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A Comparative Study of Parallel Kinematic
Architectures for Machining Applications
Philippe Wenger, Clément Gosselin, Damien Chablat
To cite this version:
Philippe Wenger, Clément Gosselin, Damien Chablat. A Comparative Study of Parallel Kinematic
Architectures for Machining Applications. Electronic Journal of Computational Kinematics, IFToMM,
2001, 1 (1), pp.23. �hal-01703831�
Architectures for Machining Applications
Philippe Wenger 1
, Clement Gosselin 2
and DamienChablat 1 1
Institut de Recherche en Communications et Cybernetiquede Nantes 1, rue de la Noe, 44321 Nantes, France
2
Departement de genie mecanique, Universite Laval Quebec,Quebec, Canada, G1K 7P4
Philippe.Wenger@irccyn.ec-nantes.fr
Abstract: Parallelkinematicmechanismsareinterestingalternativedesignsformachining ap-plications. Three 2-DOFparallelmechanismarchitecturesdedicated tomachiningapplications arestudied inthis paper. Thethreemechanismshavetwoconstantlengthstrutsglidingalong xed linear actuatedjoints with di erent relativeorientation. The comparativestudy is con-ducted onthebasis ofasameprescribedCartesianworkspacefor thethree mechanisms. The common desired workspace properties are a rectangular shape and given kinetostatic perfor-mances. The machine size of each resulting design is used as a comparative criterion. The 2-DOF machine mechanisms analyzed in this paper can be extended to 3-axis machines by addingathirdjoint.
1 Introduction
Most industrial3-axis machine toolshaveaPPPkinematicarchitecturewith orthogonaljoint axesalongthex,y,zdirections. Thus,themotionofthetoolinanyofthesedirectionislinearly relatedtothemotionofoneofthethreeactuatedaxes. Also,theperformances(e.g. maximum speeds,forces,accuracyandrigidity)areconstantinthemostpartoftheCartesianworkspace, which is a parallelepiped. In contrast, the common features of most existing PKM(Parallel KinematicMachine)areaCartesianworkspaceshapeofcomplexgeometryandhighlynon lin-ear input/output relations. FormostPKM, theJacobianmatrix which relatesthe jointrates totheoutputvelocitiesisnotconstantandnotisotropic. Consequently,theperformancesmay varyconsiderably fordi erentpointsin theCartesianworkspaceand fordi erentdirectionsat onegivenpoint, which isa seriousdrawbackfor machiningapplications [8]. Tobeof interest for machiningapplications, a parallel kinematicarchitecture should preserve good workspace properties(regularshapeandacceptablekinetostaticperformancesthroughout). Itisclearthat someparallel architecturesare moreappropriatethan others,asit hasalready been shownin previous studies [6, 7]. The aim of this paper is to compare three parallel kinematic archi-tectures. Tolimitthe analysis,thestudy isconducted for 2-DOF mechanismsbut theresults canbeextrapolatedto 3-DOFarchitectures. Thethree mechanismsstudied havetwoconstant lengthstrutsglidingalong xed linearactuatedjointswith di erentrelativeorientation. Each
IRCCyN:UMRn Æ
6597CNRS,
EcoleCentraledeNantes,UniversitedeNantes,
gular region with given kinetostatic performances, we calculate the link dimensions and joint rangesofeachmechanismforwhichtheprescribedregionisincludedinat-connectedregionof themechanismandthekinetostaticconstraintsaresatis ed. Then,wecomparethesizeofthe resultingmechanisms. Theorganisation of thispaperis asfollows. Thenextsection presents themechanismstudied. Section3isdevotedtothecomparisonofthreearchitectures. Thelast sectionconcludesthis paper.
2 Kinematic study
2.1 Serial Topology with Three Degrees of Freedom
MostindustrialmachinetoolsuseasimplePPPserialtopologywiththreeorthogonalprismatic jointaxes(Figure1).
X
Z
Y
Figure1: Typicalindustrial 3-axismachine-tool
ForaPPP topology,thekinematicequationsare:
J_ =p_ with J=1 33
where p_ = [x y z] T
is the velocity-vector of the tool center point P and _ = [ 1 2 3 ] T is thevelocity-vectoroftheprismaticjoints. TheJacobiankinematicmatrixJbeingtheidentity matrix, the ellipsoid of manipulability of velocity and of force [1] is aunit sphere for all the con gurations in the Cartesian workspace. The problem of the PPP topology is that the actuator controlling the Y axis supports at the same time the workpiece and the actuator controllingthe displacement ofthe X axis,whicha ects the dynamic performances. Tosolve this problem, itis possibletouse moresuitablekinematicarchitectureslikeparallel orhybrid topologies.
2.2 The Parallel Mechanisms Studied
Wefocusourstudyontheuseofa2-DOFparallelmechanism(Figure2)forthemotionofthe table ofthemachinetooldepictedin(Figure1).
The joint variables arethe variables 1
and 2
associated with the two actuatedprismatic joints and the output variables are the position of the tool center point P = [x y]
T . The mechanismscanbeparameterizedbythelengthsL
0 ,L 1 andL 2 ,theangles 1 and 2 andthe actuated joint ranges
1
and 2
P
A
C
B
D
r
1
a
2
q
2
q
1
a
1
r
2
L
1
L
2
Figure2: Twodegree-of-freedom paral-lelmechanism
P
A
C
B
D
r
1
r
2
Figure3: Twodegree-of-freedom paral-lel mechanism with control of the ori-entation weimposeL 1 =L 2 and 1 = 2
. Thissimpli cation alsoprovidessymmetryand, inturn, reducesthemanufacturingcosts.
Tocontrol the orientation of the reference frame attached to the toolcenter point P, two parallelogramscanbeusedwhich alsoincreasetherigidityofthestructure(Figure3).
2.3 Kinematics of the Parallel Mechanism Studied
Thevelocityp_ ofP canbewrittenintwodi erentways. Bytraversingtheclosed-loop(ACP BDP)inthetwopossibledirections,weobtain
_ p=c_+ _ 1 E(p c) (1a) and _ p= _ d+ _ 1 E(p d) (1b)
whereE istherotationmatrix,
E=
0 1 1 0
canddrepresenttheposition vectorofthepointsC andD,respectively. Moreover,thevelocityc_ and
_
dofthepointsCandD aregivenby,
_ c= c a jjc ajj _ 1 = cos( 1 ) sin( 1 ) _ 1 ; _ d= d b jjd bjj _ 2 = cos( 2 ) sin( 2 ) _ 2
The twounactuated joint rates _ 1 and _ 2
canbe eliminated from equations (1a)and (1b) by dot-multiplyingtheformerbyp candthelatterbyp d,thusobtaining
(p c) T _ p=(p c) T c a jjc ajj _ 1 (2a) (p d) T _ p=(p d) T d b jjd bjj _ 2 (2b)
Equations(2a)and(2b) cannowbecastin vectorform,namely,
A= (p c) T (p d) T ; B= (p c) T ((c a)=jjc ajj) 0 0 (p d) T ((d b)=jjd bjj)
andwith_ de nedas thevectorofactuatedjointratesandp_ de nedasthevectorofvelocity ofpointP: _ = _ 1 _ 2 ; p_ = _ x _ y
WhenAandBarenotsingular,wecanstudytheJacobiankinematicmatrixJ[2],
_ p=J _ with J=A 1 B (3a)
ortheinverseJacobiankinematicmatrixJ 1 ,suchthat _ =J 1 _ p with J 1 =B 1 A (3b) 2.4 Parallel Singularities
TheparallelsingularitiesoccurwhenthedeterminantofthematrixAvanishes[3,4],i.e. when det(A) =0. Inthis con guration, itis possibleto movelocally thetoolcenter pointwhereas the actuated joints are locked. These singularities are particularly undesirable, because the structurecannotresist anyforceandcontrolislost. Toavoidanydeterioration, itisnecessary toeliminatetheparallelsingularitiesfrom theworkspace.
Forthemechanismstudied, theparallelsingularitiesoccurwheneverthepointsC,D,andP arealigned(Figure4), i.e. when
1 2 =k,fork=1;2;:::.
P
A
C
B
D
r
1
r
2
Figure4: Parallelsingularity
P
A
C
B
D
r
1
r
2
Figure5: Structuralsingularity
TheyarelocatedinsidetheCartesianworkspaceandformtheboundariesofthejointworkspace. Moreover,structuralsingularitiescanoccurwhenL
1
isequalto L 2
(Figure5). Inthese con g-urations,thecontrolof thepointP islost.
2.5 Serial Singularities
The serial singularities occur when the determinant of the matrice B vanishes, i.e. when det(B)=0. Whenthemanipulatorisinsuchcon gurations,thereisadirectionalongwhichno Cartesianvelocitycanbeproduced. Theserialsingularitiesde netheboundaryoftheCartesian workspace[Merlet97].
For the topology studied, the serial singularities occur whenever 1 1 = =2+k, or 2 2
==2+k,fork=1;2;::: (Figure6),i.ewheneverAC is orthogonalto CP orBD is orthogonaltoDP.
P
A
C
B
D
r
1
r
2
Figure6: Serialsingularity
2.6 Application to Machining
For a machine tool with three axes as in (Figure 1), the motion of the table is performed along two perpendicular axes. The joint limits of each actuator give the dimension of the Cartesianworkspace.Fortheparallelmechanismsstudied,thistransformationisnotdirect. The resultingCartesianworkspaceismorecomplexanditssizesmaller. WewanttohaveaCartesian workspacewhichwill becloseto theCartesianworkspaceof anindustrial serial machine tool. For our 2-DOF mechanisms, we will prescribe a rectangular shape Cartesian workspace. In addition,theworkspacemustbereducedtoat-connectedregion,i.e. aregionfreeofserialand parallel singularities [9]. Finally, wewantto prescriberelativelystable kinetostatic properties in theworkspace.
2.7 VelocityAmpli cation Study
InordertokeepreasonableandhomogeneouskinetostaticpropertiesintheCartesianworkspace, westudy the manipulabilityellipsoids ofvelocity de ned by the inverse Jacobian matrixJ
1 [1]. Forthemechanismsathand,theinverseKinematicJacobianmatrixJ
1
giveninequation (3b)is simple. Inthiscase,thematricesBandJ
1
arewritten simply,
B= 1 L 1 1=c 1 0 0 1=c 2 and J 1 = 1 L 1 (1=c 1 )(p c) T (1=c 2 )(p d) T with c i =cos( i i ) i=1;2
The square roots 1 and 2 of the eigenvaluesof (JJ T ) 1
are the valuesof the semi-axes of theellipsewhich de nethe twofactorsofvelocityampli cation(fromthejointratestothe output velocities), 1 = 1= 1 and 2 = 1= 2
, accordingto these principal axes. Tolimit the variationsofthisfactorintheCartesianworkspace,weposethefollowingconstraints,
1=3< i
<3 (4)
This means that for a given joint velocity, the output velocity is either at most three times largeror,at least,threetimessmaller. Thisconstraintalsopermitstolimitthelossof rigidity (velocityampli cationlowersrigidity)andofaccuracy(velocityampli cationalsoampli esthe encoderresolution). Thevaluesinequation(4)werechosenasanexampleandshouldbede ned preciselyasafunction ofthetypeofmachiningtasks.
3.1 The Three Parallel Mechanism Architectures Studied
Thethreeparallelmechanismarchitecturesstudiedarethefollowing:
1. Thebiglide1mechanismwith 1
=0and 2
= (Fig.7)
2. Thebiglide2mechanismwith 1
==2and 2
==2(Fig.8)
3. Theorthoglidemechanismwith 1
==4and 2
=3=4(Fig.9)
Thebiglide1mechanismstudied (Fig. 7)hasbeenusedforexamplein thehexaglideandin thetriglide[10].
P
A
C
D
B
r
1
r
2
L
L
0
L
Figure7: Thebiglide1mechanism
Thebiglide2mechanism(Fig. 8)hasbeenusedintheLinapodandin [10,12].
P
A
C
B
D
r
1
r
2
L
L
0
L
Figure8: Thebiglide2mechanism
L
0
r
1
C
r
2
A
B
D
P
L
L
Figure9: Theorthoglidemechanism
The third mechanism (Fig. 9) wasintroduced in [11] and extended to 3-DOF in [10]. The main constraintof thisdesignisAC?BD, which makesitisotropic,i.e. theJacobianmatrix ofthismechanismisisotropicinsomecon gurations.
3.2 Determination of the Mechanism Dimensions
Todeterminethemechanismdimensions,weproceedinseveralstepsasfollows. LetL=L 1
=L 2 bethecommonlink lengths,letL
0
bethedistancebetweentheattachmentpointsAandBof theprismaticjointsandletbetherangeoftheactuatedjoints. ThelengthL
0
ampli cationfactor(seesection2.7). Forallmechanisms,wecanshowthatthemaximal(resp. minimal) velocity ampli cation factor is reached at the con guration for which the distance between C and D is maximal (resp. minimal). For the biglide1 and for the orthoglide, the maximal (resp. minimal) velocity ampli cationfactoris reachedat thecon guration whereC isonAandD isonB(resp. where CisonC'andDisonD') ( gures10and12).
A
r
1
L
L
0
P
B
r
2
L
C’
D’
Figure10: Thebiglide1mechanism
P
A
C’
B
r
1
r
2
L
L
L
0
D’
Figure 11: Thebiglide2mechanism
L
0
r
1
r
2
A
B
D’
P
L
L
C’
Figure12: Theorthoglidemechanism
By rstwritingthatthemaximalfactormustbesmallerthan3inthe rstcon guration,we cancalculateL
0
. Theniscalculatedbywritingthatintheoppositecon gurationthevelocity ampli cation factor must belarger than 1/3. Forthe biglide2, the maximal (resp. minimal) ampli cationfactorisreachedatthecon gurationwhereCisonAandDisonD'(resp. where CandDlieonanhorizontalline)( gure11). Inthiscase,we rstcalculateL
0
attheminimal factorcon gurationandisthencalculatedatthemaximalfactorcon guration.Thevaluesof L
0
andobtainedforallmechanismsaregiveninthe rsttworowsoftable1. Allderivations andcomputationshavebeenobtainedwithMAPLE.
Then, foreachmechanism,wedeterminethemaximumrectangular surfaceS which canbe includedin theCartesianworkspace( gures 13to 15). Wehaveused theparametricsketcher ofaCADsystemtoperformthistask. TheareaofthesurfacesSobtainedaregiveninthelast rowoftable1. Thelaststepisthescalingofthemechanismlinkdimensionsandjointrangesin
0
Biglide1 1:946L 0:547L 0:107L Biglide2 0:458L 0:529L 0:249L Orthoglide 1:961L 1:109L 0:885L
Table1: Dimensionsand rectangularCartesianworkspacesurfaceasfunction ofL
orderto haveasameCartesianworkspacerectangularforallmechanisms. The rstthreerows oftable 2givetheresultinglinkdimensionsandjointranges.
y
x
Figure 13: The Cartesian workspaceoftheBiglide1x
y
Figure 14: The Cartesian workspaceoftheBiglide2
x
y
Figure 15: The Cartesian workspaceoftheOrthoglide
3.3 Comparison of the Mechanism Size Envelopes
Table2providesthemechanismdimensionsandenvelopesizesforthethreeparallelmechanisms studied, foraprescribedrectangularCartesianworkspacesurfaceof1m
2 .
Mechanism L 0
L Mechanismenvelopesurface Biglide1 5:95 3:05 1:67 16:45
Biglide2 0:92 2:00 1:06 8:50 Orthoglide 2:08 1:06 1:18 3:91
Table2: MechanismdimensionsandenvelopesizesforasamerectangularCartesianworkspace
Figs.16,17and18showthethreemechanismsalongwiththeirCartesianworkspaceandthe samerectangularsurfaceinit. Wecannoticethattheorthoglidemechanismhassmallerlengths struts i.e. smallermass in motion and thus higher dynamic performancesthan the other two mechanisms. Thebiglide2andtheorthoglidemechanismshavesimilar valuesof . It should benoticed,also,thattheCartesianworkspaceofthebiglide2includesarectangullewhichisfar fromasquare,whereasitisanexactsquarefortheothertwomechanisms. Wehavecalculated the dimensions of the biglide2 for a square of 1 m
2
in its workspace and we have obtained L
0
=1:075,L=2:348and=1:242.
4 Conclusions
Three2-DOF parallelmechanismsdedicatedtomachiningapplicationshavebeencomparedin thispaper. Thelink dimensionsandtheactuatedjointrangeshavebeencalculatedforasame
y
x
Figure16: Workspaceofthebiglide1mechanism
x
y
Figure17: Workspaceofthebiglide2 mech-anism
x
y
Figure 18: Workspace of the orthoglide mechanism
prescribedrectangularCartesianworkspacewithidenticalkinetostaticconstraints. Themachine size of each resultingdesignwasused asacomparativecriterion. Oneofthe mechanisms,the orthoglide, was shown to havelower dimensionsthan the other twomechanisms. This result showsthat theisotropic property ofthe orthoglideinduces interestingadditional features like bettercompactness and lowerinertia. Inthe future, thecomparativestudy will be continued usingdynamicperformanceindices.
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