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A new vector theory for the analysis of spatial mechanisms
Mech. Mach. Theory Vol. 23, No. 3, pp. 209-217, 1988 Printed in Great Britain. All rights reserved
0094-114X/88 $3.00+0.00 Copyright © 1988 Pergamon Press plc
A NEW VECTOR THEORY FOR THE ANALYSIS OF SPATIAL MECHANISMS HONG-YOU LEE and CHONG-GAO LIANG Department of Mechanical Engineering, Beijing Institute of Posts and Telecommunications, Beijing, People's Republic of China (Received 27 M a y 1986)
AMtraet--This paper develops a new vector theory for the analysis of spatial mechanisms on the basis of the vector analysis theory and dual-number algebra and the recursive notation presented by J. Duffy. The new vector theory includes the displacement, velocity and acceleration equations for the open kinematic chains and loop mechanisms. The displacement equations for the loop mechanisms are classified into four groups with two forms, namely the primary, secondary, projective and the fourth-group loop/half-angle equations, which have been applied to the derivation of the 16th degree polynomial input--output equations in the tan-half-angle of the output angular displacements for the general spatial 7-1ink 7R and 6R-P mechanisms, etc. In the analysis of the velocity and acceleration, the differential calculus are changed into vectorical algebraic operations and the unified equations are expressed in the vector forms which are linear in the velocity and acceleration variables.
1. INTRODUCTION In recent years there has been considerable development in the methods for the analysis of spatial mechanisms. This was initiated by the significant contribution of Dimentberg, Denavit and Hartenberg. Duffy presented the recursive notation which greatly facilitates the resultant expressions of algebraic equations and used the method of spherical trigonometry to derive the displacement inputoutput equations for almost all the single-loop singledegree-of-freedom spatial linkages involving R, P and C pairs[2, 3], except the spatial 7-1ink 6R-P and R R P R P R R linkages. Vectors, which are well known, were used by Chace to derive closed-form displacement relations of RCCC linkage[4]. Wallace and Freudenstein used a geometric configuration method to obtain closed-form displacement relations of R R S R R linkage[5]. Yang and Freudenstein derived the closed form algebraic expressions for displacements, velocities, forces and torques of all the links of RCCC linkage using quaternions and screw operators which possess geometric meaning[6]. In the present paper, a new vector theory is developed, the derivation of equations for displacements, velocities and accelerations are performed using the vector formulae which not only possess geometric meaning but also can avoid the tedious and lengthy operations of triangle functions, and the resultant equations are expressed in the recursive notations, presented by Duffy, which are relatively concise in form and more convenient for programming. This theory has been applied to the displacement analysis of the spatial 6-1ink R R R R R C , 7-1ink 6R-P, R R P R P R R and the general 7R mechanisms[7-9]. The mathematic development of the new vector theory for the analysis of
spatial mechanisms will be described in the following sections. 2. SPATIAL KINEMATIC CHAIN AND DUAL FUNCTION Specification o f spatial k i n e m a t i c chain
Figure 1 illustrates the general spatial kinematic chain of links connected by the joints, such as R, P, C pairs, whose axes are directed by unit vectors si, s j , . . . , the link axes, which are chosen along the common perpendicular between the adjacent joint axes, are directed by unit vectors aij, ajk. • • Hence, the
209
It/
/
*m~,,.~ f
Sm
"",
(0)
/,,
~q
(b)
?ig. 1. (a) Spatial kinematic chain; (b) simplified representation.
210
HONG-You LEE and CHONG-GAoLIANG rl
r2
sj
#2
O
Fig. 2
Fig. 4
link length a~ is the shortest distance between two adjacent pair axes st and sj, measured along a 0, and the offset of joint axes Sj is the shortest distance between two adjacent link axes a,~ and ajk, measured along sj. The angle a~ is measured by the right rotation of s~ to sj about the aq, the angle 0j is measured by the right rotation of a~ to ajk about the sj. The others are analogously defined by changing the subscripts. Therefore, the three unit-vectors (a,y, s j x a0, sj or a 0, s~ x aq, s~) can form a Cartesian coordinate system. Also, the following two equations will be often used later in the derivation of the vector form equations:
where op is the position vector of arbitrary point p on the line of unit vector r, referred to a specified point O, as shown in Fig. 3. Let F be a function of the real variables x~ which are the primary components of the dual-number variables xi=xi+~x0i, That is, F = f ( x l , x2 . . . . . x,). The dual-component function corresponding to F, denoted by D(F), is defined by '.L
st x Sj = sin ocoa0
(1)
withi=l,2,...,n.
OF
"
D(F) = i~-'~jx°i~-vx~= ~ x ° ~ - - f ( x l ' x2 . . . . . xn). i= 1
a o x ajk = sin 0~s~.
(2)
By introducing the dual-angle and dual-unitvector, the parameters of the spatial chain can be expressed as: dual-angle link:
~0 = ~o + ea~
dual-angle pair:
0j = Oj+ ESj
dual-unit-vector link:
~i~= ao + Ea*
dual-unit vector pair:
g = sj + Es*
where: e is the dual operator, and e2= e 3 . . . . . 0, a* = opj x a,s, s* = op2 × sj and opl, op2 are respectively position vectors of arbitrary points pj and P2 on the lines of the unit vectors a 0 and sj, referred to a specified point O, as shown in Fig. 2. The others can be analogously defined. For the purpose of intensive analysis it is more convenient to define the D operator (or D function) which means dualization operation, i.e. to derive the corresponding dual (secondary) component from the real (primary) component.
Definition of D operator Let r represent a unit vector, the dual-component vector corresponding to r, denoted by D (r), is defined by D(r) = op × r (3)
(4)
UXi
Thus, for the real functions F, F~, F2 and unit vectors rl, r2, using (3), (4) yields =
dD(F) =
(5)
D(d \~] 5
(6)
8F D (F) = Ox---~
(7)
D(F, +_F2) = D(F 0 + D(F2)
(8)
D(F, F2) = D(FOF2 + F~D(F2) D(FI/F2) = {D(FI)F 2 -- F,D(F2)}/F ~
(9) (10)
D(ri "r2) = rl 'D(r2) + D(r0"r2 =plp2.r2 x rl,
(11)
where PiP2 is the vector from point p, to point P2, which are respectively arbitrary ones on the lines of vectors rl and r2, as shown in Fig. 4. Because of the arbitrariness, using (11) yields D(rj "r2)=p~p~'r2
x r1
thus plp2.r2 x r I = p ~ p ~ . r 2 x r 1.
(12)
In fact, equation (12) is one of the scalar components of the vector loop equation R . r z × rl = 0
where
R =PIP2 +P2P2 +P~P~ +P~P~ = 0 Fig. 3
(refer to Fig. 4)
(13)
Vector theory of spatial mechanism analysis In this paper, equations from (12) are known as secondary equations, the other scalar components of (13) are known as projective equations
3. F U N D A M E N T A L
FORMULAE
FOR
SPATIAL
KINEMATIC CHAIN F o r the kinematic chain as shown in Fig. 1, the unit vectors a0, s,, s~ × a~, sj, s~... go by the general name of elementary vectors denoted by r~ with i = 1, 2 , . . . , and the chain vectors, denoted by R with the superscripts, are defined by R ~," = Sis ~+ aoa o + S~s~ + . .
• + SnS n
R °'''' = a~a~ + S~sj + ajka~, + . • • + a,,,na,,,,,
aij "sn × anq = Vykt . . . mn
211 aq" Sq x anq = U ~I . . . .
a,q.Syx a q = - V, . . . . t~
a,q.S~x a~y= - U*,,...~j. (18)
The subscripts of the recursive notations X, X*, Y, Z and U, U*, V, W are the labels between the two vectors of the left side of the equations and indicate that the expressions are linear sines and cosines of the angular displacements with these labels. Also, these subscripts are the recursive number in sequence. It should be noted that the terms U and U* in this paper are not exactly equal to those in[2, 3]. Introducing the definitions (16)-(18) into the recursive vector formulae (14), (15) yields the primary recursive formulae. P r i m a r y recursive f o r m u l a e
etc. The expressions for the scalar product of any two elementary vectors can be derived using the scalar product law. As one knows, the scalar product of two vectors r l and r2, can be expressed by r~. r 2 = (r~" x ) (r 2 • x ) + (r 1 • y ) ( r 2 • y ) + ( r x • z ) (r 2 • z )
Xm...tkj = xm...~,cj - Y,,...tksj x*...~kj = Xm...~Sj + Ym...t,c x Y~... ~k~= X ~ . . . t,~c u -- Z . . . . t~s~
w ~ , . . . . . = u?~, . . . . s. + wj~, . . . . c.
where x, y and z are unit vectors for coordinate axes which will be replaced by a0., s~ × a~j, s~ or a0., sj x ai~,
vj~, . . . .
= u J, . . . . c . -
Sj o r . . .
U~k, . . . . .
~-" Ujkl . . . . Snq + Vjkl . . . .
Thus, the scalar product of any two elementary vectors can be expressed in the form
Ujk' . . . . . = Uy,, . . . . c . o -
s.'sj × a,~ = (s,.a~,)s~ + (s,-s~ x aj~)cy × a o. = ( s . 's~ × a~)c,~ - ( s . . s ) s ~
s,'s, = (s,' s j x a/y)s# + (s,.sy)c,~
(14)
aij.anq = (aq.s n × aran)Sn + (a~' a~,)c,
a/j'Sq = (aq'sn)Cnq -- (a,y's~ ×
anq)Cnq anq)Snq
Uj = a,~'sk = sjsjk;
Cnq
V~k,. . . . . S,q.
= ao.'s j × ajk = - s j ;
I'Vj.= a,j- ajk = cj; (15)
where, c~ = cos 0~, sj = sin 0y, c~ --- cos et0, s,~ --- sin ~t,y, etc. and in the following sections, the sines and cosines are designated in the same way, also the tangent of the half-angle 0fl2 is denoted by x~, i.e. x~ = tan(0fl2), etc. The equations (14) and (15) are known as vector recursive formulae, and it is more convenient to define the dot products as the recursive notations which are presented by Duffy[2].
U*=
-aik.si×a,j=-sic o
7j = -ajk'sj
× a~ =
ff/j=ajk'ao=cj
Si " anq "~ Y j k l . . . ran = anq " Si = Unto... kj
S i ' $ n X anq -~- - - X ~ k t . . . .
Sq" s j x a o = X*m...tk~
X * = - s , . s j × aik = - s u e j
Yj = - s ~ - s k
x ajk = - ( % s j ,
+ socjkcj)
Zj = s;sk = cijcj~ - sijsjkcj ~7i = sk" a,j =
(22)
sj, sj
A'* = sk'sj × a 0 =
-sjkcj
= sk's~ × a~ = - ( c j k s o +
sj, c~cj)
(23)
Definition o f s e c o n d a r y f o r m u l a e
(16)
Rq'nq'sq
× Si = Zojkl . . . . .
~- Z o . . . . . Ikj
RiJ'n'anq x $, = Xojkt . . . . = Uo. . . . . ,kj
S i "~q X anq = - - Yj*t . . . .
Sq" s~ x a o = Y . . . . . lkj
(21)
Processing the two sides of equations (16)-(18) with D operator yields the secondary formulae as follows.
= S q ' S i = Z n . . . . kj
aij" anq = Wjkt . . . . . = anq" aij = W . . . . . kj
-sj
x j = s,. aj~ = s,jsj
= Sk'Si = CjkCo -- SjkSqCj. Definition o f p r i m a r y f o r m u l a e S," Sq ~- Z j k I . . . . .
(20)
~ = a f s , = sjs~
U * = a u . s k × ajk= -s~cjk;
a q ' s n x anq = (a/j's n × a,rm)c n - - ( a 0 . ' a ~ , ) s .
aq.Sq X anq = (aij'Sn)Snq + (a#'s n ×
~ , , . . . . s.
All the terms X, X*, Y, Z, U, U*, V, W, if only with a single subscript, can be computed directly from equations (16)-(18), but it is necessary to distinguish the two groups by introducing the symbol " - " . They can be expressed as follows:
Sn'a O.= (s,~.ajk)c j -- (Sn'Sj X a)k)sj
s..s,
(19)
Z . . . . til= X * . . t ~ j s q + Z,~...tkC o
(17)
R j'"" a,q x a,j = W0m ..... = W0. . . . lkj
(24)
212
HONG-YOu LEE and CHONG-GAOLIANG
R'J'".(s.
x anq ) x s i = X
anq)
R),nq.sq
X
(Sj X %) = X o..... ~,j
=
--
Yojkl . . . .
RV"nq. Sq × (Si × ao. ) = Yo . . . . . lkj
R z ' " (s.
anq)
X
aq = Voj~l . . . . .
RJ'nq'(Sq x anq)
X
a O. = Uojkl . . . . .
x
vectors are %, s~ x % and sk yields
.....
RiJ'nq.(sq
X
$i
--Xoykl
F s. %
IT[
× ad (25)
Sq'% l
iSq.S+ x a,q.
(29)
L s,'sk A L Sq'Sk 3 Further, by using the transformation of vector Sq from the coordinate system ajk, Sk × ayk, sk to the coordinate system ak~, Sk x ak. Sk, the equation (29) becomes
R y' "' anq X (Sj X aij ) = -- V 0. . . . . Ikj
R q'''a"q x (s~ x %) =
--
* . . . Ikj U onm
(26)
L s~'sk d
where the chain vectors defined by
RJ'" = S~s~+ aye% + .. • + S,s.
~k
etc. And the terms (X0, X*, Yo, Zo, Uo, U*, V0, Wo), with the additional subscript 0, are the dualcomponent functions of the terms (X, X*, Y, Z, U, U*, V, W) which have the same subscripts. Thus, the recursive formulae for the terms X0, X~', Yo, Zo, Uo, U*, V0, W0 can be obtained by processing the equations (19) and (20) with D operator, and they are suitable for programming and known as secondary recursive formulae. Secondary recursive formulae
Xo.... ++q- Yom...~ksj- s i x * ..~kj
X
Xo.... +,sj+ ro~...~kq + SjXm...+kj
om
Yom
. . ~y --
. . . lkj
U&~. . . . c , - Wo~,~. . . . s . - S.~,~ . . . . . Uojkl . . . . enq -- Vojkl . . . . . Snq -- anq U ~ I . . . .
~---~
is+.s+ ×,+,l . (30)
S q ' a kl
1
0
L
Sq'Sk
-]
(Si" Sq)
/c~
-s+
/so skxa~,/
L si'sk J L o
o
L s+.sk j
=Is"s+xa'q
1T[!,
=is,-s xa, q L
s~'sk I
[
sq'akt
sk !]
o Olls,
c,
0
0
O_ILO
1
Sq'Sk
d
= ls,.skx a,q (27)
U oj*~. . . m. -- Uojkl .... Snq -~- VOjkl ..... Cnq -}- anq Ujk I ..... Uojkl. . . mn
zjk` . . . .
L
Wojkl. . . mn = Uoj,~.... s. + Wo~,~.... c, + S , V ~ ..... Vojkl . . . mn
c+
X ISq'Sk × a k l ]
= Xo .... tkjc+-- Zo .... ~ksu- aoZm...+kj
Zom.,.lkj = Xo*.... ~,~so + Zo .... a,c~ + ao'Ym...~,~
--S+ i ] [
Thus
R y'nq = S y s j + aj~ajk -I- .. • + anqanq
XOm . . . Ikj
]TICi
[si'ajk
s,.Sq=|S,.s~xaj+|
Sometimes, the secondary formulae can be derived directly from their definition (24)-(26) by using the law of mixed product of three vectors, which is most important in the algebraic displacement analysis for the loop linkages. The terms with a single subscript, (Xoj, X~., Yoj, Zoj, Uoj, U~., V0j, Wo~) can be obtained either by processing the equations (21)-(23) with D operator or directly from equations (24)-(26).
a×b=
0
Partial derivatives
Let us consider any primary formulae in (16)-(18), for example, s~.Sq= Zj, . . . . . computing it in such a Cartesian coordinate system whose three coordinate
(31)
L
o -,.z ,.yl[ .xl a.z 0 -a.x//h.y/.
(32)
-a'y a-x 0 _]Lb.zJ Performing sk x Sq in the form of the equation (32) just yields the last two matrices in (31). Thus ~--~kl ..... = ~--~k(S+'Sq) =Si'Sk X Sq.
(33)
Using the same procedure, one can take the first and higher order derivatives with respect to all angles, such as Ok, Oj, Ore, %, ~k~, ~,,,' • •, for all the primary formulae, for example, ~0--'~Vjk . . . . ~ - ~
4. DIFFERENTIAL F O R M U L A E FOR THE SPATIAL KINEMATIC CHAIN
iSq.S+ × ,,q.
Sq'Sk d As one knows, the cross product of two vectors a and b, in terms of the Cartesian coordinate system whose three coordinate vectors are x, y and z, can be expressed
.
(28)
o
L s~'sk d
± aajk
~
"
..=
(aq's n X anq ) -----ao.'$ k
a__(_,+.s+ × a~jk
X
(S n
X
anq )
a,+)
= --S i'ajk × (Sq × anq)
63 8 a0~a0, zjk . . . . - a0~a0,, (s,. Sq) =S,'Sk × {Sk × (Sin × %)}
(34)
Vector theory of spatial mechanism analysis
213
n+l
etc. It is convenient to express these derivatives in the recursive notation by using the vector formulae, for example, ~kZjk
.....
= ~kk(Si'Sq)=Si'Sk X
~en
Sq
Ooi
I
: ii/lllllllq~
xj =
- ~
0
X . ....
/k
zj
0
1
Z ~ . ... tk
Fig. 5
+ ~ x . . . . . ,~)
(35)
where the elements of the determinent are the coordinate components of s~, Sk and Sq in the system (aSk, Sk × ask, Sk), referred to the equations (16) and (17). Further, using equation (5) yields ~Z
Ojkl .....
~D ~-----~k
k + Y j X , ....
-(XojX*~
,++xjx*. .... ,+
+ YojX, . . . . ,~ + Y~Xo. . . . . ,k).
For example,
ap = D(~23) + o23 x Vp
(36)
where the point p locates at the cross point of the vector a23 and s 3 on the link a23. The scalar components of (D23, Vp, [23 and ap along a23, s 3 × a23 and s 3 are expressed in the following forms: o,x = 01X~ o r = - 0, r2 + 0~s. ¢2~z = 0 1 Z 2 dt- 02c23 v x = D((Dx)
d
(41)
= 01X02 -[- ~1X2
Vy = D ( % ) = -(01Yo2 + $1Y2) + 02a23c23 + $2s23
~- ~-~ (Si' anq )
= s,-(OA + O+sk + . . .
v~ = O(co~) = 01Z02 + .~aZ2 - 02az3s23 + 5~zCz3 (42)
+ O.s.) × a.+.
~x= O~x~- #,O~x~
Also, using equation (7) yields Ot - ' '
Olsz + 02s2
Vp= D(¢O23)
The derivative with respect to time is the complex derivative. Without loss of generality, let us consider the following examples:
=Xo,+d
=
t23 = 0"is1 + O':s2 + 0~02sl × s2
Derivatives with respect to time
dt yjkl .....
(40)
a l , n ( n + 1) ~__.SiSl --t- a12a12 + • • • + S n $ n -t- an(nO Dan(nO I)"
oJ23
Analogously, the partial derivatives for all the secondary formulae can be obtained using the formulae (5), (8) and (9).
d
= D(t,~.+ o) + oJ.(.+ i) × vp
where
Ik)
= -D(XjX*,.., =
d2 Rl.n(~+ i)
ap = ~-7i
{ ..... )= Dt'~kZJk, ..... )
(zjk,
2
St
* . •. Ik X nm
= -(x,x.*..,~
n_l
Ey = --0" 1 Y2 + 0"2s23-- 0102X2c23
.....
\dt
"
,]
Ez = 0"1Z 2 ÷ ~2C23 "4- 0 1 0 2 ~ 2
+ 0.s.) x a.q)
= D(si" (Ojsj + OkSk + " "
where D(Oj)= Sj, etc. For an open kinematic chain connected by the pairs with axes, as shown in Fig. 5, with a frame in the end, the absolute angular velocity and acceleration for any link a.(.+ i) can be shown that
a~ = D(Ex) + %v= -- cowry
= OlXo~ + ~,x~ - (01~¢~ + .¢10~)x* - 0,0~x*~ + %v= - o~=vy ay = D(Ey) + w=v~ - % V z = --(0"1Yo2 + ~t r2) +
~.(.+ 1, = L Ojs,
(37)
j=l n /j--l.
~'2a23c23 + g2s23
-- (01~ 2 ÷ ~102)X2c23 -- 0102(X02c23 -- X2a23s23 ) + (OzV x - - (OxV z
x
,++,, :,.,i o;.,+L t,-x, o,s,)× O,s,
a~ = D ( G ) + WxVy -- OyV x
The absolute linear velocity and acceleration of the referred point P, at which the a,(, + ~/intersects s(, + 1) on the link a,~.+tl, can be computed as d Rl,.(. + 1) = D(m,~. + 1))
(43)
(39)
= 0",z0~ + g,z~ - O'~a.~. + g ~ c . + 0,0~.~o~ + (01~¢2 + ~¢t0~)y~ + o~xv~ - %v~.
(44)
It is easy to transform these components, (41)-(44), referred to the moving coordinate system into those referred to as a fixed coordinate system, to say, whose
214
HONG-You LEE and CHONO-GAoLLANO
coordinate vectors are aol, Sl )< a01 and sl, as shown in Fig. 5.
Primary loop equations. The dot product of any two of the elementary vectors can be computed in terms of two chains, say chain A and B, and the two resultant expressions should be equal. For example,
zk~. . . . = z q . . ,
etc. Because only are there two chains in a single-loop linkage, it is possible to write the primary equations as the following forms:
-- (Rq'q'aq)(Rq'q'Sq)
(50)
½(R-/,~. RJ."q) (sy.Sq) - (RJ.'~.sj)(R/,"q.Sq) (51)
etc. It is very important to note that this group of equations can be reduced to be linear in the sines and cosines of the angular displacements using the vector identities [refer to equations (80)-(82)].
In the loop equations, there are many pairs of equations in the following forms: r-a o = X
r.sj x a e = - Y (45)
etc. Secondary loop equations. Deriving the D functions of the above primary equations, one obtains the secondary equations:
where r is a vector. Expanding the pair of equations in the coordinate system formed by ajk, s j x aj~ and sj yields that .4cj - Bsj = X (52) Asy + Bcj = - Y
where
Wk.... sn x sj = R~.,J.sj x sn --
= l(Rq'ii'Rq'#)(aij'Sq)
,
s,'sj × ayk, Ym...k= - X * . . . o
(49)
Four groups o f half-angle equations
sn'sj × ajkla = s~'sj x ajkls, Ym...k= -X*...ij
Zkt .... = Z q
(48)
1 ( R j, nq. R j , tlq ) (a/j" sq ) - - ( R j' nq. a/j ) ( R j' nq. Sq)
= l(Rq'u'Rq'°)(s/sq) - (Rq'°'sj)(Rq'O'Sq)
Four groups o f loop equations
Wk.... s,(sj x ajk)
= 0
R j, nq. R j, nq = Rq, #. R q. q
Figure 6 shows a single-loop linkage, the direction of which is specified counterclockwise, therefore the arrow has been omitted. For the single-loop linkage, the two forms of four groups of displacement equations are developed as follows.
s/sn,
R.r
The fourth-group loop equations. Using the loop vector equation (47) yields
5. DISPLACEMENT EQUATIONS FOR LOOP LINKAGES
sj.s~l~=s/s.l~,
must be zero, i.e.
Rnq'j'($j × ay/) x S n .
(53)
.4 = r . a j k
B = r.sj x ajk.
That is,
Introducing the trignometric identical equations q., Z o k l .~-. Z. om
Vo.... k = - x ~
sin 0s - xj cos 0i = xj
.i
o
(46)
where R jk'"~ = ajkajk + Sksk + ' ' " + a,,~a,~ etc. Projective loop equations. In the loop linkages, the loop vector is equal to zero, that is, R = R j'nq + R q'0 = 0 .
(47)
Therefore, the scalar components along any direction #
j
xj sin 0j + cos 0j = 1 where xj=tan(Oj/2), rearranging the pair of equations (52) and (53) gives a pair of half-angle equations: (.4 + X)xj + (B + Y) = 0 (54) (B - Y)xj - (A - X ) = O.
Analogously, the four groups of the half-angle equations can be derived from the corresponding loop equations. Without loss of generality, let us take a 7-1ink mechanism as an example, as shown in Fig. 7. 4
,4
m
t~
Fig. 6. Single-loop linkage.
(55)
Fig. 7. 7-1ink mechanism.
Vector theory of spatial mechanism analysis
Primary half-angle equations. Only one example is given here. Using equations (45) yields S4" 871,
X321 = X567
(56)
$4'S1 X a71 , X~21 -- -- Y567"
(57)
X32c I -- Y32Sl ~-- X567
(58)
X32Sl -{- Y32Cl = - Y567.
(59)
That is,
215
where FI = 81'4'812 = a12 -I- a23c2 "4- $3A~2 "~- a34 W32 + $4X32 F 2 - - Rl,4.sl × a12 = - $2s12 --[-a23s2Cl2 -{- $3}Y2 - a34 U~2 --[-$4Y32.
The fourth-group half-angle equations. Using
Introducing xl = tan(01/2) in (58) and (59) yields (X32 + X567)Xl + (Y32 + ]"567) = 0
(60)
(Y32 -- Ys67)xl + (X32 - )(567) = 0.
(61)
Secondary half-angle equations. Computing the D functions of the equations (58) and (59) and rearranging yields
equations (50) and (51) yields ½(86, 2 3 . 8 6, 23 ) (83' a56 ) - (R 6' 23.83) ( 8 6, 23. a 56)
= ½(R 3,56. R 3,56) (s3"856) (72)
-- (R 3. 56.S3 ) ( R 3, 56. a56 ) ~(R16'23"R6'23)(s3"s6 x a56 ) - (86,23.s3)(86,23.86 x a56 ) = 1(83'56.83'56)(s3.s6 x 856 ) _ (R 3, 56.s3 ) (83. 56.86 x 856 )
(X032 -- $1 Y32)c1 - (Yon + SIY32)Sl = X0567
(62)
(X'032 -- $1 Y32)Sl + (Y032 + SIX32)(:1 -- - Y0567.
(63)
(73)
where 86, 23 = S6s 6 -[- a67867 + $7s7 --[-a71871 -3t- SIS 1
Thus,
"Jr"a12812 + S2s 2 + a23823 (Yo32 -- 51 Y32 "[" Yo567)x1
R3'56 = $3s3 + a34a34 + $484 + a45845 + Sss5 + a86a56. + (Y032 + $1X32 + Y0567) = 0
(64)
By definition,
(Y032 "Jr-SIX32 - Yo567)Xl --
(X032
G3 == 1 (R 3, 56. R 3, 56) (s 3 . a56 ) --
8|Y32 -- Xo567) ~---0.
(65)
-- (R 3, 56. s3 ) (R 3, 56. s6.856 )
Projective half-angle equations. Let us consider the
G4 = -½(83'56"83'56)(s3-s6 × 856 )
following pair of equations: R 1,4. a71
=
_
845, 71.871
R1'4.sl x 871 = - R 45'71 .s I x 871
+ (83'56-s3)(R3'56"s6 × as6 ).
(66) (67)
R 1"4 -- S1s I + a12a12 + S2s 2 "4- a23823 + $3S 3 834834 "[- S4s 4
( F 3 + G3)x 6 + ( F 4 + G4) = 0
(76)
( F 4 -- G 4 ) x 6 - ( F 3 - G3) = 0
(77)
where
R4s, 78 = a45a45 + $5s5 + a56a56 + $686 + a67867
F3 = ½(8 6'23. R 6,23) (s3"a67)
+ STS7 + a71871 •
-- (R 6, 23. s3 ) (R 6, 23. a67 )
By definition,
(78)
F 4 = ½(86.23.R6,23)(s3.s6 x a67 ) -- (R6'23.s3)(R6'23"s6 x a67 ).
GI = - R 45,71.871 = -- (a45 W567 -~- S5X67 -[- a56 W67 -{- S 6 X 7 + a 6 7 c 7 + a71 )
(68)
G 2 = R 45'71"s I x a71 = 845 U567 • -- 55 ]"67 + 856 U~7 -- 56 ]"7 -- a67s7c71 "4- $7871.
(69)
Substituting (68) and (69) into (66) and (67) and rearranging by using the same procedure as that from (52) to (55) yields (F1 +
(75)
Substituting (74) and (75) into (72) and (73) and rearranging by using the analogous procedure yields
where
+
(74)
)X 1 -[- (F2 + G2) = 0
(70)
( F 2 - G2)x I - (F, - G , ) = 0
(71)
GI
(79)
It should be noted that the G3 and G4 are linear in the sines and cosines of the angular displacements 04 and 05, the F3 and F4 are linear in the sines and cosines of the angular displacements 07, 01 and 02. The reduction of this group half-angle equations is the most significant task for the derivation of the 16th degree polynomial input-output equation in the tanhalf-angle of the output angular displacement for the general 7R mechanism, which can be performed as follows. Let us consider the following expression: ½(R'R)(rl 'r2) - (R' r 0 ( R ' r D where R is a chain vector, such
a s R 6"23
(80) and R x56,
216
HONO-You LEE and CHONG-GAo LIANG -- a34 W45 (53 at- $4 c34 ) - 834 a45 s5 Y4
-- ($3 + $4c34 + a45X4)(S4X5 + a45e5).
R
Fig. 8
(82)
Analogously, the F3, F 4 and G4 can also be expressed in the recursive notation.
r~ and r 2 are two elementary vectors at the ends of the chain, such as %, 867 and .% x 867 etc. (refer to Fig. 8). Let R = R~ + R2, and R2"R2 is equal to constant, such as 82 = $5s5 + a56a56, thus ½(R' 8 ) (r,' r2) - (R' r]) (R. r 2) -- I ( R 1• R l ) ( r I 'r2) - (R l • r l ) ( R l 'r2)
6. VELOCITY EQUATIONS FOR LOOP LINKAGES Figure 9 shows a spatial n-link mechanism with the frame link an,, the input and output angular displacements are 0, and 0, respectively. Using equation (37) yields
+ (R, "R2)(r] 'r2) - ( R 2 - r 0 (Rj "r2)
0:,=o
- ( R . r,) (R2.r2) q- I ( R 2. R2)(r I 'r2) = ½ ( R 1 . 8 2 ) ( r l . r 2 ) - (R, . r 0 ( a , - r 2 )
-+- ( 8 1 × r l ) ' ( 8 2 × r2) - ( R . r l ) ( S 2 . r 2 ) + ½(R2' R2)(r, "r2).
(83)
j=l
(81)
In which, the terms (R 1 × r,).(R 2 x r 2 ) - ( R . r J ( R 2 . r 2 ) + ½(82.R2)(rl 'r2) can be expanded using the definition of primary and secondary formulae, the terms
Deriving the three scalar components of (83) along three vectors, say a,,, s, x a,~, s,, together with their D functions gives a total of six equations which are linear in the velocity variables. F o r single-loop singledegree-of-freedom linkages, only are there six velocity variables besides an input velocity parameter. Thus all the relative velocity variables can be obtained by solving the six equations. The absolute velocities also can be obtained by following the same procedure as that of (39), (41) and (42).
½(R,. R,)(r, .r2) - (R,. rl)(R l •r2) must be expanded reusing the same procedure, until R~.R~ also equals to constant, such as r, = 5 3 s 3 q-- a34a34. For example, G3 = ½(R 3, 56. R 3, 56) (s 3. a56) _ (R 3,56.s3 ) (R 3, 56. a56) = ½( 8 3,45.8 3, 45) (S3. a56 ) _ ( 8 3, 45 .s 3 ) ( 8 3, 45. a56 ) -]-(R 3,45
X
s 3 ) . ( R 5'56
X
856 )
_ (R 3, 56. s3 ) ( 8 5, 56. a56) + ½( R 5, 56.8 5,56) (s3' a56 ) ~-. ½(83,45. R3,45) ($3. a56) _ (83,45. $ 3 ) ( 8 3, 45. a56)
7. ACCELERATION EQUATIONS FOR LOOP LINKAGES Taking the time-derivative of equation (83) yields f i ~jsj + nfil ( j ~ , Ois,) x 0jsj = 0. j=l
(84)
j=2\jffil
Using equation (84) also yields six equations which are three scalar components of (84) and their D functions. Solving them gives the six relative acceleration variables. Further, the absolute accelerations can be obtained by using equations (40), (43) and (44).
+ X04Sss5 + r04S5cs + ½(S~ + a~6)X45 8. CONCLUSION
-- a56(S 3 + 54c34 + a45X4 + S s Z 4 + 856X45)
where ½(83, 45. R 3,45) (S 3 . a56 ) _ ( 8 3,45.$3 ) (R 3, 45. a56) --_ ½( 8 3,34. R 3,34) (s3.856 )
_
(R 3,34,s3 ) (R 3, 34. a56)
+ ( 8 3,34 × $3)'(8 4'45 X 856)
The new vector theory has provided the closedform displacement analysis for the spatial 6-1ink R R R R R C , 7-1ink 6R-P, R R P R P R R and the general 7 8 mechanisms[7-9]. Also, the theory can be applied directly to the analysis for displacement, velocity and
_ ( 8 3. 45. s3 ) (R 4, 45. a56) .q_ ½(R 4, 45.84, 45) (S3.856) • QQ
= ~($3' 2 + a]4+S24+a]5)X,5_S3(S,X,5+a34W,5)
/ -,
-
-
a34(a4585Y4 + 54c34W45)
-- ( S 3 + S4c34 -[- a45X4)(S4X 5 -t- a45c5).
j+l p
\"
•
Thus G3 = ](__ I S32 ..[_ a~4 q,_ $42 + a45 2 + S~ - a26)X45
+ Xo4S5s5 + Yo4s5c5 -- a56(S 3 + S4c34 -~" a45X 4 + $5z4)
3
n n
I Fig. 9
Vector theory of spatial mechanism analysis acceleration of general single-loop single-degree-offreedom spatial mechanisms involving R, P, C, E and S pairs. Not only are the resultant expressions relatively concise, but also the procedure of derivation possess geometric meaning. It is hoped that the new vector theory developed in this paper may offer a useful alternative in the study of the spatial mechanisms. REFERENCES
I. M. R. Spiegel, Vector Analysis. Suhaum, New York (1959). 2. J. Duffy, Analysis of Mechanisms and Robot Manipulators. Arnold, London (1980).
UNE
217
3. J. Duffy and C. Crane, Mech. Mach. Theory 15, 153-169 (1980). 4. M. A. Chace, A S M E J. Engng Indust. 85, 289-297 (1963). 5. D. M. Wallace and F. Freudenstein, A S M E J. Engng Indust. 97, 575-580 (1975). 6. A. T. Yang and F. Freudenstein, A S M E J. appl. Mech. 31, 300-308 (1964). 7. H.-Y. Lee, Kinematic analysis of the spatial mechanisms. M.Sc. Thesis, Beijing Institute of Posts and Telecommunications (1984) [in Chinese]. 8. H.-Y. Lee and C.-G. Liang, Displacement analysis of the spatial 7-1ink 6R-P linkages. Mech. Mach. Theory 22(1), l - l l 0987). 9. H.-Y. Lee and C.-G. Liang, Displacement analysis of the general spatial 7-1ink 7R mechanism. Mech. Mach. Theory. 23(3), 219-226 (1988).
NOUVELLE THEORIE VECTORIELLE POUR L'ANALYSE DES MECANISMES DANS L'ESPACE R6sum6--Dans cet article on d6velopp¢ une nouvelle th6orie vectofielle pour l'analyse des m6canismes dans l'espace en utilisant ranalyse vectorielle ainsi que l'alg6bre des hombres duals et des symboles r6cursifs de J. Duffy. Cette th6orie comprend les 6quations de d6placement, vitesse et acckl6ration pour les chaines ouvertes et ferm6es. Les &luations du d6placement d'un m6canisme ferm6 sont class6es en quatre groupes avec deux formes: primaire, secondaire, projective et tangente du demiangle. Dans l'analyse de la vitesse et de l'acckl6ration les diff6rentiations sont transform6es en op6rations alg6briques et les 6quations unifi6es sont exprim6es sous formes vectorielle qui est lin6aire en ce qui concerne la vitesse et l'acc616ration.
0094-114X/88 $3.00+0.00 Copyright © 1988 Pergamon Press plc
A NEW VECTOR THEORY FOR THE ANALYSIS OF SPATIAL MECHANISMS HONG-YOU LEE and CHONG-GAO LIANG Department of Mechanical Engineering, Beijing Institute of Posts and Telecommunications, Beijing, People's Republic of China (Received 27 M a y 1986)
AMtraet--This paper develops a new vector theory for the analysis of spatial mechanisms on the basis of the vector analysis theory and dual-number algebra and the recursive notation presented by J. Duffy. The new vector theory includes the displacement, velocity and acceleration equations for the open kinematic chains and loop mechanisms. The displacement equations for the loop mechanisms are classified into four groups with two forms, namely the primary, secondary, projective and the fourth-group loop/half-angle equations, which have been applied to the derivation of the 16th degree polynomial input--output equations in the tan-half-angle of the output angular displacements for the general spatial 7-1ink 7R and 6R-P mechanisms, etc. In the analysis of the velocity and acceleration, the differential calculus are changed into vectorical algebraic operations and the unified equations are expressed in the vector forms which are linear in the velocity and acceleration variables.
1. INTRODUCTION In recent years there has been considerable development in the methods for the analysis of spatial mechanisms. This was initiated by the significant contribution of Dimentberg, Denavit and Hartenberg. Duffy presented the recursive notation which greatly facilitates the resultant expressions of algebraic equations and used the method of spherical trigonometry to derive the displacement inputoutput equations for almost all the single-loop singledegree-of-freedom spatial linkages involving R, P and C pairs[2, 3], except the spatial 7-1ink 6R-P and R R P R P R R linkages. Vectors, which are well known, were used by Chace to derive closed-form displacement relations of RCCC linkage[4]. Wallace and Freudenstein used a geometric configuration method to obtain closed-form displacement relations of R R S R R linkage[5]. Yang and Freudenstein derived the closed form algebraic expressions for displacements, velocities, forces and torques of all the links of RCCC linkage using quaternions and screw operators which possess geometric meaning[6]. In the present paper, a new vector theory is developed, the derivation of equations for displacements, velocities and accelerations are performed using the vector formulae which not only possess geometric meaning but also can avoid the tedious and lengthy operations of triangle functions, and the resultant equations are expressed in the recursive notations, presented by Duffy, which are relatively concise in form and more convenient for programming. This theory has been applied to the displacement analysis of the spatial 6-1ink R R R R R C , 7-1ink 6R-P, R R P R P R R and the general 7R mechanisms[7-9]. The mathematic development of the new vector theory for the analysis of
spatial mechanisms will be described in the following sections. 2. SPATIAL KINEMATIC CHAIN AND DUAL FUNCTION Specification o f spatial k i n e m a t i c chain
Figure 1 illustrates the general spatial kinematic chain of links connected by the joints, such as R, P, C pairs, whose axes are directed by unit vectors si, s j , . . . , the link axes, which are chosen along the common perpendicular between the adjacent joint axes, are directed by unit vectors aij, ajk. • • Hence, the
209
It/
/
*m~,,.~ f
Sm
"",
(0)
/,,
~q
(b)
?ig. 1. (a) Spatial kinematic chain; (b) simplified representation.
210
HONG-You LEE and CHONG-GAoLIANG rl
r2
sj
#2
O
Fig. 2
Fig. 4
link length a~ is the shortest distance between two adjacent pair axes st and sj, measured along a 0, and the offset of joint axes Sj is the shortest distance between two adjacent link axes a,~ and ajk, measured along sj. The angle a~ is measured by the right rotation of s~ to sj about the aq, the angle 0j is measured by the right rotation of a~ to ajk about the sj. The others are analogously defined by changing the subscripts. Therefore, the three unit-vectors (a,y, s j x a0, sj or a 0, s~ x aq, s~) can form a Cartesian coordinate system. Also, the following two equations will be often used later in the derivation of the vector form equations:
where op is the position vector of arbitrary point p on the line of unit vector r, referred to a specified point O, as shown in Fig. 3. Let F be a function of the real variables x~ which are the primary components of the dual-number variables xi=xi+~x0i, That is, F = f ( x l , x2 . . . . . x,). The dual-component function corresponding to F, denoted by D(F), is defined by '.L
st x Sj = sin ocoa0
(1)
withi=l,2,...,n.
OF
"
D(F) = i~-'~jx°i~-vx~= ~ x ° ~ - - f ( x l ' x2 . . . . . xn). i= 1
a o x ajk = sin 0~s~.
(2)
By introducing the dual-angle and dual-unitvector, the parameters of the spatial chain can be expressed as: dual-angle link:
~0 = ~o + ea~
dual-angle pair:
0j = Oj+ ESj
dual-unit-vector link:
~i~= ao + Ea*
dual-unit vector pair:
g = sj + Es*
where: e is the dual operator, and e2= e 3 . . . . . 0, a* = opj x a,s, s* = op2 × sj and opl, op2 are respectively position vectors of arbitrary points pj and P2 on the lines of the unit vectors a 0 and sj, referred to a specified point O, as shown in Fig. 2. The others can be analogously defined. For the purpose of intensive analysis it is more convenient to define the D operator (or D function) which means dualization operation, i.e. to derive the corresponding dual (secondary) component from the real (primary) component.
Definition of D operator Let r represent a unit vector, the dual-component vector corresponding to r, denoted by D (r), is defined by D(r) = op × r (3)
(4)
UXi
Thus, for the real functions F, F~, F2 and unit vectors rl, r2, using (3), (4) yields =
dD(F) =
(5)
D(d \~] 5
(6)
8F D (F) = Ox---~
(7)
D(F, +_F2) = D(F 0 + D(F2)
(8)
D(F, F2) = D(FOF2 + F~D(F2) D(FI/F2) = {D(FI)F 2 -- F,D(F2)}/F ~
(9) (10)
D(ri "r2) = rl 'D(r2) + D(r0"r2 =plp2.r2 x rl,
(11)
where PiP2 is the vector from point p, to point P2, which are respectively arbitrary ones on the lines of vectors rl and r2, as shown in Fig. 4. Because of the arbitrariness, using (11) yields D(rj "r2)=p~p~'r2
x r1
thus plp2.r2 x r I = p ~ p ~ . r 2 x r 1.
(12)
In fact, equation (12) is one of the scalar components of the vector loop equation R . r z × rl = 0
where
R =PIP2 +P2P2 +P~P~ +P~P~ = 0 Fig. 3
(refer to Fig. 4)
(13)
Vector theory of spatial mechanism analysis In this paper, equations from (12) are known as secondary equations, the other scalar components of (13) are known as projective equations
3. F U N D A M E N T A L
FORMULAE
FOR
SPATIAL
KINEMATIC CHAIN F o r the kinematic chain as shown in Fig. 1, the unit vectors a0, s,, s~ × a~, sj, s~... go by the general name of elementary vectors denoted by r~ with i = 1, 2 , . . . , and the chain vectors, denoted by R with the superscripts, are defined by R ~," = Sis ~+ aoa o + S~s~ + . .
• + SnS n
R °'''' = a~a~ + S~sj + ajka~, + . • • + a,,,na,,,,,
aij "sn × anq = Vykt . . . mn
211 aq" Sq x anq = U ~I . . . .
a,q.Syx a q = - V, . . . . t~
a,q.S~x a~y= - U*,,...~j. (18)
The subscripts of the recursive notations X, X*, Y, Z and U, U*, V, W are the labels between the two vectors of the left side of the equations and indicate that the expressions are linear sines and cosines of the angular displacements with these labels. Also, these subscripts are the recursive number in sequence. It should be noted that the terms U and U* in this paper are not exactly equal to those in[2, 3]. Introducing the definitions (16)-(18) into the recursive vector formulae (14), (15) yields the primary recursive formulae. P r i m a r y recursive f o r m u l a e
etc. The expressions for the scalar product of any two elementary vectors can be derived using the scalar product law. As one knows, the scalar product of two vectors r l and r2, can be expressed by r~. r 2 = (r~" x ) (r 2 • x ) + (r 1 • y ) ( r 2 • y ) + ( r x • z ) (r 2 • z )
Xm...tkj = xm...~,cj - Y,,...tksj x*...~kj = Xm...~Sj + Ym...t,c x Y~... ~k~= X ~ . . . t,~c u -- Z . . . . t~s~
w ~ , . . . . . = u?~, . . . . s. + wj~, . . . . c.
where x, y and z are unit vectors for coordinate axes which will be replaced by a0., s~ × a~j, s~ or a0., sj x ai~,
vj~, . . . .
= u J, . . . . c . -
Sj o r . . .
U~k, . . . . .
~-" Ujkl . . . . Snq + Vjkl . . . .
Thus, the scalar product of any two elementary vectors can be expressed in the form
Ujk' . . . . . = Uy,, . . . . c . o -
s.'sj × a,~ = (s,.a~,)s~ + (s,-s~ x aj~)cy × a o. = ( s . 's~ × a~)c,~ - ( s . . s ) s ~
s,'s, = (s,' s j x a/y)s# + (s,.sy)c,~
(14)
aij.anq = (aq.s n × aran)Sn + (a~' a~,)c,
a/j'Sq = (aq'sn)Cnq -- (a,y's~ ×
anq)Cnq anq)Snq
Uj = a,~'sk = sjsjk;
Cnq
V~k,. . . . . S,q.
= ao.'s j × ajk = - s j ;
I'Vj.= a,j- ajk = cj; (15)
where, c~ = cos 0~, sj = sin 0y, c~ --- cos et0, s,~ --- sin ~t,y, etc. and in the following sections, the sines and cosines are designated in the same way, also the tangent of the half-angle 0fl2 is denoted by x~, i.e. x~ = tan(0fl2), etc. The equations (14) and (15) are known as vector recursive formulae, and it is more convenient to define the dot products as the recursive notations which are presented by Duffy[2].
U*=
-aik.si×a,j=-sic o
7j = -ajk'sj
× a~ =
ff/j=ajk'ao=cj
Si " anq "~ Y j k l . . . ran = anq " Si = Unto... kj
S i ' $ n X anq -~- - - X ~ k t . . . .
Sq" s j x a o = X*m...tk~
X * = - s , . s j × aik = - s u e j
Yj = - s ~ - s k
x ajk = - ( % s j ,
+ socjkcj)
Zj = s;sk = cijcj~ - sijsjkcj ~7i = sk" a,j =
(22)
sj, sj
A'* = sk'sj × a 0 =
-sjkcj
= sk's~ × a~ = - ( c j k s o +
sj, c~cj)
(23)
Definition o f s e c o n d a r y f o r m u l a e
(16)
Rq'nq'sq
× Si = Zojkl . . . . .
~- Z o . . . . . Ikj
RiJ'n'anq x $, = Xojkt . . . . = Uo. . . . . ,kj
S i "~q X anq = - - Yj*t . . . .
Sq" s~ x a o = Y . . . . . lkj
(21)
Processing the two sides of equations (16)-(18) with D operator yields the secondary formulae as follows.
= S q ' S i = Z n . . . . kj
aij" anq = Wjkt . . . . . = anq" aij = W . . . . . kj
-sj
x j = s,. aj~ = s,jsj
= Sk'Si = CjkCo -- SjkSqCj. Definition o f p r i m a r y f o r m u l a e S," Sq ~- Z j k I . . . . .
(20)
~ = a f s , = sjs~
U * = a u . s k × ajk= -s~cjk;
a q ' s n x anq = (a/j's n × a,rm)c n - - ( a 0 . ' a ~ , ) s .
aq.Sq X anq = (aij'Sn)Snq + (a#'s n ×
~ , , . . . . s.
All the terms X, X*, Y, Z, U, U*, V, W, if only with a single subscript, can be computed directly from equations (16)-(18), but it is necessary to distinguish the two groups by introducing the symbol " - " . They can be expressed as follows:
Sn'a O.= (s,~.ajk)c j -- (Sn'Sj X a)k)sj
s..s,
(19)
Z . . . . til= X * . . t ~ j s q + Z,~...tkC o
(17)
R j'"" a,q x a,j = W0m ..... = W0. . . . lkj
(24)
212
HONG-YOu LEE and CHONG-GAOLIANG
R'J'".(s.
x anq ) x s i = X
anq)
R),nq.sq
X
(Sj X %) = X o..... ~,j
=
--
Yojkl . . . .
RV"nq. Sq × (Si × ao. ) = Yo . . . . . lkj
R z ' " (s.
anq)
X
aq = Voj~l . . . . .
RJ'nq'(Sq x anq)
X
a O. = Uojkl . . . . .
x
vectors are %, s~ x % and sk yields
.....
RiJ'nq.(sq
X
$i
--Xoykl
F s. %
IT[
× ad (25)
Sq'% l
iSq.S+ x a,q.
(29)
L s,'sk A L Sq'Sk 3 Further, by using the transformation of vector Sq from the coordinate system ajk, Sk × ayk, sk to the coordinate system ak~, Sk x ak. Sk, the equation (29) becomes
R y' "' anq X (Sj X aij ) = -- V 0. . . . . Ikj
R q'''a"q x (s~ x %) =
--
* . . . Ikj U onm
(26)
L s~'sk d
where the chain vectors defined by
RJ'" = S~s~+ aye% + .. • + S,s.
~k
etc. And the terms (X0, X*, Yo, Zo, Uo, U*, V0, Wo), with the additional subscript 0, are the dualcomponent functions of the terms (X, X*, Y, Z, U, U*, V, W) which have the same subscripts. Thus, the recursive formulae for the terms X0, X~', Yo, Zo, Uo, U*, V0, W0 can be obtained by processing the equations (19) and (20) with D operator, and they are suitable for programming and known as secondary recursive formulae. Secondary recursive formulae
Xo.... ++q- Yom...~ksj- s i x * ..~kj
X
Xo.... +,sj+ ro~...~kq + SjXm...+kj
om
Yom
. . ~y --
. . . lkj
U&~. . . . c , - Wo~,~. . . . s . - S.~,~ . . . . . Uojkl . . . . enq -- Vojkl . . . . . Snq -- anq U ~ I . . . .
~---~
is+.s+ ×,+,l . (30)
S q ' a kl
1
0
L
Sq'Sk
-]
(Si" Sq)
/c~
-s+
/so skxa~,/
L si'sk J L o
o
L s+.sk j
=Is"s+xa'q
1T[!,
=is,-s xa, q L
s~'sk I
[
sq'akt
sk !]
o Olls,
c,
0
0
O_ILO
1
Sq'Sk
d
= ls,.skx a,q (27)
U oj*~. . . m. -- Uojkl .... Snq -~- VOjkl ..... Cnq -}- anq Ujk I ..... Uojkl. . . mn
zjk` . . . .
L
Wojkl. . . mn = Uoj,~.... s. + Wo~,~.... c, + S , V ~ ..... Vojkl . . . mn
c+
X ISq'Sk × a k l ]
= Xo .... tkjc+-- Zo .... ~ksu- aoZm...+kj
Zom.,.lkj = Xo*.... ~,~so + Zo .... a,c~ + ao'Ym...~,~
--S+ i ] [
Thus
R y'nq = S y s j + aj~ajk -I- .. • + anqanq
XOm . . . Ikj
]TICi
[si'ajk
s,.Sq=|S,.s~xaj+|
Sometimes, the secondary formulae can be derived directly from their definition (24)-(26) by using the law of mixed product of three vectors, which is most important in the algebraic displacement analysis for the loop linkages. The terms with a single subscript, (Xoj, X~., Yoj, Zoj, Uoj, U~., V0j, Wo~) can be obtained either by processing the equations (21)-(23) with D operator or directly from equations (24)-(26).
a×b=
0
Partial derivatives
Let us consider any primary formulae in (16)-(18), for example, s~.Sq= Zj, . . . . . computing it in such a Cartesian coordinate system whose three coordinate
(31)
L
o -,.z ,.yl[ .xl a.z 0 -a.x//h.y/.
(32)
-a'y a-x 0 _]Lb.zJ Performing sk x Sq in the form of the equation (32) just yields the last two matrices in (31). Thus ~--~kl ..... = ~--~k(S+'Sq) =Si'Sk X Sq.
(33)
Using the same procedure, one can take the first and higher order derivatives with respect to all angles, such as Ok, Oj, Ore, %, ~k~, ~,,,' • •, for all the primary formulae, for example, ~0--'~Vjk . . . . ~ - ~
4. DIFFERENTIAL F O R M U L A E FOR THE SPATIAL KINEMATIC CHAIN
iSq.S+ × ,,q.
Sq'Sk d As one knows, the cross product of two vectors a and b, in terms of the Cartesian coordinate system whose three coordinate vectors are x, y and z, can be expressed
.
(28)
o
L s~'sk d
± aajk
~
"
..=
(aq's n X anq ) -----ao.'$ k
a__(_,+.s+ × a~jk
X
(S n
X
anq )
a,+)
= --S i'ajk × (Sq × anq)
63 8 a0~a0, zjk . . . . - a0~a0,, (s,. Sq) =S,'Sk × {Sk × (Sin × %)}
(34)
Vector theory of spatial mechanism analysis
213
n+l
etc. It is convenient to express these derivatives in the recursive notation by using the vector formulae, for example, ~kZjk
.....
= ~kk(Si'Sq)=Si'Sk X
~en
Sq
Ooi
I
: ii/lllllllq~
xj =
- ~
0
X . ....
/k
zj
0
1
Z ~ . ... tk
Fig. 5
+ ~ x . . . . . ,~)
(35)
where the elements of the determinent are the coordinate components of s~, Sk and Sq in the system (aSk, Sk × ask, Sk), referred to the equations (16) and (17). Further, using equation (5) yields ~Z
Ojkl .....
~D ~-----~k
k + Y j X , ....
-(XojX*~
,++xjx*. .... ,+
+ YojX, . . . . ,~ + Y~Xo. . . . . ,k).
For example,
ap = D(~23) + o23 x Vp
(36)
where the point p locates at the cross point of the vector a23 and s 3 on the link a23. The scalar components of (D23, Vp, [23 and ap along a23, s 3 × a23 and s 3 are expressed in the following forms: o,x = 01X~ o r = - 0, r2 + 0~s. ¢2~z = 0 1 Z 2 dt- 02c23 v x = D((Dx)
d
(41)
= 01X02 -[- ~1X2
Vy = D ( % ) = -(01Yo2 + $1Y2) + 02a23c23 + $2s23
~- ~-~ (Si' anq )
= s,-(OA + O+sk + . . .
v~ = O(co~) = 01Z02 + .~aZ2 - 02az3s23 + 5~zCz3 (42)
+ O.s.) × a.+.
~x= O~x~- #,O~x~
Also, using equation (7) yields Ot - ' '
Olsz + 02s2
Vp= D(¢O23)
The derivative with respect to time is the complex derivative. Without loss of generality, let us consider the following examples:
=Xo,+d
=
t23 = 0"is1 + O':s2 + 0~02sl × s2
Derivatives with respect to time
dt yjkl .....
(40)
a l , n ( n + 1) ~__.SiSl --t- a12a12 + • • • + S n $ n -t- an(nO Dan(nO I)"
oJ23
Analogously, the partial derivatives for all the secondary formulae can be obtained using the formulae (5), (8) and (9).
d
= D(t,~.+ o) + oJ.(.+ i) × vp
where
Ik)
= -D(XjX*,.., =
d2 Rl.n(~+ i)
ap = ~-7i
{ ..... )= Dt'~kZJk, ..... )
(zjk,
2
St
* . •. Ik X nm
= -(x,x.*..,~
n_l
Ey = --0" 1 Y2 + 0"2s23-- 0102X2c23
.....
\dt
"
,]
Ez = 0"1Z 2 ÷ ~2C23 "4- 0 1 0 2 ~ 2
+ 0.s.) x a.q)
= D(si" (Ojsj + OkSk + " "
where D(Oj)= Sj, etc. For an open kinematic chain connected by the pairs with axes, as shown in Fig. 5, with a frame in the end, the absolute angular velocity and acceleration for any link a.(.+ i) can be shown that
a~ = D(Ex) + %v= -- cowry
= OlXo~ + ~,x~ - (01~¢~ + .¢10~)x* - 0,0~x*~ + %v= - o~=vy ay = D(Ey) + w=v~ - % V z = --(0"1Yo2 + ~t r2) +
~.(.+ 1, = L Ojs,
(37)
j=l n /j--l.
~'2a23c23 + g2s23
-- (01~ 2 ÷ ~102)X2c23 -- 0102(X02c23 -- X2a23s23 ) + (OzV x - - (OxV z
x
,++,, :,.,i o;.,+L t,-x, o,s,)× O,s,
a~ = D ( G ) + WxVy -- OyV x
The absolute linear velocity and acceleration of the referred point P, at which the a,(, + ~/intersects s(, + 1) on the link a,~.+tl, can be computed as d Rl,.(. + 1) = D(m,~. + 1))
(43)
(39)
= 0",z0~ + g,z~ - O'~a.~. + g ~ c . + 0,0~.~o~ + (01~¢2 + ~¢t0~)y~ + o~xv~ - %v~.
(44)
It is easy to transform these components, (41)-(44), referred to the moving coordinate system into those referred to as a fixed coordinate system, to say, whose
214
HONG-You LEE and CHONO-GAoLLANO
coordinate vectors are aol, Sl )< a01 and sl, as shown in Fig. 5.
Primary loop equations. The dot product of any two of the elementary vectors can be computed in terms of two chains, say chain A and B, and the two resultant expressions should be equal. For example,
zk~. . . . = z q . . ,
etc. Because only are there two chains in a single-loop linkage, it is possible to write the primary equations as the following forms:
-- (Rq'q'aq)(Rq'q'Sq)
(50)
½(R-/,~. RJ."q) (sy.Sq) - (RJ.'~.sj)(R/,"q.Sq) (51)
etc. It is very important to note that this group of equations can be reduced to be linear in the sines and cosines of the angular displacements using the vector identities [refer to equations (80)-(82)].
In the loop equations, there are many pairs of equations in the following forms: r-a o = X
r.sj x a e = - Y (45)
etc. Secondary loop equations. Deriving the D functions of the above primary equations, one obtains the secondary equations:
where r is a vector. Expanding the pair of equations in the coordinate system formed by ajk, s j x aj~ and sj yields that .4cj - Bsj = X (52) Asy + Bcj = - Y
where
Wk.... sn x sj = R~.,J.sj x sn --
= l(Rq'ii'Rq'#)(aij'Sq)
,
s,'sj × ayk, Ym...k= - X * . . . o
(49)
Four groups o f half-angle equations
sn'sj × ajkla = s~'sj x ajkls, Ym...k= -X*...ij
Zkt .... = Z q
(48)
1 ( R j, nq. R j , tlq ) (a/j" sq ) - - ( R j' nq. a/j ) ( R j' nq. Sq)
= l(Rq'u'Rq'°)(s/sq) - (Rq'°'sj)(Rq'O'Sq)
Four groups o f loop equations
Wk.... s,(sj x ajk)
= 0
R j, nq. R j, nq = Rq, #. R q. q
Figure 6 shows a single-loop linkage, the direction of which is specified counterclockwise, therefore the arrow has been omitted. For the single-loop linkage, the two forms of four groups of displacement equations are developed as follows.
s/sn,
R.r
The fourth-group loop equations. Using the loop vector equation (47) yields
5. DISPLACEMENT EQUATIONS FOR LOOP LINKAGES
sj.s~l~=s/s.l~,
must be zero, i.e.
Rnq'j'($j × ay/) x S n .
(53)
.4 = r . a j k
B = r.sj x ajk.
That is,
Introducing the trignometric identical equations q., Z o k l .~-. Z. om
Vo.... k = - x ~
sin 0s - xj cos 0i = xj
.i
o
(46)
where R jk'"~ = ajkajk + Sksk + ' ' " + a,,~a,~ etc. Projective loop equations. In the loop linkages, the loop vector is equal to zero, that is, R = R j'nq + R q'0 = 0 .
(47)
Therefore, the scalar components along any direction #
j
xj sin 0j + cos 0j = 1 where xj=tan(Oj/2), rearranging the pair of equations (52) and (53) gives a pair of half-angle equations: (.4 + X)xj + (B + Y) = 0 (54) (B - Y)xj - (A - X ) = O.
Analogously, the four groups of the half-angle equations can be derived from the corresponding loop equations. Without loss of generality, let us take a 7-1ink mechanism as an example, as shown in Fig. 7. 4
,4
m
t~
Fig. 6. Single-loop linkage.
(55)
Fig. 7. 7-1ink mechanism.
Vector theory of spatial mechanism analysis
Primary half-angle equations. Only one example is given here. Using equations (45) yields S4" 871,
X321 = X567
(56)
$4'S1 X a71 , X~21 -- -- Y567"
(57)
X32c I -- Y32Sl ~-- X567
(58)
X32Sl -{- Y32Cl = - Y567.
(59)
That is,
215
where FI = 81'4'812 = a12 -I- a23c2 "4- $3A~2 "~- a34 W32 + $4X32 F 2 - - Rl,4.sl × a12 = - $2s12 --[-a23s2Cl2 -{- $3}Y2 - a34 U~2 --[-$4Y32.
The fourth-group half-angle equations. Using
Introducing xl = tan(01/2) in (58) and (59) yields (X32 + X567)Xl + (Y32 + ]"567) = 0
(60)
(Y32 -- Ys67)xl + (X32 - )(567) = 0.
(61)
Secondary half-angle equations. Computing the D functions of the equations (58) and (59) and rearranging yields
equations (50) and (51) yields ½(86, 2 3 . 8 6, 23 ) (83' a56 ) - (R 6' 23.83) ( 8 6, 23. a 56)
= ½(R 3,56. R 3,56) (s3"856) (72)
-- (R 3. 56.S3 ) ( R 3, 56. a56 ) ~(R16'23"R6'23)(s3"s6 x a56 ) - (86,23.s3)(86,23.86 x a56 ) = 1(83'56.83'56)(s3.s6 x 856 ) _ (R 3, 56.s3 ) (83. 56.86 x 856 )
(X032 -- $1 Y32)c1 - (Yon + SIY32)Sl = X0567
(62)
(X'032 -- $1 Y32)Sl + (Y032 + SIX32)(:1 -- - Y0567.
(63)
(73)
where 86, 23 = S6s 6 -[- a67867 + $7s7 --[-a71871 -3t- SIS 1
Thus,
"Jr"a12812 + S2s 2 + a23823 (Yo32 -- 51 Y32 "[" Yo567)x1
R3'56 = $3s3 + a34a34 + $484 + a45845 + Sss5 + a86a56. + (Y032 + $1X32 + Y0567) = 0
(64)
By definition,
(Y032 "Jr-SIX32 - Yo567)Xl --
(X032
G3 == 1 (R 3, 56. R 3, 56) (s 3 . a56 ) --
8|Y32 -- Xo567) ~---0.
(65)
-- (R 3, 56. s3 ) (R 3, 56. s6.856 )
Projective half-angle equations. Let us consider the
G4 = -½(83'56"83'56)(s3-s6 × 856 )
following pair of equations: R 1,4. a71
=
_
845, 71.871
R1'4.sl x 871 = - R 45'71 .s I x 871
+ (83'56-s3)(R3'56"s6 × as6 ).
(66) (67)
R 1"4 -- S1s I + a12a12 + S2s 2 "4- a23823 + $3S 3 834834 "[- S4s 4
( F 3 + G3)x 6 + ( F 4 + G4) = 0
(76)
( F 4 -- G 4 ) x 6 - ( F 3 - G3) = 0
(77)
where
R4s, 78 = a45a45 + $5s5 + a56a56 + $686 + a67867
F3 = ½(8 6'23. R 6,23) (s3"a67)
+ STS7 + a71871 •
-- (R 6, 23. s3 ) (R 6, 23. a67 )
By definition,
(78)
F 4 = ½(86.23.R6,23)(s3.s6 x a67 ) -- (R6'23.s3)(R6'23"s6 x a67 ).
GI = - R 45,71.871 = -- (a45 W567 -~- S5X67 -[- a56 W67 -{- S 6 X 7 + a 6 7 c 7 + a71 )
(68)
G 2 = R 45'71"s I x a71 = 845 U567 • -- 55 ]"67 + 856 U~7 -- 56 ]"7 -- a67s7c71 "4- $7871.
(69)
Substituting (68) and (69) into (66) and (67) and rearranging by using the same procedure as that from (52) to (55) yields (F1 +
(75)
Substituting (74) and (75) into (72) and (73) and rearranging by using the analogous procedure yields
where
+
(74)
)X 1 -[- (F2 + G2) = 0
(70)
( F 2 - G2)x I - (F, - G , ) = 0
(71)
GI
(79)
It should be noted that the G3 and G4 are linear in the sines and cosines of the angular displacements 04 and 05, the F3 and F4 are linear in the sines and cosines of the angular displacements 07, 01 and 02. The reduction of this group half-angle equations is the most significant task for the derivation of the 16th degree polynomial input-output equation in the tanhalf-angle of the output angular displacement for the general 7R mechanism, which can be performed as follows. Let us consider the following expression: ½(R'R)(rl 'r2) - (R' r 0 ( R ' r D where R is a chain vector, such
a s R 6"23
(80) and R x56,
216
HONO-You LEE and CHONG-GAo LIANG -- a34 W45 (53 at- $4 c34 ) - 834 a45 s5 Y4
-- ($3 + $4c34 + a45X4)(S4X5 + a45e5).
R
Fig. 8
(82)
Analogously, the F3, F 4 and G4 can also be expressed in the recursive notation.
r~ and r 2 are two elementary vectors at the ends of the chain, such as %, 867 and .% x 867 etc. (refer to Fig. 8). Let R = R~ + R2, and R2"R2 is equal to constant, such as 82 = $5s5 + a56a56, thus ½(R' 8 ) (r,' r2) - (R' r]) (R. r 2) -- I ( R 1• R l ) ( r I 'r2) - (R l • r l ) ( R l 'r2)
6. VELOCITY EQUATIONS FOR LOOP LINKAGES Figure 9 shows a spatial n-link mechanism with the frame link an,, the input and output angular displacements are 0, and 0, respectively. Using equation (37) yields
+ (R, "R2)(r] 'r2) - ( R 2 - r 0 (Rj "r2)
0:,=o
- ( R . r,) (R2.r2) q- I ( R 2. R2)(r I 'r2) = ½ ( R 1 . 8 2 ) ( r l . r 2 ) - (R, . r 0 ( a , - r 2 )
-+- ( 8 1 × r l ) ' ( 8 2 × r2) - ( R . r l ) ( S 2 . r 2 ) + ½(R2' R2)(r, "r2).
(83)
j=l
(81)
In which, the terms (R 1 × r,).(R 2 x r 2 ) - ( R . r J ( R 2 . r 2 ) + ½(82.R2)(rl 'r2) can be expanded using the definition of primary and secondary formulae, the terms
Deriving the three scalar components of (83) along three vectors, say a,,, s, x a,~, s,, together with their D functions gives a total of six equations which are linear in the velocity variables. F o r single-loop singledegree-of-freedom linkages, only are there six velocity variables besides an input velocity parameter. Thus all the relative velocity variables can be obtained by solving the six equations. The absolute velocities also can be obtained by following the same procedure as that of (39), (41) and (42).
½(R,. R,)(r, .r2) - (R,. rl)(R l •r2) must be expanded reusing the same procedure, until R~.R~ also equals to constant, such as r, = 5 3 s 3 q-- a34a34. For example, G3 = ½(R 3, 56. R 3, 56) (s 3. a56) _ (R 3,56.s3 ) (R 3, 56. a56) = ½( 8 3,45.8 3, 45) (S3. a56 ) _ ( 8 3, 45 .s 3 ) ( 8 3, 45. a56 ) -]-(R 3,45
X
s 3 ) . ( R 5'56
X
856 )
_ (R 3, 56. s3 ) ( 8 5, 56. a56) + ½( R 5, 56.8 5,56) (s3' a56 ) ~-. ½(83,45. R3,45) ($3. a56) _ (83,45. $ 3 ) ( 8 3, 45. a56)
7. ACCELERATION EQUATIONS FOR LOOP LINKAGES Taking the time-derivative of equation (83) yields f i ~jsj + nfil ( j ~ , Ois,) x 0jsj = 0. j=l
(84)
j=2\jffil
Using equation (84) also yields six equations which are three scalar components of (84) and their D functions. Solving them gives the six relative acceleration variables. Further, the absolute accelerations can be obtained by using equations (40), (43) and (44).
+ X04Sss5 + r04S5cs + ½(S~ + a~6)X45 8. CONCLUSION
-- a56(S 3 + 54c34 + a45X4 + S s Z 4 + 856X45)
where ½(83, 45. R 3,45) (S 3 . a56 ) _ ( 8 3,45.$3 ) (R 3, 45. a56) --_ ½( 8 3,34. R 3,34) (s3.856 )
_
(R 3,34,s3 ) (R 3, 34. a56)
+ ( 8 3,34 × $3)'(8 4'45 X 856)
The new vector theory has provided the closedform displacement analysis for the spatial 6-1ink R R R R R C , 7-1ink 6R-P, R R P R P R R and the general 7 8 mechanisms[7-9]. Also, the theory can be applied directly to the analysis for displacement, velocity and
_ ( 8 3. 45. s3 ) (R 4, 45. a56) .q_ ½(R 4, 45.84, 45) (S3.856) • QQ
= ~($3' 2 + a]4+S24+a]5)X,5_S3(S,X,5+a34W,5)
/ -,
-
-
a34(a4585Y4 + 54c34W45)
-- ( S 3 + S4c34 -[- a45X4)(S4X 5 -t- a45c5).
j+l p
\"
•
Thus G3 = ](__ I S32 ..[_ a~4 q,_ $42 + a45 2 + S~ - a26)X45
+ Xo4S5s5 + Yo4s5c5 -- a56(S 3 + S4c34 -~" a45X 4 + $5z4)
3
n n
I Fig. 9
Vector theory of spatial mechanism analysis acceleration of general single-loop single-degree-offreedom spatial mechanisms involving R, P, C, E and S pairs. Not only are the resultant expressions relatively concise, but also the procedure of derivation possess geometric meaning. It is hoped that the new vector theory developed in this paper may offer a useful alternative in the study of the spatial mechanisms. REFERENCES
I. M. R. Spiegel, Vector Analysis. Suhaum, New York (1959). 2. J. Duffy, Analysis of Mechanisms and Robot Manipulators. Arnold, London (1980).
UNE
217
3. J. Duffy and C. Crane, Mech. Mach. Theory 15, 153-169 (1980). 4. M. A. Chace, A S M E J. Engng Indust. 85, 289-297 (1963). 5. D. M. Wallace and F. Freudenstein, A S M E J. Engng Indust. 97, 575-580 (1975). 6. A. T. Yang and F. Freudenstein, A S M E J. appl. Mech. 31, 300-308 (1964). 7. H.-Y. Lee, Kinematic analysis of the spatial mechanisms. M.Sc. Thesis, Beijing Institute of Posts and Telecommunications (1984) [in Chinese]. 8. H.-Y. Lee and C.-G. Liang, Displacement analysis of the spatial 7-1ink 6R-P linkages. Mech. Mach. Theory 22(1), l - l l 0987). 9. H.-Y. Lee and C.-G. Liang, Displacement analysis of the general spatial 7-1ink 7R mechanism. Mech. Mach. Theory. 23(3), 219-226 (1988).
NOUVELLE THEORIE VECTORIELLE POUR L'ANALYSE DES MECANISMES DANS L'ESPACE R6sum6--Dans cet article on d6velopp¢ une nouvelle th6orie vectofielle pour l'analyse des m6canismes dans l'espace en utilisant ranalyse vectorielle ainsi que l'alg6bre des hombres duals et des symboles r6cursifs de J. Duffy. Cette th6orie comprend les 6quations de d6placement, vitesse et acckl6ration pour les chaines ouvertes et ferm6es. Les &luations du d6placement d'un m6canisme ferm6 sont class6es en quatre groupes avec deux formes: primaire, secondaire, projective et tangente du demiangle. Dans l'analyse de la vitesse et de l'acckl6ration les diff6rentiations sont transform6es en op6rations alg6briques et les 6quations unifi6es sont exprim6es sous formes vectorielle qui est lin6aire en ce qui concerne la vitesse et l'acc616ration.