1993 a Sequential Integration Method for Inverse Dynamic Analysis of Flexible Link Manipulators

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A Sequential Integration Method for Inverse Dynamic

Analysis of Flexible Link Manipulators

F. Xi R. G . Fenton

AbstractIn thas paper a sequentaa l zn tegra tzon method 2s pro-

posed f o r t h e inverse dynamac ana lyszs of flexzhle l z i i ki n a n z p u la t o r s . D u e t o li n k d e fl e c t i o ns , t h c Ll i ie ina l icand dyiiair izc equa tzons of f lexib le l inki na i i i pu la lo r 5 a r ccoupled . Based o n the p roposed n ie ihod , u ih ile solr-z i i g t h e d y n a m z c e q u a t z o n f or l ink def lec i io i i s erplic-a l ly, jo in t m o t z o n s c a n be zniplicat ly ohtat ired w i t h t h ehe lp of t h e kzne ina tzcs of flexable lznk n i n i , t p n l a f o r s

T h e n s u b s t i tu t i o n o f Ih e co inpuled j o i i i i i i i o ~ l o i i ~ r i i dh k e j lec lzo i i s i n to t h e d y i i a i i i i c equa l?o i if o r I l i t j o i n t

to rques l ends t o t h e solutioii of t h e prohleirr of i h f I I I -verse dyna inzcs The proposed m e t h o d I S c o i i c e p l u a l l ya n d conipu ta tzo i ia l ly s t ra igh t forward , c o i i i p a i c d io ! l i ~o t h e r m e t h o d s , and z t s e ffec l iveness 1s de inor r s t r a t ed bya n U n erz c n l exa np le

1 IntroductionIt is advantageous to use flexible link manipula-

t,ors in terms of higher opera.tion speed, great,er rat . ioof payload-to-weight,, lower power requirement, iiiorocompact struct ure, etc. However, d ue to l i n l t flcxihil-it.y, flexible link mani pula t,or s might. untlergo \:il)rat o i i sduring operation. Est.eiisive researcli h a s h ~ i i ( J I I P II

recent years on reducing t.liese viliratioiih, ii h s ~ i i i i i i i i i -

rized in tlre paper by 13001; ( 1 9 9 0 ) . Oii r of t h e iiii1)or-taiit probleiiis i n this research is th e inverse dyi ia rn icsanalysis as it provides t lie comput,ed joint t.orqiies fori n ani pu la t.or control .

The inverse dyiiamics of a flexible l i n k ~i~aii ipulat ,ormay be stated as: for a given traject.ory of a. flexiblelink manipul at.or, it is required to tleterniiiie t.lie join ttorques required to drive t he manipula.t,or alon g it.s de-sired trajectory. A complete iiiodel for t.he inverse cl),-iia.mic analysis consists of tlie kinematic and dyiiaiiiicequations of a flexible link ma.iiipulator (Xi 1 9 9 2 ) . 1 3 -note by 0 he joint variables and by 11 t.he clastic l i i i l t

deflect,ions. Th e ltinema.t,ic equations can he sy i i i lml i -cally expressed a s follows. The displaceincnt fyua tioi iis

D = D ( O , U ) ( 1 )

tlie velocity equation is

the acceleration equation is

D = D ( 0 , 6 , 0 , 1, 11, ti) ( 3 )

Th e clyiiamic equat,ioiis ca i i lie expressed as

T = T ( O , 0 , 0 , U , i, i i ) ( 4 )

where T = [ r , IT ; an t l T rttpresciit,s t , l ic joint. t,orqucs.

It is worth noting t,lia.t. luc, to (.lie l i i i l i dcflect.ious U

a n d t.lieir derivatives ii, U , t.lre kinematic a n d dynainiceqiiat,ioiis of a flexible l i l i l i manipulator are coupled.Convent.ionally, tlie iiiverse dyiiiuiiics of a. flexible liiikiiianipulat.or is simplified by dec oupl ing these two setsof equat.ioiis based on t,lie concept of nominal jo int mo-t,ioiis. These joint motioiis are det ermined using t,he1;iiiemat.ics fo r the rigid link counterpart of a flexiblel ink manipulator, neglecting the influence of th e l inktleflec ions on th e manip ulato r Itiiicnia t.ics.

The inethods puli l ishecl ii i the literature t o solve thisprol)lem ma y lie categorized i i r t . 0 t.Iiiwfgroups. 111 t.lrefirst gro~11). lir joiiit torques, T , a t r t l the link cleflec-1,ioiis. 1 1 . ar c s i i n ~ r l t ~ i r t c ~ o u ~ l yI i ~ l ~ ( ~ r i i i i i i ( ~ ( 1y i t (>ra t w l yw l v i i i g t l i e t l y r i a i i i i c q u a t oiih c,ori,c,sl~oil(liiig t~ 1 I i riioiiii1i;iI joint iiiotioris (Hayo P(. a l 1<)89)> A s a t l a et

al 1 9 9 0 ) . 111 t.lie second group, thr dynaiiiic equationsar e siiiiplifiecl by neglecting tlie influelice of th e deflec-Lioiis of th e precediiig links OI I tlic kineniat,ics of (liesucceediiig linlts (Chang a n d I-Ianiiltoii 1990). As il re-s u l t , the tlynamic equations ar e linearized with respectt.o t.he linli deflec tion s. 111 tlie t,liird group, tlie linkcleflect~ioiis a r e split. i1it.o tw o c o m p o n e n t s : on e clup toslow motion ancl the other clue to fast iiiot,ion (Sicil-iaiio ancl Book 1988). Tlie foriner is the tleformatioiicausetl Iiy t.he manipulator's payload a n t l tho iwigIit,s oflililis an t1 a.ct.uat,ors. Tlie latt,ei, s t,hc vil)rat,ion excitcdh y t.lie t,iiiie-varying forces act.ing oii t h e linl;s duringinanipulator motioii. Pfeiffer i i i i c l C;c:l)lcr ( 1988) pro-postd a nirt I io t l t.o det,eriiiiiie t l i e corrwt,ioii values oft l i t : joint, tlisplacemeiits for coinpensating for tlre m a -nipulator's flexible deviati on caused by tlie link tlefor-mat,ions. This method is Inset1 011 t l ~ e y ii am ic equa.-tioiis liy iieglect.ing th e iimtia.1 forces, iianic:ly consid-er ing quasi-sta.ttic ta.slts. L a t e r , Si a n d Feiitoii (I991a)showed t.liat, t.lie correct,ioi i va. lues of t,he joiiit. displace-n i en t , s for qiiasi-stat,ic t.axlts ca i i actually l i e obtainc.tl

1050-4729/93 3.00 0 1993 EEE143

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by the inverse kinematic analysis of flexible link nia-nipulators.

In this paper, a sequent,ial integrat ion me thod isproposed to analyze the inverse dynamics by consid-ering the complete model of a flexible link manipula-tor. While solving explicitly the dynamic equation for

the link deflections, th e joint motions can be implicitlyobtained with the help of the kinematics for a flexiblelink manipulator. Substitution of the computed jointmotions and link deflections into the dynamic equa-tion for the joint torques leads t o th e solution for theinverse dyn amics of the flexible link mani pulato r. Th ismethod is conceptually and computationally straight-forward, compared t o the other methods.

2 Kinematics of Flexible Link Manipu-lators

A flexible link is modeled here as cla.mped a t, th ehub and free at the tip. Based on a truncated modalapproximation (Book 1984 ), the link deRect.ions ca.n heexpressed as

where matrices, * an d 9 , epresent t,he modal eigeti-functions of the link deflections and t,he t,ime-varyingamplit udes of the modes of the link deflect,ions, respec-tively. a ca.n be predetermined according t.o the l i n kgeometry an d sh ape by cla.ssic approximat.ioii iiietliotls.such as the Rayleigh-Ritz met.liod. I i o w e v t ~ r. 1ia.s 1.0

be determined based on the dynamic equations ancl i tis used in the paper to indicate U , a s U is virt,ually afunction of 9 .

Based on the kinema.tic forinulation (X i and Fent.on1992a), the displacement equat.ion of a flexible link ina-nipulator with respect to the ba.se frame as shown i nFig. 1 can be expressed as the conibinat,ion of tha t . ofthe rigid link counte rpart of the flexible link manipula-tor, D, , and the manipulator's flexible devia.t,ion, D J ,caused by the link deflections rehtive t,o the rigid l i d icounterpart,. Th at is

U = * @ (5 )

By taki ng t he t ime deriva.tive of t,lie clisplacenient,equation twice, the nmnipuhtor's velocity and acceler-ation equa,tion ca.n be derived a.s

D = j O + j i ( 7 )

i j = J O + j i ; j + j & + , j & ( 8 )an d

where J is the Jacobian associa.ted with the joint, mo-tions and J is the Ja.cobian associa.t.ed wit,li t h c l i n l i d +flections. J is a. function of the joitit. variables a i i t l t l iclink deflections, i.e. J ( 0 , a ) , nd i t ca n Iw t . s p r c ~ s s ~ ~ las a combination of rigid a n d flexi1,le coiiipoiient,s as

( 9 )

Obviously, J, is a special ca.se of 5 when t.he I i n I i de-flections are negligiI>le. 3 is a function of the joint

J = 5,. + J,

variables only, i.e. J ( 0 ) . he detailed expressions forJ an d J are given i n (Xi 1992).

Since Ij..is linear i n terms of th e joint. velocit.ies 6and so is D in terms of the joint a.ccelera.tions, 6 , fJacobian J is invertible, the joi nt velocity an d acceler-

ation equations can be expressed as

0 = j - I ( D - j&) (10)

an d

6 = j - I ( i j - - j& - j& ) ( 11 )

If J is not square, 5- ' is repiacecl by J + , th e general-ized inverse.

3 Dynamics of Flexible Link Manipula-tors

A dynamic equat,ion simi1a.r to that developed hyBook (1984) is used here a n d it . is expressed as

Mil+ C i l + W = T (12)

where q is t.he vector of t,he gcneralizcd coordinatcs;M is t,lie inert.ia matrix; C is thc, iiiatrix coiit,ainiiig(.'oriolis, cent,rifugal arid gyrobi'ic Torcr.s; i l l l ( l W is t , l i c

I 1 la t r x coli .a I I i I ig gra v i t a t.io 1 a forces a id t he for cc'sd u e t.o the interaction of l i n k tlcflec(.iona. T h e y aregiveti as

q = [O , a y

w = [WO. W J T ( 16 )

E q n . ( 1 2 ) can he split into t h e following two p a r t s

T = ~ 0 0 6 M O + & + cO~O+ C O + & + W O (17 )

E q n . (17) is t h e dynamic equation for obt,ainitig t.liejoint, torques an d e q n . (18) is t h e d y n a m i c equationfor solving t,liP l i n l i deflections.

4 The Sequential Integration MethodA scciiitwtial iiit,egiatioii i i i e t . l roc1 is proposed Iicr(: to

solve [,lie prol)lem of' th e inverse dynaiiiics of flexiblelirik niatiipulators by using the Iii~1et~la.ti~ ii d dynamic

oquat.ions presented i n t.lie previous sections. The pro-cedure i n a y b e sta.ted as : first the joint inotions andl i 11 defect oiis a.re determined a n d the i s U lis t i tu t ionof these results into th e dyna mic equation for the jointt,orques leads to the solut.ion for the iiiversp dyna.niics.

744




4.1 Determinat ion of the Link Deflections

Recall that the joint a.ccelerations O a.re a funct.ionof the second derivative of the link deflect.ioiis a!, a.sshown in eqn. (11). Subst itut ion of eqn. (11) ititoeqn. (18) yields a second order non1inea.r tlifferent.ialequation for the link deflections

@= [M,sJ - ' J - M,b]- ' [M+eJ- ' (D - J6 - h )

+ c,e6 +c,& +W,]

i = H ( G ~ & G ~ OE)

(19)

(20)

which may be expressed in a compact forin as

By virtue of the stat.e spa.ce, eqn . ( 2 0 ) c a n lw r r -duced to a first order different.ial equat.ion as

& = G @ & G o 0 + E (2.5)

where

an d

& = [a!, i ] T ( 2 6 )

0 = [ O , b]T ( 2 7 )

G , = [HG, , 0IT ( 2 8 )

E = [HE, 0IT

Obviously, & is a function of & a.nd 0 ; .e . h = f ( 6 ,

( 2 5 ) 6 a.re t,he st.ate vect,ors to b e deter-mined. 0 may be considered as t.he int.rinsic parame-ters which can be related to the st,at*e ect,ors t81iroughthe displacement and velocity equat,ion of a flexible linlimanipulator. For given init,ia.l cond iti ons of t,lie l i n k tle-flections &', eqn. ( 2 5 ) can be integra.tetl by a nuinl>erof metho ds. In this paper the R.unge-1l;utt.a met.liodis used. Denote by h. the integral tim e int,erva.l, th efourth order solution at th e bt,h cotnput.a.tion can bepresented as

0 ) .In eqn.

S k & k - l + (K : + 2 K i + 2K: + K:i)h/(i ( 3 1 )

where

Kk-

( 6 k - 1 , @ - ' )K 2 - ( a k - 1 + K:h/2, 0 k - l )

K! = f ( Q k - ' + K $ h / 2 , ak - ' )

( 3 2 )( 3 3 )

( 3 4 )

K: = f ( a k - ' K i h . O k - ' ) ( 3 5 )

It is worth noting from th e last, four equat,iotis thatthe link deflections of the curreiit computation arefunctions of the link deflections and the joint mqtionsobtained from the previous computa.t.ion, i.e. O k -4 .2 Determinat ion of the Joint Mot ions

O k [ak , kITcan be now determined using thekinematic relation between t he joint motions and thelink deflections described before. Fo r determining thejoint displacements O k t, the bt h comput.ation, an in-finitesimal relation can be given hy linearizing t.he ina-nipu lator 's displacement ecluat.ion as

( & k - l , O k - l 1.

AD^ = J ( o k - 1 a k - 1 ) a o . " + . j ( o . " - - ' ) ~ @ k 36)

where( 37 )

(33)

AD E = ~k - k - 1

AOk = a! k - ak-1

Then El k can lie determined by

a k= o k - - ' + J ( a k - l @ k - l ) - 1 [ 1 ~ k _ j ( a t - l ) 4 a k ]

(39)In view of eqn. ( 1 0 ) .~3~ a ii ol>t.iiiiic~cI > y

0" = *l((-P, ! k ) - ' [ D k - ;((-I.")@."] ( 4 0 )

I t is worth not.ing from t.lie foregoing t.wo equationstha t . the joint motions a t t,he current computation arefunct.ions of t h e l ink deflections at the current and pre-vious coinputa.t.ion as well as of the joint inotions at tlie

4 .3 Determinat ion of the Joint Torques

Th e joint torques, r k , .t the Iith computation canhe rea.dily determined by subst,it,ut.ing 6' an d G k ntoeqn. ( 1 7 ) , i.e. t.he dynaniic equat.ion for the joiiit.torques. Since 6 k s 0111 , c ~ I a t . t ~ I.0 .Iic prvvioiis C O I I I -

put.at.ion, t. is first drt.eriiiiiietl at. t . he ktlr computation,t,lien G k s oht.ained a n d f ina l ly T." s acquired. A p p a r-ent, ly the problem of t,he inverse dynaniics of a flexiblelink tna.nipulat,or is solved i n a sequent,ia.l fashion andt h i s t,he proposed inet,liotl is ca.lletl a sequential inte-gration method.

4 .4 Description of Algor i thm

Based 011 the foregoing analyses, tlie a.lgorithm oftlie sequential integration met.hod caii tie present .ed asfol lo\vs:

,\ ' ttp I : clefiniiig t.1ie nia.nipuIat.or t.ra.jectory. D , D , Da i i t ~ tie iriitial contlitions of t ~ i coint rriotioiis, 0 0 . iiti

t,Iiose of t . I r e I i t i I i tI(:flect,ioiis: &'.,;, S k p 2: select,irig the t,inie inkgral interval, h , ac-cording t,o t.he required accuracy a n d discretizing themanipulator's t.rajectory i n terms of time segments,t" = b h ! b = 1 , .. , n,) , n t,he duration of manipula-t.or opelat ion, T = t n = n h ;

previous coniput,ation, i.e. O k- (6k, k - 1 1 k - l .

745




Step 3: computigg the link deflections, 6'. l l view ofeqn. (25) using and 0';

Step 4 : computing the joint displaceinelits. 8 ' y eqn .(39) and th e joint velocities by eqn. (40) using 6'. 4 'an d 6' ;Step 5: computing the joint torques, T " ? y eqn. (17)

using 6' an d 0 ' ;Repeating-step 3 to compute ( f k , repeating step 4 tocompute O k and repea.ting step 5 to comput,e T ~ ,nt, i lL = n . As a r esu lt, th e link deflections, t he joint, 1110-tions an d finally the joint torques can be determinedin sequence.

On e of the iinportaiit problems pertaining t o the im-pleinent.ation of the proposed algorithm is sta.l>ilit,y. Bythe nature of the algorithm presented here, the stabil-ity inay be categorized a s kinematic and the dynamicstabi l i ty. The former is associated with t,lie singiilar-ities occurring i n the inversion of .Jacobian. J , a i t d i tw a s discussed previously (Xi antl Fenton 1992l>) .Tlielatter is meant to be the st.abilit,y of a nonlinear tly-namic sys tern. Due to t,he liniit.ed space io t , l i e p a p e r,this problem wil l be discussed separa.t,ely

5 SiiiiulatioiiTo illustrate the proposed method, a flexible ma-

nipul ator wit,li t.wo pa.ralle1 revolute joint,s is sirnulat~etlfor the inverse dynamic analysis. Tlie tw o lilllis of t . 1 1 ~manipulator are of tlie same length. l ( m ) , and of t liesame uniform cross-section. Tlie elastic. const .a t i t . s i in t lthe moments of inertria of t he two linlis ar c ident.icol.E = 2 x 10'O(iVn1-~)), = x I U - ~ ( ( ~ U " )rilf! iotioiiis constrained i n th e yo-z, plane. as sliown i n Fig.1. Accordingly, t,he manipula.tor motion a n d t,lie l i nkdeflections are considered in this plane.

Jacobians J an d J are first derived a n d Iiased onthem the derivatives can be readily obtained. To sho\vthe main theme of the proposed inet.hod, only thetransverse bending deflections in th e x,-yo plane a r e

considered and they are a.ssumed to 1ia.i.e a mode oforder one fo r each link, tha t is

J l obt,aiiied by Xi and Fenton (19911~) .s present.eclhere for completeness. Matrix 3,. is t.Iie conventionalJacobian for the rigid "upu la to r an t l i t is given as

( 4 2 )

where ci , si represent cosOi, s i n O i , respect,ivelF, c , ] , s , j

indicate cos(8i + Oj) , sin(Oi + O j ) , respectively; a n d I

is the length of the linlis. Matrix, . J , , is t.Iw Jacohiaiiassociated with the link deflections ant1 i t is

TIE comI)iiiat.ion of Jr a11t1 , , forills .J . ~ a c o ~ j i a n , ,is derived as

(44)

T h e coefficient matri ces of tlie dynami c equation can

be obtained by using a symbo1ica.l computation pack-age, such as Macsyina (Xi 1992) .

In view of the algoritlit n described in Sec. 4.4, acomputational code w a s writtsen i n Fort ran 77 incor-porating a subrou tine of the Runge-I iutta method inthe numerical computa.tion pa.cliage I M S L . The initialconfigurations of the manipulat~or s given as

[O;'. O:] = [*15". J(J"] (4.5)

Th e initial joint velocit,ies, th e initial l i i r k deflectionsand tlreir derivat,ives ar e al l zero. Th e manipula.tor'sclisplacement, is given a s

D = [ ~ . s ( T ) , ] (46)

where /I = 1(7/$); = f / T ; iid A ( T ) is defined as

s ( t ) = ~ O T ~~ T " (jr5 ( 4 7 )

where S ( T )a n d S ( T )can h e readily derived as

T I I ~na.xiniuiii vpIoci t ,y along (Iirc;c.tioii xo . fir , , i d . r =1 5 6 / S 7 ' = 0 . 9 3 i 5 ( , ? / A ) . occurring at . r = 1 /2 - is a.bout.I lie t o p s l e w ve1ocit.y of iiitlust,rial inaltipulat80rs (Leea n d \ \ 'ang 1088). I n other wortls, the nia.nipulator con-sidered here is operat.ed a t a. fast speed a.nd the effectof t.he l i i i l i flexil>ilit,y on it.s dyn ami c be1ia.vior could besignificant. I n comput,at.ion, h is selected as 0.033s andT is given as 2 s . Accordingly n = G O . In this particu-l a r case. tlie execution time of t.he computation is 0 . 2 s ,wli ic l i is rea.sona.bly eficient. Tlie simula.t.ion resultsare report,ed i n Figs. 2 t,o 5 .

Tlre results of t . l ie j o i n t tlisl)la.c.eiiic-1lt.s ar e plott.edi n Figs. 2 ( a ) a n d ( b ) for joii1t.s 1 a n d 2 , respectjvely.I t, cat1 lie seen that. botli joint. tIisplacement.s oscillatearoi i i id t .liose of t.he rigid c.ortiitt,rl)art. This indicat,esthe e ff ec t of t.llc, l i n k tleflc,ct,ions oii t.hp .joint motions.I'ig. 3 sliotvs t,lie restr1t.s of t,he l i n k tleflect.iotis. I t canlw oliserved t h a t . tlie l i n k tlefiect,ions v a I y and deviatewi t , l i t.ime. The variation is due to change i n the dy-namic forces during the manipulator's operation. The

74 6




deviation is du e to change of the effect, of t,lie gr av ih -tion force on the links. Th e simulat,ion rrsu1t.s of th ejoint torques are plot,ted in Fig. 4 .

Th e pos ition error of th e manipula.t.or is clet.erminedby considering the difference between the tlesirable 110-

sition and the actual position. The a c t . t l i l l posit,ioii iscalculated by using the computed joint displacements

and link deflections. T h e result is given i n Vig. 5 andthe error range is within [-0.03, 0.031 ( n im) .

Moreover, it m a y be observed from Fig. 3 that tlirlink defledions still exists even after the joiiit motmioilster min ate. Thi s is t,lie residual respon se, which how-ever will event,ually va.nish du e to damping effects. T h eduration of the residual response is the sett.ling time.which is one of the performance indices for flexible l i n kmanipulators .

6 C o ~ i c l u s ~ o n sI t is sliown i n this p a p e r t.liat (,lie joiiil tiiolioiis of

flexible link "iip ula tor s are affect,ed by t lit, l i l l l i (I(>-flections. This effect, 1ia.s t.o be t,alien ii1t.o co1isid(,riit.iotifor the inverse d y n a m i c s analysis of flc.sil)l~~ i n k i i i c i -

nipulators. For t,Iiis purpose, a sequeiitial iiitegrat ioii

method is presenbed in the paper wliicli is straiglit.for-ward and effective i n the sense that it.vrat.ioiis a r c i iot

required. Based on this method , t .he problem of t h cinverse dynatnics can lie solved sequeiit.iallj., first. d c > -termining the l i n k deflections. tlieii t h e join1 motionsand finally the joint torques.

7 References

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Bayo, B . , Papadopoulos, P. , St~ubbe. J. a i i c l S e r n a ,M . A . , 1989. Inverse Dynarnics a n d I\inwiat,icsof Multi-Link Elastic Rob0t.s: A n 1terat.ive: Fre-quency Doimin Approach. I t t f I . J . Robol. K c s . ,Vol. 8, No. 6, pp. 49-62.

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DE-Vol. 25 , pp . 75-81, Chicago.Lee, J.D . arid Wa.ng, B., 1988. Opt,imal Control of R

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1993 a Sequential Integration Method for Inverse Dynamic Analysis of Flexible Link Manipulators

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