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Displacement analysis of the RCRCR five-link spatial mechanism
Jnl. Mechanisms Volume 6, pp. 119-134/Pergamon Press 1971/Printed in Great Britain
Displacement Analysis of The RCRCR Five-Link Spatial Mechanism* Mark S. C. Yuan~ Received 30 March 1970 Abstract Using the method of line coordinates, the input-output displacement equation of the RCRCR five-link spatial mechanism is obtained as a fourth-order algebraic equation. For each set of the input and output angles obtained from the equation, all other variable parameters of the mechanism are uniquely determined. A numerical example is presented, and the accuracy of each set of solutions is verified. Zusammenfassung- Bewegungsanalyse des f0nfgliedrigen r¨ichen RCRCRGetriebes: S. C. Yuan. Mittels der Methoden der Linienkoordinaten wird for die Beziehung zwischen der Eingangs- und der Ausgangsgr6sse des f0nfgliedrigen r¨ichen RCRCRGetriebes eine algebraische Gleichung vierten Grades abgeleitet. Mit jedem Paar von Eingangs- und Ausgangswinkeln, die aus der Gleichung gewonnen werden, sind alle 0brigen ver&nderlichen Parameter des Getriebes eindeutig bestimmt. Ein numerisches Beispiel wird vorgef0hrt und die Genauigkeit aller L6sungspaare wird 0berpr0ft. P e 3 m M e - AHa.r1143 l-lepe~etueHH~l I-IpocTpaHCTBeHHOFO l-lflXH3BeHtlOrO MexaHn3xla T u n a BFIBFIB: MapK C. IOan l-Io~lb3y~tcb Mero/lo,~t nHHefiHbtX KOOpi1aHar, nonyqeHo aJlre6paHaecKoe ypaaHeHHe HeTBepToa cxeneHH a.an nepeMemeHH~ B e a y ~ e r o 14 ae~oMoro 3aeHbeB npocTpaHCTBeHHOrO naTH3aeHHoro ,x~exaHH3Ma rHna BFIBFIB. ~Jln Ka~zloro 14a60pa yF.rlOB noBopoxa Be/lyulero H BeI1OMOFO 3BCHbeB, Ho,qyHCHHblX 143 ypaaHeHHn, CnHHCTBeHHOonpe~eneHbt Bce/apyrHe nepeMeHHble napaMeTpb[ MexaHH3Ma. 1-IpHBeaeH H14CJ'ICHHbI~npHMep H noaxBep~lleHa TOttHOCTb Ka}K~OFOHa60pa peuaeHH~.
Introduction FOR THE displacement analysis of spatial mechanisms, there are basically two approaches, i.e., the numerical approach and the algebraic approach; each one has advantages, and both are highly useful. Examples of the numerical approach are: the vector method of Chace[2, 3], the tensor method of Ho[7], and iterative method of Uicker, Denavit, and Hartenberg[9], etc. This paper is concerned with the algebraic approach. * Based on the doctoral dissertation of the a u t h o r in partial fulfillment of the requirements for the degree of Doctor of Engineering Science in the School of Engineering and Applied Science. Columbia University, N.Y., 10027, U.S.A. ~Senior Engineer, T h e Singer C o m p a n y , Industrial Product Division. Elizabeth, N.J. 07206, U.S.A. Presently. Staff Engineer, Remington Rand Office M a c h i n e s Division. Sperry Rand Corporation, Blue Bell, Pa. 19422. U.S.A.
119
120
The displacement of the R C C C * four-link spatial mechanism has been investigated by numerous researchers, e.g.. Dimentberg[5, 6] bv the method of dual vectors. Denavit[4] by dual matrices, Yang[13] and Yang and Freudenstein[14] by dual quaternions, etc. Dimentberg also investigated the displacements of the RSSR tour-link and the R R C C R and R C R C R five-link spatial mechanisms. For each of the five-link spatial mechanisms Dimentberg obtained an input-output displacement equation in a determinant form which could be reduced to an algebraic equation of eighth-order. The investigation, however, was not carried further, and no numerical example was obtained. Later on, Yang[15] obtained a fourth-order algebraic equation for the R C R C R five-link spatial mechanism by the method of dual matrices. The intermediate variable parameters of the mechanism were also solved, but only the input-output displacement curves were presented in a numerical example. Gt.pta (the discussor of Yang's paper) however obtained three different solutions for the numerical example, and, consequently, raised a question concerning the order of the input-output displacement equation. For other types of five-link spatial mechanisms, Wallace[10l and Wallace and F r e u d e n s t e i n [ l l ] obtained fourth-order algebraic equations l'or the input-output displacement equations of the R R S R R and R R E R R mechanisms by the method of geometric-configurations. Dimentberg's eighth-order solution for the R R C C R mechanism was confirmed by the author[16, 17] by the method of line coordinates. In this paper the solution of the R C R C R mechanism by the same method is presented. The displacement equation of the mechanism is obtained as a fourth-order algebraic equation which confirms Yang's solution. The coefficients of the equation are obtained explicitly in terms of the constant parameters of the mechanism. Moreover, for each set of the input and output angles obtained from the equation, all other variable parameters of the mechanism are uniquely determined, and a method of verification is presented. A computer program is developed for the purpose, and a complete solution of Yang's numerical example is obtained. The solution shows that most of Gupta's results are correct. Line C o o r d i n a t e s
The position of a point in space can be uniquely defined by the ratios of four coordinates ,r, 3,, z, t. and. conversely, these ratios are uniquely determined by the point, The tour coordinates are called the homogeneous coordinates of the point. Only three of the coordinates are independent since (,v, 3', :., t) and A(,v,>', :, :) determine the same point. Let (&,),,, :.~. t~), (.r._..)'e, z,_,,t._,) be the homogeneous coordinates of two distinct points P~, P,,, respectively. Define the six n u m b e r s ; L j = .v~t.,_ - - . r ~ t ~ ,
L., = y~t._,-- y.,t~,
L) = 3'~z:-
l,a = z ~ . v e - zz.r~.
l,:: = ~ q t , - - 2._,tj.
(3) Y,_,::~.
I,,; = -r~ 3', - . v . , 3 ' z .
The ratios of the six numbers uniquely determine the line joining the two points, and, conversely, the ratios of the six numbers are uniquely determined by the line. In line * R . C . S . E d e n o t e revolute, cylindrical, spherical, and p h m a r pairs respective[~. ~-See M a x w e l l [8]. pp. 6 5 - 7 3 . and W o o and F r e u d e n s t e i n [ 12I.
121
geometry, the six numbers are referred to as the coordinates of the line, or line coordinates. Only four of the six numbers are independent because (L~. L~, L:,, L4. L.~. L0 and X(L,. L~. L:,, L4, Ls, Ln) determine the same line. and they satisfy the relation
(2)
LtL4 + L:L5 + L~L~ = 0. The line coordinates are said to be normalized if LI ~+ L._,"+ L f = 1.
(3)
In this case the number of independent coordinates is increased to five. General line coordinates can be expressed as a multiple of normalized line coordinates. If the homogeneous coordinates of the two points. P1. P._,. are expressed as (x~, Yr, z t , - 1). (x~, y..,, z e , - 1), then (xt, yr. zt), (xz. y.,., z.,_) become the Cartesian coordinates of Pt, P._,. respectively. The first three numbers of the line coordinates, (Lz, L._,, L:0, then become the Cartesian components of the vector PtP._,, and the last three numbers. (L~, L.~, L~), become those of the moment vector of PtP~ about the origin, i.e., OPt x PtP~. Therefore, the line coordinates (L,. Lz, L:~. L,. L~. L6) can be interpreted as the line vector* (L1, L._,,L3) + •(L4, L.~. L,;), and vice versa. In this expression, E is an algebraic unit having the property (-' = 0.* Normalized line coordinates correspond to a unit line vector.
Transformation of Line Coordinates If the Cartesian frame of reference is transformed from x - y - z that
to x t - y ~ - z t
such
OO1 = (tl, t.,, t:~),
(4)
ry,xl]--|r,,l-r"r.,_,q" rI:'Ii -x] r._,,q/y l,
(5)
LZ,J
Lr:~t r:r_, r:~3JLZ_l
the line coordinates of L with reference to x t - Y l - zl can be obtained from those of L with reference to x - - y - - z. Consider L as a unit line vector L ---- (L,, L.,, L3) + E(L4, L~, L6), with reference to x - - y - z . line v e c t o r s 3
(6)
Transform L in accordance with the transformation law of
*See Brand[I]. pp, 5 5 . 5 6 . 6 3 - 6 7 . -~See Brand [ 1]. pp. 63-67.
122
IL,I=Lr , r.:.: r: n r:,.: raa
=
{I r
L:,,
i -- 0t:l 0,
L
t.2 - - t t
t~
L.,
0
L
re i
r.,.2
r2a
r
o oo 1 ~-L l-' 0
0
0
r::l
r::.,
1.:~:~
0
0
0
~ r , : d . _ , - - r v , t:,, flit: ~ - rlatl
rv:t,-
rid.,
rtl
rL.: r,:,q]
I r:l:lL_, - - r:l-_,t:l r31t:,, -- r3:l/'t
1.a._,[i
1.:ilL,
r:l I r:;._, 1.:1:'
L_, i L:~ I (7) L:,I
LL,d where (L)~ denotes the line coordinates of L with reference to x~ -Yz - :.~. By combining the primary and dual parts. (L)l = T1L.
(8)
in which. --
Ti =
r,I
r12
rt:~
0
0
0
r.e T
r.,.:
r_,:~ r:~a
0 0
0 0
0 0
r:lt
ra.e
Fl:IL2--1"I.212:~
/'ll[3--
7
i i
/'[;it I 1 [ 2 [ 1 - - 1 " 1 1 [ 2 111 /'12 rl; ~ [
(9)
/"_:;L2 - - r,,£~ r.,t r.,., 1..,., I . . r.:tt:,-. . . r.,_at~ . . .r.,.,lt . .--1..,iL, . . ra:d._, - - r:~.eta r:nta - - r;~atl r:r_,tl - - r:ut., ral r:r,_ 1.aa
Equation (8) is the transformation equation of line coordinates for the finite transformation of the Cartesian frame of reference. Dual Angles and Line Coordinates T h e scalar product of two unit line vectors is equal to the cosine of the dual angle between the lines.* Therefore, for two unit line vectors L and M, [(L~,L._,.La)+e(L4, L:,,L6)]" [ ( M , . M . , . M a ) + e ( M 4 , M~ M~)] = cos 0 - e z s i n
0,
(10)
where 0 is the angle, and z is the c o m m o n perpendicular distance between the lines. Hence. in terms of line coordinates. LtS'l~ + L,,M.. + LaMa = cos O,
(1t)
L~M4 + L._,M~+ LaMs + L4M~ + L~M.2 + L~5la = -- z sin 0.
(12)
Line Coordinates of Axes of Mechanism Pairs T h e R C R C R five-link spatial mechanism is schematically shown in Fig. 1. Link 5 is fixed. Links 1 and 4 are the input and output links respectively. S,, Se, Sa, $4. Sa are the axes of the five m e c h a n i s m pairs whose line coordinates are St = ( S , ~ . Sj._,. S j a . Sj4, Sa.~. Si~),j = 1 to 5.
*See Brand[l],
pp. 68-70.
123 -S3
Link 3 ~
z ~
~
~4
Link 2
23
z2
a45 Link
Yl
~5 0151
- Link I
~5
_s5 ,
~"
s1
Figure 1. The RCRCR five-link spatial mechanism. with reference to an arbitrary Cartesian frame. T h e parameters of the m e c h a n i s m are: zl = c o m m o n perpendicular distance between a~ and at.o, measured from as~ to at..,, and positive in the direction of St; 0t = angle between a~ and at.,; at., = c o m m o n perpendicular distance between St and $2; at-. = angle between St and S_o; and similarly for other parameters. All angles are measured according tb a right-handedscrew convention. T h e input and output parameters are 0t and 05 respectively. O t h e r variable p a r a m e t e r s are 0.,, z~, 03, 04, and z4. T h e line coordinates of S~ can be expressed as functions of the mechanism parameters in the following manner. xj
;'x
Figure 2.
/--L,nk ,
124
Let two sets of c a r t e s i a n c o o r d i n a t e axes x - v i - . z i , x ~ - y j - . z j be a t t a c h e d to L i n k s k. i. r e s p e c t i v e l y , as s h o w n in Fig. 2. T h e line c o o r d i n a t e s of S: with r e f e r e n c e to .v~ - Ya - zj are
(Sj)j = (0. O, l, O, O. 0).
(13)
T o o b t a i n the line c o o r d i n a t e s of Sj with r e f e r e n c e to .r~ - y~ - .~. c o n s i d e r the transf o r m a t i o n of .r0 - )'j - za to .vi0- Yu - zu. followed b y a n o t h e r t r a n s f o r m a t i o n to .r~ - v, - c.. In the first step. by e q u a t i o n s (8) a n d (9). V
1 0 i ! 0 (S~)i~=i 0
0 % ,% 0
l
0
- - ~lijSij
0
L_
0 - s,o c'~o 0
0 0 0 I
- - (dijCi j 0
0 0 0 0
0 0 0 0
(S~)~.
(14)
Cij - - Sij
(lijCij - - {1i)S o 0 Sij
('ij
w h e r e (So)u d e n o t e s the line c o o r d i n a t e s of Sj with r e f e r e n c e to . r 0 - ) ' ~ j - . h i . and c,: = cos - u . su = sin a~j. Similarly. in the s e c o n d step. c, si 0
(S,),. =
-s, 0 0 <'~ 0 0 0 1 0
- - Ci3"i ~.iCi
0
0 0 0 0 0 0
- - ~.i('i
0
C i -- S i 0
--"~iSi
0
5' i
Ci 0
0
0
0
0
( S~)~;.
(15)
1
w h e r e (S~)~ d e n o t e s the line c o o r d i n a t e s of S~ with r e f e r e n c e to . v ~ - y i - c . ~ . a n d c'~ = cos 0i, si = sin O, S u b s t i t u t i n g e q u a t i o n (14) into e q u a t i o n (15). a n d m u l t i p l y i n g out the m a t r i c e s , the line c o o r d i n a t e s of Sj with r e f e r e n c e to . r i - ) ' ~ - zi are o b t a i n e d as (S~)i = Tu(Sjb.
(16)
where c~
s, rij
~-
-- s i c u
sis~
0
0
tic u
-- c'is o
0
0
0
0
0
0
0
su
cu
- - ~iSi
( a i j S i S i j - - ~iCiCij )
( (.lijSiCij AU ~.iciaij )
~,iCi
( - - a i j c i s i j - - .~,isicij)
0
auci5
('i - - SiCtj
( - - ~ i j c i c i j -JF Zisisij ) S i - - aiJSij
0
0
SiSij
CiC 0 - - CiSij Sij
Cij
In the s a m e m a n n e r the line c o o r d i n a t e s of Si with r e f e r e n c e to x , - y ~ - z j o b t a i n e d as
(Si)j=
Tji(Si
i.
(17)
c a n be
(18)
125
where
i
c~ - - SiCij
Z;i
=
si
0
0
CiCij
Sij
0
SiSij
-- cisij
- - ZiSi
ZiCi
[ ( a i j s i s i j - - Zicicij)
(-- aifl.'isij-- ZiSiCo)
cij
0
0
Ci
0 0 0 si
(loc 0 - - SiCij
[_ (ausicu + Zicisu) (-- aucicu+ zisiso) -- ausij
0 0 0 0
(19)
CiCo Si t
SiSi j - - CiSi j Cij
Note that Tu is the inverse of Tji. i.e.. TuTji = I, the identity matrix. Input-Output
Displacement
Equation
The input-output displacement equation of the mechanism is obtained in the following manner: Referring to Fig. 1, with reference to the Cartesian frame xt - y t Zt attached to the fixed link, the line coordinates of So are expressed as functions of 01. and those of $4 as functions of 0s. Next, with reference to the Cartesian frame xa-y:t-z:~ attached to Link 2. the line coordinates of S..,and $4 are expressed as functions of 0a. Since the scalar product of two line vectors is an invariant with respect to the transformation of coordinate axes, the scalar product of S., and $4 with reference to Xl - Y t - - q is equated to that of S, and $4 with reference to x3-ya--z:~. This yields a dual equation which involves all the parameters of the mechanism except 0.,, z.,, 04, z4. Finally, the dual equation is separated into two real equations, and 0a is eliminated between the two equations; this results in the input-output displacement equation. With xt - y ~ - z z as the frame of reference. -
-
S, = (0, 0, 1,0, 0, 0).
(20)
By equations (16) and (17), S._, = TtzSt =
( S I S I 2 , - - C I S I 2 , C12, Z I C I S I 2 "q- a r 2 S t C v 2 , Z I S I S I 2 - - a l 2 C t C l 2 "
--a,..,s,.,).
(21)
By equations (18) and (19), 54
TIzTz4St ES45S5, $45C5C51 -]- C45S51, - - $45C5S51 71- C45C51, $45Z5C5 ~ ~45C45S5,
- - $45 (a51c5s51 + Z5S5C51 ) + c4.5a51c51 + a45c45cscst - - a45s45s51 , S45 ( - - azlCr, Cs! + Z5S5S51 ) - - c45a51s5 t - - a45c45c5s51 - - a45s45c51 ].
(22)
Similarly, with xa -- Ya -- Z3 as the frame of reference, $3 = (0,0, 1 , 0 , 0 , 0 ) ,
(23)
$4 = Ta4Sa (S3S34 , - - C3S34, C34, Z3C3S34 -{- a 3 4 s 3 c 3 4 , Z3S3S34 - - a34C3C34 , - - a34S3.1) ,
(24)
126 S.,_ = Ta._,S:~
= (0, S, a, C.,_:,, O. a_,:,(,_:~, --a_,as.,_:~).
25)
By equating the scalar product of equations (21) and (22) to that of equations (24) and (25). a dual equation can be obtained. The primary part of the equation is - seacaaa4 - c..,aCa4 =
s45Y,5stsL2
--
{, $45C5C51 -~ (_'45S51 )('iSle
& ( - - s45cJ51 + c45ca~ )c~2:
(26)
and the dual part is $2:; (Z,:;S:~S:,t - - a 3 4 c 3 c 3 4
) -- c23a34s34
- - (.123(.23c3s3.1 - - a 2 3 s 2 3 c : l
= s ~ : , s : , ( z < , s : _ + a:_s~c~.,) + (s~::,c:,~ + c~sa~ ) (zlst.s,~ - a,._,c~c~2) - - C45C5l )(/12Sl2 +
-~- ( $ 4 5 C 5 S 5 t
($45Z5C 3 + (J.45ClsSs,)SI,S'I2
-~- [ $45 ( (151 C5S51 -L S5S5C51 ) __ C45,~151 C51 - - a 4 5 C 4 5 C 5 C 5 1 -~- &t45s45s51 ] C l S l 2
+ [S45(--a.-,icacal+Z..asas:,t)
--c4aa:,ls.-,,--a4ac4:,cas:,l--a45S45Cal]Cv_,.
(27)
These two equations can be rearranged to C::c:~ = A isa + B,c.-, - - C I .
(28)
C~sa + Ca{'a = Aes:, + B.c.-, - - C.,_.
(29)
Ca = - se:~s:.,
t30)
A l = s~:,sr_,s~.
(31)
B~ = - s,:,ca~.s'r,_ct - - sc, s:,tc'r,,
(32)
C~ = cc, s:,:v_,Cl - cc, c:,tc:_ + c2::c:H,
(33)
C4 = s.,:~z:::l,.
(34)
C a = - - $2:1gl34C34 - - C/.23C2:1S34,
(3 5 )
Ae =
($45(/12C12 "At-{145C45S12)S1 -}-,~'t5S12 ( ~,1 "~- ~,5C5l) Cl ÷ $45"~5S51('12"
(36)
Be =
$45-Y12 (CsIZ1 -~- ~,5)SI -~- (--$45C51Cl12Cl2 q - $45(151S51512 - - /l-t5C45('51SI2) C1
where
(37)
- ~ $45S51LI12S12 - - $45C/51 C51C12 - - a45('.15S5 IC12 ,
C 2 =
- - c45.'Q~1.7..is12s I -}- ( C 4 5 S 5 1 L l 1 2 C 1 2 @- C 1 5 C 5 1 a l 2 S I 2
-}- C45(151C51S12 - - a45-~'U)S513"I2 )(--'t
-4- C45(.151S51CI2 -~- a 4 5 S 4 5 C 5 l C 1 2
- - ~123.~'2;;(7:Ii - - C 2 3 a 3 4 S 3 4 .
(38)
From equations (28) and (29), c:~ = ( A : : , +
Blca-
C,)/C>
.% = [ (A ~sa + B t or, - - (7~) Ca - - C a ( A .,_& + B.,ca - - C., ) ] / -
(391 C:~C ~.
(40)
127
Eliminate 05 between equations (39) and (40). The result is Vs.,,"- + W s s c s + Xs~, + Yc~, + Z = O.
(41)
V = (Ca" + Cs'-)DII + CaO-D._,,_-- 2CaC-,,Dr_,,
(42)
W = (Ca 2 + Cs")E11 + C3"-E.,.., - 2C3CsEI:,
(43)
X = - (Ca" + C5") F i t - C3"-F.,..,.+ 2C.~C:,Ft..,,
(44)
Y = - (Ca" + Cs'-) G it - C3"G.,_.,. + 2C3C:,G r,.
(45)
Z = (Ca" + C~,")Htz + C3"H.,_.,_ - 2C3CsHv,. - C3"-C4".
(46)
where
in which Di~ = A i A ~ - - BiB~, Eij = A~Bj + BIAs, Fo = AiCj+ CiAj,
(47)
Gi.i = B i C j + CiB~, Hi~ = BiBj + CiCi.
By substituting x = tan (OJ2), or s~ = (Zr)/(l +x~), c.~= ( l - - x " - ) / ( l + x " - ) , equation (41 ), a fourth-order polynomial
into
4
Z P,x" = 0
(48)
0
is obtained as the input-output displacement equation. The coefficients of the polynomial are Po = Y + Z ,
P~ = 2 ( W + X ) , Pe = 4 V + 2Z,
(49)
P:~ = 2 ( - W + X ) , Pa
=
--
Y+Z.
For a given input angle, 01, a maximum of four output angles. 0.~, may be obtained, since it is the maximum number of real roots of equation (48). Other Variable Parameters of the M e c h a n i s m For each set of 0t and Os obtained from the input-output displacement equation, 03 is uniquely determined by equations (39) and (40). To compute 0._,and z,_,for each set of 0~, 0:3, and 05, equate the expression of $4. with x ~ - y ~ - z, as the frame of reference, obtained by equation (16) to that of St obtained by equation (18): Tt:~ T2a Ta4Sx = TIs T~S1. JM Vol. 6. No. I - - I
(50)
128 Premultiply both sides by T.,_, T.,_:;Ta4SI =
T.q T I s T a 4 S
I
{51}
since T._,~Tv,_ = 1. Let T.,,Tt~ = T._,,~, and multiply out the last two matrices of both sides of equation (5 1). $3S3 4 -- C'33a4 [ T.-a ('34 ] = T215 C1:;4S3C34 ~ ~3C3S34 i -- tl:~4C3C34 ~ ~,3S3S3] ~ -- a34S34
0 s4s i i
C45
(52)
i" I
a ~5C4.5 i CZ~3S ~3~
Multiply out both sides, and equate the c o r r e s p o n d i n g elements of { 1, 1), (2, I). (4, 1). (c.,.acasa4 + s.,aca4) s., + sas:]4c.: = (T.,_~a) p,s ~5 + (T._qa) t3c ~.
(53)
SaSa4S.,_ -- {c.,_ac,as:,4 + s,,.:,ca4)c., = ( T.,,~)._,..,s45+ ( T,,ta).,:]c~:,,
{54)
(-- S.,SaSa4 -t- C=,C.:.~C3Sa4 -+- C._,S.,_.3Ca4)7.._ q- a2:~s.,(C.,_3Ca4-- s.2:~casa4 ) + c._, ( 7.acas:}4+ aa4s:lca4 ) -- s._,C.,:] ( ZaSa&4
--
aa4CaCa4
) --
S._,Seaa:~4S:~4
~- (T2,.5) 4es45 Jr" ( T:~t~,)4ac4:,-~ (T215) 45a4ac4~,-- ( Teta)4~a4as45,
(55)
w h e r e (T,_,~)~j is the (i,j) element of T..,~5. T.,~.~ is completely d e t e r m i n e d for each set of 0~ and 05; hence, from equations (53) and (54), 0., is uniquely determined:
s., = [ ( c23 cas34 + s,_,aCa4 ) { ( T.,15 ) 12s4.5 + ( T.., la )la c4:, } + s:,sa4 { ( T,_,,5 )._,._,s45 + (;r~,.~) ~:}c~} ] / [ (c.,:,c~s:, + s.,_:,c:,)~ + (s:,s:,.,) q,
t56)
c., = [sa&4{ (T,,_~s) p,.s45 + ( T._,,5)nc4a} - ( c._,acas34 + &aca4){ ( T.,,,s).,_.,_s4:, + (T,,_,:,) ._,:,c+~} ] / [ (c,,.~c.~s~ + s.,_:}c:, ) ' + (&s.~,)'- ].
(57)
By substituting the values ofs., and c., into equation (55), z.., is uniquely determined. T h e remaining variable parameters, 04 and z4, can be c o m p u t e d in a similar manner. F r o m equation (50), T15Ta~ = TI.,T._,:~T:~4.
(58)
Premultiply both sides of the equation by T~, and let T5, Tr,_T,_,aTa4 = Ta~2a4T54 = T5 p_,.]4
{5 9 )
since T.~Tt:, = 1. T h e right-hand side of equation (59) is c o m p l e t e l y d e t e r m i n e d for a given set of 0t, 0._,. :.~, 0a. and 0a. T h e left-hand side is given by equation (19). H e n c e , by equating the ( 1, 1), ( 1,2), and (4. 1) elements of the left-hand side to the c o r r e s p o n d i n g
129
elements of the right-hand side, S4 = (T51234)1.,,
(60)
c4 = (Tsl~34) u,
(6 1)
Z4 = -- (T~r_,.34) ~1/s4,
(62)
where (Ts~z34)u is the ( i , j ) element ofT5,234. T h e numerical values of each set of the variable parameters thus obtained can be verified by substituting back to the matrices of T o, and computing the matrix product T~.T~_3T34T~sT~1, which should be equal to the identity matrix, 1. Computer
Program
and Numerical
Example
A computer program has been developed for the purpose; the outputs of the program for a numerical example are shown in the appendix. The mechanism proportions of the numerical example are the same as those of the example given by Yang [ 15]. The input-output displacement curves of the example are plotted in Fig. 3. Note that there are two closed circuits in the figure, which means that there are two distinct ways of assembling the same mechanism, and both are of the double-rocker type. The displacements of AssemblyA are plotted in Figs. 4 and 5, and Assembly B in Figs. 6 and I
I
I
I
I
I
I
I
I
I
180
1201
60
C~ ~g -60
o -1 20
-180
I 60
120 180 240 300 Input Angle @I (Degrees)
Figure 3. I n p u t - o u t p u t Mechanism
displacement proportions:
curves
I 360
of RCRCR
a~_~ = 1.0, a._,3= 4 . 0 , a34 = 3 . 0 , a~5 = 2 . 5 , a51 = 3 . 2 , a12 = 3 0 °, a34 = 4 5 ° , a45 = 6 0 °, as~ = 1 0 ° , z i = 0, z~ = 2 . 5 , zs = 3 . 0 .
a.,3 = 3 5 °,
130
180
~-
12O
"~
/
~,
-6o
/
\
@
o4
-120
-180
-240
4
-300
I
I
I
I
I
I
60
120
180
240
300
360
Input Angle @1 (Degrees)
Figure 4. Displacement curves of RCRCR-A (0,,, 0:~, 0~, 0J. --
6.0
I
-
I
I
I
1
~
3.0 -
r
4
-\~\
~
\
-
O'
v
-3,0 ~q
-6.0
-9.0
-12.0
0
I 60
I I I I 120 180 240 300 Input Angle @I (Degrees)
I 360
Figure 5. Displacement curves of RCRCR-A (z._,,z4).
t
I
I
I
I
I
131
I
180
120 2
Q
60
~
O'
-60
-120 ¢ -180
I 60
I I I I 120 180 240 300 Input Angle @I (Degrees)
I 360
Figure 6. Displacement curves of RCRCR-B (~, ~:,, ~4, ~). I
I
I
9.O --
I
/f-
I
I
~.\
6.0
1
3.0
t
~-4
J -5.0 2
-6.0
-9.0
0
I 60
I I I I 120 180 240 300 Input Angle @1 (Degrees)
I 560
Figure 7. Displacement curves of RCRCR-B (z._,, z4).
132
7. + To move the mechanisms around the closed circuits, the input and output have to be interchanged at the extreme-range positions of 0~. The number of solutions obtained from the input-output displacement equation agrees with those obtained by Yang [ 15]. however, the numerical values do not. On the other hand, most of the numerical values of the output angles obtained by Gupta (the discussor of Yang's paper) are in good agreement with the results presented here. but the number of solutions differs. Both Yang's and Gupta's results can not be verified because no intermediate variables are obtained. The results presented here are believed to be correct since each set of solutions has been verified by substituting back to the transformation matrices of line coordinates, and multiplying out to obtain the identity matrix.
Conclusions Using the method of line coordinates, an algebraic equation of fourth-order has been obtained for the input-output displacement of the RCRCR five-link spatial mechanism which confirms Yang's [ 15] solution. The method provides an efficient way for the computations of the intermediate variable parameters of the mechanism. It also provides a simple way of verifying the accuracy of numerical computations. For the purpose of illustration. Yang's numerical example has been reworked, and a complete and correct solution is presented. A description of computer program can be obtained from the author upon request.
Appendix: Computer Program Outputs (abbreviated) R C R C R 5 - L i n k Spatial M e c h a n i s m D i s p l a c e m e n t A n a l y s i s ( M e c h a n i s m p r o p o r t i o n s are t a k e n from A. T. Y a n g s R C R C R 5-1ink) 12 23 34 45 51 IJ 1.00(X) 4.0000 3-0000 2"5000 3.21)00 (Inch) A (IJ) 60.0t)0 10.000 ( D e g r e e ) A l p h a ( IJ ) 30"000 35"000 45"000 Z (1~ 0"0 2" 5000 3.(1000 +Inch) Numberofrealroots = 2 T h e t a (I) = 0, (Degree) Theta(3) Theta(4) T h e t a (2) ZI2) (Degree) (Inch) (Degree) (Degree) - 146.978 - 3-7584 41-281 - 162.208 Check forT(51)*Tl12)~T(231*T(34)+T(45) =l 1-000000
-- 0-000000
-
0-000000
0.0
1-000000
-
0.000001
0.0
0.000000
-0-000000 0.0 1-000000 0-0 0.000002 0-000005 -- 0.000006 1-000000 - 0.000003 - 0-000t)01 - 0.000003 0.000000 0.000010 0.000001 -- 0.000002 - - 0 . 0 0 0 0 0 0 --131-177 --8.7370 149-812 147.538 Check forT(51)~T(12)*Ti23)*T(34)*T(45) =1 1-000016 - 0 . 0 0 ( R K ~ - 0 - 0 0 0 0 0 0 0.0 0-000003 1.000005 -- 0-000010 0-0 0.000004 - 0 - 0 0 0 0 0 6 1.000010 0.0 --0,000009 0-000012 --0-00t)008 1.000016 0-000028 - 0.000060 - 0-000025 0-000003 0.000039 0-000030 0-000072 0.000004 T h e t a ( 1) = 60. T h e t a (1) = 120. T h e t a !2)
(Degree) (Degree) Z(21
ZI4)
(Inch) - 0.5349 0-0 0-0 0-0
0.0
0-0 0-0 - 0-000000 1.000000 0.0 4.1163 0-0 0.0 0-0 --0-000000 1.000005 - 0.000006
Thetal5) (Degree) - 78.752
-
0.000000
-
0.000001 1.000000
- 169-184 0.0 0.0 0.0 -
0.0001300
- 0.000010 1-000010
N u m b e r of real roots = 0 N u m b e r of real r o o t s = 2 T h e t a (3) T h e t a (4)
ZI4t
T h e t a (5 )
* T h e d i s p l a c e m e n t c u r v e s h a v e b e e n p l o t t e d with the aid of a c o m p u t e r p r o g r a m d e v e l o p e d by the a u t h o r in Ref. [16], w h i c h also has s h o w n that the c u r v e s are c l o s e c i r c u i t e d : h o w e v e r t h e s e c o u l d h a v e b e e n d o n e . t h o u g h m o r e difficult, by c o m p u t a t i o n s w i t h s m a l l s p a c i n g s o f the input angle.
133 (Degree) 18-489 - 146-343
(Inch) - 7.3300 - 2-6262
(Degree) 82.794 117-516
(Degree) 107.196 150-318
(Inch) - 2.9976 - 5.7428
(Degree) - 0.422 I 11508
T h e t a ( 1) = 180. (Degree) T h e t a (2) Z(2) (Degree) (Inch) 96-887 -7.8744 21.153 -2.8398 - 0"767 - 7.4271 146,419 -2-1367
N u m b e r of real T h e t a (3) (Degree) - 115.760 - 107.312 99.701 82-780
roots = 4 T h e t a (4) (Degree) - 135.895 - 112-388 111 '059 161.180
Z(4) (Inch) 5.7326 0.2202 0.2780 -6-8129
T h e t a (5) (Degree) - 15-340 39-036 - 53.129 78.626
T h e t a ( 1 ) = 240. (Degree) T h e t a (2) Z(2) (Degree) (Inch) -29-050 -0.7319 82.003 -8.9711 - 135.796 - 1:9979 -41-642 -7.3968
N u m b e r of real T h e t a (3) (Degree) -75-879 - 126.521 42.565 126.398
roots = 4 T h e t a (4) (Degree) - 110.387 - 115-324 - 179.278 115-033
Z(4) (Inch) 0.3215 8-8636 -5.4172 2-9055
T h e t a (5) (Degree) 7.678 -63.397 38.355 -92,071
T h e t a (I) = 300. (Degree) T h e t a (2) Z (2) (Degree) (Inch) - 121.255 - 1.7903 - 30.550 - 1.7797 27.019 -6.7199 -90-487 -8-5731
N u m b e r of real T h e t a (3) (Degree) 11.814 - 80.544 - 120.878 145.319
roots = 4 T h e t a (4) (Degree) - 154.381 - 96-846 - 8 8 " 157 130.602
Z(4) (Inch) - 1.8252 3-5524 7-7129 5-0047
T h e t a (5) (Degree) - 16-539 - 54.733 -91.292 - 129,962
-
Acknowledgements-The author is greatly indebted to his thesis adviser, Professor Ferdinand Freudenstein of Columbia University. w h o s e studies, in collaboration with Mr. Lin S. Woo of the IBM New York Scientific Center, led to the d e v e l o p m e n t of this method. T h e author is grateful to the National Science Foundation for the support of this research through G r a n t N S F - G K- 1693 ; to Columbia university for the use of computational facilities; and to the Singer C o m p a n y for their assistance and cooperation.
References [ I] B R A N D L., Vector and Tensor Analysis. John Wiley, N e w York (1962). [2] C H A C E M. A., M e c h a n i s m analysis by vector m a t h e m a t i c s , Trans. Serenth Conf. Mechanisms Purdue University. 100-113 (1962). [3] C H A C E M. A.. Vector analysis of linkages, J. Eng. Ind., Trans. ASME85, 289-297. (Aug. 1963). [4] D E N A V I T J., Description and Displacement Analysis of Mechanisms Based on (2 × 2) Dltal Matrices. Doctoral Dissertation, N o r t h w e s t e r n University (1956). [5] D I M E N T B E R G F. M., T h e Determination of the Positions of Spatial M e c h a n i s m s . (Russian) Akad. Nauk., M o s c o w . (1950). [61 D I M E N T B E R G F. M., A General Method for the Investigation of Finite Displacements of Spatial Mechanisms and Certain Cases of Passive Joints. Purdue Translation, Purdue University (May 1959). [7] H O C. Y. A n analysis of spatial four-bar linkage by tensor method, IBM J. Res. Dev., 10, 3, 2 0 7 212 (May 1966). [8] M A X W E L L E. A. General Homogeneous Coordinates in Space of Three Dimensions. Cambridge University Press. (1967). [9] U I C K E R J. J., D E N A V I T J, and H A R T E N B E R G R. S. A n iterative method for the displacement analysis of spatial m e c h a n i s m s , J. appl. Mech.. Trans. A S M E 8 6 , 3 0 9 - 3 14 (June 1964). [I0] W A L L A C E D. M., Displacement Analysis of Spatial Mechanisms With More Than Four Links, Doctoral Dissertation. Columbia University (1968). [1 I] W A L L A C E D. M. and F R E U D E N S T E I N F. T h e displacement analysis of the generalized tracta coupling. J. app/. Mech.. Trans. A S M E 37, Series E, 3, 7 1 3 - 7 1 9 (Sept. 1970). [12] W O O L. and F R E U D E N S T E I N F. Application of Line Geometry to Theoretical Kinematics and tile Kinematic Analysis of Mechanical Systems, IBM N e w York Scientific C e n t e r Technical Report no. 3 2 0 - 2 9 8 2 , p. 103 (Nov. 1969). [ 131 Y A N G A. T. Application ofDuaI-Namber Quaternion Algebra to the Analysis of Spatial Mechanisms, Doctoral Dissertation, Columbia University (1963). [14] Y A N G A. T. and F R E U D E N S T E I N F. Application of d u a l - n u m b e r quaternion algebra to the analysis of spatial m e c h a n i s m s , J. appl. Mech., Trans. A S M E 86, 300-308. (June 1964).
134 [15j YANG A. T.. Displacement analysis of spatial five-link mechanisms using i3 × 3) matrices with dual-number elements, d. Ene. Ind.. Trarls. A S M E V o [ . 91, pp. 152-157. iFeb. 1969). [ 161 Y U A N M. S. C. Kinematic" Anal.vsis o f Spatial Mechanisms by Means ~g-Screw Coordinates'. Doctoral Dissertation. Columbia University (1970). [17] YUAN M. S. C. Displacement analysis of the RRCCR five-link spatial mechanism. J. appl. Mech.. Trails. ,4S3,1E 37. Series E. 3,689-696 (Sept. 1970).
Displacement Analysis of The RCRCR Five-Link Spatial Mechanism* Mark S. C. Yuan~ Received 30 March 1970 Abstract Using the method of line coordinates, the input-output displacement equation of the RCRCR five-link spatial mechanism is obtained as a fourth-order algebraic equation. For each set of the input and output angles obtained from the equation, all other variable parameters of the mechanism are uniquely determined. A numerical example is presented, and the accuracy of each set of solutions is verified. Zusammenfassung- Bewegungsanalyse des f0nfgliedrigen r¨ichen RCRCRGetriebes: S. C. Yuan. Mittels der Methoden der Linienkoordinaten wird for die Beziehung zwischen der Eingangs- und der Ausgangsgr6sse des f0nfgliedrigen r¨ichen RCRCRGetriebes eine algebraische Gleichung vierten Grades abgeleitet. Mit jedem Paar von Eingangs- und Ausgangswinkeln, die aus der Gleichung gewonnen werden, sind alle 0brigen ver&nderlichen Parameter des Getriebes eindeutig bestimmt. Ein numerisches Beispiel wird vorgef0hrt und die Genauigkeit aller L6sungspaare wird 0berpr0ft. P e 3 m M e - AHa.r1143 l-lepe~etueHH~l I-IpocTpaHCTBeHHOFO l-lflXH3BeHtlOrO MexaHn3xla T u n a BFIBFIB: MapK C. IOan l-Io~lb3y~tcb Mero/lo,~t nHHefiHbtX KOOpi1aHar, nonyqeHo aJlre6paHaecKoe ypaaHeHHe HeTBepToa cxeneHH a.an nepeMemeHH~ B e a y ~ e r o 14 ae~oMoro 3aeHbeB npocTpaHCTBeHHOrO naTH3aeHHoro ,x~exaHH3Ma rHna BFIBFIB. ~Jln Ka~zloro 14a60pa yF.rlOB noBopoxa Be/lyulero H BeI1OMOFO 3BCHbeB, Ho,qyHCHHblX 143 ypaaHeHHn, CnHHCTBeHHOonpe~eneHbt Bce/apyrHe nepeMeHHble napaMeTpb[ MexaHH3Ma. 1-IpHBeaeH H14CJ'ICHHbI~npHMep H noaxBep~lleHa TOttHOCTb Ka}K~OFOHa60pa peuaeHH~.
Introduction FOR THE displacement analysis of spatial mechanisms, there are basically two approaches, i.e., the numerical approach and the algebraic approach; each one has advantages, and both are highly useful. Examples of the numerical approach are: the vector method of Chace[2, 3], the tensor method of Ho[7], and iterative method of Uicker, Denavit, and Hartenberg[9], etc. This paper is concerned with the algebraic approach. * Based on the doctoral dissertation of the a u t h o r in partial fulfillment of the requirements for the degree of Doctor of Engineering Science in the School of Engineering and Applied Science. Columbia University, N.Y., 10027, U.S.A. ~Senior Engineer, T h e Singer C o m p a n y , Industrial Product Division. Elizabeth, N.J. 07206, U.S.A. Presently. Staff Engineer, Remington Rand Office M a c h i n e s Division. Sperry Rand Corporation, Blue Bell, Pa. 19422. U.S.A.
119
120
The displacement of the R C C C * four-link spatial mechanism has been investigated by numerous researchers, e.g.. Dimentberg[5, 6] bv the method of dual vectors. Denavit[4] by dual matrices, Yang[13] and Yang and Freudenstein[14] by dual quaternions, etc. Dimentberg also investigated the displacements of the RSSR tour-link and the R R C C R and R C R C R five-link spatial mechanisms. For each of the five-link spatial mechanisms Dimentberg obtained an input-output displacement equation in a determinant form which could be reduced to an algebraic equation of eighth-order. The investigation, however, was not carried further, and no numerical example was obtained. Later on, Yang[15] obtained a fourth-order algebraic equation for the R C R C R five-link spatial mechanism by the method of dual matrices. The intermediate variable parameters of the mechanism were also solved, but only the input-output displacement curves were presented in a numerical example. Gt.pta (the discussor of Yang's paper) however obtained three different solutions for the numerical example, and, consequently, raised a question concerning the order of the input-output displacement equation. For other types of five-link spatial mechanisms, Wallace[10l and Wallace and F r e u d e n s t e i n [ l l ] obtained fourth-order algebraic equations l'or the input-output displacement equations of the R R S R R and R R E R R mechanisms by the method of geometric-configurations. Dimentberg's eighth-order solution for the R R C C R mechanism was confirmed by the author[16, 17] by the method of line coordinates. In this paper the solution of the R C R C R mechanism by the same method is presented. The displacement equation of the mechanism is obtained as a fourth-order algebraic equation which confirms Yang's solution. The coefficients of the equation are obtained explicitly in terms of the constant parameters of the mechanism. Moreover, for each set of the input and output angles obtained from the equation, all other variable parameters of the mechanism are uniquely determined, and a method of verification is presented. A computer program is developed for the purpose, and a complete solution of Yang's numerical example is obtained. The solution shows that most of Gupta's results are correct. Line C o o r d i n a t e s
The position of a point in space can be uniquely defined by the ratios of four coordinates ,r, 3,, z, t. and. conversely, these ratios are uniquely determined by the point, The tour coordinates are called the homogeneous coordinates of the point. Only three of the coordinates are independent since (,v, 3', :., t) and A(,v,>', :, :) determine the same point. Let (&,),,, :.~. t~), (.r._..)'e, z,_,,t._,) be the homogeneous coordinates of two distinct points P~, P,,, respectively. Define the six n u m b e r s ; L j = .v~t.,_ - - . r ~ t ~ ,
L., = y~t._,-- y.,t~,
L) = 3'~z:-
l,a = z ~ . v e - zz.r~.
l,:: = ~ q t , - - 2._,tj.
(3) Y,_,::~.
I,,; = -r~ 3', - . v . , 3 ' z .
The ratios of the six numbers uniquely determine the line joining the two points, and, conversely, the ratios of the six numbers are uniquely determined by the line. In line * R . C . S . E d e n o t e revolute, cylindrical, spherical, and p h m a r pairs respective[~. ~-See M a x w e l l [8]. pp. 6 5 - 7 3 . and W o o and F r e u d e n s t e i n [ 12I.
121
geometry, the six numbers are referred to as the coordinates of the line, or line coordinates. Only four of the six numbers are independent because (L~. L~, L:,, L4. L.~. L0 and X(L,. L~. L:,, L4, Ls, Ln) determine the same line. and they satisfy the relation
(2)
LtL4 + L:L5 + L~L~ = 0. The line coordinates are said to be normalized if LI ~+ L._,"+ L f = 1.
(3)
In this case the number of independent coordinates is increased to five. General line coordinates can be expressed as a multiple of normalized line coordinates. If the homogeneous coordinates of the two points. P1. P._,. are expressed as (x~, Yr, z t , - 1). (x~, y..,, z e , - 1), then (xt, yr. zt), (xz. y.,., z.,_) become the Cartesian coordinates of Pt, P._,. respectively. The first three numbers of the line coordinates, (Lz, L._,, L:0, then become the Cartesian components of the vector PtP._,, and the last three numbers. (L~, L.~, L~), become those of the moment vector of PtP~ about the origin, i.e., OPt x PtP~. Therefore, the line coordinates (L,. Lz, L:~. L,. L~. L6) can be interpreted as the line vector* (L1, L._,,L3) + •(L4, L.~. L,;), and vice versa. In this expression, E is an algebraic unit having the property (-' = 0.* Normalized line coordinates correspond to a unit line vector.
Transformation of Line Coordinates If the Cartesian frame of reference is transformed from x - y - z that
to x t - y ~ - z t
such
OO1 = (tl, t.,, t:~),
(4)
ry,xl]--|r,,l-r"r.,_,q" rI:'Ii -x] r._,,q/y l,
(5)
LZ,J
Lr:~t r:r_, r:~3JLZ_l
the line coordinates of L with reference to x t - Y l - zl can be obtained from those of L with reference to x - - y - - z. Consider L as a unit line vector L ---- (L,, L.,, L3) + E(L4, L~, L6), with reference to x - - y - z . line v e c t o r s 3
(6)
Transform L in accordance with the transformation law of
*See Brand[I]. pp, 5 5 . 5 6 . 6 3 - 6 7 . -~See Brand [ 1]. pp. 63-67.
122
IL,I=Lr , r.:.: r: n r:,.: raa
=
{I r
L:,,
i -- 0t:l 0,
L
t.2 - - t t
t~
L.,
0
L
re i
r.,.2
r2a
r
o oo 1 ~-L l-' 0
0
0
r::l
r::.,
1.:~:~
0
0
0
~ r , : d . _ , - - r v , t:,, flit: ~ - rlatl
rv:t,-
rid.,
rtl
rL.: r,:,q]
I r:l:lL_, - - r:l-_,t:l r31t:,, -- r3:l/'t
1.a._,[i
1.:ilL,
r:l I r:;._, 1.:1:'
L_, i L:~ I (7) L:,I
LL,d where (L)~ denotes the line coordinates of L with reference to x~ -Yz - :.~. By combining the primary and dual parts. (L)l = T1L.
(8)
in which. --
Ti =
r,I
r12
rt:~
0
0
0
r.e T
r.,.:
r_,:~ r:~a
0 0
0 0
0 0
r:lt
ra.e
Fl:IL2--1"I.212:~
/'ll[3--
7
i i
/'[;it I 1 [ 2 [ 1 - - 1 " 1 1 [ 2 111 /'12 rl; ~ [
(9)
/"_:;L2 - - r,,£~ r.,t r.,., 1..,., I . . r.:tt:,-. . . r.,_at~ . . .r.,.,lt . .--1..,iL, . . ra:d._, - - r:~.eta r:nta - - r;~atl r:r_,tl - - r:ut., ral r:r,_ 1.aa
Equation (8) is the transformation equation of line coordinates for the finite transformation of the Cartesian frame of reference. Dual Angles and Line Coordinates T h e scalar product of two unit line vectors is equal to the cosine of the dual angle between the lines.* Therefore, for two unit line vectors L and M, [(L~,L._,.La)+e(L4, L:,,L6)]" [ ( M , . M . , . M a ) + e ( M 4 , M~ M~)] = cos 0 - e z s i n
0,
(10)
where 0 is the angle, and z is the c o m m o n perpendicular distance between the lines. Hence. in terms of line coordinates. LtS'l~ + L,,M.. + LaMa = cos O,
(1t)
L~M4 + L._,M~+ LaMs + L4M~ + L~M.2 + L~5la = -- z sin 0.
(12)
Line Coordinates of Axes of Mechanism Pairs T h e R C R C R five-link spatial mechanism is schematically shown in Fig. 1. Link 5 is fixed. Links 1 and 4 are the input and output links respectively. S,, Se, Sa, $4. Sa are the axes of the five m e c h a n i s m pairs whose line coordinates are St = ( S , ~ . Sj._,. S j a . Sj4, Sa.~. Si~),j = 1 to 5.
*See Brand[l],
pp. 68-70.
123 -S3
Link 3 ~
z ~
~
~4
Link 2
23
z2
a45 Link
Yl
~5 0151
- Link I
~5
_s5 ,
~"
s1
Figure 1. The RCRCR five-link spatial mechanism. with reference to an arbitrary Cartesian frame. T h e parameters of the m e c h a n i s m are: zl = c o m m o n perpendicular distance between a~ and at.o, measured from as~ to at..,, and positive in the direction of St; 0t = angle between a~ and at.,; at., = c o m m o n perpendicular distance between St and $2; at-. = angle between St and S_o; and similarly for other parameters. All angles are measured according tb a right-handedscrew convention. T h e input and output parameters are 0t and 05 respectively. O t h e r variable p a r a m e t e r s are 0.,, z~, 03, 04, and z4. T h e line coordinates of S~ can be expressed as functions of the mechanism parameters in the following manner. xj
;'x
Figure 2.
/--L,nk ,
124
Let two sets of c a r t e s i a n c o o r d i n a t e axes x - v i - . z i , x ~ - y j - . z j be a t t a c h e d to L i n k s k. i. r e s p e c t i v e l y , as s h o w n in Fig. 2. T h e line c o o r d i n a t e s of S: with r e f e r e n c e to .v~ - Ya - zj are
(Sj)j = (0. O, l, O, O. 0).
(13)
T o o b t a i n the line c o o r d i n a t e s of Sj with r e f e r e n c e to .r~ - y~ - .~. c o n s i d e r the transf o r m a t i o n of .r0 - )'j - za to .vi0- Yu - zu. followed b y a n o t h e r t r a n s f o r m a t i o n to .r~ - v, - c.. In the first step. by e q u a t i o n s (8) a n d (9). V
1 0 i ! 0 (S~)i~=i 0
0 % ,% 0
l
0
- - ~lijSij
0
L_
0 - s,o c'~o 0
0 0 0 I
- - (dijCi j 0
0 0 0 0
0 0 0 0
(S~)~.
(14)
Cij - - Sij
(lijCij - - {1i)S o 0 Sij
('ij
w h e r e (So)u d e n o t e s the line c o o r d i n a t e s of Sj with r e f e r e n c e to . r 0 - ) ' ~ j - . h i . and c,: = cos - u . su = sin a~j. Similarly. in the s e c o n d step. c, si 0
(S,),. =
-s, 0 0 <'~ 0 0 0 1 0
- - Ci3"i ~.iCi
0
0 0 0 0 0 0
- - ~.i('i
0
C i -- S i 0
--"~iSi
0
5' i
Ci 0
0
0
0
0
( S~)~;.
(15)
1
w h e r e (S~)~ d e n o t e s the line c o o r d i n a t e s of S~ with r e f e r e n c e to . v ~ - y i - c . ~ . a n d c'~ = cos 0i, si = sin O, S u b s t i t u t i n g e q u a t i o n (14) into e q u a t i o n (15). a n d m u l t i p l y i n g out the m a t r i c e s , the line c o o r d i n a t e s of Sj with r e f e r e n c e to . r i - ) ' ~ - zi are o b t a i n e d as (S~)i = Tu(Sjb.
(16)
where c~
s, rij
~-
-- s i c u
sis~
0
0
tic u
-- c'is o
0
0
0
0
0
0
0
su
cu
- - ~iSi
( a i j S i S i j - - ~iCiCij )
( (.lijSiCij AU ~.iciaij )
~,iCi
( - - a i j c i s i j - - .~,isicij)
0
auci5
('i - - SiCtj
( - - ~ i j c i c i j -JF Zisisij ) S i - - aiJSij
0
0
SiSij
CiC 0 - - CiSij Sij
Cij
In the s a m e m a n n e r the line c o o r d i n a t e s of Si with r e f e r e n c e to x , - y ~ - z j o b t a i n e d as
(Si)j=
Tji(Si
i.
(17)
c a n be
(18)
125
where
i
c~ - - SiCij
Z;i
=
si
0
0
CiCij
Sij
0
SiSij
-- cisij
- - ZiSi
ZiCi
[ ( a i j s i s i j - - Zicicij)
(-- aifl.'isij-- ZiSiCo)
cij
0
0
Ci
0 0 0 si
(loc 0 - - SiCij
[_ (ausicu + Zicisu) (-- aucicu+ zisiso) -- ausij
0 0 0 0
(19)
CiCo Si t
SiSi j - - CiSi j Cij
Note that Tu is the inverse of Tji. i.e.. TuTji = I, the identity matrix. Input-Output
Displacement
Equation
The input-output displacement equation of the mechanism is obtained in the following manner: Referring to Fig. 1, with reference to the Cartesian frame xt - y t Zt attached to the fixed link, the line coordinates of So are expressed as functions of 01. and those of $4 as functions of 0s. Next, with reference to the Cartesian frame xa-y:t-z:~ attached to Link 2. the line coordinates of S..,and $4 are expressed as functions of 0a. Since the scalar product of two line vectors is an invariant with respect to the transformation of coordinate axes, the scalar product of S., and $4 with reference to Xl - Y t - - q is equated to that of S, and $4 with reference to x3-ya--z:~. This yields a dual equation which involves all the parameters of the mechanism except 0.,, z.,, 04, z4. Finally, the dual equation is separated into two real equations, and 0a is eliminated between the two equations; this results in the input-output displacement equation. With xt - y ~ - z z as the frame of reference. -
-
S, = (0, 0, 1,0, 0, 0).
(20)
By equations (16) and (17), S._, = TtzSt =
( S I S I 2 , - - C I S I 2 , C12, Z I C I S I 2 "q- a r 2 S t C v 2 , Z I S I S I 2 - - a l 2 C t C l 2 "
--a,..,s,.,).
(21)
By equations (18) and (19), 54
TIzTz4St ES45S5, $45C5C51 -]- C45S51, - - $45C5S51 71- C45C51, $45Z5C5 ~ ~45C45S5,
- - $45 (a51c5s51 + Z5S5C51 ) + c4.5a51c51 + a45c45cscst - - a45s45s51 , S45 ( - - azlCr, Cs! + Z5S5S51 ) - - c45a51s5 t - - a45c45c5s51 - - a45s45c51 ].
(22)
Similarly, with xa -- Ya -- Z3 as the frame of reference, $3 = (0,0, 1 , 0 , 0 , 0 ) ,
(23)
$4 = Ta4Sa (S3S34 , - - C3S34, C34, Z3C3S34 -{- a 3 4 s 3 c 3 4 , Z3S3S34 - - a34C3C34 , - - a34S3.1) ,
(24)
126 S.,_ = Ta._,S:~
= (0, S, a, C.,_:,, O. a_,:,(,_:~, --a_,as.,_:~).
25)
By equating the scalar product of equations (21) and (22) to that of equations (24) and (25). a dual equation can be obtained. The primary part of the equation is - seacaaa4 - c..,aCa4 =
s45Y,5stsL2
--
{, $45C5C51 -~ (_'45S51 )('iSle
& ( - - s45cJ51 + c45ca~ )c~2:
(26)
and the dual part is $2:; (Z,:;S:~S:,t - - a 3 4 c 3 c 3 4
) -- c23a34s34
- - (.123(.23c3s3.1 - - a 2 3 s 2 3 c : l
= s ~ : , s : , ( z < , s : _ + a:_s~c~.,) + (s~::,c:,~ + c~sa~ ) (zlst.s,~ - a,._,c~c~2) - - C45C5l )(/12Sl2 +
-~- ( $ 4 5 C 5 S 5 t
($45Z5C 3 + (J.45ClsSs,)SI,S'I2
-~- [ $45 ( (151 C5S51 -L S5S5C51 ) __ C45,~151 C51 - - a 4 5 C 4 5 C 5 C 5 1 -~- &t45s45s51 ] C l S l 2
+ [S45(--a.-,icacal+Z..asas:,t)
--c4aa:,ls.-,,--a4ac4:,cas:,l--a45S45Cal]Cv_,.
(27)
These two equations can be rearranged to C::c:~ = A isa + B,c.-, - - C I .
(28)
C~sa + Ca{'a = Aes:, + B.c.-, - - C.,_.
(29)
Ca = - se:~s:.,
t30)
A l = s~:,sr_,s~.
(31)
B~ = - s,:,ca~.s'r,_ct - - sc, s:,tc'r,,
(32)
C~ = cc, s:,:v_,Cl - cc, c:,tc:_ + c2::c:H,
(33)
C4 = s.,:~z:::l,.
(34)
C a = - - $2:1gl34C34 - - C/.23C2:1S34,
(3 5 )
Ae =
($45(/12C12 "At-{145C45S12)S1 -}-,~'t5S12 ( ~,1 "~- ~,5C5l) Cl ÷ $45"~5S51('12"
(36)
Be =
$45-Y12 (CsIZ1 -~- ~,5)SI -~- (--$45C51Cl12Cl2 q - $45(151S51512 - - /l-t5C45('51SI2) C1
where
(37)
- ~ $45S51LI12S12 - - $45C/51 C51C12 - - a45('.15S5 IC12 ,
C 2 =
- - c45.'Q~1.7..is12s I -}- ( C 4 5 S 5 1 L l 1 2 C 1 2 @- C 1 5 C 5 1 a l 2 S I 2
-}- C45(151C51S12 - - a45-~'U)S513"I2 )(--'t
-4- C45(.151S51CI2 -~- a 4 5 S 4 5 C 5 l C 1 2
- - ~123.~'2;;(7:Ii - - C 2 3 a 3 4 S 3 4 .
(38)
From equations (28) and (29), c:~ = ( A : : , +
Blca-
C,)/C>
.% = [ (A ~sa + B t or, - - (7~) Ca - - C a ( A .,_& + B.,ca - - C., ) ] / -
(391 C:~C ~.
(40)
127
Eliminate 05 between equations (39) and (40). The result is Vs.,,"- + W s s c s + Xs~, + Yc~, + Z = O.
(41)
V = (Ca" + Cs'-)DII + CaO-D._,,_-- 2CaC-,,Dr_,,
(42)
W = (Ca 2 + Cs")E11 + C3"-E.,.., - 2C3CsEI:,
(43)
X = - (Ca" + C5") F i t - C3"-F.,..,.+ 2C.~C:,Ft..,,
(44)
Y = - (Ca" + Cs'-) G it - C3"G.,_.,. + 2C3C:,G r,.
(45)
Z = (Ca" + C~,")Htz + C3"H.,_.,_ - 2C3CsHv,. - C3"-C4".
(46)
where
in which Di~ = A i A ~ - - BiB~, Eij = A~Bj + BIAs, Fo = AiCj+ CiAj,
(47)
Gi.i = B i C j + CiB~, Hi~ = BiBj + CiCi.
By substituting x = tan (OJ2), or s~ = (Zr)/(l +x~), c.~= ( l - - x " - ) / ( l + x " - ) , equation (41 ), a fourth-order polynomial
into
4
Z P,x" = 0
(48)
0
is obtained as the input-output displacement equation. The coefficients of the polynomial are Po = Y + Z ,
P~ = 2 ( W + X ) , Pe = 4 V + 2Z,
(49)
P:~ = 2 ( - W + X ) , Pa
=
--
Y+Z.
For a given input angle, 01, a maximum of four output angles. 0.~, may be obtained, since it is the maximum number of real roots of equation (48). Other Variable Parameters of the M e c h a n i s m For each set of 0t and Os obtained from the input-output displacement equation, 03 is uniquely determined by equations (39) and (40). To compute 0._,and z,_,for each set of 0~, 0:3, and 05, equate the expression of $4. with x ~ - y ~ - z, as the frame of reference, obtained by equation (16) to that of St obtained by equation (18): Tt:~ T2a Ta4Sx = TIs T~S1. JM Vol. 6. No. I - - I
(50)
128 Premultiply both sides by T.,_, T.,_:;Ta4SI =
T.q T I s T a 4 S
I
{51}
since T._,~Tv,_ = 1. Let T.,,Tt~ = T._,,~, and multiply out the last two matrices of both sides of equation (5 1). $3S3 4 -- C'33a4 [ T.-a ('34 ] = T215 C1:;4S3C34 ~ ~3C3S34 i -- tl:~4C3C34 ~ ~,3S3S3] ~ -- a34S34
0 s4s i i
C45
(52)
i" I
a ~5C4.5 i CZ~3S ~3~
Multiply out both sides, and equate the c o r r e s p o n d i n g elements of { 1, 1), (2, I). (4, 1). (c.,.acasa4 + s.,aca4) s., + sas:]4c.: = (T.,_~a) p,s ~5 + (T._qa) t3c ~.
(53)
SaSa4S.,_ -- {c.,_ac,as:,4 + s,,.:,ca4)c., = ( T.,,~)._,..,s45+ ( T,,ta).,:]c~:,,
{54)
(-- S.,SaSa4 -t- C=,C.:.~C3Sa4 -+- C._,S.,_.3Ca4)7.._ q- a2:~s.,(C.,_3Ca4-- s.2:~casa4 ) + c._, ( 7.acas:}4+ aa4s:lca4 ) -- s._,C.,:] ( ZaSa&4
--
aa4CaCa4
) --
S._,Seaa:~4S:~4
~- (T2,.5) 4es45 Jr" ( T:~t~,)4ac4:,-~ (T215) 45a4ac4~,-- ( Teta)4~a4as45,
(55)
w h e r e (T,_,~)~j is the (i,j) element of T..,~5. T.,~.~ is completely d e t e r m i n e d for each set of 0~ and 05; hence, from equations (53) and (54), 0., is uniquely determined:
s., = [ ( c23 cas34 + s,_,aCa4 ) { ( T.,15 ) 12s4.5 + ( T.., la )la c4:, } + s:,sa4 { ( T,_,,5 )._,._,s45 + (;r~,.~) ~:}c~} ] / [ (c.,:,c~s:, + s.,_:,c:,)~ + (s:,s:,.,) q,
t56)
c., = [sa&4{ (T,,_~s) p,.s45 + ( T._,,5)nc4a} - ( c._,acas34 + &aca4){ ( T.,,,s).,_.,_s4:, + (T,,_,:,) ._,:,c+~} ] / [ (c,,.~c.~s~ + s.,_:}c:, ) ' + (&s.~,)'- ].
(57)
By substituting the values ofs., and c., into equation (55), z.., is uniquely determined. T h e remaining variable parameters, 04 and z4, can be c o m p u t e d in a similar manner. F r o m equation (50), T15Ta~ = TI.,T._,:~T:~4.
(58)
Premultiply both sides of the equation by T~, and let T5, Tr,_T,_,aTa4 = Ta~2a4T54 = T5 p_,.]4
{5 9 )
since T.~Tt:, = 1. T h e right-hand side of equation (59) is c o m p l e t e l y d e t e r m i n e d for a given set of 0t, 0._,. :.~, 0a. and 0a. T h e left-hand side is given by equation (19). H e n c e , by equating the ( 1, 1), ( 1,2), and (4. 1) elements of the left-hand side to the c o r r e s p o n d i n g
129
elements of the right-hand side, S4 = (T51234)1.,,
(60)
c4 = (Tsl~34) u,
(6 1)
Z4 = -- (T~r_,.34) ~1/s4,
(62)
where (Ts~z34)u is the ( i , j ) element ofT5,234. T h e numerical values of each set of the variable parameters thus obtained can be verified by substituting back to the matrices of T o, and computing the matrix product T~.T~_3T34T~sT~1, which should be equal to the identity matrix, 1. Computer
Program
and Numerical
Example
A computer program has been developed for the purpose; the outputs of the program for a numerical example are shown in the appendix. The mechanism proportions of the numerical example are the same as those of the example given by Yang [ 15]. The input-output displacement curves of the example are plotted in Fig. 3. Note that there are two closed circuits in the figure, which means that there are two distinct ways of assembling the same mechanism, and both are of the double-rocker type. The displacements of AssemblyA are plotted in Figs. 4 and 5, and Assembly B in Figs. 6 and I
I
I
I
I
I
I
I
I
I
180
1201
60
C~ ~g -60
o -1 20
-180
I 60
120 180 240 300 Input Angle @I (Degrees)
Figure 3. I n p u t - o u t p u t Mechanism
displacement proportions:
curves
I 360
of RCRCR
a~_~ = 1.0, a._,3= 4 . 0 , a34 = 3 . 0 , a~5 = 2 . 5 , a51 = 3 . 2 , a12 = 3 0 °, a34 = 4 5 ° , a45 = 6 0 °, as~ = 1 0 ° , z i = 0, z~ = 2 . 5 , zs = 3 . 0 .
a.,3 = 3 5 °,
130
180
~-
12O
"~
/
~,
-6o
/
\
@
o4
-120
-180
-240
4
-300
I
I
I
I
I
I
60
120
180
240
300
360
Input Angle @1 (Degrees)
Figure 4. Displacement curves of RCRCR-A (0,,, 0:~, 0~, 0J. --
6.0
I
-
I
I
I
1
~
3.0 -
r
4
-\~\
~
\
-
O'
v
-3,0 ~q
-6.0
-9.0
-12.0
0
I 60
I I I I 120 180 240 300 Input Angle @I (Degrees)
I 360
Figure 5. Displacement curves of RCRCR-A (z._,,z4).
t
I
I
I
I
I
131
I
180
120 2
Q
60
~
O'
-60
-120 ¢ -180
I 60
I I I I 120 180 240 300 Input Angle @I (Degrees)
I 360
Figure 6. Displacement curves of RCRCR-B (~, ~:,, ~4, ~). I
I
I
9.O --
I
/f-
I
I
~.\
6.0
1
3.0
t
~-4
J -5.0 2
-6.0
-9.0
0
I 60
I I I I 120 180 240 300 Input Angle @1 (Degrees)
I 560
Figure 7. Displacement curves of RCRCR-B (z._,, z4).
132
7. + To move the mechanisms around the closed circuits, the input and output have to be interchanged at the extreme-range positions of 0~. The number of solutions obtained from the input-output displacement equation agrees with those obtained by Yang [ 15]. however, the numerical values do not. On the other hand, most of the numerical values of the output angles obtained by Gupta (the discussor of Yang's paper) are in good agreement with the results presented here. but the number of solutions differs. Both Yang's and Gupta's results can not be verified because no intermediate variables are obtained. The results presented here are believed to be correct since each set of solutions has been verified by substituting back to the transformation matrices of line coordinates, and multiplying out to obtain the identity matrix.
Conclusions Using the method of line coordinates, an algebraic equation of fourth-order has been obtained for the input-output displacement of the RCRCR five-link spatial mechanism which confirms Yang's [ 15] solution. The method provides an efficient way for the computations of the intermediate variable parameters of the mechanism. It also provides a simple way of verifying the accuracy of numerical computations. For the purpose of illustration. Yang's numerical example has been reworked, and a complete and correct solution is presented. A description of computer program can be obtained from the author upon request.
Appendix: Computer Program Outputs (abbreviated) R C R C R 5 - L i n k Spatial M e c h a n i s m D i s p l a c e m e n t A n a l y s i s ( M e c h a n i s m p r o p o r t i o n s are t a k e n from A. T. Y a n g s R C R C R 5-1ink) 12 23 34 45 51 IJ 1.00(X) 4.0000 3-0000 2"5000 3.21)00 (Inch) A (IJ) 60.0t)0 10.000 ( D e g r e e ) A l p h a ( IJ ) 30"000 35"000 45"000 Z (1~ 0"0 2" 5000 3.(1000 +Inch) Numberofrealroots = 2 T h e t a (I) = 0, (Degree) Theta(3) Theta(4) T h e t a (2) ZI2) (Degree) (Inch) (Degree) (Degree) - 146.978 - 3-7584 41-281 - 162.208 Check forT(51)*Tl12)~T(231*T(34)+T(45) =l 1-000000
-- 0-000000
-
0-000000
0.0
1-000000
-
0.000001
0.0
0.000000
-0-000000 0.0 1-000000 0-0 0.000002 0-000005 -- 0.000006 1-000000 - 0.000003 - 0-000t)01 - 0.000003 0.000000 0.000010 0.000001 -- 0.000002 - - 0 . 0 0 0 0 0 0 --131-177 --8.7370 149-812 147.538 Check forT(51)~T(12)*Ti23)*T(34)*T(45) =1 1-000016 - 0 . 0 0 ( R K ~ - 0 - 0 0 0 0 0 0 0.0 0-000003 1.000005 -- 0-000010 0-0 0.000004 - 0 - 0 0 0 0 0 6 1.000010 0.0 --0,000009 0-000012 --0-00t)008 1.000016 0-000028 - 0.000060 - 0-000025 0-000003 0.000039 0-000030 0-000072 0.000004 T h e t a ( 1) = 60. T h e t a (1) = 120. T h e t a !2)
(Degree) (Degree) Z(21
ZI4)
(Inch) - 0.5349 0-0 0-0 0-0
0.0
0-0 0-0 - 0-000000 1.000000 0.0 4.1163 0-0 0.0 0-0 --0-000000 1.000005 - 0.000006
Thetal5) (Degree) - 78.752
-
0.000000
-
0.000001 1.000000
- 169-184 0.0 0.0 0.0 -
0.0001300
- 0.000010 1-000010
N u m b e r of real roots = 0 N u m b e r of real r o o t s = 2 T h e t a (3) T h e t a (4)
ZI4t
T h e t a (5 )
* T h e d i s p l a c e m e n t c u r v e s h a v e b e e n p l o t t e d with the aid of a c o m p u t e r p r o g r a m d e v e l o p e d by the a u t h o r in Ref. [16], w h i c h also has s h o w n that the c u r v e s are c l o s e c i r c u i t e d : h o w e v e r t h e s e c o u l d h a v e b e e n d o n e . t h o u g h m o r e difficult, by c o m p u t a t i o n s w i t h s m a l l s p a c i n g s o f the input angle.
133 (Degree) 18-489 - 146-343
(Inch) - 7.3300 - 2-6262
(Degree) 82.794 117-516
(Degree) 107.196 150-318
(Inch) - 2.9976 - 5.7428
(Degree) - 0.422 I 11508
T h e t a ( 1) = 180. (Degree) T h e t a (2) Z(2) (Degree) (Inch) 96-887 -7.8744 21.153 -2.8398 - 0"767 - 7.4271 146,419 -2-1367
N u m b e r of real T h e t a (3) (Degree) - 115.760 - 107.312 99.701 82-780
roots = 4 T h e t a (4) (Degree) - 135.895 - 112-388 111 '059 161.180
Z(4) (Inch) 5.7326 0.2202 0.2780 -6-8129
T h e t a (5) (Degree) - 15-340 39-036 - 53.129 78.626
T h e t a ( 1 ) = 240. (Degree) T h e t a (2) Z(2) (Degree) (Inch) -29-050 -0.7319 82.003 -8.9711 - 135.796 - 1:9979 -41-642 -7.3968
N u m b e r of real T h e t a (3) (Degree) -75-879 - 126.521 42.565 126.398
roots = 4 T h e t a (4) (Degree) - 110.387 - 115-324 - 179.278 115-033
Z(4) (Inch) 0.3215 8-8636 -5.4172 2-9055
T h e t a (5) (Degree) 7.678 -63.397 38.355 -92,071
T h e t a (I) = 300. (Degree) T h e t a (2) Z (2) (Degree) (Inch) - 121.255 - 1.7903 - 30.550 - 1.7797 27.019 -6.7199 -90-487 -8-5731
N u m b e r of real T h e t a (3) (Degree) 11.814 - 80.544 - 120.878 145.319
roots = 4 T h e t a (4) (Degree) - 154.381 - 96-846 - 8 8 " 157 130.602
Z(4) (Inch) - 1.8252 3-5524 7-7129 5-0047
T h e t a (5) (Degree) - 16-539 - 54.733 -91.292 - 129,962
-
Acknowledgements-The author is greatly indebted to his thesis adviser, Professor Ferdinand Freudenstein of Columbia University. w h o s e studies, in collaboration with Mr. Lin S. Woo of the IBM New York Scientific Center, led to the d e v e l o p m e n t of this method. T h e author is grateful to the National Science Foundation for the support of this research through G r a n t N S F - G K- 1693 ; to Columbia university for the use of computational facilities; and to the Singer C o m p a n y for their assistance and cooperation.
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