The necessity of optimal design for parallel machines and a possible certified methodology

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The necessity of optimal design for parallel machines and a possible certified methodology

Jean-Pierre Merlet

To cite this version:

Jean-Pierre Merlet. The necessity of optimal design for parallel machines and a possible certified

methodology. Robotic Systems for Handling and Assembly, SFB 562, May 2005, Braunschweig. �inria-

00000008�

and a possible ertiedmethodology

J-P.Merlet

INRIASophia-Antipolis,Frane

Abstrat: Although theyhave many advantages in term of positioning a-

uray, stiness, load apaity parallel mahines have also a main drawbak:

theirperformanesareverysensitivetotheirdimensioning. Henealthoughthe

hoieofagivenmehanialstrutureamongthenumerouspossibilitiesthatare

oeredforparallelmahinesmayinuenetheperformanesofthemahinethe

ruleof thumbis: amehaniallyappropriatebut poorlydimensionedmahine

willpresentingenerallargelylowerperformanesthanawelldesignedmahine

withamehanialarhitetureapriorilessadequate.

Optimal dimensioning of a parallel mahine is hene a ritial issue but

alsoa omplexone, espeially if unertainties in themanufaturing are taken

intoaount. Wewill presentapossibledesignmethodologybasedoninterval

analysisandwillillustratethismethodologyonrealistiexamples.

1 Introdution

It is not so well known that the performanes of parallel robots are highly

sensitive to their geometri design. Consider for example a Gough platform

whoseattahmentpointsloatedonthebasehaveasoordinates:

A

1

:( 9;9) A

2

:(9;9) A

3

:(12; 3) A

4

:(3; 13) A

5

:( 3; 13) A

6

:( 12; 3)

Theattahmentpointsontheplatformhavealassialrepartitiononairleof

radiusr

1

with 2adjaentpointsseparatedbyanangleof30degrees. Wethen

onsiderthestinessmatrixKoftherobot, assumingaunitvalueforthelink

stinesswhih willleadto K =J T

J 1

at apartiularpose fortheplatform

(base and platform are parallel and the enter of the base and platform are

loatedonthevertialaxis). Figure1presentsthevariationofk

x

asafuntion

ofr

1

. It maybeseenthat foravariationofr

1

from3to9thevariationofk

x

is roughlyfrom 20 to 200. It mayalso been intuitively understood that suh

salefatorwill get worsewhen weonsider all poses within theworkspaeof

therobot. Thisexampleshowslearlythe importane ofawelltought design

proess.

Designsynthesisisatwo-stepproess:

struturesynthesis: determinethegeneralarrangementofthemehanial

struturesuhasthetypeandnumberofjointsandthewaytheywillbe

onneted

dimensional synthesis: determine the length of the links, the axis and

loationofthejoints,:::. Inthispapertheworddimensionwillhavethe

3.0 4.0 5.0 6.0 7.0 8.0 9.09.0 25

55 85 115 145 175

k

x

r1

Figure 1: Variationof thestiness element k

x

as afuntion of the platform

radiusr

1

broadsense of anyparameterthat will inuene therobot behaviorand

isneededforthemanufaturingoftherobot

For serial robots general trends for the robot performanes may be dedued

from thestruture. For examplewemayompare thereahableworkspaeof

3 d.o.f. robot of type PPP and RRR: assuming a stroke of L for the linear

atuator and a length L for the links the PPP workspaevolume will be L 3

while it will be 40L 3

for the RRR robot. But even for serial robot suh

trendswill notbesuÆientto fullydeterminetheoptimalrobot: indeed many

performaneshavetobetakenintoaountfordeninganoptimalrobot,some

ofthembeinghighlydependentuponthedimensionsoftherobot(forexample

the load apaity). The ase is worse for losed-hain robots for whih suh

generaltrendsannotbeeasilyderived

Hene optimal design for a robot implies both type of synthesis but our

experienein thedesignof losed-hainrobotshasled usto thefollowingrule

ofathumb: arobotwithan a-priorimoreappropriatemehanial struturebut

whosedimensionshavebeenpoorlyhosenwillexhibitlargelylowerperformanes

thanawelldimensionallydesignedrobotwithana-priorilessappropriate stru-

ture. Weare notlaiming that strutural synthesis is not an important area

but that it annot be disonneted from dimensional synthesis. The point is

thatstruturalsynthesis,althoughstillinprogress,hasstrongtheoretialbak-

grounds(suh as srewand grouptheories)while, aswe will see, dimensional

synthesislakofsuhbakground.

Dimensionalsynthesisisaproblemthathasattratedalotofattentionbutmost

oftheworksfousondesignforaspeirobot'sfeaturesuhasworkspae[1,

5,9,10,11℄orauray[7,18,19℄.

Theusualwaytosolvetheoptimaldesignproblemistodeneareal-valued

funtion C as a weighted sum of performane indies P

i

[4℄ with a value in

the range [0,1℄. These indies measure how muh eah of the performane

requirement is violated for a given robot. A value equal to 0 indiates that

therequirementis fullysatisedwhile avalue1isusedwhen therequirement

isfully violated. These performaneindies arelearlyfuntions ofthe design

parameterssetP. Theostfuntion isthendenedas

C= X

i w

i P

i (P)

wherew

i

areweights. Itisassumedthattheoptimaldesignsolutionisobtained

forthevalueoftheparametersinPthatminimizeCandanumerialproedure

isusedtondthevaluesofP whihminimizeC,usuallystartingwithaninitial

guessP

0

. But thismethod hasmanydrawbaksand wewill mentionafewof

them.

First it is assumed that the requirement indies an be dened and that

theyanbealulatedeÆiently(indeedthenumerialoptimizationproedure

requiresalargenumberofevaluationoftheseindies). Alltheseassumptionsare

diÆulttorealizeinpratieforrobots: forexamplewhatouldbethedenition

ofanindex thatindiatesthat aubeof givenvolumemustbeinludedin the

robot's workspae? Evaluation of someindies may also be aquite diÆult

problem: forexamplewemaydeneasindextheworstpositioningerroralonga

givenaxisforanyposeoftherobotwithinapresribedworkspaeandevaluating

thisindexisbyitselfadiÆultonstrainedoptimizationproblem.

A seond drawbak of the ost-funtion approah is the diÆulty in the

determination of the weights. These weights are present in the funtion not

onlytoindiatethepriorityoftherequirementsbutalsototaklewiththeunits

problemintheperformaneindies. Forexampleiftheusedperformaneindies

are the workspae volume and positioning auray for a 3-dof translational

robotwearedealingwithquantitieswhoseunitsdierbyaratioof10 3

: hene

theweightsmustbeusedtonormalizetheindies. Thehoieoftheweightsis

thereforeessentialwhilethereisnotintuitiverulesfordeterminingtheirvalues.

Furthermore asmall hange in theweightsmayleadto verydierentoptimal

designs.

Eveniftheost-funtioniseetiveitmayleadtoinonlusiveresult. This

wasexempliedbyStoughton[20℄whowaswantingtodeterminespeialkindof

Gough platformwith improveddexterityand areasonableworkspaevolume.

Hene Stoughtonhas onsidered tworiteria in his ost-funtion: the dexter-

ity andtheworkspaevolume. He ndoutthat these riteriawerevarying in

opposite ways: the dexterity was dereasingwhen the workspaevolume was

was in fat to determine an aeptable ompromisebetweenthe two require-

ments. This advoates thepointthat in optimal design we should not tryto

maximizeoneperformanewithoutimposing onstraintontheminimal values

ofotherperformanes(forexampleGosselin[6℄showsthattheGoughplatform

havingthelargestworkspaefor agivenstrokeoftheatuator willhaveage-

ometry suh that it annot be ontrolled). It may also be onsidered that in

someasessomerequirementsareimperative i.e. theymust neverbeviolated

whilesomeothers maybesomewhatrelaxed. Butimposing an imperativere-

quirementsintheost-funtionisdiÆultwithoutviolatingthedierentiability

onstraintand/orallowinglargeviolationontheotheronstraints.

Athird drawbakoftheost-funtionapproahisthatitprovidesonlyone

solution. Thisausesthreemainproblems:

manufaturingtoleraneswill besuhthat therealrobotwill dierfrom

the theoretial one. Hene with only one theoretial design solutionwe

annotguaranteethat therealrobotwillfullls therequirements

providingonlyonesolutiondoesnotallowtoonsider seondaryrequire-

ments that may have not been used in the ost-funtion but may be a

deision fator if two robots satisfy in a similar way the main require-

ments

forprovidingonlyonesolutionwehavetoassumethat thedesignermas-

ters all the riterion that will lead the end-user to a solution. This is

seldom the ase in pratie: for example eonomi onsiderations will

usually play arole although the designer annot be fully awareof their

levelofimpliation

Wewill proposenowanotherdesignmethodology.

3 Another design methodology: the parameters

spae approah

Wewillrstdenetheparameters spae S n

asan-dimensionalspaeinwhih

eah dimension orresponds to one of the n design parameters of the robot.

Heneapointin thatspaeorrespondto oneuniquedesignoftherobot.

Consider now a list of m requirements fR

1

;:::;R

m

g that dene minimal

ormaximalallowedvaluesofsomerobot'sperformane(suhasauray,sti-

ness, :::) or some required properties (for example that a set of pre-dened

trajetorieslie within therobot's workspae) and assume that weare able to

designanalgorithmthatisabletoalulatetheregionZ

i

denedastheregion

oftheparameterspaeS n

thatinludes alltherobot'sdesignthat satisesthe

requirementR

i

. Thenthe intersetion ofall theZ

i

will dene allthe robot's

designthatsatisesalltherequirements. Withthisapproahwewillhavefound

allpossibledesignsolutions.

TomakethisapproahpratialweareonfrontedtotwodiÆulties:

1. alulatingtheregionZ

i

2. omputingtheintersetionoftheregions

The alulation of the region is indeed quite diÆult aswe havebasially to

determine regions whose borders are determined by a set of omplex highly

non-linearrelations (but in some asesthis may be possible if the numberof

design parameters is not too high, see [13, 16℄). But a good point is that it

isnotneessaryto determinethese regionsexatly. Indeeddeterminingpoints

ofthe regionlose to theborder doesnotmakesense asifthey arehosenas

nominalparametervalue,thentherealrobot,whoseparameterareaetedby

manufaturingtoleranes,mayinfathavearepresentativepointintheparam-

etersspaethat is outsidethe Z

i

regions. Hene omputinganapproximation

oftheregionswhoseborderissuÆientlylosetotherealborder issuÆient.

It isnowwellknown that interval analysis maybeanappropriatetoolfor

omputingthis approximation. Indeedeah of the ndesign parameters P

i in

P represents aphysial quantity (e.g. link length, rotationaxis,:::) that an

usually be bounded i.e. weanassign arange foreahparameter and weare

looking only for design solutions suh that the values of the parameters lie

within theirassignedrange. Thenintervalanalysis willbeableto provide,for

example,anapproximationofallvaluesofthedesignparameterssuhthataset

ofinequalitiesF(P)0. Theresultwillbealistofm elements,eahelement

being onstituted of n ranges R j

P

i

, j 2 [1;m℄;i 2 [1;n℄, one for eah design

parameter. Choosing asvalue of the design parameters an arbitrary number

within the ranges of a given element ensures that the set of inequalities will

be satised. Thequality oftheapproximationonlydependupon theminimal

widthof therangesthat areallowedin theelement. Examples ofappliations

of suh algorithm are desribed in [2℄(fore transmissionin a3d.o.f. robot),

[8℄(workspaerequirements),[14℄(singularitydetetion).

4 Optimal design

Wehaveseenthat ouroptimaldesignapproahrequiresthealulationof the

regionsZ and then their intersetion. Intervalanalysis seems to be quite ap-

propriateforthe seond part. Indeed ifwe assumethat weare ableto obtain

theregionsZ asasetofboxes,thenalulatingtheirintersetionisalassial

probleminomputationalgeometrythat anbesolvedeasily.

We are now onfronted to the problem of alulating the region Z using

intervalanalysis. Asmentionedpreviouslythereisnoneedto alulateexatly

these regionsas pointson theborder annot beonsidered asnominal design

parametervalues beausethe eet of manufaturing toleranes may put the

valueoftherealrobotparameteroutsidetheregionZ. Thispointmaybeused

asanadvantageforintervalanalysis-basedmethodbyusing thefollowingrule:

the range for eah design parameter has a widthwhih is at least equal to the

manufaturingtoleranefor thisparameter

Therationalbehindthisrulemaybeillustratedonanexample. Assumethat

for a given parameter whose manufaturing tolerane is [ ;℄ the algorithm

provides theresult range[a;b℄. If b a 2then wemay hoose asnominal

valuefortheparameteranyvalueintherange[a+;b ℄: indeed toanysuh

valuewemayaddanarbitrarymanufaturingtoleraneintherange[ ;℄with

aresultstillin[a;b℄. Inotherwordstheparametervalueforthereal robotwill

stillbesuhthat its representativepointsin theparametersspae willbelong

toZ.

Intervalanalysis-basedmethodmaybethoughtasamethodtoomputean

approximationoftheregionZ inwhihtheparts ofZ thataretooloseto the

borderareeliminated.

We havenow to explainhowwemaydesign analgorithm to alulate the

regionZ.

4.1 Calulating Z: an example

Up to now we haveassumed that the performane requirement hasa losed-

form that an be intervalevaluated. This is notalwaysthe ase in robotis.

ForexampleassumethatweonsiderthepositioningaurayXoftherobot

withrespettothejointmeasurementerrors. Bothquantities arelinearly

relatedby

X=J(X)

where J is the Jaobianmatrix of therobot, whose elements are funtions of

theposeXandofthedesignparameters.

The following requirement is lassial: being given bounds M

on the

jointerrorsdetermine thedesignparameterssuhthat therobot'spositioning

errorsarelowerthangiventhresholdsX M

,whateveristheposeoftherobot

in a given workspaeW. Unfortunately for losed-hain robots the matrix J

maybequiteomplex(orevenmaynotbeavailable)whileitsinverseJ 1

may

haveasimpleform. ButitispossibletostatetheproblemusingonlyJ 1

: nd

thedesignparametersP suhthatforallXinWallthesolutionsinXofthe

linearsystemJ 1

(X;P)X=with

M

areinludedinX M

.

Wehavethus to solvealassialproblem of interval analysis: beinggiven

anintervalmatrixAandanintervalvetorbdetermineanenlosureofallthe

solutionsof the linearinterval systemAx =bi.e. aregion that inludes the

solutionofAx=b forall A;b inludedin A;b[15, 17℄. It anbeshown that

lassialmethods of linear algebra(suh as the Gauss elimination algorithm)

maybeextendedtodealwiththisproblem. Wemaydiretlyusethesemethods

toomputeanenlosureofX andstoreasresulttheparametersboxessuh

thatthis enlosureis inludedin X M

. ButwemayimprovetheireÆieny:

indeed these methods assume no dependeny between the elements of A i.e.

valuewithintheirrangesinA. Inourasetherearedependeniesbetweenthe

elementsofJ 1

andnotallpossiblevaluesareallowed.

OurbasimethodistheGausseliminationsheme. Weomputeaninterval

evaluation A (0)

ofAand anintervalevaluationb (0)

ofb(usingthederivatives

of theomponentsof A;b to improve these interval evaluations). The Gauss

eliminationshememaybewritten as[17℄

A (j)

ik

=A (j 1)

ik

A (j 1)

ij A

(j 1)

jk

A (j 1)

jj

8iwithj>k (1)

b (j)

i

=b (j 1)

i

A (j 1)

ij b

(j 1)

j

A (j 1)

jj

(2)

TheenlosureofthevariableX

j

anthenbeobtainedfromX

j+1

;:::;X

n by

X

j

=(b (j 1)

j

X

k >j A

(j 1)

jk X

k )=A

(j 1)

jj

(3)

Wehaveimprovedtheintervalevaluationofthequantitiesappearingin the

shemebytakingintoaountthederivativesoftheelementsofA (0)

;b (0)

with

respetto thepose anddesignparametersandpropagating thembyusing the

derivativesoftheelementsofA (j 1)

toalulatethederivativesoftheelements

ofA (j)

and usethemfortheintervalevaluation. Ourexperimentshaveshown

thatthisleadtoadrastiinreaseintermofthetightnessoftheenlosure.

Note also that this method maybeused to determine what should be the

design parametersso that anywrenh in a set may be produed at anypose

of W while the joint fores/torquesare bounded. By duality the method an

also solve the veloity problems (for bounded joint veloities nd the design

parameterssuh that anyend-eetor twistin agiven set may be realized at

anyposeinW).

4.2 A ritial analysis of the zone alulation

Wehavepresentedin theprevioussetionvariousmethods toomputean ap-

proximationoftheregionZ. Howeveritisnotpossibleto laimthat weguar-

anteeto getanapproximationoftheregionthat inludesallpossiblevaluesof

thedesignparameters,uptothemanufaturingtoleranes,that willsatisfythe

performaneindex. Indeedforomplexperformanesindextheoverestimation

ofintervalarithmetismaybesolargethatonlyforverysmallboxes(i.e. whose

width is lowerthan the manufaturing toleranes) we anguarantee that the

performane index is satised. But the union of suh small boxes, that may

existintheintersetionoftheZ

i

,mayonstituteboxeswhosenal widthmay

belargerthanthemanufaturingtoleranes.

Ourexperienehoweveristhatforrobotisproblemthisisnotthease. But

apossibility to taklethis problem is to assumethat the toleranes aremuh

andtheirintersetionwemaythendereasetheresultbytherealtoleranesto

getasafedesignregion.

4.3 Calulating the intersetion of the Z

As soon as an approximation of the regions Z

i

have been determined as a

setofboxesintheparametersspaealulatingtheirintersetionisalassial

problemofomputationalgeometrywithomplexityO(nlogn)fornboxes. But

alulatingtheintersetionmaybeavoidedinawaythatevenspeed-upthetotal

alulation. Indeedassumethat theregionZ

1

hasbeenomputedfortherst

requirement,leadingto alistofboxesL 1

. Fortheseondrequirementinstead

ofusingP

0

assingleelementofthelistL

P

(andthuslookingforallparameters

that satisfy the seond requirement) we may use L 1

as L

P

, thereby looking

onlyfortheparametersthatsatisesbothrequirements. Proeedingalongthis

line for all requirements will lead to a result that satisfy all requirements. A

drawbakhoweveristhatifoneofthealgorithmfailtoprovidedesignsolutions

(orifwewanttomodifyarequirement)wemayhavetorestartalargepartof

thealulation.

4.4 The algorithm in pratie

AsmentionedpreviouslythealgorithmareimplementedinC++usingBIAS/Profil

forintervalarithmetisand ourownintervalanalysis libraryALIAS 1

that oer

high-level modules that are ombined for implementing the alulationof the

regionZ.

4.5 Choosing the optimal design

Assume now that we havesueeded in omputing theregions forall require-

mentsandthentheirintersetionZ

\

=\Z

i

. Clearlyweannotproposeto the

end-user an innite set of solutions and our purpose is now to propose vari-

ousdesignsolutionswhoserepresentativepointsliein Z

\

(i.e. theysatisfythe

requirements). But arobot presentsvarious performanes, denoted seondary

requirements, that may not be partof the main requirements but whih an

be used to help hoosing the best design. Ideally the presented design solu-

tions should be representative of various ompromises between the seondary

requirements. Unfortunatelythere isnoknownmethodtodealwiththisprob-

lem. Hene we just sample the region Z

\

using a regular grid, ompute the

seondaryrequirementsatthenodesofthegridsandretainthemostrepresen-

tativesolutions.

Note that the algorithms for omputing the region Z may also be used

to verify that a given design(or asmall family of designas, for example, the

familyofrobotswhosedesignparametershavevaluesaroundnominalvaluesand

1

www.inria-sop.fr/oprin/logiiel/ALIAS/ALIAS.html

arequirement,inwhihasetheywillbemuhmorefaster. Usingthisproperty

andaswewill providenally onlyaniteset of designsolutionwemayrelax

therequirementswhen omputingtheregions. Forexamplefor theworkspae

algorithminsteadofspeifyingawhole 6Dregionasdesiredworkspaewemay

speifyonlyanitesetofposes: thiswillallowafasteralulationoftheregion

intheparametersspaeandwewillonlyhavetoverifythat theproposednal

designsolutionindeedinludes thewhole 6workspae.

Similarly it may happen that for a spei requirement an algorithm for

omputingtheregionZ is notavailable. Butassoonasanalgorithm forver-

ifying the requirement for the family of manufatured robots is available our

designmethod maystillbeapplied.

4.6 Appliations

As mentionedpreviously we havedevelopedalgorithms for omputing the re-

gionZ for thefollowingrequirements: workspae,singularitydetetion, au-

ray, veloity and statianalysis. Suh requirements are the most frequently

enounteredforpratialappliations. Thedesignmethodologyhasthenbeen

usedforvariouspratialappliations: designofourownprototypes(forexam-

plethemiro-robotMIPSformedialappliation[12℄),nepositioningdevies

fortheEuropeanSynhrotronRadiationFaility(ESRF)withaloadoverone

tonsandanabsoluteauraybetterthanamirometer [3℄, theCMWmilling

mahineforhigh-speedmanufaturing[21℄. Weare urrentlyusingthis design

methodologyapproahwith AlatelSpae Industry forthedevelopmentofan

innovativedeployablespaetelesope.

Ineahofthese asestheon-the-shelfalgorithmsforalulatingtheregion

Z hasto beadapted to dealwith speiitiesof the appliation (for example

thelargeworkspaefortheCMWmillingmahineimpliesthatwehavetodeal

withpassivejointlimitswhile theESRF one,with areduedworkspae,suh

limitsdonotplayarole). Buttheexibilityofintervalanalysisislargeandhas

allowedustosolvetheproblem.

5 Conlusion

Theproposeddesignmethodologyhasthemainadvantagesofprovidingalarge

panelofdesignsolutionwithaguaranteeonthesatisfationofthemainrequire-

ments,eventakingintoaountmanufaturingtoleranes. Howeveritspratial

implementation needs someexpertise in intervalanalysis for the algorithm to

beeÆient. Aurrentrestritionisthatonlynontime-dependentrequirements

(i.e. requirementsthatarenotsolutionofadierentialequations)maybetaken

intoaount: forexampleweannot dealwithdynamis. Howeverthereisno

theoretialimpossibilitiestodealwiththeserequirementswithintervalanalysis

andthisisaprospetiveforourwork.

verydierentdomains: manufaturing,nepositioning,spaeand medialap-

pliations.

Finallythemethodologyhasbeendevelopedtodealwithrobotsandmeh-

anismsdesignbutmaybeextendedtoproblemsin otherareaaswell.

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