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MechanismandMachineTheory35( )1287±1298
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The static balancing of the industrial robot arms
Part I: Discrete balancing
IonSimionescu*,LiviuCiupitu
MechanicalEngineeringDepartment,POLITEHNICAUniversityofBucharest,SplaiulIndependentei313,RO-77206,
Bucharest6,Romania
Received2October1998;accepted19May1999
Abstract
The paper presents some new constructional solutions for the balancing of the weight forces of the
industrial robot arms, using the elastic forces of the helical springs. For the balancing of the weight
forces of the vertical and horizontal arms, many alternatives are shown. Finally, the results of solving a
numericalexamplearepresented.7 ElsevierScienceLtd.Allrightsreserved.
Keywords:Industrialrobot;Staticbalancing;Discretebalancing
1.Introduction
The mechanisms of manipulators and industrial robots constitute a special category of
mechanical systems, characterised by big mass elements that move in a vertical plane, with
relatively slow speeds. For this reason the weight forces have a high share in the category of
resistance that the driving system must overcome. The problem of balancing the weight forces
is extremely important for the play-back programmable robots, where the human operator
mustdriveeasilythemechanicalsystemduringthetrainingperiod.
Generally, the balancing of the weight forces of the industrial robot arms results in the
decrease of the driving power. The frictional forces that occur in the bearings are not taken
*Correspondingauthor.
E-mailaddress:(I.Simionescu).
0094-114X/00/$-seefrontmatter7 ElsevierScienceLtd.Allrightsreserved.
PII: S0094-114X(99)00067-1
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I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
into consideration because the frictional moment senses depend on the relative movement
senses.
In this work, some possibilities of balancing of the weight forces by the elastic forces of the
cylindricalhelicalspringswithstraightcharacteristicsareanalysed.
This balancing can be made discretely, for a ®nite number of work ®eld positions, or in
continuous mode for all positions throughout the work ®eld. Consequently, the discrete
systemsrealisedonlyanapproximativelybalancingofthearm.
The use of counterweights is not considered since they involve the increase of moving
masses,overallsize,inertiaandthestressesofthecomponents.
2.Thebalancingoftheweightforceofarotatinglinkaroundahorizontal®xedaxis
There are several possibilities of balancing the weight forces of the manipulator and robot
armsbymeansofthehelicalspringelasticforces.
The simple solutions are not always applicable. Sometimes an approximate solution is
preferred,leadingtoaconvenientalternativefromconstructionalpointofview.
The simplest balancing possibility of the weight force of a link 1 (the horizontal robot arm,
for example) which rotates around a horizontal ®xed axis is schematically shown in Fig. 1. A
helical spring 2, joined between a point A of the link and a ®xed B one, is used. The equation
thatexpressestheequilibriumoftheforcesmoments[1],whichacttothelink1 ,is
wheretheelasticforceofthehelicalspringis:
Á
m1OG1cosji?m2AXA g?Fsa?0, i?1,...,6,
?1?
Fs?F0?k?AB l0?,
and
Fig.1
I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
1289
a? XBYA XAYB;
AB
X
Y
x
y
cosji sinj
A
1A
i ;
?Rji
; Rji ?
sinji
cosji
A
1A
q
BG2
2
2
B
AB? ?X X ? ??Y Y ? ; m2A?
m2:
A
B
A
AB
ThegravitycentreG2 ofspring2iscollinearwithpairscentresAandB.
The sti?ness coecient of the spring is denoted by k, m1 is the mass of the link 1, m2 is the
massofthehelicalspring2,andgrepresentsthegravityaccelerationmagnitude.
Thus,theunknownfactors:x1A,y1A,XB,YB,F0 andkmaybecalculatedinsuchawaythat
theequilibriumoftheforcesisobtainedforsixdistinctvaluesoftheangleji:Themovableco-
ordinate axis system x1Oy1 attached to the arm 1 was chosen so that the gravity centre G1 is
upontheOx1 axis.Theco-ordinatesx1A andy1A de®nedthepositionofpointAofthearm1.
in®niteIn thenumberparticularofsolutions,case, characterisedwhichverifytheby equation:y1A?XB?l0?F0?0, the problem allows an
?m1OG1?m2Ax1A?g,
k?
x Y
1A
B
foranyvalueofanglej:
Since in this case, Fs?kAB (see line 1, Fig. 2),some diculties arise in the constructionof
this system where it is not possible to use a helical extension spring. The compression spring,
which has to correspond to the calculated feature, must be prevented against buckling.
Consequently, the friction forces that appear in the guides make the training operation more
dicult.
Eveninthegeneralcase,wheny1A 0?andXB 0?,resultsareducedvalueoftheinitiallength
l0 of the spring, corresponding to the forces F ?0: The modi®cation of the straight
0
characteristic position to the necessary spring for balancing (line 2, Fig. 2), i.e. to obtain an
acceptableinitiallengthl0 fromtheconstructionalpointofview,maybeachievedbyreplacing
the ®xed point B of spring articulation by a movable one. In other words, the spring will be
articulatedwithitsBendofamovablelink2,whosepositiondependsonthatofthearm1.
Link 2 may have a rotational motion around a ®xed axis, a plane-parallel or a translational
one,anditisdrivenbymeansofanintermediarykinematicschain(Figs.3±5).
FurtherpossibilitiesareshowninRefs.[2±7].
Fig.2
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I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
Fig.3.Balancingelasticsystemwithfourbarmechanism.
Fig.3showsakinematicsschemainwhichlink2isjoinedwiththeframeatpointC,andit
is driven by means of the connecting rod 3 from the robot arm 1. The balancing of the forces
systemthatactsonthearm1isexpressedbythefollowingequation:
Á
fi? m1OG1cosji?m4AXA g?Fs?YAcosyi XAsinyi??R31XYE R31YXE?0,
i?1,...,12,
?2?
Y YA
BG
4
B
where:yi?arctanX X ;m4A?AB m4;m4B?m4 m4A;
B
A
X
Y
x
X
X
Y
BC
0
cosci sinc
sinci cosci
E
1E
B
C
i
?Rji
;
?
?Rci
; Rci ?
:
y
Y
B
E
1E
C
Thecomponentsofthereactionforcebetweentheconnectingrod3andthearm1,ontheaxes
of®xedco-ordinatesystem,are:
Á
T?XD XE??m3 XD XG3 ?XC XE?g
R31X
? YD?X X ? Y ?X X ? Y ?X XD?
;
C
E
C
D
E
E
C
Á
R31X?YE YD? m3 XG XD g
?
3
,
R31Y
X X
D
E
where:
h
Â
Ã
Á
Á
T?Fs ?XB XC?sinyi ?YB YC?cosyi ? m2 XG XC ?m3 XG XC ?m4B?XB
2
3
i
XC? g,
I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
1291
X
Y
X
Y
x
y
X
Y
X
Y
x
y
D
C
2D
G2
C
2G2
?
?Rci
;
?
?Rci
;
D
C
2D
G
C
2G
2
2
X
X
Y
x
y
G
C
3G
3
?
?R
3
,
xi
Y
G
C
3G
3
3
cosxi sinx
sinxi cosxi
i
Rxi ?
:
Thevalueofangleci:
p
c ?arctan U U2?V2 W2 VW
p
representsthesolutionoftheequation:
a
i
V U2?V2 W2 UW
Á
Á
Ucos ci?a ?Vsin ci?a ?W?0,
where:
U?2CD?XC XE?; V?2CD?YE YC?;
W?OE2?CD2?OC2 DE2 2?XEXC?YEYC?;
y
a?arctanx2D:
2D
Similartothepreviouscase,theangleoftheconnectingrod3is:
Á
xi?arccosCDcos ci?a ?XC XE
DE
ThedistancesOG1 andBG4,andtheco-ordinates:x2G2,y2G2,XG3,YG3 givethepositionsofthe
masscentresoflinks1,4,3and2 ,respectively.
The unknowns of the problem: x1A, y1A, x1E, y1E, x2D, y2D, XC, YC, ED, BC, F0 and k are
foundbysolvingthesystemmadeupthroughreiteratedwritingoftheequilibriumequation(2)
for 12 distinct values of the position angle ji of the robot arm 1, which are contained in the
work ®eld. The masses mj, j?1,...,4, of the elements and the positions of the mass centres
are assumed as known. The static equilibrium of the robot arm is accurately realised in those
12 positions according to angles ji, i?1,...,12 only. Due to continuity reasons, the
unbalancingvalueisnegligiblebetweenthesepositions.
Infact,theproblemissolvedin aniterativemanner,becauseatthebeginningofthedesign,
themassesofthehelicalspringandlinks2and3areunknown.
The maximum magnitude of the unbalanced moment is inverse proportional to the number
ofunknownsofthebalancingsystem.Byassemblingthetwohelicalspringsinparallelbetween
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I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
Fig.4.Elasticsystemwithslider-crankmechanismI.
arm 1 and link 2, the balancing accuracy is increased, since 18 distinct values of angle ji may
beimposedwithinthesamework®eld.
In Fig. 4, another possibility for the static balancing of a link that rotates around a
horizontal ®xed axis is shown. The point B belongs to slide 2 which slides along a ®xed
straight line and is driven by means of the connecting rod 3 by the robot arm 1. The system,
formedbyfollowingequilibriumequations:
Á
fi? m1OG1cosji?m4AXA g?Fs?YAcosy XAsiny? R13XYE?R13YXE?0,
i?1,...,11,
?3?
where
R13X? Â
Ã
cosci;
?m2?m3?m4B?gsina Fscos?y a? DE m3gDG
3sina
Á
DEcos a c
i
Â
Ã
m3gDG3cosacosci ?m2?m3?m4B?gsina Fscos?y a? DEsinc
R13Y
?
Á
i;
DEcos a c
i
c ?a?arcsinXEsina YEcosa b e;
DE
i
XB?esina??Si?d?cosa; YB??Si?d?sina ecosa,
aresolvedwithrespecttotheunknowns:x1A,y1A,x1D,y1D,CD,d,b,e,a,F0 andk.
ThedisplacementSi ofthesliderhasthevalue:
I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
1293
Fig.5.Elasticsystemwithslider-crankmechanismII.
Si? XE?DEcosci ?b?e?sina
cosa
p
, if a ?,
2
or
Si? YE?DEsinci??b?e?cosa, if a 0?:
sina
If the work ®eld is symmetrical with respect to the vertical axis OY, the balancing
mechanismhasaparticularshape,characterisedbyy1A?y1D?b?e?0,anda?p= 2[5].
Thenumberoftheunknownsdecreasedtosix,butthebalancingaccuracyishigher,because
itispossibletoconsiderthatthepositionanglesji verifytheequality:
ji?6?p ji, i?1,...,6:
?4?
Likewise, the balancing helical spring 4 can be joined to the connecting rod 3 at point B
(Fig.5).Eq.(3)wherethecomponentsofthereactionforcebetweenthearm1andlink3are:
? Â
?Â
Ã
?m2?m3?m4B?gsina?Fscos?y a? cosci
R13X
Á
cos a c
i
Ã
Â
Ã
m3?XG3 XD??m4B?XB XD? g?Fs ?XB XD?siny ?YB YD?cosy
Á
sina;
DEcos a c
i
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I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
Ã
? Â
?m2?m3?m4B?gsina?Fscos?y a? sinci
R13Y
Á
cos a c
i
Â
Ã
Â
Ã
cosa;
m3?XG3 XD??m4B?XB XD? g?Fs ?XB XD?siny ?YB YD?cosy
Á
DEcos a c
i
XEsina YEcosa e,
X
Y
X
Y
x
y
B
D
3B
?
?Rci
, ci?a?arcsin
DE
B
D
3B
issolvedwithrespecttotheunknowns:x1A,y1A,x1D,y1D,x3B,y3B,CD,e,a,F0 andk.
Fig. 6 shows another variant for the balancing system. The B end of the helical spring 4 is
joinedtotheconnectingrod3whichhasaplane-parallelmovement.Thefollowingunknowns:
x1A,y1A,x1E,y1E,x3B,y3B,XC,YC,d,F0 andkarefoundassolutionsofthesystemmadeup
ofequilibriumequation(3),where:
? Usinci V?XE XC?; R13Y? V?YC YE? Ucosc
R13X
i;
W
W
and:
h
Â
Ã
Á
Á
U?Fs ?XB XC?siny ?YB YC?cosy ? m2 XG XC ?m3 XG XC ?m4B?XB
2
3
i
XC? g;
Á
V?Fscos ci y ?m3gsinci;
Fig.6.Balancingelasticsystemwithoscillating-slidermechanism.
I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
1295
W??YC YE?sinci??XC XE?cosci;
c ?arctanYC YE arcsin d
;
i
X X
CE
C
E
q
2
2
E
CE? ?X X ? ??Y Y ? :
C
E
C
In the same manner as the constructive solution shown in Fig. 4, the balancing accuracy is
higher, if the work ®eld is symmetrical with respect to the vertical OY axis ?y1A?y1E?y3B
?
d?X ?0?[5],becausethepositionanglesj verifytheequality(4).
C
i
Fig.7.Balancingelasticsystemsforverticalandhorizontalrobotarms.
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I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
3.Thestaticbalancingoftheweightforcesoffourbarlinkageelements
The static balancing of a vertical arm of a robot presents some particularities, considering
that it bears the horizontal arm. For this reason, most of the robot manufacturers use a
parallelogram mechanism as a vertical arm (Fig. 7). Therefore, the link 3 has a circular
translational movement. At point K is joined the elastic system that is used for balancing the
weight of the horizontal robot arm. For balancing of the weight forces of the four-bar linkage
elements,anyoneoftheconstructivesolutionsmentionedabovecanbeused.Forexample,the
elastic system schematised in Fig. 3 is considered. The unknown dimensions of the elastic
systemarefoundbysimultaneouslysolvingthefollowingequations:
m2dYG
dYC
dYG4
dt
dYG5
dt
dYG6
dt
dt ??m3?m8?m9?m10?m11? dt
2
?m4
?m5
?m6
dt g?Fs dt
?
m7 dYI dYJ
dIJ
? 2
dt
?0,
?5?
whicharewrittenfor12distinctvaluesofthepositionanglej2i oftheverticalarm.
These equations result from applying on the virtual power principle to force system which
acts on the linkage. The equality (5) is valid when the horizontal arm does not rotate around
the axis of pair C, and consequently the velocity of the gravity centre of the ensemble formed
by the elements 3, 8, 9, 10 and 11 is equal to the velocity of point C. The masses of the links
andthepositionsofthegravitycentresaresupposedtobeknown.
Eq.(5)maybesubstitutedbyEq.(6),ifitisassumedthatdj2=dt?1:
m2dYG
dYC
dYG
dYG
dYG
2
??m3?m8?m9?m10?m11?
dj2 ?m4 dj ?m5 dj ?m6 dj
4
5
6
dj2
2
2
2
g?F
m7 dYI dYJ
dIJ
sdj2
? 2 dj2 ?dj2
?0,
?6?
where:
q
2
2
J
Fs?F0?
?X X ? ??Y Y ? l0 k;
I
J
I
YG ?x2G2sinj2i?y2G2cosj2i;
2
YG ?x4G4sinj2i?y4G4cosj2i;
4
YG ?YF?x5G5sinj5i?y5G5cosj5i;
5
I.Simionescu,L.Ciupitu/MechanismandMachineTheory35( )1287±1298
1297
YG ?YH?x6G6sinj6i?y6G6cosj6i;
6
YI?YH?x6Isinj6i?y6Icosj6i;
YJ?x2Jsinj2i?y2Jcosj2i;
XF?x2Fcosj2i y2Fsinj2i;
YF?x2Fsinj2i?y2Fcosj2i;
YC?BCsinj2i;
p
VW?U U2?V2 W2
j5i?arctan
p;
UW V U2?V2 W2
U?2FG?XF XH?; V?2FG?Y YH?;
F
W?GH2 FG2 ?XF XH? ? YF YH?2;
2
p
j6i?arctanST R R2?S2 T2
p;
RT S R2?S2 T2
R?2GH?XH XF?; S?2GH?YH YF?;
T?FG2 GH2 ?XF XH? ? YF YH?2:
2
Theunknownsoftheproblemare:
. thelengthsFG
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