˜noz
UniversidaddeGuanajuato,DICIS,DepartamentodeIngenieríaMecánica,36885Salamanca,Guanajuato,Mexico Received4August2015;accepted26January2016
Availableonline3March2016
Abstract
ThispaperpresentsasimpleandsystematicapproachtoformulatetheinversepositionproblemofaSchönfliesparallelmanipulator.Asa result,theinversepositionproblemissolvedinclosedformandleadsdirectlytotheautomaticgenerationoftheworkspaceofthemanipulator.
Additionally,asystematicvelocityanalysisisalsopresented,whichallowstodetectandcharacterizeallthesingularitiesrelatedtothismanipulator.
AllRightsReserved©2016UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisan openaccessitemdistributedundertheCreativeCommonsCCLicenseBY-NC-ND4.0.
Keywords:Workspace;Singularity;Schönfliesparallelmanipulator
1. Introduction
Althoughseveralstudiesofworkspacehavebeenperformedfor manytypesofmanipulators(Abdel-Malek, Adkins,Yeh,&
Haug,1997;Bohigas,Manubens,&Ros,2012;Bonev&Ryu,2001;Davidson&Hunt,1987;Gosselin,1990;Gupta&Roth,1982;
Lee&Lee,2012;Macho,Altuzarra,Pinto,&Hernández,2013;Merlet,1999;Pernkopf&Husty,2006),thesearchofageneral andprecisedefinitionoftheworkspaceofarobotisasubjectivetask.Perhapsitisbecausetheworkspaceofamanipulatormaybe describedwithrespecttoitsabilitytoreachpoints,lines,planesorthree-dimensionalbodiesattachedtothemobileplatform.Hence thesimplestdefinitionofworkspaceisthatrelatedtopositioningmanipulators.Forthissimplecase,theworkspaceisdefinedas thevolumeofspacethatapointoftheendeffectorcanreach.However,whenthephysicalentityattachedtothemobileplatform isaline,aplaneora3Dobject,theproblemrelatedtothegraphicalvisualizationofthecorrespondingworkspaceisnotaneasy task.Therefore,duetotheparticularfeaturesofaSchönfliesmotion,namely,aspatialtranslationandarotationaboutafixedaxis, thispaperdealswiththeso-calledreachableworkspace,i.e.,thevolumeofspacewithineverypointcanbereachedbythemobile platforminatleastoneorientation.
Furthermore,qualitative andquantitative studiesof workspacesare importantbecause theymaybeused to:(a)yielduseful insightsaboutthekinematicarchitectureofthemanipulatorinthedesignstage,(b)leadtocriteriafortheevaluationofdifferent typesofmanipulators,(c)assistintheplanningofdesiredtasksinfavorablezones,and(d)avoiddangerouscollisionswithobjects.
Moreover,evenforthesimplestroboticsystem,therobotcontrollerprogrammustcontrolthemotionsofthemanipulatorandmobile platformtocarryoutataskinthespecificworkspace.
Ontheotherhand,afirst-ordersingularityanalysisdealswiththoseproblemsencounteredduringthesolutionstageofthevelocity analysisofamanipulator(Hao&McCarthy,1998;Gosselin&Angeles,1990;Altuzarra,Pinto,Avilés,&Hernández,2004;Amine, Masouleh,Caro,Wenger,&Gosselin,2012;Ghosal&Ravani,2001;Zlatanov,Fenton,&Benhabib,1995).Asaresultofsuch problems,degreesoffreedommaybeinstantaneouslygainedorlost.Particularlydangerousarethosemanipulator’sconfigurations
∗Correspondingauthor.
E-mailaddresses:[email protected](J.M.Rico-Martínez),[email protected](V.H.Pérez-Mu˜noz).
PeerReviewundertheresponsibilityofUniversidadNacionalAutónomadeMéxico.
http://dx.doi.org/10.1016/j.jart.2016.01.004
1665-6423/AllRightsReserved©2016UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccess itemdistributedundertheCreativeCommonsCCLicenseBY-NC-ND4.0.
O
O1
O2
A2
B2
C2
D2
O3
A3
B3
C3 D3
A1
B1
C1
D1
O4
A4
B4
D4
C4
P
2 3
1 4
11
12 13 43
44
41 42
10
14
32 33
34
30 40
31
20 21 22 23
24
0 25 5
15
35 45
X0
Y0
Z0
X5
Y5
Z5
Fig.1.LayoutoftheSchönfliesparallelmanipulator.
wheredegrees offreedomaregainedinanunexpectedway.There,themanipulatormaydangeritsownenvironment,including adjacentequipmentandhumanbeings.Moreover,certaintypesofsingularitiesdividethewholeworkspaceintoseveralregions.
Henceitisimportanttodetectallthesingularitiesandtoknowabouttheirdistributionintheworkspaceofthemanipulator.
In particular, inAmine,Masouleh,Caro, Wenger,andGosselin (2012)it isreportedasingularityanalysis of 3T1Rparallel manipulators withidentical limb structures, wherea specificcase study is fully detailed. However, the kinematicstructure of themanipulatordescribedinthatcasestudyisnotequaltothekinematicarchitectureofthemanipulatorreportedinthepresent paper.Moreover,the singularityanalysisreportedin(Amineetal.,2012)is basedonGrassmann-Cayley Algebra,whereasthe singularity analysis introducedin the present paper is based onlyon classicalconcepts of vectors andLinear Algebra, which resultsinasimplerapproach.Furthermore,duetotheexhaustivenatureoftheapproachproposedinthepresentpaper,asetof95 singularityconfigurationsaremathematicallyidentifiedandgeometricallycharacterized.Finally,thesingularitiesareplottedinto themanipulator’sworkspace,thusenlighteningtheirgeometricmeaning.
Fromtheforegoingdiscussion,thecontributionofthispaperwillbefocusedonfourdirections:(a)aclosedformsolutionof theinversepositionproblem,(b)aworkspacegenerationscheme,(c)asystematicvelocityanalysis,and(d)characterizationand detectionofallthesingularitiesandtheirdistributioninthereachableworkspaceofaSchönfliesparallelmanipulator.Itisexpected thatthesecontributionsmaybeusefulforanadequateplanningoftasks.
2. TheSchönfliesparallelmanipulator
Figure1showsaspatial4-dofparallelmanipulatorwhosemobileplatformgeneratesaSchönfliesmotion.
ReferringtoFigure1,itmaybenotedthatthemovingplatform(link5)isconnectedtoafixedbase(link0)byfournonidentical legs.TheobjectiveistohaveaSchönfliesparallelmanipulatorwithdifferentactuationschemes,i.e.,usingtworotatoryactuators andtwoprismaticactuators.Asaresult,theJacobianmatriceswillnotbehomogeneousintermsofunits,seeEq.(41).Itisexpected
O3: (R3x, R3Y, 0)
O4: (R4x, R4Y, 0)
O1: (R1x, 0, 0) O2: (R
2x, R
2Y, 0)
3: (ρ3x, ρ3Y, 0)
2: (ρ2x, ρ2Y, 0)
4: (ρ4x, ρ4Y, 0)
I: (ρ1x, 0, 0) Y0
Y5
X0 X5 Y5
Y0
X0 X5
O
P
φ
Fig.2.Generalgeometryofthefixedandmobileplatforms.
z1 z0
A1 b1
B1 h1
L1 D1 θ1
β1
a1
O1
X1 X0
Y1 Y0 C1 C1
g1
g1 D1
f1
e1 P
γ1
e1 e1
1
1 d1
ϕ1
f1 f1
f1
×
≡
Fig.3. Geometryofthefirstleg.
toconductananalysisofthisparticulartypeofJacobianmatricesandtheirrelationwithdexterityindices.Thiswillbetheresearch topicinaforthcomingpaper.
2.1. Kinematicarchitectureofthelegs
Thekinematicarchitecture1ofthelegsinvolvestwotypesoflegs:
(a) Thefirsttypeoflegismadeupoffiverevolutejoints,seethefirst(O1-A1-B1-C1-D1-1)andthethird(O3-A3-B3-C3-D3-3)legs showninFigure1.Inthisleg,thesecondandfifthjointaxesareparalleltothefirstjointaxis,whereasthefourthjointaxisis parallelthethirdjointaxis.Moreover,thethirdjointaxisintersectsthesecondperpendicularly,andthefifthjointaxisintersects thefourthperpendicularly.Furthermore,thereisanoffsetdistancebetweenthefirst andthesecond jointaxes.Arotational actuatorisusedtodrivethefirstjointofthelegwherethemotorisinstalledonthefixedplatform.
(b) Thesecondtypeoflegismadeupofoneprismaticjointandfourrevolutejoints,seethesecond(O2-A2-B2-C2-D2-2)andthe fourth(O4-A4-B4-C4-D4-4)legsshowninFigure1.Inthisleg,thesecondandfifthjointaxesareparalleltothefirstjointaxis, whereasthefourthjointaxisisparallelthethirdjointaxis.Moreover,thethirdjointaxisintersectsthesecondperpendicularly, andthefifthjointaxisintersectsthefourthperpendicularly.Furthermore,thereisanoffsetdistancebetweenthefirstandthe secondjointaxes.Thefirstmovinglinkofthislegisdrivenbyatranslationalactuatormountedonthefixedplatform.
2.2. Geometryofthemanipulator
ForthespatialparallelmanipulatorshowninFigure1,thefourfixedpointsO1,O2,O3,andO4definethegeometryofthefixed platform,andthefourmovingpoints1,2,3,and4definethegeometryofthemobileplatform.Althoughtheparticularmanipulator’s platformsshowninFigure1 aresymmetrical,itshouldbenoted thatboth, thefixedplatformandthemobileplatform,maybe arbitraryplanarquadrilaterals,seeFigure2.
Additionally,Figures3–6showthelinklengthsandjointvariablesrelatedtothefourlegs.Itisimportanttomentionthatunit vectorse1,e2,e3ande4denotethejointaxesofthoserevolutejointsthatjoinlinks12and13,22and23,32and33,and42and43, respectively.
1 Accordingtotheapproachproposedinthemanipulatorunderstudywasobtainedbyassemblingfourlegs.Thesefourlegsincludetwotypesofbasiclegs proposedinKongandGosselin(2007),whichweredesignedtogenerateaSchönfliesmotion.However,itisimportanttomentionthatthisparticularmanipulator, asawhole,isnotexplicitlyreportedinKongandGosselin(2007).
f2
d2
D2
C2 g2 e2
f2 e2
e2
f2
C2 Z2
p2
O2
X2
g2
D2 P
2
X2 X0
Y2
Y2
Y0
h2
b2
B2 A2
f2
L2
β2
γ2
α2
ϕ2
≡ 2
×
Fig.4. Geometryofthesecondleg.
Z3
C3
C3 h3
L3 D3 d3
3
e3
g3 f3
α3
β3
γ3
θ3
B3 b3 A3
a3
O3
g3≡ e3 f3
Y3
Y3
Y0
X3 X0
P
3
D3 X3
f3
f3 e3
ϕ3
×
Fig.5.Geometryofthethirdleg.
Insummary,theposeofthemobileplatformcanbespecifiedintermsofthepositionofpointP,andanorientationangle,namely, φ,seeFigure2.Moreover,theoriginofthefixedcoordinateframeX0Y0Z0islocatedatpointO.
3. Kinematicpositionanalysis
Theobjectiveofthissectionistoformulatetheinversepositionproblemassociatedwiththemanipulatorunderstudy.Ontheone hand,itshouldbenotedthatanglesϕ1,ϕ2,ϕ3,ϕ4,β1,β2,β3,β4,γ1,γ2,γ3andγ4arepassivejointvariables,whereasθ1,p2,θ3
andp4areactivejointvariables,seeFigures3–6.Ontheotherhand,rP/O=(x,y,z)TisthepositionvectorofmovingpointPwith respecttofixedpointO,whichismeasuredintheX0Y0Z0coordinateframe,andφdenotestherotationofthemobileplatformabout theZ0axis,seeFigure2.
f4
f4 e4
C4
d4 D4
C4 Z4
b4
L4 4
h4
B4 A4
p4
O4 X4
Y4
f4
Y4
Y0
X0
g4
D4
P 4
e4 X4
α4
γ4 ϕ4
β4
g4≡ e4×f4
Fig.6.Geometryofthefourthleg.
3.1. Constraintequations
Inordertoobtaintheso-calledconstraintequations,theprocedurebeginsbywritingaloop-closureequationforeachleg:
rOi/O+rAi/Oi+rBi/Ai+rCi/Bi+rDi/Ci+ri/Di =rP/O+ri/P,i=1,2,3,4. (1) whererj/kstandsforthepositionvectorofpointjwithrespecttopointk.
Writingequation(1)fori=1,2,3,4,andtakingtheX0Y0Z0coordinateframeasareference,itisobtainedthat:
R1X+b1cosθ1+L1sin(θ1+ϕ1)cosβ1=x+ρ1Xcosφ (2)
b1sinθ1−L1cos(θ1+ϕ1)cosβ1=y+ρ1Xsinφ (3)
a1+h1+L1sinβ1+d1=z (4)
R2X−L2cosϕ2cosβ2cosα2+(b2+L2sinϕ2cosβ2)sinα2=x+ρ2Xcosφ−ρ2Ysinφ (5) R2Y−L2cosϕ2cosβ2sinα2−(b2+L2sinϕ2cosβ2)cosα2=y+ρ2Xsinφ+ρ2Ycosφ (6)
p2+h2+L2sinβ2+d2=z (7)
R3X+{b3cosθ3+L3sin(θ3+ϕ3)cosβ3}cosα3−{b3sinθ3−L3cos(θ3+ϕ3)cosβ3}sinα3
=x+ρ3Xcosφ−ρ3Ysinφ (8)
R3Y+{b3cosθ3+L3sin(θ3+ϕ3)cosβ3}sinα3+{b3sinθ3−L3cos(θ3+ϕ3)cosβ3}cosα3
=y+ρ3Xsinφ+ρ3Ycosφ (9)
a3+h3+L3sinβ3+d3=z (10)
R4X+L4cosϕ4cosβ4cosα4−(b4+L4sinϕ4cosβ4)sinα4=x+ρ4Xcosφ−ρ4Ysinφ (11) R4Y+L4cosϕ4cosβ4sinα4+(b4+L4sinϕ4cosβ4)cosα4=y+ρ4Xsinφ+ρ4Ycosφ (12)
p4+h4+L4sinβ4+d4=z (13)
whicharetheconstraintequationssought.
3.2. Handlingoftheconstraintequations
Theapproachcanbestartedbyfocusingonthefactthatequations(2)–(13)arelinearinthesinesandcosinesofpassivejoint variablesϕ1,ϕ2,ϕ3,andϕ4.Thus,fromsimultaneoussolutionofequations(2),and(3),(5),and(6),(8),and(9),(11),and(12), respectively,itisfoundthat:
sinϕ1= ρ1Xcos(φ−θ1)+(x−R1X)cosθ1+ysinθ1−b1
L1cosβ1
(14)
cosϕ1= −ρ1Xsin(φ−θ1)+(x−R1X)sinθ1−ycosθ1
L1cosβ1
(15)
sinϕ2= −ρ2Xsin(φ−α2)−ρ2Ycos(φ−α2)−R2Xsinα2+R2Ycosα2+xsinα2−ycosα2−b2
L2cosβ2
(16)
cosϕ2= −ρ2Xcos(φ−α2)+ρ2Ysin(φ−α2)+R2Xcosα2+R2Ysinα2−xcosα2−ysinα2
L2cosβ2
(17)
sinϕ3= ρ3Xcos(φ−θ3−α3)−ρ3Ysin(φ−θ3−α3) L3cosβ3
−R3Xcos(θ3+α3)+R3Ysin(θ3+α3)−xcos(θ3+α3)−ysin(θ3+α3)+b3
L3cosβ3
(18)
cosϕ3=−ρ3Xsin(φ−θ3−α3)+ρ3Ycos(φ−θ3−α3) L3cosβ3
+−R3Xsin(θ3+α3)+R3Ycos(θ3+α3)+xsin(θ3+α3)−ycos(θ3+α3) L3cosβ3
(19)
sinϕ4= ρ4Xsin(φ−α4)+ρ4Ycos(φ−α4)+R4Xsinα4−R4Ycosα4−xsinα4+ycosα4−b4
L4cosβ4
(20)
cosϕ4= ρ4Xcos(φ−α4)−ρ4Ysin(φ−α4)−R4Xcosα4−R4Ysinα4+xcosα4+ysinα4
L4cosβ4
(21) Introducingthetrigonometricidentitiessin2ϕi+cos2ϕi=1,fori=1,2,3,and4,Eqs.(14)–(21)become:
2ρ1X{xcosφ+ysinφ−b1cos(φ−θ1)−R1Xcosφ+ρ1X/2}−2b1(xcosθ1+ysinθ1−R1Xcosθ1)+(R1X−x)2
+y2+b21−L21cos2β1=0 (22)
2ρ2X{xcosφ+ysinφ+b2sin(φ−α2)−R2Xcosφ−R2Ysinφ+ρ2X/2}−2ρ2Y{xsinφ−ycosφ−b2cos(φ−α2)
−R2Xsinφ+R2Ycosφ−ρ2Y/2}−2b2{xsinα2−ycosα2−R2Xsinα2+R2Ycosα2−b2/2}
+(R2X−x)2+(R2Y−y)2−L22cos2β2=0 (23)
2ρ3X{xcosφ+ysinφ−b3cos(φ−θ3−α3)−R3Xcosφ−R3Ysinφ+ρ3X/2}−2ρ3Y{xsinφ−ycosφ
−b3sin(φ−θ3−α3)−R3Xsinφ+R3Ycosφ−ρ3Y/2}−2b3{xcos(θ3+α3)+ysin(θ3+α3)
−R3Xcos(θ3+α3)−R3Ysin(θ3+α3)−b3/2}+(R3X−x)2+(R3Y −y)2−L23cos2β3=0 (24)
2ρ4X{xcosφ+ysinφ−b4sin(φ−α4)−R4Xcosφ−R4Ysinφ+ρ4X/2}−2ρ4Y{xsinφ−ycosφ+b4cos(φ−α4)
−R4Xsinφ+R4Ycosφ−ρ4Y/2}+2b4{xsinα4−ycosα4−R4Xsinα4+R4Ycosα4+b4/2}
+(R4X−x)2+(R4Y −y)2−L24cos2β4=0 (25)
Additionally,Eqs.(4),(7),(10),and(13)aresolvedforsinβi,andthensquared.Next,Eqs.(22)–(25)aresolvedfor cos2βi. Then,byintroducingthetrigonometricidentitiessin2βi+cos2βi=1,fori=1,2,3,and4,thefollowingequationsareobtained:
ε1sinθ1+2σ1cosθ1+κ1=0 (26)
p22+2σ2p2+κ2=0 (27)
ε3sinθ3+2σ3cosθ3+κ3=0 (28)
p24+2σ4p4+κ4=0 (29)
where
ε1≡−2b1(y+ρ1Xsinφ) σ1≡b1(R1X−x−ρ1Xcosφ)
κ1≡x2+y2+(a1+d1+h1−z)2−2R1X(x+ρ1Xcosφ)+2ρ1X(xcosφ+ysinφ)+b21+R21X+ρ1X2 −L21 σ2≡d2+h2−z
κ2≡(R2X−x)2+(R2Y−y)2+(d2+h2−z)2−2R2X(ρ2Xcosφ−ρ2Ysinφ)−2R2Y(ρ2Xsinφ+ρ2Ycosφ) +2ρ2X{xcosφ+ysinφ+b2sin(φ−α2)+ρ2X/2}−2ρ2Y{xsinφ−ycosφ−b2cos(φ−α2)−ρ2Y/2}
−2b2(xsinα2−ycosα2−R2Xsinα2+R2Ycosα2−b2/2)−L22
ε3≡2b3{xsinα3−ycosα3−ρ3Xsin(φ−α3)−ρ3Ycos(φ−α3)−R3Xsinα3+R3Ycosα3} σ3≡−2b3{xcosα3+ysinα3+ρ3Xcos(φ−α3)−ρ3Ysin(φ−α3)−R3Xcosα3−R3Ysinα3} κ3≡(R3X−x)2+(R3Y−y)2+(a3+d3+h3−z)2+2ρ3X(xcosφ+ysinφ+ρ3X/2)
−2ρ3Y(xsinφ−ycosφ−ρ3Y/2)−2R3X(ρ3Xcosφ−ρ3Ysinφ)−2R3Y(ρ3Xsinφ+ρ3Ycosφ)+b23−L23
σ4≡d4+h4−z
κ4≡(R4X−x)2+(R4Y−y)2+(d4+h4−z)2−2R4X(ρ4Xcosφ−ρ4Ysinφ)−2R4Y(ρ4Xsinφ+ρ4Ycosφ) +2ρ4X{xcosφ+ysinφ+ρ4X/2}−2ρ4Y(xsinφ−ycosφ−ρ4Y/2)−L24+2b4{(x−R4X)sinα4
+(R4Y −y)cosα4−ρ4Xsin(φ−α4)−ρ4Ycos(φ−α4)+b4/2}
Atthispoint,itshouldbementionedthatequations(26)–(29)canbesolvedfortheinputdisplacements,namely,θ1,p2,θ3and p4,respectively,whichisaprocedureusuallyknownasinversepositionproblem.
4. Workspacegeneration
TheworkspaceofthemanipulatorwillbedefinedhereasthevolumeofspacethatpointPofthemobileplatformcanreachinat leastoneorientation.Thus,themanipulator’sworkspacewillbecomposedbyalargesetofpointsPi,whoseCartesiancoordinates aregivenbyxi,yi,zi.AteachpointPi,themobileplatformwillhaveacommonorientationangle,namely,φG.
InordertodetectwhichpointPiiscontainedwithinthemanipulator’sworkspace,thefollowingapproachisproposed.Firstly, iftheworkspaceisgivenintermsoftheCartesiancoordinatesx,y,zandtheorientationangleφ,thenequations(26)–(29)canbe solvedfortheinputdisplacements,namely,θ1,p2,θ3andp4,respectively.Analyzingequations(26)–(29),itcanbeobservedthat equations(27)and(29)arequadraticinp2andp4,respectively.Moreover,equations(26)and(28)aretrigonometricexpressions thatcanbeconvertedintoquadraticequationsinτ1≡tan(θ1/2)andτ3≡tan(θ3/2),respectively.Thus,thesolutionprocesscanbe summarizedasfollows:
θ1=2arctan
−ε1±√ δ1
κ1−2σ1
, δ1≡ε21+4σ12−κ21 (30)
p2=−σ2±
δ2, δ2≡σ22−κ2 (31)
θ3=2arctan
−ε3±√ δ3
κ3−2σ3
, δ3≡ε23+4σ32−κ23 (32)
400
350
z
300
250
100
400
350
z
y
x 300
250
100 50
0 –50 0
–50
50 50
0
–50 –50
0 50
x Y
a b
c d
400
350
z
300
250
100 50
0 –50
–50 0
50 x y
400
350
z
300
250
100 50
0 50
0
–50 –50
y
x
Angle φG = –30°. Angle φG = –20°.
Angle φG = –10°. Angle φG = 0°.
Fig.7.WorkspacesfordifferentvaluesofthegivenorientationangleφG.
p4=−σ4±
δ4, δ4≡σ42−κ4 (33)
Then,apointPi(accompaniedwithagivenorientationangleφ=φG)willbepartoftheworkspaceifandonlyifthefollowing constraints:
δ1≥0, δ2≥0, δ3≥0, and δ4≥0. (34)
aresimultaneouslysatisfied.Suchconditionsguaranteethatatleastonesetofrealinputdisplacementsexistforthatpoint.
Insummary,theworkspaceisgeneratedbyconsideringathreedimensionalgridofpointsPiequippedwithconditions(34).As aresult,theplotsshowninFigures7and8wereobtained.
Figures7and8weregeneratedbyconsideringthefollowingnumericalvaluesofthedesignparameters:R1X=180,R2X=180, R2Y=0,R3X=0,R3Y=−180,R4X=−180,R4Y=0,ρ1X=181.10,ρ2X=13.25,ρ2Y=119.26,ρ3X=176.69,ρ3Y=39.75,ρ4X=−17.67, ρ4Y=−159.02,a1=a3=113,b1=b2=b3=b4=40,h1=h2=h3=h4=123,L1=L2=L3=L4=100,d1=d2=d3=d4=103,whichare giveninanarbitrarysystemofunits.Moreover,numericalvaluesofconstantangleswerechosenasα2=0◦,α3=0◦andα4=0◦.
5. Velocityanalysis
Avelocityanalysisrelatedtotheparallelmanipulatorunderstudyisintroducedinthissection.Inordertopresentasystematic approach,thecorrespondingmathematicalformulationisdividedintothefollowingthreeparts.
400
350
300
250
100
400
350
300
250
100
400
350
300
250
100 50
0
–50 –50
0 50
x y
50 0
–50 –50
0 50
x y
50 0
–50 –50
0 50
x y
z z
400
350
300
250
100 50
0
–50 –50
50
x
y 0
z z
Angle φG = 10°.
a
Angle φG = 30°. Angle φG = 40°.
c d
Angle φG = 20°.
b
Fig.8.WorkspacesfordifferentvaluesofthegivenorientationangleφG.
5.1. Velocityanalysisrelatedtothejointmotions
Duetotheseveralclosedloopsthatcomposethekinematicarchitectureoftheparallelmanipulator,thejointmotionsarenot independent.Hence,theobjectiveofthissectionistoobtainthelinearrelationshipsthatexistbetweenjointvelocities.Tothisend, theprocedurebeginsbyformulatingthevelocitystate2ofthemobileplatformwithrespecttofixedplatformintermsofthejoint motionsofeachmanipulator’sleg.Thus,byresortingtoscrewtheory(Rico,Gallardo,&Duffy,1999),itisobtainedthat:
V15/0 =
ω15/0
vP/O
=
k0 k0 0 k0
k0×rP/O1 k0×rP/B1 e1×rD1/C1 k0×rP/D1
⎡
⎢⎢
⎢⎢
⎢⎣
˙θ1
˙ϕ1
˙β1
γ˙1
⎤
⎥⎥
⎥⎥
⎥⎦
(35)
2 ThevelocitystateofbodyiwithrespecttobodyjisdenotedbyVi/j≡(ωi/j,vPi/Oj)T.Thisisasix-dimensionalvectorcomposedoftwothree-dimensional vectors:(a)theangularvelocityvectorofbodyiwithrespecttobodyj,namely,ωi/j,and(b)thevelocityvectorvPi/OjofapointPi(fixedonbodyi)withrespectto anypointofbodyj,suchaspointOj.
V25/0=
ω25/0
vP/O
=
0 k0 0 k0
k0 k0×rP/B2 e2×rD2/C2 k0×rP/D2
⎡
⎢⎢
⎢⎢
⎢⎣ p˙2
˙ϕ2
˙β2
γ˙2
⎤
⎥⎥
⎥⎥
⎥⎦
(36)
V35/0=
ω35/0
vP/O
=
k0 k0 0 k0
k0×rP/O3 k0×rP/B3 e3×rD3/C3 k0×rP/D3
⎡
⎢⎢
⎢⎢
⎢⎣
˙θ3
˙ϕ3
˙β3
γ˙3
⎤
⎥⎥
⎥⎥
⎥⎦
(37)
V45/0=
ω45/0
vP/O
=
0 k0 0 k0
k0 k0×rP/B4 e4×rD4/C4 k0×rP/D4
⎡
⎢⎢
⎢⎢
⎢⎣ p˙4
˙ϕ4
˙β4
γ˙4
⎤
⎥⎥
⎥⎥
⎥⎦
(38)
Ontheotherhand,sinceeachlegsharesacommonfixedandmobileplatform,i.e.,links5,15,25,35and45representthemobile platform,whereaslinks0,10,20,30and40representtothefixedplatform,itcanbestatedthat:
V15/0=V25/0, V15/0=V35/0, V15/0 =V45/0 (39)
whichyieldsthefollowingmatrixarray:
⎡
⎢⎢
⎣
C1 D1 E1 F1
C2 D2 E2 F2
C3 D3 E3 F3
⎤
⎥⎥
⎦
˙qI
˙qP
=0,
C D E F
˙qI
˙qP
=0 (40)
where:
C1≡
1 0 1 0
k0×rP/O1 −k0 0 0
C2≡
1 0 −1 0
k0×rP/O1 0 −k0×rP/O3 0
C3≡
1 0 0 0
k0×rP/O1 0 0 −k0
D1≡
1 0 1 −1
k0×rP/B1 e1×rD1/C1 k0×rP/D1 −k0×rP/B2
D2≡
1 0 1 0
k0×rP/B1 e1×rD1/C1 k0×rP/D1 0
D3≡
1 0 1 0
k0×rP/B1 e1×rD1/C1 k0×rP/D1 0
E1≡
0 −1 0 0
−e2×rD2/C2 −k0×rP/D2 0 0
E2≡
0 0 −1 0
0 0 −k0×rP/B3 −e3×rD3/C3
E3≡
⎡
⎢⎢
⎢⎣
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎤
⎥⎥
⎥⎦
F1≡
⎡
⎢⎢
⎢⎣
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎤
⎥⎥
⎥⎦
F2≡
−1 0 0 0
−k0×rP/D3 0 0 0
F3≡
0 −1 0 −1
0 −k0×rP/B4 −e4×rD4/C4 −k0×rP/D4
˙qI≡
˙θ1 p˙2 ˙θ3 p˙4
T
˙qP ≡
˙ϕ1 ˙β1 γ˙1 ˙ϕ2 ˙β2 γ˙2 ˙ϕ3 ˙β3 γ˙3 ˙ϕ4 ˙β4 γ˙4
T
Eq.(40)willbereferredtoasstructuralvelocitymodel.Itshouldbenotedthatequation(40)doesnotcontainanyparameter (e.g.,theCartesiancoordinates˙x,˙y,˙zofvelocityvectorofpointP)relatedtotheoutputmotionofthemobileplatform,butitonly containsinputjointvelocities ˙qIandpassivejointvelocities ˙qP.
5.2. Velocityanalysisrelatedtotheinputandoutputmotions
Inaparallelmanipulator, thearchitectureof thelegsdeterminesthe transformationofthe joint motionsintomotions of the mobileplatform.Thus,themobileplatformacquiresacertainvelocitystatethroughtheactuationofthelegscomposingtheparallel manipulator.Hencetheobjectiveofthissectionistorelatetheoutputmotionofthemobileplatformwiththeinputmotionsgenerated bymanipulator’sactuators.
Forthepurposesofthispaper,theoutputmotionisdefinedasthevelocitystateofthemobileplatformwithrespecttothefixed platform,namely,V5/0≡(ω5/0,vP/O)T.Ontheotherhand,theinputmotionisdefinedbyafour-dimensionalvector ˙qI,whichinvolves theinputjointvelocitiesoftheparallelmanipulator,i.e., ˙qI ≡(˙θ1,p˙2, ˙θ3,p˙4)T.
Inordertoreachtheobjectiveformulatedpreviously,theoryofreciprocalscrewscanbeusedtoprovideanelegantformulation.
Thus,computingtheKleinformofbothsidesofEqs.(35)–(38),andaftersomealgebra,itfollowsthat:
⎡
⎢⎢
⎢⎢
⎢⎣
μ1 rTD1/C1 μ2 rTD
2/C2
μ3 rTD
3/C3
μ4 rTD4/C4
⎤
⎥⎥
⎥⎥
⎥⎦
φ˙ vP/O
=
⎡
⎢⎢
⎢⎢
⎢⎣
λ1 0 0 0
0 λ2 0 0
0 0 λ3 0
0 0 0 λ4
⎤
⎥⎥
⎥⎥
⎥⎦
⎡
⎢⎢
⎢⎢
⎢⎣
˙θ1
p˙2
˙θ3
p˙4
⎤
⎥⎥
⎥⎥
⎥⎦, A˙s=B ˙qI (41)
where:
μ1≡(k0×r1/P)·rD1/C1, λ1≡(k0×rB1/A1)·rD1/C1, μ2≡(k0×r2/P)·rD2/C2, λ2≡k0·rD2/C2,
μ3≡(k0×r3/P)·rD3/C3, λ3≡(k0×rB3/A3)·rD3/C3, μ4≡(k0×r4/P)·rD4/C4, λ4≡k0·rD4/C4.
Giventhe rolerepresentedbyEq. (41),itwill bereferredtoas input–outputvelocitymodel.Itshouldbenotedthat Eq.(41) directlyrelatestheoutputmotion,vP/O, ˙φ,withtheinputmotion, ˙qI.Inotherwords,theoutputmotionisdecoupledfromthepassive jointmotions, ˙qP.