Vibration Control of a Flexible Cartesian Robot: Extension of a Preshaping Input Method

e

Copyrigtb IFAC Motion Control for Intelligent Automation Perugia, Italy, October 27-29, 1992

VIBRATION CONTROL OF A FLEXIBLE CARTESIAN ROBOT: EXTENSION OF A PRESHAPING INPUT METHOD E. D'AMATO·, F. DURANTE· and P. RISSONE

* UniversitA de L'Aquila, Dipartimcnlo di Energetica, I 67040 Roio Poggio, Italy •• UniversitA di Firenze. Dipartimemo di M~nica e Tecnologie Indlistriali, Via Santa Marta. Firenze. Italy

, ABSTRACT

Moreover, the _thod has be,," nll .... rl"ally ."del"r:I to evaluate the senslt.lvlt.y {rohustnessl to tt>-. accuracy of the MOdal pnrll."..ters (MOdal f,·""uencles and structural da.plng).

This paper dpals with the extension of a pr'eshaplng Input method 1.0 the flexibility control of a cartesian robot. The ...,thod Is the original one recently proposed by the authors and already MOdeled and discussed for a polar robot geometry . Besides the above extension, the method has be"n numeri c ally simulal.ed to analyze the behavlor of a one 11 nk robot under different mode I parameters condlt Ions . This acll v\t.y allowed the characterization of the method for what concerning the robustness : the effectiveness of the method under structural damping factors and modal frequencies variations . The observed behavior has been physi~ally interpreted and presented in quanti tat! ve terms.

EXTENSION OF THE METHOD The method here consir:lpred alloW!'; I. he reduction of resldulll oscl ))a\.lons of " f1exibl" link by the application of a sllit.able ac('plp"atlon law to the base . In figure I, the "e Is a sket.ch "r the physical model here consld"red and th~ shape of the driving signal. The model Is referred to ... c"rteslan geolllf>t,·y were th" single lInk I", constrained to translate on t.he horlzont.al plane using a single degr .. e of freedom. The ot.he,· end of the beam Is the end effector' hut payload Is not considered here.

INTRODUCTION The flexibility. control of robot links Is of capital Importance when t.he finite stiffness of robot members has to be taken Into account. This fa~t can't be neglected when the amplitude of vibration of the end effector is unr:leslr'able out of very small val ues, or the stl ffness of the robot links can't be high in appllcat Ions needing light weight structures. The reduction of residual oscillations, assuming a flexlbllit.y of the structure, can be obtained using very sma,· t control techniques (11. (2). These have to est.abllsh t.he Instantaneous end effector position and produce actions to avoid oscillations (closed loop control); It is also pos1'
°1

()

I,

X~i

Jr 1wl' \.

i I

I

tI

I

I

I

I

I

I

('

I

I I

•i'

0

~~

~ \

I

Figure For what concerning the driving signal. It Is double step shaped: the durat Ion Is 11 nked to t "'" period of the first .adal shape, whi If' the _plltude Is co.puted on the basis of the Inet·t1a of the syste•. Here following. the analyt\r.al MOdel will ~ developed, then the nu_rlcal .ode I to verify the effectiveness of the technique and the robustness analysis will be described . AnaI7!lcal ....el We shall now consider the above described syste. whose response tl) base excitation can be

153

D'AMATO E.• DURANTE F.• RISSONE P. exa.inp.d in terms of beam oisplacements In relation to a noninertial or·thognnal reference (w.x). with the X axis tangent to the constrained point (Figure I) . The beam. ch\mped with resped to the reference. is subjected to a distributed force whose expression Is: '(t.x) = - pA let) .: 2

where ;jet) Is the acceleration of the non inertial reference In relation to a fixed one. With ;jet). we can derive the positIon of the non Inertial reference. thereby unlvocally determinIng the motion of all the points of the beam. Considering a base excitation characterized by a constant acceleratlon ;jet) = ;j we must subject t.he beam In the noninertial r'efer'ence to a uniform constant (in time). distributed force as follows: '(x) = - pA;j .: = , (x) = 2

pAa:

(3)

2

N

Sine .. the t.able reveal,; r!lffer' lnR "I.atl" "nr! fIrst IIOda J deformatt nn,;. t h'" "pp II eAU nn ('Of t h" ,;tep forclnR In (3) me"",; e"rJtln~ hlgh"r' vlhr'''! lOll modes in adr!ltlon to t.hp fIrst albeit In a I",;" .. r d"!!ree. Hence, we can now der'lve thp. fordng r .. sponse of a beam sub.l.. cted to the st"I' force In (3). having Inlt.lal conditions '1(0) = o. iJ(O) ~ n. WP. st art from th .. e'lual\on nf mnt\ nn d",scr'l bl nR the flp.xural vIbration for a unIform beam [SI: El a·w/ax· + pA a~w/ax~= f(t,x) wh .. re Young's IIIOdu IllS be"m's section "",_nt of Iner·t\" p MaSS density A cross-section arpa f(t,x) : external force per unit length

F. I

The solution is gIven by a linear combln"Uon of mode shapes and gener'all zed fllnct Ions of t I me :

A clamped-free beam subjected to such force has a static deformation differing only slightly from the first modal shape. I.e. (see Ftgur'e 2). , = C [COSh k x - cos k x 1

1

1

(l

1

co

w(t,x) =[ q,~, ,

(1\)

~ 1

Ry substituting In th .. equation of motion.

(s i nh k x 1

co

r

El s =

co

q

L1 "

I

r

,." + pA

iJ'~

L,

~)

mnfi,,1 "nd

f(t.x)

=

To tr'ansform th"se equ"U on roor·dinates. we multiply bot.h sides

24

get:

I

, - I

~

we

t" toy

Integr 'ate over' beam Ipneth "I" as follnws [7): s

(S)

J> L

..

satisfy modes The normal orthngonal i ty conditions [6) :

-,

I

J')',

dx : 0

)

I

,

J 'I'~" o

position static modal rercenl a 1 ~~?8lhe dlapt ace.enl dlsp I ace.enl error

the Eq .

0 . 000000 0 . 016774 0 . 063871 O. 136483 0.229884 0.339523 0.461135 0.590876 0 . 725477 0.862400 1.000000

0 . 000 0 . 100 0 . 080 0.070 0 . 050 0.040 0.030 0.019 0.011 0 . 005 0 . 000

(S)

J~~~dX 0

are uncoupled. yielding:

,

rlx

o

Considering the same force as In Eq. (3) . we get: f(t.x) : -pAa

{,~

dX)c'J;+[EI

o

{'~?dX)q,=

(3)

-PAa{"

0

fix = F,

(6)

0

The modal shapes for a uniform. clamped-free beam are:

Thus, with a constant acceleration on the constraint, the beam In the reference exhibits static deformation resembling the first modal shape (as visible In the table) and since the shape of the motion of the beam subjected to a step of such force only resembles the first mode, it's possible to apply the method In question. Assigning appropriate values to the following variables (5), we find :

, ,

,

,

; =C [COSh k x - cos k x where (l

,

-4

, [slnh k,x

,

The values of k

a

I

- sin k,X)](7)

cosh k I + cns k I I

,

sinh k I + sin k 1 I

:constralnt acceleration :frequency of the first modal motion T = 2 n/w:osclllatlon period :generallzed time - dependent coordinate x = q describing the beam motion Xl = Xl :posltlon of the relative reference (x,w) with respect to fixed reference Using previous results [5) and assigning a constant acceleration 4 to the constraint point for a free oscillation time T and Immediately afterward a constant acceleration -4 for the S8Jle time, we find that the beam lacks stiff IIOtlon and vibration. and is d~splaced with respect to the fixed reference by aT In time 2T . c

J'If(t.x)

lI,:

Table I

A

''')

I

dx =

--

0.000000 0.018700 0.069866 O. 147670 0 . 243200 0 . 354166 0 . 475200 0 . 602700 0.733866 0 . 866700 1.000000

0

0

Furthermore. since:

The static and modal dlsplacements (normalIzed) are compared In Table i.

0.0 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

following

the

J ,; ,", ox

dx : 0

0

0

Ftgur'e 2

, J " ....,

and

(l

I

are obtained from Table 11 .

By substitutlng Eq. (7) in the Intf'gral on the right hand side of Eq . (S) and Integrating by pnrt.s we find the generalized forces :

W

(8)

k

I

The integral on the left hand side of Eq. (6) Is:

J;~ dx : C: I

o

154

VIBRAnON CONlROL OF A FLEXIBLE CARTESIAN ROBOT: EXTENSION OF A PRESIIAPING INPUT METIlOD We .ay set

the nor_lIzation constant C,

so

that C~ Is I, which In the case of (6), dividing by PAJ

•

~~dX

pA,

o

yields:

a ex

C ,

=f

I

I

where:

J J~:dX

J~;2dX

I

W

2 I

.

El

I

El

_;2dX

0

pA

T

"' F/pA

I

-

MInimurn_gy

~tu\AM VVVVVIII

0

f

WP. find that, "t. 1.. ",,1. f,,.. " ell"pl""~mrfll nr amplitude I, the beam tip's r'esldual vihr·"t.\on given by the s e c ond moel .. has an ampllt\ld .. or 0.00222 at a most. Th"t value Is rpl .. v"nl. to the c".se occurr'lng wh .. n lhe period of oscill"llofl" of the first mode Is a whole odd flumbe. · of I.Im .. " of the second mode half period. The minimum of thp. amplitude ocurrs, on lhe contrary , If It. Is an .. ven "nd whole n\llllber . In fact, with r·efp.ren" e 10 tll", figure 3, In the first case t.he energy st.or·eel hy

-""-

0

I

pA

whe"e : (w 1",,,,< / Ll = 0 . 019

Having deterMined the forces, we can now compute the global response which ' is given by F.q . (4) where the generalized functions ql In the

--

T

1(01

J

1(0 1

FI gure 4

FIgure 3

undaMped case are : f , Iw?I [1 - cos (wit») In this context, knowing the highest values of the gp,,*,rallzed funct Ions Is .are h'portant than knowing the whole expression of the law of motion: q

= 2 f /w

I_x

I

2

t

Frolll Table 11, we can see that the first and the second modes significantly concur with the IIIOtlon, while the contributions of other II\Odal shapes are very small .

,

ex

1. 875 4.694 7.855 10 . 996 14 . 137 17 . 279

0 . 1.341 1. 0185 0 . 9992 1. 0000 0 . 9999 1.0000

mode

1 2 3 4 5 6

k I

,

,

f If

,

1.000 0 . 554 0 . 325 0 . 234 0 . 176 0 . 081

--

1

1.000 0.014 0 . 001 1. 97E-4 5 . 44E-5 I. 12E- 5

Returning to the systeM with the base excitation given at the beginning of this section, the fact that q, for I = 2, ... , _ are different frolll zero .eans that after elimination of both the rigid DIOtion and the elastic deformation of the first DIOde still re_In residual vibration frOM higher modes . In co.parlng t.he residual oscillation with the structure's global dlsplace.ent, we can view the global motion as the bea. tip describing a path of length: T .. 2 I( where : 1

where: ~I (l) .. 2 Cl

=2

W

2n

laking amo ng d

the nearest to <1

,

2

var 'yi ng from

<11

1

Table 11

The highest first-mode the bea. tip Is given by:

Is dlffer'ent from that which minimize lh .. residual nscillation due to the sf!cond mode (d =n(2" l w».

tollt. Increases that of the first one; the contrary happens assuming a time va"ylng from d " to d . As l 2 consp.quence, exists a duration of t.h.. dr' lvlng signal which minimize the sum of the ampl I t.udes and this will be nearest to d because this is refe'Ted

Ql •• x

1. 0000 39 . 280 308 . 02 1182.9 3231 . 6 7212 . 2

to the oscillation period of the flr'st mod .. , Tll,

1 2n to d2nthe ampll tude due to the sec ond mode r'ed""es

q'aax

(k Ik )4 1

the s'ystem vibrating In thp. second mode when t.he the excitation stops Is maximum while In the second rase the same eneq~y Is 7ero ( f I glJre "). Nevertheless, the pr'obablllty for the two ""ses described is very small; moreover, "I"n t.he remaining modes would verify the same rundit.1on, rut this Is occasional. r... ner·ally, the I.lme dllr"tlon of the steps which minimize Ih .. ,·.. sld\lal oscillation relevant with the flr 'st mo d .. (d, equal

to t.he first II\Ode having the biggest. amptlturle . Extending this consideration to the modp. over t.he sp.cond, It can be assumed the exlst.ence of a duration which mlnlml7e the residual global oscillation due to the considered nat.ural modes . For what concerning the damping, It reducps the aJIIplltude with an exponential clecr.. as I ng correction. This Is an undesirable effecl. on th.. first IIIOde because at the stop of the E;'xcltlng thp displacement function has a non z ero value (figurE;' 5); as consequence, thp.re Is a residual osclllat.\on whose 8J\Ipll tude Is grp.at.er when the damp I ng fa c t.or Increases.

Law motion variation with damping factor

1

oscillation val ue

at

Cl '

By considering the expression of ql_x '

2 f q

I_x

z

W

we find:

4 aex

1

2

k

1

2 8 a ex C 1 ....

,

1

w

k

1

W

2

1

1

C

1

W

1

2

1

8 a ex k 1 1

Figure 5

1 2

w

For what concerning the left over modes, t.h", clamping can decrease or Inc r'ease thp. global residual oscillation . In figure 6 Is plot.tf!d the difference betwep.n t.he p"ak-peak rp.sI dlTAI oscillation versus steps duratinn, In damp"d "nd

1

Now, since: w~xlL •

(w1.... /L) 0 . 014, for L =1, w2-.

0 . 00111

155

T)'AMATO E., DURANTE F., RISSONE P. undaape d conditio ns, for a single degree of freedoM oscilla tor under a double st.ep driving action (lIke In figure 1) . It can be seen that In correspo ndence of a step duratio n of a whole nUMbpr of oscilla tion there Is a MaximUM of the anallzed periods , a quantit y ; on the contrary , there Is a minimum for step duratio n of a whole odd number of half oscilla tion periods .

After r:oMputa tlon of the elgp.nve cto!·s hl\"p.d on the followin g express ion

a, [Slnh k1x -

"'1= Cl [COSh k,x - cos k1x -

sin klX)]

.. ar the mot Ion of t he beam can be dp.r I ved as I I n ns combina tion of the In-th.e general lzed functio and the modal shapes

=r1='

w(t,x)

ql"'l

By thp. sUpP.rp osltlon of t.he motions In the two of assumed r'eferen ce systems , the absolut e mol.lon the of r'eslllt The obtalnp d . Is beam the the computa tions Is then post-pro cessed to obtain the Il\w motion of the beam's tip, tt.s zoom, and n of the beaM. visuall~tlon of the In-time vibratio In the flow chl\rt depicte d In flgllrp 7 t.he fwocedu re l'.nd t.he poss I hie resul ts are shown.

I : .. . .

ii iiii'll.

l.._. _l '. i;FO.i piii.~:,i:Ai .

.("T S TFM ' ·IIARM ·fFRI ..... nl s

1

FIgure 6 Numeric al IDOdel The proposp. d numeric al model is based on the solutio n of the followin g uncoupl ed differe ntial equatio ns system

q' (t) I

+ 2

<1

W

q

1 1

2 w q (t) = f

(t) +

1

1

1= 1,2 ...

I

III

n derived from the equati o n of the flexura l vibratio of of a uniform section beam and solved by means the separat ion of variabl es method . It has also the been transfor med In modal coordin ates using to orthogo nallty propert ies and adding the term due the damping . The solutio n under step forcing conditio ns is the followin g [BJ :

!·IJ.II! :~' 71·. 1 rlO .\ '

t) +

cos(w 01

sln(w

dt

dl

tl]

the first Assumin g two referen ce systems , the fixed to the ground and the second attached to to one ' end of the link, that equatio n permits ql (t) dlsplace ments general ized the derive Introdu cing the general ized forces and the phase, f

=1

a a

C 1_1 2 ___

k

'" =

1

atn

[~)

the IIOdal frequen cies

and

structu ral damping The general ized time

w

d

w

1

1t_<2

';..1 ~ 'fI),v !

!EI/' :,Ino,\' I

II ",,,:~:;':,,,,, ";<'1

,. III i /l 'I\I: : i',, "IOIl " !

\,.I...i

L

' Ill) ; - 1. .n

'

t

~("i qj

1

I \\ ( ,\,' ) t .

I

, n't., 1I1 _1 Iono.\ I t/lJ)I1'/f/'"

i

I, ,/

;::.,:,/1,' "

c '

I' -] [-

TlI'

" '(III .tIN'" , 11/" ,ILlZ 11/l1.v

!

.+

-

i I"' "" "",

.l I

/!I "'1. II oMF.\ 7

~

I/q ',lll l ~7JOI\

"

-1r

, nFu/ ....... (:I,n"".. .IIIIIT.I/F,\1

I

1/ ,I",·' ·.I/. / 7 .lno ,\

t

I!

'\\'\ ) , i . ,\ \ 111/ .' r \ \, !, , i I

I

Ill '

.

1

I FI glll'e 7

On the basis of t.hl't flow chat·t. a comput.e r ion pro gram has been devplnp ed and IIsed for simulat s are of an expet' lmental system whose char'ac terlstlc the followln gs beam length : L = 1 m density : p = 7B60

3

kg / m

beam section : A = 2 . 4 10Young' s modulus : E section Inertia : I

w = k 2 j _E_ _I_ P A 1 I

('IJ .I/I'I

-,

fj

I I

!' ,'

+ e-
I l rllF.fl'F\l IFS!

f'IJIU'E

I

I

5

2

m

= 2 . 1E+II = 2 . 88£- 12

[n figure B is globa l displace ment

the plol and the

of the t. 1 p' s beam zoom of residua l

<.

functio ns ql have been obtaine d In the non Inertia l referen ce system dividin g the motion In three followi ng phases : of - the first Is relevan t with the appllca tlon the first step of driving signal whose Initial conditio ns are: ql(O) = 0

m8If .. 0 .9171

"'

ql(O) =0

the second describ es the DOtlon during the applica tion of the second step using as initial conditio ns the ending of the previou s phase; the - the third is the free motion derived by residua l oscilla tion at the end of the second Initial the also, here step; acceler ation conditio ns are assumed accordi ng to the ahovp. criterio n.

2T.=1. 9965

t[5]

FIgure B oscilla tions at thp. end of the dt' \ vlng excitati on;

156

VIBRAnoN CONTROL OF A A1!XIBLE CAR1F.S1AN ROBOT: EX"mNSION OF A PRFSHAPiNG INPUT MEllIOD U",s., r..sults are relevant to the descrIbed syste. assu.lng a step rorclng duration or Tt = 0.998 s and a daJlplng ractor 1;= 0 . 02.

for a ther..

It can be seen that

belUl base displace_nt L of af!= 0.996 • Is a value of the noraal ized peaJc-peaJc

resIdual oscillation aaplitude of 2 . 11 10-

3

•.

ANALYSI S The described nu_rlcal .odel has been used to perfor. dlffer..nt analyses of t.he _chanlcal syste. Introducing variations of the paraaeters or the dyna.lc ldent lficat Ion. In this context Is also possl ble to search the val ues of dlUlplng and the driving action duration which .Inl.ize the residual oscillation. Moreover, It Is possible to evaluate the Influence or the nu.ber of the considered vibration aodes or the bea•. This robustness analysis Is referred to the _chanlcal syste. d ..scribed In the nu_rical .ode I section assualng, .oreover, the IUlplltude of the

I.. the tota I nu.ber of "''"'" I d",· .. ri rood .. ~;) fro. ,hP v"l...., 1 to 6 uslnll! for thfo step" thfo t I _ ri ... ·""on which .lnl.l7.e the ".,sidual osclllaUon. 'lh~se values have been derlvp~ for. IhP analysis of lable Ill. and as In that tabl .. , thP _xl.u. no"roall7ed of the oscil lation ha", been co.puted ror difre"ent da.plng factors .

~

AOIIUSJlI[SS

2

driving step equal to 1 a/s . In table III are the values of the noraallzed peaJc-peaJc oscillation 3

aaplltude (.ultlplled by 10 U 4U

aax

)

- U

_In

L

where

2

L = aT displacement at the base; T duration of the steps by assuaing the variation of lhe damping factor froN 0 . 00 to 0 . 03 and the duration of the excitation fro. 0.950 to 1.050 s being that of the first ...,de equal 1.0 0.998 s. In this table, the mi nlmuN amp 11 tude of the residual oscillations Is pointed out for various damping factors.

~ 0.950 0 . 955 0 . 960 0 . 965 0.970 0.975 0 . 980 0 . 983 0.984 0.985 0 . 990 0 . 991 0 . 992 0.993 0 . 994 0.995 1.000 1. 005 1. 010 1. 015 1.020 1. 025 1.030 1.035 1.040 1.045 1.050

0 . 00

0.01

0 . 02

0 . 03

8.2309 6.4506 4.9759 3 . 8385 3.0259 2.4914 2 . 2123 2 . 1627 [2. 1619'j 2.1683 2.3072

7 . 6568 6 . 1412 4 . 8161 3.7009 2.8788 2.2977 1.8958

8.0173 6.5779 5.3703 4 . 3850 3.5723 2.8979 2.3519

8 . 4832 7 . 1821 6.0287 5.1242 4 . 3804 3 . 7254 3.2910

2.5939 2.9991 3 . 4259 3.9736 4.6244 5 . 3828 6.2590 7.2238 3 . 2474 9 . 3361 10.454 11.592

2 . 9503 2.0242 1. 7178 1.8778 2.6121 1. 6040 2.5960 1. 5864 [!.57@ 1. 8186 j2 . 581;o 1. 6043 [1.81@ 2.6478 1. 8652 2.7738 1. 9387 1. 7132 3 . 0374 2 . 2424 1.9943 3.2216 2 . 4498 2.3191 3.4890 2 . 7738 2.6166 3 . 8208 3 . 0639 3. 1330 4.0938 3.6773 3.5211 4.6161 4. 1833 4.3904 5 . 1559 4 . 8699 5 . 2050 5.9038 5.5731 6 . 0442 7 . 2687 6.2975 6.8814 7.5645 7 . 0726 7 . 8319 8.4424 8.0620 8.9045 Table III

As already discussed and physically Interpreted, It can be seen that for a zero daaplng the t l _ duration of the excitation which .Inl.lze the r..sldual osci Ilation is co.prlsed between the t l _ given by a whole nu.ber (6 In thIs case) of periods of the second aode (. 1592 • 6 = . 955 s) and the period of the first .ode (.998 s). As the daapl ng factor I ncreases there I s a fI rst benefl t due to a good co.binatlon of the second and next aedes while for daaplng factors over 0.03, the bad Influence on the first .ode aaJce neglectable the effect of the other aodes . In tabl., IV Is shown the Influence of IIOde truncatlon. It has been developed adding .odes (n

0.00

I

2 3 4 5 6

0 . 01

O. 02

0 . 03

_._- t--.

~.

0 . 6356 1.94?O 2.1592 2 . 1591 2.1591 2 . 1619

0 . 3901 1. 53?9 1. 5713 1 . 5708 I. 5700 1. 5701

I. 0Sl91 I. 8244 I. R041

1. R101 I. 8100 1. 8109

2. ?lOO

2.6082 2 . 5R36 2.5R12 2 . 5812 2 . 5812

Table IV It can be noted lhat the effectivenpss or the _t.hod Is I.portant also in a conl.lmY>us sysle. having an I nrlnl te nuat...r of d ... grees or rreedo., being ll.lted to the second .ode the inrluence on t.he ,aaplltllde of the residual oscillations. Moreover, that Influence Is less i.portanl when the syste. Is da.ped at the real values of the applications .

cotD.USIONS The application or an original ('onl.,·ol strategy to a cartesl"n robot has been desrl'ibed . It Is based on a procedure recently developed by the authors for a po I ar robot geomelry which hM' e has been IIIOdeled In analytical and nu_rl c al t.er .... . The numerical IIOdel has allowed to perro"m analy,;es of the system under different physical configurations and different driving law parameters . It has been demonstrated the efre (, liveness of the method for the c ontrol of a distrlbuled parameter system also assuming Important e'Tors on the modal parameters Identification. ACIOIOIILEDGKENTS This work was supported by the Consigllo Na7.lonale delle Rlcerche (CNRl under' lhe Prog... tto Flnallzzato Robotlca . REFERENCES III P . H. Meckl and W. P . Seering, "Reducing Rf>sinll"l Vibration In Systems with Ti..,.-Varylng Resonance" . ProCE' .. cll ngs of 1987 I FEE International Conference on Robotics and Auto_tlon, pp. 1690-1695. Ralelgh, Ha,-ch 1987 . [2) P.lI . Meckl and W.P.Seerlng, "Controlling Veloclty-Lillited Sy,;te.... to Reduce Residual Vibration". Proceedings of 1988 IF.F.E International Conference on Robol.ics and Auto_lion, pp . 1428-1433. Philadelphia, Apr-ll

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P. H. Meckl and W. P . Seeri ng, "Exp<>ri men! "I Evaluation of Shaped Inputs to Reduce Vibration for a Cartesian Robot". Trans. of ASHF:, .Iournal of DynaNlc Syst eas , Measure_nt, and Contro I. Vol. 112, June 1990, p. 159-165. (4) S.P. Bhat H. Tanaka and D. K.Hiu "Experio.ents on Point-to-Point Position Control of a Flexlhle Seaa Using l.aplace Transfor. Technique - Part I: Open-Loop" . Trans. of ASHE, Jot ... na I of Dynaalc Systeas, Measure_nt, and Control. Vol. 113, Septe.ber 1991, pp . 432-437 . (5) E. D' A_to and F. Durante and P . Rlssone, " A Preshaplng Input Method ror the Vi brat.! on Control of flexible Robot Links". International Conference on Control and Robotics . Vancouver, CANADA, August 1992. (6) E . D'A_to and P. Dl Gregorlo and P.Rlssone, "DlnaMlc aodellng of a flexible robot .anlpulator" . International Conference on Control and Robotics . Vancouver, CANADA. August

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