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Derivative synthesis of plane mechanisms to generate functions of two variables
Jnl. MechanismsVolume5, pp. 249-271/PergamonPress1970/Printedin GreatBritain
Derivative Synthesis of Plane Mechanisms to Generate Functions of Two Variables K. Lakshminarayana*
and R. G. Narayanamurthi~:
Received 1 October 1969 Abstract Successful procedures for five precision derivative (third order) and nine precision derivative (fourth order) synthesis, with transmission angle control, of the seven link two-degree-of-freedom mechanism with a double joint are presented along with the results of computational experiments to evaluate the procedures developed. Some important special configurations of the mechanism are also considered.
Zusammenfassung-Ableitungs-Synthese Ebener Getriebe zur Erzeugung yon Funktionen zweier Ver&nderlichen: K. Lakshminarayana und R. G. Narayanamurthi. Es werden for siebengliedrige Getriebe mit zwei Freiheitsgraden und einem Doppelgelenk erfolgreiche Syntheseverfahren for f/dnf bzw, neun 0bereinstimmende Ableitungen (d.h, Synthese dritter bzw. vierter Ordnung) bei Kontrolle der 0bertragungswinkel angegeben. Numerische Rechenergebnisse ermrglichen die Bewertung der Verfahren. Zus&tzlich werden einige wichtige spezielle Konfigurationen des Getriebes betrachtet. Pe311oMe~rlpOH3BO~Hblll~ CHHTe3 rl.qOCKHX MeXaHH3MOB J1EIR HpOH380~CT~3. (~yHKHH~ ~ayx nepCMeHHblX" K. JlaKl~MHHapagHa H P. l-, HapaAuaMypTH. VcnemHble npHeMbt CHHTe3a ngTH TOqHblX NpOH3SORHblX (TpeTb~lrO nopRara) H ~]eBRTH TOHHblX npoH3BO~HblX (qCTBCpTaro nop~nKa) C ronTponeM yrna nepc~.a,~H CCMH3BeHHaI'o C /].ByM~I CTeneHflMH CBO~Orlbl MeXaHH3Ma C OIIHHM /].BO~HblM lllapHHpOM, /],alOTCR C pe3yJlbTaTaMH qHC~eHHblX 3KCnepHMCHTOB ~t]~l OHeHKH pa3BHTblX npoueJ1yp. HeroTop~,le Ba.)KHble o¢06ble KOHTypb! MCXaHH3Ma TaK)Ke pa3cMaTpxaa~TCR.
1. Introduction THE SEVEN-link mechanism (Figs. 1 and 2) is the simplest two-degree-of-freedom mechanism that permits rotary inputs and output. Its synthesis has been the subject of a few investigations [ 1-4]. Confining ourselves to the case where the input and output scale factors are free to be chosen, Allen[2] provided simple graphical procedures for four precision displacement and four precision derivative synthesis of the type-I mechanism (Fig. 1). Freudenstein and Sandor[3] showed how Burmester theory can be used to obtain an arbitrary set of four precision displacements for the type-I mechan*Lecturer in MechanicalEngineering.Indian Institute of Technology, Madras-36, India. :~Professorand Head, Department of Mechanical Engineering,lndian Institute of Technology, Madras36, India.
249
250
/NPUT I ~ A
N~UT-2 L?
F i g u r e 1. The type-I m e c h a n i s m . F
B
E
NPUT- 2
Figure 2. The type-II m e c h a n i s m . ism. More recently, Philipp and Freudenstein [4J investigated the use of the parameter perturbation procedure[5] for a 12-precision displacement synthesis of the type-II mechanism (Fig. 2) by algebraic methods. T h e i r investigation revealed the difficulties involved in the numerical solution of the equations of precision displacement synthesis. This paper is mainly concerned with five precision derivative (third order) and nine precision derivative (fourth order) syntheses of the type-I mechanism by algebraic methods. For functions of two variables, precision derivative approximation may be considered to be particularly advantageous, as shown in the following. Let (0.40) and ,h be respectively the rotations of the input shafts and the output shaft from the design position of the mechanism. T h e following notation is introduced: At the design position. a ,h aN =
O,l, ; gg =
0 " ,~ 0 -'~I, ; oo = < ; 5-g =
O"-'llJ ; ooo
0 a~
0 a~O
Oa,h Oallj 002040 -- Ba; 0006"- = / 3 4 ;
[,!J]`* = first directional derivative of ,/J at 0 = (h = 0, in the direction making an angle y with the 0-axis in the (0. ,b) plane; [to]re = second derivative; [~b]`*a = third derivative. It can be easily shown that: [¢]`* = oJl cos -/+ aJ_~sin y [tO],,._, --- oel cos.' 7 + oe._,sin" y, + 2o~acos "y sin 7 [~] .-,a =/3, cos a 7 +/3.2 sin a 7 + 3/3:1cos" 7 sin y + 3/34 cos 3' sin.' T.
251 From the above expressions for the directional derivatives it is seen that to achieve, for instance, fourth order approximation between the desired and generated functions in A L L the directions of the ( 0 . 6 ) plane, it is only necessary to equate the corresponding partial derivatives of the first, second and third orders for the two functions.
2. Displacement Equations T o obtain the relation between the input rotations and the output rotation, the coordinate projections of the link length-vectors are chosen as the parameters most suitably defining the mechanism. Figure 3 shows the reference or design position of the general seven-link mechanism (type-II) with O A D as the fixed link and O Z as the output link. T h e length O Z is taken equal to unity for convenience, without any loss of 2
+ B (a t÷ b t , a 2 ÷ b 2 )
~ ~ - ( 1 + C
F
t, C2)
(!+f l ,
f2 )
" E (d s + et~d 2 ÷ e2 )
Figure 3. Reference or design position of the type-II mechanism. O, 6 = input angular displacements from the design position (positive counterclockwise); ~ = output angular displacement from the design position (positive counterclockwise); r / = e - O; p = ~ - 6; k = clockwise angular displacement of link ZCF with respect to OZ;--~: with respect to OADT-",: with respect to OZ. generality. Axis-1 is chosen along OZ and attached to it. This choice of axes considerably simplifies the equations and their derivation without, in any way, affecting the general applicability of the result. T h e legend attached to Fig. 3 explains the notation. The co-ordinates of the various joints are indicated in the figure within parentheses. Indicating the co-ordinate projections of the links in the displaced position by means of primes, we have the following relations between the projections in the displaced and reference positions: a~ = b'l = c't = d't = e~ = f't =
at cos t0 + a., sin ~; b, cos rt + b.,_sin "0; c, cos h + c_. sin h; d~ cos q~+d2 sin tO; et cos p + e 2 sin/9; f, cos X + A sin X;
a" = b" = c2 d" =
a2 cos O - a, sin tO b2 cos "0 - bt sin "0 c2 cos k - ct sin ~, d2 cos $ -- d~ sin 0 e~ = e2 c o s p - - e l sin p f " = f , cos h -- ft sin h.. P
(1)
252
Using the fact that the length of the link B C does not change, ta~--bt-l-c~)e+tae+b,,-ce)
~=iai+b'~-l-c~)e+(a;--b'.-4lL
which, on expansion, gives: (alb~ + a.,b.,) - ( a l c l + a.,c.,) - ( b l c l + bece) + ct - at - bl = = (a'~b[ + a'b'.,) -
(a;c'~ + a'c',,_) -
(b;c; + b'c')
+ c[ - a ; -
b't
2~
T h e following notation is introduced: alb~+a.,b._,=C~;
gztcl - ~ - a e Q = A t :
htc~÷b._,c.,_=Bl:
a . , b l - a~b.,_ = C._,;
a.,c~ -- aLce = A.,;
becl-
blc._, = Be:
d t e ~ + 4 e . _ , = F,;
d,j; +de¢.'_, = D,;
eJ;+
ee); = & :
d.,_el - - d t e e = F.,;
d . , _ j ; - - d , f e = De;
e , , / ; - - e J ; = Ee;
F r o m equations ( 1 ) a n d ( 3 ) , a'~b't + a.',b.'_, = Cz cos O+ C.e sin O.
i4)
By analogy, a',c't+a:,.c; = A t
+A._, sin ( t o - X )
sin ,/*)cos a + {A, sin to--Me cos 0 ) s i n X
(5)
b'tc'~+b.'c.',_= ( B , c o s v l + B . , s i n r l ) c o s X + ( B t s i n v l - B . , _ c o s ~ ) s i n h .
(6)
=
(,.4 1 C O S
cos ( 0 - , \ )
to-}-A.,
and
T h e following additional notation is introduced: X~ = A t c o s to + A._, sin ~) + B~ c o s rl + B.,_ sin ~ -- c ~
X2 = A t sin to - A., cos ~/i + B 1 s i n r / - - Be cos r / - - c., Xa =
(C~-At-Bt)+(ct-a,-b,)-(CtcosO+C.,_sinO)
+ a t cos t o + a e sin ~ + b , cos ~ + b ~ sinr~ Yt = D t cos qJ + De sin 6 + Et cos p + Ee sin p - J;
(7)
Ye = D~ sin to-- D., cos to+ Et sin p - Ee cos p - - f , Ya = ( F t - D t - E t )
+ ( ) " ~ - - d t - et) - (F~ cos 4~+ Fe sin 6 )
+ d t cos ~0+ de sin tO+ e, cos p + e., sin p. Substituting from (4) to (6) in (2) and using (7), the following displacement equation for the loop " 0 . 4 B C Z " is obtained: X~ cos ~ + X . , _ s i n h + X a
= 0.
(8)
25,3. The displacement equation for the loop " O D E F Z ' " is written down readily by analogy: (9)
Yt cos h + Y~ sin h + Y3 = 0.
Elimination of h from (8) and (9) gives the displacement equation for the whole mechanism:
(lO)
(X~,Y3--X3Y.,): + ( X 3 Y , - - X , Y 3 ) " = (X,Y.,_-- X~.YO"-.
This is probably the simplest possible form suitable for synthesis with a large number of precision conditions.
3. Derivative Equations for the Individual Loops in the Design Position For the purpose of order approximation it is necessary to form the successive derivatives of either the displacement equation for the mechanism, (10), or the displacement equations for the individual loops, (8) and (9). The latter will be used here. The following notation is introduced: At the design position, OX~ = Xto; OzXl = X,~,_; O"X, _ OsX, aO O0z 000(5 - X , ~ ; O02O(b = Xt~,,; X1 = X,o;
Oh , azh 02K a--O= ~,0; ~ = h~,; OOa(b = ho, ; similarly others. On repeated differentiation of the displacement equation for the loop " O A B C Z " and setting all angular displacements equal to zero,
Xto + Xao + X2oh0 = 0
(lla)
X t , + X.~ + X.,oA, = 0
(12a)
X,~., + X3e.,_+ 2X.,_oXo - Xtoho 2 + X2oh~, = 0
(13a)
Xt,., + X3,., + 2Xz,h,t,-- X , o h , z + Xzoh,.,. = 0
(14a)
X l ~ + X 3 ~ + X._,e~ho+ X.,_oh.n-- Xlo,koh~ + X.,,oh~ = 0
(15a)
X ,o3 + X3o:~ + 3 X2~:~o - 3 X ,oXo2 + 3 ( X,,_o- X ~oho) he2 + X2o( Xoa - X fl )
=
0
(16a) X,~a + X3,3 + 3X~,.,h, -- 3Xt,,k, 2 + 3 (X._,~-- X,0,k,~) h,2 + X2o( ;%3 -- h~a) = 0 (17a) Xt~,, + X3~,¢,+ 2X2o, ho + X2~2k, -- X,,ho" -- 2XtoXoh, + (X.,, - - X , o ~ ) X~. + 2 ( X 2 0 - - Xloho) ho~ --{-X2o ( ho,2.~ -- ho2ho) = 0
Xlo~2 "~ X30~2 + 2X~.~h, + X.,_,2ko--
Xloh,
2 --
(18a)
2 X l , h o k , + ( X~_o-- gtoho) ks.,_
+ 2 (X.,, -- X,,,h,) h ~ + X2o ( h~., -- h0h, 2) = 0.
(19a)
Coffesponding equations (llb)-(19b) for the second loop " O D E F Z " will be obtained by replacing X by Y in the above set of equations. The X,Y-derivatives at the
254
design position, occurring in the above equations, are evaluated from I7). They are presented in [8]. 4. Third Order Synthesis of the Type-I Mechanism For the purpose of order synthesis, it has been found very advantageous to replace the parameters a~, a..,. d~ and d._, by the parameters g~. ge. h~ and It.,_. The latter are the co-ordinate projections of links CB and FE (CE for type-l) respectively I Fig. 3). thus satisfying the following relations for type-I: = g, -= cl + 1"
at-b1
d~'e~=h~+cz+
I;
a., + b.,_ = ;,., + c., d.,~-e., = / t , - - c . , .
The following notation is introduced: n
=
('l (-"2
--
{
20)
n being the cotangent of the output transmission angle in the design position. (Transmission angles are discussed later.) Also, c,_,t ,% -
coz) =
~o~ x
and c.,( X,~ - ~o.,) = co..,3".
defining two new quantities x and y. With the above indicated changes and notation, equations (l la)-(12b) reduces to the following on substituting for the values of the X, Y-derivatives [8] : (provided~o., ¢ 0 )
;,~=(tl-})~q,:
(21
(provided~o~ ~ 0 )
h~=(n--1)h._,
~22)
(providedq., ~ 0)
bl
(provided he # 0 )
e,=i,,-l)ee+co..(l-V).
be+co~
123)
y
,24)
Equations (t3a-15b) take the form given in the APPENDIX. on substitution from equations I20-24) and using the following notation: Z I =
(/~02-- (.yl)c.,:
2.2 ~--- { ~a5,2-- 0~.2)4',2'
Z:I =
(~,9,,5- (_~:;)c.,.
Eliminating Z,, Z_~ and Z a from the relations appearing in the ,-\PPEND[X arid introducing the following notation:
255 I
1
c.*_ - (l+nZ)(x--Y)--C.2
± g~,=
[
y 1+
(--'yl-' n
Y/ ]g2
and
/,-T=x i+ . - 7
'
I(
')]'
we get: ( o~., v)-' - (~=y)" (~,x)i~._, y) - (b._~+ ~,,~-) (~,_,y)
(e,, + ~ y ) " /in/g?/ (~,x) (e., + ~._o') JL ]I/7.Z J
=
I
2
(25)
L (x - y ) (~:, - ~o, o~._,n ). This is a system of three linear equations in the three unknowns 1/c.,_, 1/g.,. and l/h.,, if the values of n, x, y, b._,and e._,are chosen. Thus, the problem of third order synthesis has been thoroughly simplified first by replacing the parameters at, a._,,dj, and d., by the parameters gt, g._,,hx and tt._,and secondly by a proper choice of the parameters whose values are to be assumed and of those parameters whose values are to be determined. T h e above choice, at the same time, enables us to have control over the transmission angles in the design position. {Transmission angle control is dealt with in a later section.) Solving the equations (25) and using the following notation:
P = (x--y)[oq-wlZn+wl ( n - ~ ) ] - Y [l + (n-~)"-]b._, Q= (x-y)[a.,-oJ.,'-n+oJ.,(n-l)]+x[l+(n-x) R = (x--y)
1
" ]e.,
(a:~-- ~otco._,n)
(26) (27) (28)
S = (oJ.,_y)P- (oax)R T = ( o J t x ) Q - (~.,_y)R U = b.,e.,+ (oJ2y)b.,_+ (oJtx)e.,_.
(29) (30) (31)
We have:
(o~lx) (oJ.,.y)b.,e.,. c., = -S(e.,_ + oJ.,_y)+ T(b., + oax) - R (b~_e.,_) (1 + g.,_=-S(e.,+~.,_y)+T(t.o,x)y
1+ n -1
h.,=+S(~.,y)+T(b.z+OJtx)X
1+ n--
n 2) ( x - - y )
(32) (33)
' .
(34)
256
A f t e r calculating c'z. ,e~ and h..,. the remaining p a r a m e t e r s are o b t a i n e d from (20) ~o 124). It m a y be n o t e d that this s o l u t i o n a s s u m e s that ¢m and co~ are n o n - z e r o . C o n s i d e r a tions like the choice of values for n . x . y , b., and e., are not dealt with here but taken up later in c o n n e c t i o n with fourth order synthesis, as the main use for third o r d e r synthesis is in the solution of the fourth o r d e r synthesis equations.
5. Equations for Fourth Order Synthesis of the Type-I Mechanism T h e following notation is introduced: K~-- toh.r)(.r--v) II÷nZ) 1-('.,
,
L.,:
-
I +n
. n-- v
o~.,_
y ) .r
[ ( ')l' 1 4- tt
(o~,,y)g,,
0)2
2 --
09,2
1] I c.,
--
h.,
i38)
c.,
n-
+ o:._, n - -
-,q.,
!39)
Ir~,
( 40 )
1'
¢ol 2 4 - 0L t
x
37
~
v +cq
(,'
~t -
36
.'.,"_'
(e..,+oJ._,y)/T'
[ [
( e,_, 4-
--
~°'x)
.12"
"~1 : X
135
_r ) t I -'- n-' ) - -
¢°l~-°);(l--r,
=
--,k'_,
,/'L,
I ( b . , + o h . r ) g'-'
M'=)'[
~'~4.,
,
[ ( ')1'
+ Io).:y)y [
.'f
1
( o~._,y : ( y
Lt =
I--n
[ l + n ttl . . '1. ) . '
+ (ohx).r K., :
Ib.,÷oh.v)v "
-/r,
~41l
1
(42)
{43)
P., :
.r
m t m . 2 4- CZ:;
44)
. . . . . X
[1,, "
P r o c e e d i n g on the s a m e lines as in the p r e v i o u s section and using the a b o v e n o t a n o n . equations I 16a- 19b) lead to the following four equations for fourth o r d e r synthesis: 3Z~ Kx + 3 (co,r)"L~ + 3 ( b._,+ = [x-y)[(/3,-co,
:~)-o,,t3a,~t-
3Z._,K., + 3 (co,, y )"-L, + 3 ( e.,_ +
=(y-x)
co~x),'vl~ - 3 ( c o l . r ) N ~
co,,
1)]
i45)
y )M.,_ - - 3 ( co.,_y ) N.,_
[ (/3~ - o~._,:{ ) - o,._.(3oc, n -
t)]
E46)
257
2Z:~Kt-ZtK.,+2(wtx)(o~.,.y)Lt--
(cutx)"L.,_+ (oJo_y)Mt- (e~ + co._,y)NL
+ 2 (b.., + mix) Pz - 2 (cotx)P., = ( x - y ) [(/3:; - cot2co._,) -- n (arm._, + 2aacot ) ] (47) 2ZaK., -- Z.,_Kt + 2(oJlx) (~._,y) L: - (w._,y )ZL, + (~ix)M., - (b., + colx) N., + 2(e.,.+co.,.y)P.,_--2(oJ2y)Pt
= ( y--x)[(~4--wlcoo."-)--n(o~.,wl
+ 2a:~oa.,) ].
(48) The above arrangement of the equations is aimed at minimization of computation. These equations are, effectively, four highly non-linear simultaneous equations in the five unknowns t,, x, y. b.., and e., since I/c.,, l/g._, and 1/h.,. are directly expressible in terms of n, x. y, b., and e2 by equations (32), 133) and t34). Expressions for Z t , Z.,_ and Z~ are available in the APPENDIX. The practical solution of these equations is dealt with in the succeeding sections.
6. Method of Solution For the numerical solution of the fourth order synthesis equations (45-48), it was decided to exclude methods that (a) require a close initial approximation, or (b) show evidence of lack of success on problems of comparable difficulty, or (c) seem promising but for which sufficient numerical evidence is lacking, or (d) depend on simplifying assumptions. On account of (a), only minimization methods come into consideration. Because of the extreme non-linearity of the equations, it was expected that the minimization function will be extremely " m i s b e h a v e d " with very narrow "ridges"* in the contour "'map" of the function. The "direct" search methods, that depend only on the qualitative comparison of function values, are essentially of the "ridge tracking" type and also permit easy incorporation of constraints. T h e direct search minimization method of Rosenbrock [6] was selected on the basis of its reported [6, I 0] comparative success with very difficult ridges and also since it takes the most pessimistic view of the nature of the minimization function. T o obtain a suitable starting point for the minimization procedure adopted, a short random search, that takes constraints into account was carried out. This serves two purposes: (a) With the method of dealing with constraints adopted (see next section), it is necessary to start from a feasible solution (i.e. from a point where the constraints are not violated). (b) Since the technique can lead to a non-zero minimum, it was considered desirable to make a short random search and start with the best point obtained in an effort to minimize such a possibility.
7. Constraints-Transmission Angles In order to obtain practically useful mechanisms, it is necessary to prescribe limits on such quantities as transmission angles and maximum to minimum link length ratio. With a view to be able to investigate the effectiveness of the chosen minimization technique proper, it was decided to control the transmission angles alone. Their control was obtained by limiting the values of the transmission angles and their maximum rates of change in the design position. The transmission angles in the design position are controlled by the method of transformation of variables [7, p. 156]. Let the function to be minimized be F ( q i , q., . . . . ) subject to: - q , o <~ q , <~ q,o, for instance. By replacing *This is a term more appropriate for a maximising problem. It is thought unnecessary to introduce new terminology.
258 q,, by a new variable t, in the following m a n n e r : q, =- q,,o sin t,. the constraint is eliminated. C o n s t r a i n t s eliminated in this m a n n e r will be referred to as implicit constraint',. T h e m a x i m u m rates of change of the t r a n s m i s s i o n angles are controlled by simpl,, rejecting the points that violate the constraints, as "'failures". C o n s t r a i n t s taken care of in this m a n n e r will be referred to as explicit constraints.* Figure 4 s h o w s the type-l linkage with the j u n c t i o n t r a n s m i s s i o n angle./a.j and the
(g)
C~
Figure 4. Transmission angles in type-1 m e c h a n i s m . output t r a n s m i s s i o n angle/xo. It is easily seen that COt
(49)
P-o ----- tl
cot/a.j =
l
1
X
V
= k (say)
51
,90 = to1 I +
0.,, Oa5
-
r).u.i _ O0 Ota.j a6
_
(50)
( +:'t
52
c._,/
b._,+ ¢ o , x g._,
co,x /t.,
w.,_y
e., + to.ey
g._,
h._,
5
~
c54)
T h e m a x i m u m rates of c h a n g e of the t r a n s m i s s i o n angles /a.o and /x~ are r e s p e c t i v e l y given by: ro r~
\ &b /
= ,/(0,,,7
(0,,q-'
~/\ aOl + \ a(b /
" O t h e r e x p l i c i t c o n s t r a i n t s can be ea~,il_v i n c o r p o r a t e d into the c o m p u t e r p r o g r a m m e s [8]
56
259
The maximum allowable limits on these values will be referred to as rot and r~ respectively. 8. The Objective Function For the solution of the equations (45-48), an objective function F, to be minimized, is to be formed. Writing the four equations as ,/'1 = 0: f_, = 0, f~ = 0 and f4 = 0, it is usual to have:
F = f ? + A " - + A " - + L 2. This gives equal weightage to all the four derivatives/3~,/3.,,/33 and/34. Such a formulation will not matter as long as a zero is reached. However, in case it is not possible to make F zero*, it will be desirable that the errors in the derivatives/3~,/3._,,/33 and/34 are so distributed that, considering the error derivative (/3~) in all the directions of the (0, 6) plane, the maximum value of the error derivative is minimized. However, since the determination of the maximum value leads to the solution of a cubic equation, the root mean square value of the error derivative has been used as the objective function. Using the subscript e for the derivatives of the "error surface" at the design position, it follows from the expression for third order directional derivative, given in Section 1, that /3e =/3t~ cos a Y+/3.,.e s ina 3'+ 3/3.~ecos'-' 3' sin 3'+ 3/34~cos y sin" y.
(57)
Considering the variation o f y from 0 to 2rr, the root mean square value of/3e is obtained aS;
I /3~r.m.s= 4--X72~,/[ (/3~- 3/34~)'-'+(/3.,~- 3/3.~e): + 9 (/3~+/34~)-' + 9 (/3._,~+/3.~)-'].
(58) The objective function actually used in the computations is: F=32
[/3er.m.s.] 2.
(59)
It is easily seen that any point that makes F zero will be a solution of the original system of equations. 9. Random Search Instead of making the random search with the variables n, x, y, b., and e2, the procedure given in the next paragraph is followed, in order to be able to control the transmission angles at the design position by the method of transformation of variables. This procedure also avoids the problem of having to choose the values o f x or y whose ranges of variation are not known. A value of the link length 'b' is randomly chosen between certain lower and higher limits b~ and bh. Also, b2 is chosen less than 'b' in magnitude, by multiplying 'b' by a random number lying between - 1 a n d + 1. The value ofbt is found from: b~ = - ~
- b._,"
* Either because of constraints or because of the obtaining of a relative minimum or because of the failure of the minimization technique.
260
where the sign before the radical is randomly chosen. The values of n ti.e. cot/.<) and k (i.e. cot ~.~) are obtained from: n~ S i l l t l
60
k = kt sin r,
61
tz ~
and
where nz and kt represent the specified upper limits on the magnitudes of cot g,~ and cot/xj respectively and tt and t._, are randomly chosen between - = / 2 and + =/2. Using the notation: 1 l l t ~--" I 7 - - - X
and 1 Ul~r/
--
Y
,
we have, from equation (50): l +u~v~ Ul --
=
(621
k
14 l
and from equation (_.~): "~'
/ 63 )
Eliminating v~ from (62) and (63), (a2, -- b, - kb., )u(-' + [be ( n k -
1 ) -4- b I (tl + ]c.) ]it, + [to I - - / 7 ( k b ~ - - be) ] = O.
(64) Equation (64) is solved for u~, choosing randomly the sign before the radical in the expression for tq. T h e value of c'~ is then obtained from equation (62). Knowing tq and th, we have: l -
=
n -- u t
(65)
=
n--
(66)
X
and -
1
v~.
Y T h e value of e.z is now chosen randomly between certain limits - e.,z and + e~t, so that el can be calculated from equation (24) and e from: e = ~ / e l e + e.F. The value of e thus found is checked to be within specified limits. (If it is not, briefly stated, the process is repeated from some stage. Details are given in the c o m p u t e r p r o g r a m m e in [8].)
261
The third order synthesis is now carried out and the explicit constraints (r0 ~< rot; r~ <~ rig) are checked for violation. If a constraint is violated, the procedure is repeated from the beginning. If no constraint is violated, the objective function F is evaluated and the procedure is repeated from the beginning. After the specified number of function evaluations, the feasible solutions thus obtained are sorted out to the extent necessary so that the best 20 (for example) solutions are available in the order of increasing function values. These feasible solutions are used as starting points for the sequential search considered in the next section. A working computer programme in Fortran-IV for the random search described here is presented in [8].
10. Sequential Search The method adopted, as already indicated, is largely the method of rotating coordinates of Rosenbrock. It will be briefly described here. A detailed description is to be found in [6, 8]. The independent variables for the search are t~, to, l/y, b._,and eo-. This set of five parameters is equivalent to the set of n, x, y, b,, and eo- as shown below and is chosen because, in that way, the transmission angles in the design position can be controlled: With the s a m e limits nt and kt as in the preceding random search, n and k are first calculated from equations (60) and (61), at any point (h,t.,_, l/y, b2,e.,). Then, vl is found from (66) so that u~ is obtained from (62) and hence x from (65). Knowing thus the values of n, x, y, b., and e2, third order synthesis is carried out, violation of explicit constraints is checked and the objective function F is evaluated. Even though the number of equations to be solved is only four, equations (45-48), it was decided to leave all the five parameters as unknowns to increase the probability of finding a solution at the expense of possibly increased computation. Briefly, the search is made in mutually orthogonal directions, making only one trial at a time in each direction and returning to the original direction after exhausting all the five directions. When a step is taken in the chosen direction, if the explicit constraints are violated, the step is considered a "failure". If the constraints are not violated, F is evaluated. If this value of F is more than the value at the " b a s e " point, the step is a "failure". if it is less than or equal to the value at the base point, it is a "success". If it is a success, the basepoint is shifted to the succesful test point and the step size in the direction is doubled. If the step is a failure, we return to the base point and the step size in the direction is halved and changed in sign. When a success followed by a failure has occurred in all the five directions, the orthogonal search directions are rotated such that one of them (direction 1) is in the direction of total progress made since the last rotation of directions and the initial step sizes are restored. It may be noted that direction 1 provides an initial guess of the direction of the "ridge". Since it is difficult to determine from an internal check [6, p. 67] whether the minimum has been reached or not, unless the minimum happens to be zero, the sequential search programme is arranged such that the search will terminate after a specified number of function evaluations or after a specified function value (Ft) has been reached, whichever is earlier. This specified function value can be a small positive quantity based on an estimate of the extent of round-off i.e. searching for function values below this level is rendered meaningless by round-off error. In order to be able to examine the results for convergence, the following information is printed each time before the search directions are rotated: F; tl, t.,_, l/y, b.,_, e.,; "Pgt'" i.e. the square of the total progress made since the search directions were last rotated; "r'" i.e. the square of the sine of the angle between the former direction 1 and the vector of total progress. If
262
p,j, is small for at least a few successive rotations and r is always large, then the minimum may be considered to be reached, though this is not certain [[6], pp. 68. 105: [8] pp. 6-29). At the end of the search, the latest direction cosines are also printed so that, if it is desired to resume the search after examining the results, the latest estimate of the direction of the ridge is available [6, p. 68]. A working c o m p u t e r p r o g r a m m e in FortranIV for the sequential search described here is presented in [8]. Numerical examples and a discussion of the results are presented in Section I2 of this paper.
11. Displacement Analysis Referring to Fig. 5, the analysis p r o g r a m m e [8] computes 6e for values of y and obtained by uniformly incrementing from zero. T h e values of 0 and ~b are found from: 0 =---- E COS T
and ~5 : esin 7. Choice of (0, do) in this way allows the plotting of error curves with constant 7. T h e s e curves represent "'vertical" sections of the error surface through the origin (0 = 4, = 0) and are like the error curves of single variable order synthesis, so that the obtaining of order approximation could be verified easily. Moreover, if there are no restrictions on the relative limits of variation of 0 and ~b, it seems rational to limit the range of variation of (0.4~) in all directions through a constant e .... . The percentage error can then be defined as: E~ = ( I';'~ ]m~,~) 100/(Range of variation of q, within a radius of e ..... )" A less rational but more convenient definition is: E=
. . . . )100/(2
....
i.
T h e computation of ~. involves the solution of two quadratic equations sequentially, i.e. given the positions of B and E (Fig. 4), there are two possible positions for C and
~ CON "/'OURS OF I CONSTANT
~(
t, ~jO.(c~O
)
• , \
SECTION X-X OF
\
Figure
5. The
error
surface.
263
given the position of C, there are two possible positions for Z. The analysis programme [8] automatically selects the positions of C and Z such that the same branches of the loops are obtained in the displaced position as in the design position. For this purpose, the programme utilizes the signs of sin/_ CBE and sin / COZ in the design position as criteria. 12. Numerical Examples and Discussion of the Results For numerical verification of the procedures described, the following two functional relationships were chosen:
O=
0 + ~b 2
(Adder)
and 0 = 0 + ~b + 0q~ 2
(Multiplier)
where 0, ~b and qJ are in radians. The second relationship represents the function z = xy if we let: x = k(O+ 1),y = l(~b+ 1) andz = kl(2t0+ 1), where the values of k and l are free to be chosen. For the adder, a random search of 2000 function evaluations gave the following best ten values for F: 1-37 (SA); 1-62 (SC); 1-90 (SG); 2.19 (SB); 2.23; 2.23 (SE); 2.30: 2-31 (SF); 2.33; 2-44 (SD). Those used as starting points for the sequential search are marked within parentheses with the corresponding example names for the sequential searches. For the multiplier, a random search of I000 function evaluations gave the following best ten values of F: 6-66 (MA); 8.15 (MB); 8.58 (MC); 8.78 (MD); 9.63 (ME); 12-65; 12.74; 14-11; 14-34; 15-11. it may be mentioned that even though these values are much higher than those for the adder, the former were reduced to a level comparable to that of the latter in a few hundred function evaluations, in the sequential search. The following limits were used in the random and sequential searches: 0-5 ~
264
SB and M D will be presented along with comments on all the examples to provide an appreciation of the merits and limitations of the synthesis procedure. Tables 1 and 2 provide a summary of the results of examples SA, SB and MD. The corresponding mechanisms are shown in their design positions in Figs. 6 - 8 respectively. Example SA led to an implicitly constrained minimum ,,vith k] = k~. Example SC, starting from an entirely different point, led to the same solution in about 8000 function evaluations. Example S G ted, in about 3000 evaluations, to a solution that is practically same as SA or SC if 0 and 4~ interchange their roles. Example SE, after 2500 evaluations, was very closely approaching the mirror image of solution SG on the output link OZ, The search was therefore discontinued, even though the minimum was not yet reached. Table 1 Example
SB
,',.4
Starting Search variables
;, 0"3209 t, 0.5468 I/y - 1 . 8 6 7 1 b, - 1-2088 e._, 0-1836
17 /3. . . . . . .
Final
MD
Starting
Final
Starting
Final
0-2396 1-5708 -1-4963 - 2-0564 0-1092
-0-2033 -0-8142 -1-5351 -0,4487 0,4275
0.6011 - 1,5136 -15.6574 -I)-6858 0-2800
1.1383 1.3802 0-5517 0.4032 -0.9717
1.5659 1-5693 0-9953 0.6713 -0.5011
1-0601
2,1936
0.00(R)
8.7795
0-316-4
1-3719 0" 18
0.00
0"099
(final) Number of function evaluations to minimum
]tz I [ k] r,~ rj
About 2800
(t:~ c r o s s e d ) 1461
About 7000
0.11866184 O. 5 0 0 0 0 0 0 0 0.92 0-50
0.28276356 O. 499 t 8140 ---
0-49999407 0.49999946 0"584 {)'365
On Axis- I
Link projections
a b c et e
6.3266 0.7471 1.1215 2.1637 0.560I
On Axis-2
On Axis- 1
On Axis-2
On Axis- t
On Axis-2
14.5739 -2-0564 9.4511 2.7237 0. I092
%0000 -- 1.3591 --0.1548 0.(.R)00 0.3651
- 0-0000 --0.6858 -0-5476 0-0000 0.2800
0.1801 0-1656 - 0-3116 0.0665 --I).0828
--0.6029 0-6713 -0.6233 -0.1195 --0.5011
Table 2. Example
SA
SB MD
% Error E, at 2e ..... = 30 ~
5W
70 °
90:
0'31 0"07 0"16
0"87 0"35 0"52
1"8 1"06 1-51
4-7 2-47 3"38
265
~9 8
c
Figure 6. Design position of solution SA. E
8f
"e
Figure 7. Design position of solution SB. z
Figure 8. Design position of solution MD. Example SB led to an unconstrained zero on the boundary of the implicit constraint lkl = k~. The inputs and the output are co-axial. Example SF also led to an unconstrained zero in 950 evaluations, giving co-axial inputs and output. In example SD, the minimization technique failed (because of an extremely narrow ridge) but only after achieving a value of/3er.m.s. = 0"06. It could be concluded with reasonable certainty that, otherwise, a co-axial mechanism would again have been obtained.
266
Example M D led to an unconstrained minimum very close to the boundaries of the implicit constraints Inl ~< nF and Jkl <~ kt. Example M B gave an unconstrained minimum close to Ik ~ k~ in about 5000 evaluations, with/3~r.m.~. = 0"12. In examples 3,1,4 and M C , the minimum had not been reached after 15,000 function evaluations, which was set as the maximum limit to be tried for a single example. The searches were close to implicitly constrained minima and the slow progress was due to very narrow ridges. T h e ¢?~r.~.,~.values were 0-12 and 0-14 for M A and M C , respectively. Example M E started with F---9.6334736 and reached an explicitly constrained minimum of F = 8.4264888 in about 1000 function evaluations. Very narrow ridges were encountered in most of the examples, thus establishing the necessity to use a good ridge tracking procedure, as was done in the present investigation. It may also be mentioned that satisfactory control over the transmission angles in the displaced position was obtained in all cases, It is to be noted that the first and second order derivative values are always exactly obtained whatever the value o f F may be. Some of the possible lines of improvement of the procedures are indicated in the following: {I) It seems advisable to reduce the possibility of obtaining a relative minimum by a long non-sequential search (preferably the grid type[9]) to be followed by a sequential search. This will have the additional advantage that starting points that will lead to previously obtained solutions can be spotted and avoided to a larger extent than was possible in these examples. (2) Since transformation of the variables alters the nature of the function, it may be worthwhile to try suitable transformation functions other than the sine.
13. Double Precision Computation The accuracy of single precision computation, as determined by comparing the values of the objective function obtained by single precision arithmetic with those by double precision, for the same set of values of the variables, varied between 4 and 6 significant digits. Apart from fixing a lower limit for the function value below which it will be meaningless to continue the search, this small round-off error usually has no significant effect on the effectiveness of the sequential search, This will be clear from the realization that the effect of the round-off will be merely to "'distort". to a "'small" extent, the contour surfaces of the objective function. H o w e v e r , sometimes a very narrow ridge may be "'blurred" by single precision round-off and a double precision search may be able to locate the same. This is not necessarily disadvantageous since an unwanted relative minimum may thus be avoided by the single precision search. Numerical trials have confirmed the above. It may be concluded that there is usually no need to evaluate the objective function with double precision except perhaps for refining the final result. 14. Computation Time For the IBM 7044 computer, on which the programmes[8] were run, the running time for the sequential search consists of about I rain and 25 sec for compilation and about I min and 45 sec of actual computation time for every 10,000 function evaluations in single precision. These values are small enough to make the procedure economically feasible in most cases. 15. Special Configurations-The Co-axial mechanism As against the general procedures of the preceding sections, another line of investigation consists of the study of special configurations suitable for special classes of
267
functions. When such a configuration can be found, the synthesis procedure can be considerably simpler. The co-axial mechanism (Fig. 9) and the cyclically symmetric mechanism (Fig. 10) represent two such cases. The co-axial mechanism (Fig. 9) can be designed to a high accuracy, comparable to. or even better than, the highest accuracy obtainable with the general mechanism, for a particular class of functions. A subset of this class consists of all those functions F
C
~OAD Figure 9. The co-axial type-II mechanism.
A
b=l B
~'~c .,,(=o
E -
Figure 10. The cyclically symmetric mechanism. which can be expressed as a relation between two of the following three differences: (~b-0), (@-(~) and ( 0 - ~ ) . The problem is then reduced to that of the synthesis of a six-link Watt mechanism with one degree of freedom. The latter provides a maximum of nine precision points and the resultant synthesis can be easily seen to give "'9 precision-line" synthesis of the seven-link mechanism. The accuracy may, therefore, be expected to compare favourably with that obtained by 13-point synthesis of the general mechanism. For order approximation consider, for instance, the generation of the function: (~b-th) = F ( 0 - - ~ ) . Writing ~ - - ( ~ = p and 0--(h=-0, we have p = F ( ~ ) and the derivatives of p with respect to "0at p = 77 = 0 may be designated as to, a, fl, etc. Expansion of F(,/) in Taylor's series about the point p = ~ = 0 gives on rearranging:
~ = 0~, + ~(1 - °4 + T ~ + - y ~ - 0 ~ + ~ ~---g- B - _ _
~-~--y- ~ + . . . . (67)
268 E x p a n s i o n of t0(0, 4~), on the o t h e r hand. gives:
,b= 0~,,+~¢o.,_+7~,,+--c~._,+0e0~:~+
~+
/:3.,+--/3:,+~---/3,+ .... i68)
C o m p a r i s o n of (67) and (68) leads to the c o n c l u s i o n that synthesis of the six link m e c h a n i s m to any o r d e r o f a p p r o x i m a t i o n provides synthesis of the seven link m e c h a n ism to the S A b l E o r d e r o f a p p r o x i m a t i o n . This m e a n s that up to 9th order a p p r o x i m a tion is possible as against only partial 5th o r d e r a p p r o x i m a t i o n in the most general case.
T h e following compatibility conditions, derivable from (67) and (68), are n e c e s s a r y and sut~cient to base the synthesis on the six link m e c h a n i s m , even it the desired function does not belong to the subset originally c o n s i d e r e d : oot "4- co., =
l
(Id I ~
--(,~:]
(~? =
(69)
13, = / 3 , = --/3~ = -/3:~. etc. Equations for fourth o r d e r synthesis o f the type-I six link m e c h a n i s m with transmission angle control are given in [8]. T h e p r o c e d u r e d e p e n d s on the solution o f a quadratic equation only. E x a m p l e S B o f Section 12 was verified using this procedure.
16. The Cyclically Symmetric Mechanism T h e suitability o f the cyclically s y m m e t r i c m e c h a n i s m (Fig. 10) for functions that are cyclically s y m m e t r i c in ~h, 0, and & will be investigated. T h e m e c h a n i s m has the following p r o p o r t i o n s : I
1
bl =--%-;
b.,=-
#~ =
~/3 2
" e.,= -
n-
V'3
,-); g l -
~
W/3
1 + n ~/3
2
' ge-
Q;
"~
/t~
n +
=-
¢.. h . , -
V'3 7~
c:~:
1 - n V'3 2
c.,. -
T h e values of the derivatives are: col = co., = - l
(70)
O~l = ~2 =
(71)
C~:] =
tt H - -
2LY:/ 1
!72)
C2
,8, = -- 6eea'-'+ 2 K
(73)
fie = - 6oqle -- 2 K
(74)
/3:, = - 4c~f + K -- ( k + ] )
! 75 )
134 = --4eLa'-'-- K - - (L + 1)
/76)
269 where
K = X/3(a.~-n)[1-+ n(3-n'-)(cta-n)
(77)
and L = 3(l-3n~)(a:~-n) ( l + n ~ ) '-'
~
(78).
E q u a t i o n s ( 7 3 - 7 8 ) a s s u m e that ( 7 0 - 7 2 ) are satisfied. F o r a cyclically s y m m e t r i c function f ( 0 , qS, 0) = 0, it can be s h o w n that the following relations are valid: o91 = co,, = -- 1
(70a)
al = a._, = 2c~3
(71a)
/3~ -----ft., = -- 6a3"
(73a)
f13 = f14 =--40~,.," + [ A - 3B + 2C]
(75a)
where
A
=
a.y= a62 a y _ a~ ay~
aO.3
ay
ay
B = a02aq, = ~ ,
(for numerical values only)
etc.
c = (aflao) = (ofla6)
= (oflaq,)
and C-
O3f
OOO4~OqJ It is evident that third o r d e r synthesis will be p r o v i d e d , for a cyclically s y m m e t r i c function, by a cyclically s y m m e t r i c m e c h a n i s m that satisfies (72). T o a c h i e v e fourth o r d e r synthesis, it is n e c e s s a r y that K = 0, f r o m (73) and (74) and also f r o m (75) and (76):
1+
n(3 - n 2 ) ( a 3 - - n) (1 + n")" =0
(79)
excluding the case c._, = ~. T h e o t h e r condition follows f r o m (75), (76) and (78): /33 + 4aa2 + 1 -I- 3 ( 1 --3n2) (as--n)-' " (I + n ~ ) "-' = 0.
(80)
We have thus two equations (79) and (80) to be satisfied by the only u n k n o w n n. Full fourth o r d e r s y n t h e s i s is therefore, in general, not o b t a i n a b l e and only least s q u a r e s solutions m a y be a i m e d at. Elimination of n b e t w e e n (79) and (80) leads to a c o m patibility condition c o n n e c t i n g f13 and a3. T h i s condition has been f o u n d to be too
270 c o m p l i c a t e d to be useful. If the d e s i r e d f u n c t i o n satisfies this c o n d i t i o n , full f o u r t h o r d e r s y n t h e s i s is p o s s i b l e .
17. Conclusion T h e p r e s e n t a r t i c l e u t i l i z e d o r d e r a p p r o x i m a t i o n for the s,,, n t h e s i s or" the p l a n e s e v e n link m e c h a n i s m . D e p e n d i n g o n the f u n c t i o n to be g e n e r a t e d , p r e c i s i o n p o i n t o r m i x e d p o i n t - o r d e r a p p r o x i m a t i o n s m a y s o m e t i m e s be m o r e d e s i r a b l e . It is b e l i e v e d t h a t the d i r e c t s e a r c h m i n i m i z a t i o n p r o c e d u r e s will be u s e f u l in t h e s e c a s e s also, p r o v i d e d that the n u m b e r o f i n d e p e n d e n t v a r i a b l e s is r e d u c e d b y first s a t i s f y i n g s o m e o f the p r e c i s i o n c o n d i t i o n s , as w a s d o n e in this article.
References [ I J SVOBODA A. Computing 3,1echanisms and Linkm,,es. McGraw Hill. New York I [ 948i. [2] ALLEN C. W. The design of linkages to generate function of t,,~,o variables. J. E,'t:,'m,,[nJ. 8lB. 23-2'~ (1959). [3] F REU D ENSTRIN F. and SANDOR G. N. On the Burme,,ter points of a plane. J. april Mech. 2& 41 et seq. (1961). [4] PHILIP R. E. and FREUDENSTRAIN F. Synthesis of t~o-degree-of-freedom Linkages tor the maximum number of precision positions. J. Mech. 1, 9-21 i 1966). [5] FREUDENSTEIN F. and ROTH B. Numerical solutions of systems of non-linear equations. J. Ass comput. Math. 10, 550-556 (1963). [61 ROSEN BROCK H. H. and STOREY C. Computationa[ Techniquesfbr Chemical En,~,ineers. Pergamon Press, Oxford (1966). [7] POWELL M. J. D. Minimization of Functions of Several \'ariables. ,An article in Numerical,4mdysi.~. Academic Press, New York (1966). [81 LAKSHMINARAYANA K. Synthesis of plane lower pair mechanisms for bi-variate function generation. Doctoral Dissertation. Department of Mechanical Engineering, Indian Institute of Technology, Madras (1969). [9] SPANG H. A. A review of minimization techniques for re)n-linear functions. SIAM Rev. 4. 343-365 t 1962). [I01 Wl LDE D. J. Optimum Seekim,, Methods. Prentice Hall, New York (1964).
APPENDIX
--Y[l-t-n(n--~)](~'x):~-L--v[l-l-("t-~):~l(b"+~ax):~. ~ -~ - ' , c . , " Z~--x
[
a,-~,
( 1)J[
~ n-
x
-x
l+n
(')] -7 n
.,,
,
(a~,.r)'-'--c,
1
-- y [ l --{-(/z - 1 ):z ] (oJix) 2 h.~ " Z.,=y
a~
_
n--
_•_[. --y[l +(n--~:)'-'l(oJ.,y)e g~
Z., = x [a,_, + aJ._, ( n -- 1 ) ( 1
--x[l+n(n--l)](o~._,y)
__
"-' 1 _ x I l + ( n _ l ) - ' l ( e . _ , + ~ o . , _ y ) . , C,>
l
7.,.
271
1
--V "
n--
1+
\
b.,.+cotx)(co.,_y)--; gz
x/
j
h.,'
Derivative Synthesis of Plane Mechanisms to Generate Functions of Two Variables K. Lakshminarayana*
and R. G. Narayanamurthi~:
Received 1 October 1969 Abstract Successful procedures for five precision derivative (third order) and nine precision derivative (fourth order) synthesis, with transmission angle control, of the seven link two-degree-of-freedom mechanism with a double joint are presented along with the results of computational experiments to evaluate the procedures developed. Some important special configurations of the mechanism are also considered.
Zusammenfassung-Ableitungs-Synthese Ebener Getriebe zur Erzeugung yon Funktionen zweier Ver&nderlichen: K. Lakshminarayana und R. G. Narayanamurthi. Es werden for siebengliedrige Getriebe mit zwei Freiheitsgraden und einem Doppelgelenk erfolgreiche Syntheseverfahren for f/dnf bzw, neun 0bereinstimmende Ableitungen (d.h, Synthese dritter bzw. vierter Ordnung) bei Kontrolle der 0bertragungswinkel angegeben. Numerische Rechenergebnisse ermrglichen die Bewertung der Verfahren. Zus&tzlich werden einige wichtige spezielle Konfigurationen des Getriebes betrachtet. Pe311oMe~rlpOH3BO~Hblll~ CHHTe3 rl.qOCKHX MeXaHH3MOB J1EIR HpOH380~CT~3. (~yHKHH~ ~ayx nepCMeHHblX" K. JlaKl~MHHapagHa H P. l-, HapaAuaMypTH. VcnemHble npHeMbt CHHTe3a ngTH TOqHblX NpOH3SORHblX (TpeTb~lrO nopRara) H ~]eBRTH TOHHblX npoH3BO~HblX (qCTBCpTaro nop~nKa) C ronTponeM yrna nepc~.a,~H CCMH3BeHHaI'o C /].ByM~I CTeneHflMH CBO~Orlbl MeXaHH3Ma C OIIHHM /].BO~HblM lllapHHpOM, /],alOTCR C pe3yJlbTaTaMH qHC~eHHblX 3KCnepHMCHTOB ~t]~l OHeHKH pa3BHTblX npoueJ1yp. HeroTop~,le Ba.)KHble o¢06ble KOHTypb! MCXaHH3Ma TaK)Ke pa3cMaTpxaa~TCR.
1. Introduction THE SEVEN-link mechanism (Figs. 1 and 2) is the simplest two-degree-of-freedom mechanism that permits rotary inputs and output. Its synthesis has been the subject of a few investigations [ 1-4]. Confining ourselves to the case where the input and output scale factors are free to be chosen, Allen[2] provided simple graphical procedures for four precision displacement and four precision derivative synthesis of the type-I mechanism (Fig. 1). Freudenstein and Sandor[3] showed how Burmester theory can be used to obtain an arbitrary set of four precision displacements for the type-I mechan*Lecturer in MechanicalEngineering.Indian Institute of Technology, Madras-36, India. :~Professorand Head, Department of Mechanical Engineering,lndian Institute of Technology, Madras36, India.
249
250
/NPUT I ~ A
N~UT-2 L?
F i g u r e 1. The type-I m e c h a n i s m . F
B
E
NPUT- 2
Figure 2. The type-II m e c h a n i s m . ism. More recently, Philipp and Freudenstein [4J investigated the use of the parameter perturbation procedure[5] for a 12-precision displacement synthesis of the type-II mechanism (Fig. 2) by algebraic methods. T h e i r investigation revealed the difficulties involved in the numerical solution of the equations of precision displacement synthesis. This paper is mainly concerned with five precision derivative (third order) and nine precision derivative (fourth order) syntheses of the type-I mechanism by algebraic methods. For functions of two variables, precision derivative approximation may be considered to be particularly advantageous, as shown in the following. Let (0.40) and ,h be respectively the rotations of the input shafts and the output shaft from the design position of the mechanism. T h e following notation is introduced: At the design position. a ,h aN =
O,l, ; gg =
0 " ,~ 0 -'~I, ; oo = < ; 5-g =
O"-'llJ ; ooo
0 a~
0 a~O
Oa,h Oallj 002040 -- Ba; 0006"- = / 3 4 ;
[,!J]`* = first directional derivative of ,/J at 0 = (h = 0, in the direction making an angle y with the 0-axis in the (0. ,b) plane; [to]re = second derivative; [~b]`*a = third derivative. It can be easily shown that: [¢]`* = oJl cos -/+ aJ_~sin y [tO],,._, --- oel cos.' 7 + oe._,sin" y, + 2o~acos "y sin 7 [~] .-,a =/3, cos a 7 +/3.2 sin a 7 + 3/3:1cos" 7 sin y + 3/34 cos 3' sin.' T.
251 From the above expressions for the directional derivatives it is seen that to achieve, for instance, fourth order approximation between the desired and generated functions in A L L the directions of the ( 0 . 6 ) plane, it is only necessary to equate the corresponding partial derivatives of the first, second and third orders for the two functions.
2. Displacement Equations T o obtain the relation between the input rotations and the output rotation, the coordinate projections of the link length-vectors are chosen as the parameters most suitably defining the mechanism. Figure 3 shows the reference or design position of the general seven-link mechanism (type-II) with O A D as the fixed link and O Z as the output link. T h e length O Z is taken equal to unity for convenience, without any loss of 2
+ B (a t÷ b t , a 2 ÷ b 2 )
~ ~ - ( 1 + C
F
t, C2)
(!+f l ,
f2 )
" E (d s + et~d 2 ÷ e2 )
Figure 3. Reference or design position of the type-II mechanism. O, 6 = input angular displacements from the design position (positive counterclockwise); ~ = output angular displacement from the design position (positive counterclockwise); r / = e - O; p = ~ - 6; k = clockwise angular displacement of link ZCF with respect to OZ;--~: with respect to OADT-",: with respect to OZ. generality. Axis-1 is chosen along OZ and attached to it. This choice of axes considerably simplifies the equations and their derivation without, in any way, affecting the general applicability of the result. T h e legend attached to Fig. 3 explains the notation. The co-ordinates of the various joints are indicated in the figure within parentheses. Indicating the co-ordinate projections of the links in the displaced position by means of primes, we have the following relations between the projections in the displaced and reference positions: a~ = b'l = c't = d't = e~ = f't =
at cos t0 + a., sin ~; b, cos rt + b.,_sin "0; c, cos h + c_. sin h; d~ cos q~+d2 sin tO; et cos p + e 2 sin/9; f, cos X + A sin X;
a" = b" = c2 d" =
a2 cos O - a, sin tO b2 cos "0 - bt sin "0 c2 cos k - ct sin ~, d2 cos $ -- d~ sin 0 e~ = e2 c o s p - - e l sin p f " = f , cos h -- ft sin h.. P
(1)
252
Using the fact that the length of the link B C does not change, ta~--bt-l-c~)e+tae+b,,-ce)
~=iai+b'~-l-c~)e+(a;--b'.-4lL
which, on expansion, gives: (alb~ + a.,b.,) - ( a l c l + a.,c.,) - ( b l c l + bece) + ct - at - bl = = (a'~b[ + a'b'.,) -
(a;c'~ + a'c',,_) -
(b;c; + b'c')
+ c[ - a ; -
b't
2~
T h e following notation is introduced: alb~+a.,b._,=C~;
gztcl - ~ - a e Q = A t :
htc~÷b._,c.,_=Bl:
a . , b l - a~b.,_ = C._,;
a.,c~ -- aLce = A.,;
becl-
blc._, = Be:
d t e ~ + 4 e . _ , = F,;
d,j; +de¢.'_, = D,;
eJ;+
ee); = & :
d.,_el - - d t e e = F.,;
d . , _ j ; - - d , f e = De;
e , , / ; - - e J ; = Ee;
F r o m equations ( 1 ) a n d ( 3 ) , a'~b't + a.',b.'_, = Cz cos O+ C.e sin O.
i4)
By analogy, a',c't+a:,.c; = A t
+A._, sin ( t o - X )
sin ,/*)cos a + {A, sin to--Me cos 0 ) s i n X
(5)
b'tc'~+b.'c.',_= ( B , c o s v l + B . , s i n r l ) c o s X + ( B t s i n v l - B . , _ c o s ~ ) s i n h .
(6)
=
(,.4 1 C O S
cos ( 0 - , \ )
to-}-A.,
and
T h e following additional notation is introduced: X~ = A t c o s to + A._, sin ~) + B~ c o s rl + B.,_ sin ~ -- c ~
X2 = A t sin to - A., cos ~/i + B 1 s i n r / - - Be cos r / - - c., Xa =
(C~-At-Bt)+(ct-a,-b,)-(CtcosO+C.,_sinO)
+ a t cos t o + a e sin ~ + b , cos ~ + b ~ sinr~ Yt = D t cos qJ + De sin 6 + Et cos p + Ee sin p - J;
(7)
Ye = D~ sin to-- D., cos to+ Et sin p - Ee cos p - - f , Ya = ( F t - D t - E t )
+ ( ) " ~ - - d t - et) - (F~ cos 4~+ Fe sin 6 )
+ d t cos ~0+ de sin tO+ e, cos p + e., sin p. Substituting from (4) to (6) in (2) and using (7), the following displacement equation for the loop " 0 . 4 B C Z " is obtained: X~ cos ~ + X . , _ s i n h + X a
= 0.
(8)
25,3. The displacement equation for the loop " O D E F Z ' " is written down readily by analogy: (9)
Yt cos h + Y~ sin h + Y3 = 0.
Elimination of h from (8) and (9) gives the displacement equation for the whole mechanism:
(lO)
(X~,Y3--X3Y.,): + ( X 3 Y , - - X , Y 3 ) " = (X,Y.,_-- X~.YO"-.
This is probably the simplest possible form suitable for synthesis with a large number of precision conditions.
3. Derivative Equations for the Individual Loops in the Design Position For the purpose of order approximation it is necessary to form the successive derivatives of either the displacement equation for the mechanism, (10), or the displacement equations for the individual loops, (8) and (9). The latter will be used here. The following notation is introduced: At the design position, OX~ = Xto; OzXl = X,~,_; O"X, _ OsX, aO O0z 000(5 - X , ~ ; O02O(b = Xt~,,; X1 = X,o;
Oh , azh 02K a--O= ~,0; ~ = h~,; OOa(b = ho, ; similarly others. On repeated differentiation of the displacement equation for the loop " O A B C Z " and setting all angular displacements equal to zero,
Xto + Xao + X2oh0 = 0
(lla)
X t , + X.~ + X.,oA, = 0
(12a)
X,~., + X3e.,_+ 2X.,_oXo - Xtoho 2 + X2oh~, = 0
(13a)
Xt,., + X3,., + 2Xz,h,t,-- X , o h , z + Xzoh,.,. = 0
(14a)
X l ~ + X 3 ~ + X._,e~ho+ X.,_oh.n-- Xlo,koh~ + X.,,oh~ = 0
(15a)
X ,o3 + X3o:~ + 3 X2~:~o - 3 X ,oXo2 + 3 ( X,,_o- X ~oho) he2 + X2o( Xoa - X fl )
=
0
(16a) X,~a + X3,3 + 3X~,.,h, -- 3Xt,,k, 2 + 3 (X._,~-- X,0,k,~) h,2 + X2o( ;%3 -- h~a) = 0 (17a) Xt~,, + X3~,¢,+ 2X2o, ho + X2~2k, -- X,,ho" -- 2XtoXoh, + (X.,, - - X , o ~ ) X~. + 2 ( X 2 0 - - Xloho) ho~ --{-X2o ( ho,2.~ -- ho2ho) = 0
Xlo~2 "~ X30~2 + 2X~.~h, + X.,_,2ko--
Xloh,
2 --
(18a)
2 X l , h o k , + ( X~_o-- gtoho) ks.,_
+ 2 (X.,, -- X,,,h,) h ~ + X2o ( h~., -- h0h, 2) = 0.
(19a)
Coffesponding equations (llb)-(19b) for the second loop " O D E F Z " will be obtained by replacing X by Y in the above set of equations. The X,Y-derivatives at the
254
design position, occurring in the above equations, are evaluated from I7). They are presented in [8]. 4. Third Order Synthesis of the Type-I Mechanism For the purpose of order synthesis, it has been found very advantageous to replace the parameters a~, a..,. d~ and d._, by the parameters g~. ge. h~ and It.,_. The latter are the co-ordinate projections of links CB and FE (CE for type-l) respectively I Fig. 3). thus satisfying the following relations for type-I: = g, -= cl + 1"
at-b1
d~'e~=h~+cz+
I;
a., + b.,_ = ;,., + c., d.,~-e., = / t , - - c . , .
The following notation is introduced: n
=
('l (-"2
--
{
20)
n being the cotangent of the output transmission angle in the design position. (Transmission angles are discussed later.) Also, c,_,t ,% -
coz) =
~o~ x
and c.,( X,~ - ~o.,) = co..,3".
defining two new quantities x and y. With the above indicated changes and notation, equations (l la)-(12b) reduces to the following on substituting for the values of the X, Y-derivatives [8] : (provided~o., ¢ 0 )
;,~=(tl-})~q,:
(21
(provided~o~ ~ 0 )
h~=(n--1)h._,
~22)
(providedq., ~ 0)
bl
(provided he # 0 )
e,=i,,-l)ee+co..(l-V).
be+co~
123)
y
,24)
Equations (t3a-15b) take the form given in the APPENDIX. on substitution from equations I20-24) and using the following notation: Z I =
(/~02-- (.yl)c.,:
2.2 ~--- { ~a5,2-- 0~.2)4',2'
Z:I =
(~,9,,5- (_~:;)c.,.
Eliminating Z,, Z_~ and Z a from the relations appearing in the ,-\PPEND[X arid introducing the following notation:
255 I
1
c.*_ - (l+nZ)(x--Y)--C.2
± g~,=
[
y 1+
(--'yl-' n
Y/ ]g2
and
/,-T=x i+ . - 7
'
I(
')]'
we get: ( o~., v)-' - (~=y)" (~,x)i~._, y) - (b._~+ ~,,~-) (~,_,y)
(e,, + ~ y ) " /in/g?/ (~,x) (e., + ~._o') JL ]I/7.Z J
=
I
2
(25)
L (x - y ) (~:, - ~o, o~._,n ). This is a system of three linear equations in the three unknowns 1/c.,_, 1/g.,. and l/h.,, if the values of n, x, y, b._,and e._,are chosen. Thus, the problem of third order synthesis has been thoroughly simplified first by replacing the parameters at, a._,,dj, and d., by the parameters gt, g._,,hx and tt._,and secondly by a proper choice of the parameters whose values are to be assumed and of those parameters whose values are to be determined. T h e above choice, at the same time, enables us to have control over the transmission angles in the design position. {Transmission angle control is dealt with in a later section.) Solving the equations (25) and using the following notation:
P = (x--y)[oq-wlZn+wl ( n - ~ ) ] - Y [l + (n-~)"-]b._, Q= (x-y)[a.,-oJ.,'-n+oJ.,(n-l)]+x[l+(n-x) R = (x--y)
1
" ]e.,
(a:~-- ~otco._,n)
(26) (27) (28)
S = (oJ.,_y)P- (oax)R T = ( o J t x ) Q - (~.,_y)R U = b.,e.,+ (oJ2y)b.,_+ (oJtx)e.,_.
(29) (30) (31)
We have:
(o~lx) (oJ.,.y)b.,e.,. c., = -S(e.,_ + oJ.,_y)+ T(b., + oax) - R (b~_e.,_) (1 + g.,_=-S(e.,+~.,_y)+T(t.o,x)y
1+ n -1
h.,=+S(~.,y)+T(b.z+OJtx)X
1+ n--
n 2) ( x - - y )
(32) (33)
' .
(34)
256
A f t e r calculating c'z. ,e~ and h..,. the remaining p a r a m e t e r s are o b t a i n e d from (20) ~o 124). It m a y be n o t e d that this s o l u t i o n a s s u m e s that ¢m and co~ are n o n - z e r o . C o n s i d e r a tions like the choice of values for n . x . y , b., and e., are not dealt with here but taken up later in c o n n e c t i o n with fourth order synthesis, as the main use for third o r d e r synthesis is in the solution of the fourth o r d e r synthesis equations.
5. Equations for Fourth Order Synthesis of the Type-I Mechanism T h e following notation is introduced: K~-- toh.r)(.r--v) II÷nZ) 1-('.,
,
L.,:
-
I +n
. n-- v
o~.,_
y ) .r
[ ( ')l' 1 4- tt
(o~,,y)g,,
0)2
2 --
09,2
1] I c.,
--
h.,
i38)
c.,
n-
+ o:._, n - -
-,q.,
!39)
Ir~,
( 40 )
1'
¢ol 2 4 - 0L t
x
37
~
v +cq
(,'
~t -
36
.'.,"_'
(e..,+oJ._,y)/T'
[ [
( e,_, 4-
--
~°'x)
.12"
"~1 : X
135
_r ) t I -'- n-' ) - -
¢°l~-°);(l--r,
=
--,k'_,
,/'L,
I ( b . , + o h . r ) g'-'
M'=)'[
~'~4.,
,
[ ( ')1'
+ Io).:y)y [
.'f
1
( o~._,y : ( y
Lt =
I--n
[ l + n ttl . . '1. ) . '
+ (ohx).r K., :
Ib.,÷oh.v)v "
-/r,
~41l
1
(42)
{43)
P., :
.r
m t m . 2 4- CZ:;
44)
. . . . . X
[1,, "
P r o c e e d i n g on the s a m e lines as in the p r e v i o u s section and using the a b o v e n o t a n o n . equations I 16a- 19b) lead to the following four equations for fourth o r d e r synthesis: 3Z~ Kx + 3 (co,r)"L~ + 3 ( b._,+ = [x-y)[(/3,-co,
:~)-o,,t3a,~t-
3Z._,K., + 3 (co,, y )"-L, + 3 ( e.,_ +
=(y-x)
co~x),'vl~ - 3 ( c o l . r ) N ~
co,,
1)]
i45)
y )M.,_ - - 3 ( co.,_y ) N.,_
[ (/3~ - o~._,:{ ) - o,._.(3oc, n -
t)]
E46)
257
2Z:~Kt-ZtK.,+2(wtx)(o~.,.y)Lt--
(cutx)"L.,_+ (oJo_y)Mt- (e~ + co._,y)NL
+ 2 (b.., + mix) Pz - 2 (cotx)P., = ( x - y ) [(/3:; - cot2co._,) -- n (arm._, + 2aacot ) ] (47) 2ZaK., -- Z.,_Kt + 2(oJlx) (~._,y) L: - (w._,y )ZL, + (~ix)M., - (b., + colx) N., + 2(e.,.+co.,.y)P.,_--2(oJ2y)Pt
= ( y--x)[(~4--wlcoo."-)--n(o~.,wl
+ 2a:~oa.,) ].
(48) The above arrangement of the equations is aimed at minimization of computation. These equations are, effectively, four highly non-linear simultaneous equations in the five unknowns t,, x, y. b.., and e., since I/c.,, l/g._, and 1/h.,. are directly expressible in terms of n, x. y, b., and e2 by equations (32), 133) and t34). Expressions for Z t , Z.,_ and Z~ are available in the APPENDIX. The practical solution of these equations is dealt with in the succeeding sections.
6. Method of Solution For the numerical solution of the fourth order synthesis equations (45-48), it was decided to exclude methods that (a) require a close initial approximation, or (b) show evidence of lack of success on problems of comparable difficulty, or (c) seem promising but for which sufficient numerical evidence is lacking, or (d) depend on simplifying assumptions. On account of (a), only minimization methods come into consideration. Because of the extreme non-linearity of the equations, it was expected that the minimization function will be extremely " m i s b e h a v e d " with very narrow "ridges"* in the contour "'map" of the function. The "direct" search methods, that depend only on the qualitative comparison of function values, are essentially of the "ridge tracking" type and also permit easy incorporation of constraints. T h e direct search minimization method of Rosenbrock [6] was selected on the basis of its reported [6, I 0] comparative success with very difficult ridges and also since it takes the most pessimistic view of the nature of the minimization function. T o obtain a suitable starting point for the minimization procedure adopted, a short random search, that takes constraints into account was carried out. This serves two purposes: (a) With the method of dealing with constraints adopted (see next section), it is necessary to start from a feasible solution (i.e. from a point where the constraints are not violated). (b) Since the technique can lead to a non-zero minimum, it was considered desirable to make a short random search and start with the best point obtained in an effort to minimize such a possibility.
7. Constraints-Transmission Angles In order to obtain practically useful mechanisms, it is necessary to prescribe limits on such quantities as transmission angles and maximum to minimum link length ratio. With a view to be able to investigate the effectiveness of the chosen minimization technique proper, it was decided to control the transmission angles alone. Their control was obtained by limiting the values of the transmission angles and their maximum rates of change in the design position. The transmission angles in the design position are controlled by the method of transformation of variables [7, p. 156]. Let the function to be minimized be F ( q i , q., . . . . ) subject to: - q , o <~ q , <~ q,o, for instance. By replacing *This is a term more appropriate for a maximising problem. It is thought unnecessary to introduce new terminology.
258 q,, by a new variable t, in the following m a n n e r : q, =- q,,o sin t,. the constraint is eliminated. C o n s t r a i n t s eliminated in this m a n n e r will be referred to as implicit constraint',. T h e m a x i m u m rates of change of the t r a n s m i s s i o n angles are controlled by simpl,, rejecting the points that violate the constraints, as "'failures". C o n s t r a i n t s taken care of in this m a n n e r will be referred to as explicit constraints.* Figure 4 s h o w s the type-l linkage with the j u n c t i o n t r a n s m i s s i o n angle./a.j and the
(g)
C~
Figure 4. Transmission angles in type-1 m e c h a n i s m . output t r a n s m i s s i o n angle/xo. It is easily seen that COt
(49)
P-o ----- tl
cot/a.j =
l
1
X
V
= k (say)
51
,90 = to1 I +
0.,, Oa5
-
r).u.i _ O0 Ota.j a6
_
(50)
( +:'t
52
c._,/
b._,+ ¢ o , x g._,
co,x /t.,
w.,_y
e., + to.ey
g._,
h._,
5
~
c54)
T h e m a x i m u m rates of c h a n g e of the t r a n s m i s s i o n angles /a.o and /x~ are r e s p e c t i v e l y given by: ro r~
\ &b /
= ,/(0,,,7
(0,,q-'
~/\ aOl + \ a(b /
" O t h e r e x p l i c i t c o n s t r a i n t s can be ea~,il_v i n c o r p o r a t e d into the c o m p u t e r p r o g r a m m e s [8]
56
259
The maximum allowable limits on these values will be referred to as rot and r~ respectively. 8. The Objective Function For the solution of the equations (45-48), an objective function F, to be minimized, is to be formed. Writing the four equations as ,/'1 = 0: f_, = 0, f~ = 0 and f4 = 0, it is usual to have:
F = f ? + A " - + A " - + L 2. This gives equal weightage to all the four derivatives/3~,/3.,,/33 and/34. Such a formulation will not matter as long as a zero is reached. However, in case it is not possible to make F zero*, it will be desirable that the errors in the derivatives/3~,/3._,,/33 and/34 are so distributed that, considering the error derivative (/3~) in all the directions of the (0, 6) plane, the maximum value of the error derivative is minimized. However, since the determination of the maximum value leads to the solution of a cubic equation, the root mean square value of the error derivative has been used as the objective function. Using the subscript e for the derivatives of the "error surface" at the design position, it follows from the expression for third order directional derivative, given in Section 1, that /3e =/3t~ cos a Y+/3.,.e s ina 3'+ 3/3.~ecos'-' 3' sin 3'+ 3/34~cos y sin" y.
(57)
Considering the variation o f y from 0 to 2rr, the root mean square value of/3e is obtained aS;
I /3~r.m.s= 4--X72~,/[ (/3~- 3/34~)'-'+(/3.,~- 3/3.~e): + 9 (/3~+/34~)-' + 9 (/3._,~+/3.~)-'].
(58) The objective function actually used in the computations is: F=32
[/3er.m.s.] 2.
(59)
It is easily seen that any point that makes F zero will be a solution of the original system of equations. 9. Random Search Instead of making the random search with the variables n, x, y, b., and e2, the procedure given in the next paragraph is followed, in order to be able to control the transmission angles at the design position by the method of transformation of variables. This procedure also avoids the problem of having to choose the values o f x or y whose ranges of variation are not known. A value of the link length 'b' is randomly chosen between certain lower and higher limits b~ and bh. Also, b2 is chosen less than 'b' in magnitude, by multiplying 'b' by a random number lying between - 1 a n d + 1. The value ofbt is found from: b~ = - ~
- b._,"
* Either because of constraints or because of the obtaining of a relative minimum or because of the failure of the minimization technique.
260
where the sign before the radical is randomly chosen. The values of n ti.e. cot/.<) and k (i.e. cot ~.~) are obtained from: n~ S i l l t l
60
k = kt sin r,
61
tz ~
and
where nz and kt represent the specified upper limits on the magnitudes of cot g,~ and cot/xj respectively and tt and t._, are randomly chosen between - = / 2 and + =/2. Using the notation: 1 l l t ~--" I 7 - - - X
and 1 Ul~r/
--
Y
,
we have, from equation (50): l +u~v~ Ul --
=
(621
k
14 l
and from equation (_.~): "~'
/ 63 )
Eliminating v~ from (62) and (63), (a2, -- b, - kb., )u(-' + [be ( n k -
1 ) -4- b I (tl + ]c.) ]it, + [to I - - / 7 ( k b ~ - - be) ] = O.
(64) Equation (64) is solved for u~, choosing randomly the sign before the radical in the expression for tq. T h e value of c'~ is then obtained from equation (62). Knowing tq and th, we have: l -
=
n -- u t
(65)
=
n--
(66)
X
and -
1
v~.
Y T h e value of e.z is now chosen randomly between certain limits - e.,z and + e~t, so that el can be calculated from equation (24) and e from: e = ~ / e l e + e.F. The value of e thus found is checked to be within specified limits. (If it is not, briefly stated, the process is repeated from some stage. Details are given in the c o m p u t e r p r o g r a m m e in [8].)
261
The third order synthesis is now carried out and the explicit constraints (r0 ~< rot; r~ <~ rig) are checked for violation. If a constraint is violated, the procedure is repeated from the beginning. If no constraint is violated, the objective function F is evaluated and the procedure is repeated from the beginning. After the specified number of function evaluations, the feasible solutions thus obtained are sorted out to the extent necessary so that the best 20 (for example) solutions are available in the order of increasing function values. These feasible solutions are used as starting points for the sequential search considered in the next section. A working computer programme in Fortran-IV for the random search described here is presented in [8].
10. Sequential Search The method adopted, as already indicated, is largely the method of rotating coordinates of Rosenbrock. It will be briefly described here. A detailed description is to be found in [6, 8]. The independent variables for the search are t~, to, l/y, b._,and eo-. This set of five parameters is equivalent to the set of n, x, y, b,, and eo- as shown below and is chosen because, in that way, the transmission angles in the design position can be controlled: With the s a m e limits nt and kt as in the preceding random search, n and k are first calculated from equations (60) and (61), at any point (h,t.,_, l/y, b2,e.,). Then, vl is found from (66) so that u~ is obtained from (62) and hence x from (65). Knowing thus the values of n, x, y, b., and e2, third order synthesis is carried out, violation of explicit constraints is checked and the objective function F is evaluated. Even though the number of equations to be solved is only four, equations (45-48), it was decided to leave all the five parameters as unknowns to increase the probability of finding a solution at the expense of possibly increased computation. Briefly, the search is made in mutually orthogonal directions, making only one trial at a time in each direction and returning to the original direction after exhausting all the five directions. When a step is taken in the chosen direction, if the explicit constraints are violated, the step is considered a "failure". If the constraints are not violated, F is evaluated. If this value of F is more than the value at the " b a s e " point, the step is a "failure". if it is less than or equal to the value at the base point, it is a "success". If it is a success, the basepoint is shifted to the succesful test point and the step size in the direction is doubled. If the step is a failure, we return to the base point and the step size in the direction is halved and changed in sign. When a success followed by a failure has occurred in all the five directions, the orthogonal search directions are rotated such that one of them (direction 1) is in the direction of total progress made since the last rotation of directions and the initial step sizes are restored. It may be noted that direction 1 provides an initial guess of the direction of the "ridge". Since it is difficult to determine from an internal check [6, p. 67] whether the minimum has been reached or not, unless the minimum happens to be zero, the sequential search programme is arranged such that the search will terminate after a specified number of function evaluations or after a specified function value (Ft) has been reached, whichever is earlier. This specified function value can be a small positive quantity based on an estimate of the extent of round-off i.e. searching for function values below this level is rendered meaningless by round-off error. In order to be able to examine the results for convergence, the following information is printed each time before the search directions are rotated: F; tl, t.,_, l/y, b.,_, e.,; "Pgt'" i.e. the square of the total progress made since the search directions were last rotated; "r'" i.e. the square of the sine of the angle between the former direction 1 and the vector of total progress. If
262
p,j, is small for at least a few successive rotations and r is always large, then the minimum may be considered to be reached, though this is not certain [[6], pp. 68. 105: [8] pp. 6-29). At the end of the search, the latest direction cosines are also printed so that, if it is desired to resume the search after examining the results, the latest estimate of the direction of the ridge is available [6, p. 68]. A working c o m p u t e r p r o g r a m m e in FortranIV for the sequential search described here is presented in [8]. Numerical examples and a discussion of the results are presented in Section I2 of this paper.
11. Displacement Analysis Referring to Fig. 5, the analysis p r o g r a m m e [8] computes 6e for values of y and obtained by uniformly incrementing from zero. T h e values of 0 and ~b are found from: 0 =---- E COS T
and ~5 : esin 7. Choice of (0, do) in this way allows the plotting of error curves with constant 7. T h e s e curves represent "'vertical" sections of the error surface through the origin (0 = 4, = 0) and are like the error curves of single variable order synthesis, so that the obtaining of order approximation could be verified easily. Moreover, if there are no restrictions on the relative limits of variation of 0 and ~b, it seems rational to limit the range of variation of (0.4~) in all directions through a constant e .... . The percentage error can then be defined as: E~ = ( I';'~ ]m~,~) 100/(Range of variation of q, within a radius of e ..... )" A less rational but more convenient definition is: E=
. . . . )100/(2
....
i.
T h e computation of ~. involves the solution of two quadratic equations sequentially, i.e. given the positions of B and E (Fig. 4), there are two possible positions for C and
~ CON "/'OURS OF I CONSTANT
~(
t, ~jO.(c~O
)
• , \
SECTION X-X OF
\
Figure
5. The
error
surface.
263
given the position of C, there are two possible positions for Z. The analysis programme [8] automatically selects the positions of C and Z such that the same branches of the loops are obtained in the displaced position as in the design position. For this purpose, the programme utilizes the signs of sin/_ CBE and sin / COZ in the design position as criteria. 12. Numerical Examples and Discussion of the Results For numerical verification of the procedures described, the following two functional relationships were chosen:
O=
0 + ~b 2
(Adder)
and 0 = 0 + ~b + 0q~ 2
(Multiplier)
where 0, ~b and qJ are in radians. The second relationship represents the function z = xy if we let: x = k(O+ 1),y = l(~b+ 1) andz = kl(2t0+ 1), where the values of k and l are free to be chosen. For the adder, a random search of 2000 function evaluations gave the following best ten values for F: 1-37 (SA); 1-62 (SC); 1-90 (SG); 2.19 (SB); 2.23; 2.23 (SE); 2.30: 2-31 (SF); 2.33; 2-44 (SD). Those used as starting points for the sequential search are marked within parentheses with the corresponding example names for the sequential searches. For the multiplier, a random search of I000 function evaluations gave the following best ten values of F: 6-66 (MA); 8.15 (MB); 8.58 (MC); 8.78 (MD); 9.63 (ME); 12-65; 12.74; 14-11; 14-34; 15-11. it may be mentioned that even though these values are much higher than those for the adder, the former were reduced to a level comparable to that of the latter in a few hundred function evaluations, in the sequential search. The following limits were used in the random and sequential searches: 0-5 ~
264
SB and M D will be presented along with comments on all the examples to provide an appreciation of the merits and limitations of the synthesis procedure. Tables 1 and 2 provide a summary of the results of examples SA, SB and MD. The corresponding mechanisms are shown in their design positions in Figs. 6 - 8 respectively. Example SA led to an implicitly constrained minimum ,,vith k] = k~. Example SC, starting from an entirely different point, led to the same solution in about 8000 function evaluations. Example S G ted, in about 3000 evaluations, to a solution that is practically same as SA or SC if 0 and 4~ interchange their roles. Example SE, after 2500 evaluations, was very closely approaching the mirror image of solution SG on the output link OZ, The search was therefore discontinued, even though the minimum was not yet reached. Table 1 Example
SB
,',.4
Starting Search variables
;, 0"3209 t, 0.5468 I/y - 1 . 8 6 7 1 b, - 1-2088 e._, 0-1836
17 /3. . . . . . .
Final
MD
Starting
Final
Starting
Final
0-2396 1-5708 -1-4963 - 2-0564 0-1092
-0-2033 -0-8142 -1-5351 -0,4487 0,4275
0.6011 - 1,5136 -15.6574 -I)-6858 0-2800
1.1383 1.3802 0-5517 0.4032 -0.9717
1.5659 1-5693 0-9953 0.6713 -0.5011
1-0601
2,1936
0.00(R)
8.7795
0-316-4
1-3719 0" 18
0.00
0"099
(final) Number of function evaluations to minimum
]tz I [ k] r,~ rj
About 2800
(t:~ c r o s s e d ) 1461
About 7000
0.11866184 O. 5 0 0 0 0 0 0 0 0.92 0-50
0.28276356 O. 499 t 8140 ---
0-49999407 0.49999946 0"584 {)'365
On Axis- I
Link projections
a b c et e
6.3266 0.7471 1.1215 2.1637 0.560I
On Axis-2
On Axis- 1
On Axis-2
On Axis- t
On Axis-2
14.5739 -2-0564 9.4511 2.7237 0. I092
%0000 -- 1.3591 --0.1548 0.(.R)00 0.3651
- 0-0000 --0.6858 -0-5476 0-0000 0.2800
0.1801 0-1656 - 0-3116 0.0665 --I).0828
--0.6029 0-6713 -0.6233 -0.1195 --0.5011
Table 2. Example
SA
SB MD
% Error E, at 2e ..... = 30 ~
5W
70 °
90:
0'31 0"07 0"16
0"87 0"35 0"52
1"8 1"06 1-51
4-7 2-47 3"38
265
~9 8
c
Figure 6. Design position of solution SA. E
8f
"e
Figure 7. Design position of solution SB. z
Figure 8. Design position of solution MD. Example SB led to an unconstrained zero on the boundary of the implicit constraint lkl = k~. The inputs and the output are co-axial. Example SF also led to an unconstrained zero in 950 evaluations, giving co-axial inputs and output. In example SD, the minimization technique failed (because of an extremely narrow ridge) but only after achieving a value of/3er.m.s. = 0"06. It could be concluded with reasonable certainty that, otherwise, a co-axial mechanism would again have been obtained.
266
Example M D led to an unconstrained minimum very close to the boundaries of the implicit constraints Inl ~< nF and Jkl <~ kt. Example M B gave an unconstrained minimum close to Ik ~ k~ in about 5000 evaluations, with/3~r.m.~. = 0"12. In examples 3,1,4 and M C , the minimum had not been reached after 15,000 function evaluations, which was set as the maximum limit to be tried for a single example. The searches were close to implicitly constrained minima and the slow progress was due to very narrow ridges. T h e ¢?~r.~.,~.values were 0-12 and 0-14 for M A and M C , respectively. Example M E started with F---9.6334736 and reached an explicitly constrained minimum of F = 8.4264888 in about 1000 function evaluations. Very narrow ridges were encountered in most of the examples, thus establishing the necessity to use a good ridge tracking procedure, as was done in the present investigation. It may also be mentioned that satisfactory control over the transmission angles in the displaced position was obtained in all cases, It is to be noted that the first and second order derivative values are always exactly obtained whatever the value o f F may be. Some of the possible lines of improvement of the procedures are indicated in the following: {I) It seems advisable to reduce the possibility of obtaining a relative minimum by a long non-sequential search (preferably the grid type[9]) to be followed by a sequential search. This will have the additional advantage that starting points that will lead to previously obtained solutions can be spotted and avoided to a larger extent than was possible in these examples. (2) Since transformation of the variables alters the nature of the function, it may be worthwhile to try suitable transformation functions other than the sine.
13. Double Precision Computation The accuracy of single precision computation, as determined by comparing the values of the objective function obtained by single precision arithmetic with those by double precision, for the same set of values of the variables, varied between 4 and 6 significant digits. Apart from fixing a lower limit for the function value below which it will be meaningless to continue the search, this small round-off error usually has no significant effect on the effectiveness of the sequential search, This will be clear from the realization that the effect of the round-off will be merely to "'distort". to a "'small" extent, the contour surfaces of the objective function. H o w e v e r , sometimes a very narrow ridge may be "'blurred" by single precision round-off and a double precision search may be able to locate the same. This is not necessarily disadvantageous since an unwanted relative minimum may thus be avoided by the single precision search. Numerical trials have confirmed the above. It may be concluded that there is usually no need to evaluate the objective function with double precision except perhaps for refining the final result. 14. Computation Time For the IBM 7044 computer, on which the programmes[8] were run, the running time for the sequential search consists of about I rain and 25 sec for compilation and about I min and 45 sec of actual computation time for every 10,000 function evaluations in single precision. These values are small enough to make the procedure economically feasible in most cases. 15. Special Configurations-The Co-axial mechanism As against the general procedures of the preceding sections, another line of investigation consists of the study of special configurations suitable for special classes of
267
functions. When such a configuration can be found, the synthesis procedure can be considerably simpler. The co-axial mechanism (Fig. 9) and the cyclically symmetric mechanism (Fig. 10) represent two such cases. The co-axial mechanism (Fig. 9) can be designed to a high accuracy, comparable to. or even better than, the highest accuracy obtainable with the general mechanism, for a particular class of functions. A subset of this class consists of all those functions F
C
~OAD Figure 9. The co-axial type-II mechanism.
A
b=l B
~'~c .,,(=o
E -
Figure 10. The cyclically symmetric mechanism. which can be expressed as a relation between two of the following three differences: (~b-0), (@-(~) and ( 0 - ~ ) . The problem is then reduced to that of the synthesis of a six-link Watt mechanism with one degree of freedom. The latter provides a maximum of nine precision points and the resultant synthesis can be easily seen to give "'9 precision-line" synthesis of the seven-link mechanism. The accuracy may, therefore, be expected to compare favourably with that obtained by 13-point synthesis of the general mechanism. For order approximation consider, for instance, the generation of the function: (~b-th) = F ( 0 - - ~ ) . Writing ~ - - ( ~ = p and 0--(h=-0, we have p = F ( ~ ) and the derivatives of p with respect to "0at p = 77 = 0 may be designated as to, a, fl, etc. Expansion of F(,/) in Taylor's series about the point p = ~ = 0 gives on rearranging:
~ = 0~, + ~(1 - °4 + T ~ + - y ~ - 0 ~ + ~ ~---g- B - _ _
~-~--y- ~ + . . . . (67)
268 E x p a n s i o n of t0(0, 4~), on the o t h e r hand. gives:
,b= 0~,,+~¢o.,_+7~,,+--c~._,+0e0~:~+
~+
/:3.,+--/3:,+~---/3,+ .... i68)
C o m p a r i s o n of (67) and (68) leads to the c o n c l u s i o n that synthesis of the six link m e c h a n i s m to any o r d e r o f a p p r o x i m a t i o n provides synthesis of the seven link m e c h a n ism to the S A b l E o r d e r o f a p p r o x i m a t i o n . This m e a n s that up to 9th order a p p r o x i m a tion is possible as against only partial 5th o r d e r a p p r o x i m a t i o n in the most general case.
T h e following compatibility conditions, derivable from (67) and (68), are n e c e s s a r y and sut~cient to base the synthesis on the six link m e c h a n i s m , even it the desired function does not belong to the subset originally c o n s i d e r e d : oot "4- co., =
l
(Id I ~
--(,~:]
(~? =
(69)
13, = / 3 , = --/3~ = -/3:~. etc. Equations for fourth o r d e r synthesis o f the type-I six link m e c h a n i s m with transmission angle control are given in [8]. T h e p r o c e d u r e d e p e n d s on the solution o f a quadratic equation only. E x a m p l e S B o f Section 12 was verified using this procedure.
16. The Cyclically Symmetric Mechanism T h e suitability o f the cyclically s y m m e t r i c m e c h a n i s m (Fig. 10) for functions that are cyclically s y m m e t r i c in ~h, 0, and & will be investigated. T h e m e c h a n i s m has the following p r o p o r t i o n s : I
1
bl =--%-;
b.,=-
#~ =
~/3 2
" e.,= -
n-
V'3
,-); g l -
~
W/3
1 + n ~/3
2
' ge-
Q;
"~
/t~
n +
=-
¢.. h . , -
V'3 7~
c:~:
1 - n V'3 2
c.,. -
T h e values of the derivatives are: col = co., = - l
(70)
O~l = ~2 =
(71)
C~:] =
tt H - -
2LY:/ 1
!72)
C2
,8, = -- 6eea'-'+ 2 K
(73)
fie = - 6oqle -- 2 K
(74)
/3:, = - 4c~f + K -- ( k + ] )
! 75 )
134 = --4eLa'-'-- K - - (L + 1)
/76)
269 where
K = X/3(a.~-n)[1-+ n(3-n'-)(cta-n)
(77)
and L = 3(l-3n~)(a:~-n) ( l + n ~ ) '-'
~
(78).
E q u a t i o n s ( 7 3 - 7 8 ) a s s u m e that ( 7 0 - 7 2 ) are satisfied. F o r a cyclically s y m m e t r i c function f ( 0 , qS, 0) = 0, it can be s h o w n that the following relations are valid: o91 = co,, = -- 1
(70a)
al = a._, = 2c~3
(71a)
/3~ -----ft., = -- 6a3"
(73a)
f13 = f14 =--40~,.," + [ A - 3B + 2C]
(75a)
where
A
=
a.y= a62 a y _ a~ ay~
aO.3
ay
ay
B = a02aq, = ~ ,
(for numerical values only)
etc.
c = (aflao) = (ofla6)
= (oflaq,)
and C-
O3f
OOO4~OqJ It is evident that third o r d e r synthesis will be p r o v i d e d , for a cyclically s y m m e t r i c function, by a cyclically s y m m e t r i c m e c h a n i s m that satisfies (72). T o a c h i e v e fourth o r d e r synthesis, it is n e c e s s a r y that K = 0, f r o m (73) and (74) and also f r o m (75) and (76):
1+
n(3 - n 2 ) ( a 3 - - n) (1 + n")" =0
(79)
excluding the case c._, = ~. T h e o t h e r condition follows f r o m (75), (76) and (78): /33 + 4aa2 + 1 -I- 3 ( 1 --3n2) (as--n)-' " (I + n ~ ) "-' = 0.
(80)
We have thus two equations (79) and (80) to be satisfied by the only u n k n o w n n. Full fourth o r d e r s y n t h e s i s is therefore, in general, not o b t a i n a b l e and only least s q u a r e s solutions m a y be a i m e d at. Elimination of n b e t w e e n (79) and (80) leads to a c o m patibility condition c o n n e c t i n g f13 and a3. T h i s condition has been f o u n d to be too
270 c o m p l i c a t e d to be useful. If the d e s i r e d f u n c t i o n satisfies this c o n d i t i o n , full f o u r t h o r d e r s y n t h e s i s is p o s s i b l e .
17. Conclusion T h e p r e s e n t a r t i c l e u t i l i z e d o r d e r a p p r o x i m a t i o n for the s,,, n t h e s i s or" the p l a n e s e v e n link m e c h a n i s m . D e p e n d i n g o n the f u n c t i o n to be g e n e r a t e d , p r e c i s i o n p o i n t o r m i x e d p o i n t - o r d e r a p p r o x i m a t i o n s m a y s o m e t i m e s be m o r e d e s i r a b l e . It is b e l i e v e d t h a t the d i r e c t s e a r c h m i n i m i z a t i o n p r o c e d u r e s will be u s e f u l in t h e s e c a s e s also, p r o v i d e d that the n u m b e r o f i n d e p e n d e n t v a r i a b l e s is r e d u c e d b y first s a t i s f y i n g s o m e o f the p r e c i s i o n c o n d i t i o n s , as w a s d o n e in this article.
References [ I J SVOBODA A. Computing 3,1echanisms and Linkm,,es. McGraw Hill. New York I [ 948i. [2] ALLEN C. W. The design of linkages to generate function of t,,~,o variables. J. E,'t:,'m,,[nJ. 8lB. 23-2'~ (1959). [3] F REU D ENSTRIN F. and SANDOR G. N. On the Burme,,ter points of a plane. J. april Mech. 2& 41 et seq. (1961). [4] PHILIP R. E. and FREUDENSTRAIN F. Synthesis of t~o-degree-of-freedom Linkages tor the maximum number of precision positions. J. Mech. 1, 9-21 i 1966). [5] FREUDENSTEIN F. and ROTH B. Numerical solutions of systems of non-linear equations. J. Ass comput. Math. 10, 550-556 (1963). [61 ROSEN BROCK H. H. and STOREY C. Computationa[ Techniquesfbr Chemical En,~,ineers. Pergamon Press, Oxford (1966). [7] POWELL M. J. D. Minimization of Functions of Several \'ariables. ,An article in Numerical,4mdysi.~. Academic Press, New York (1966). [81 LAKSHMINARAYANA K. Synthesis of plane lower pair mechanisms for bi-variate function generation. Doctoral Dissertation. Department of Mechanical Engineering, Indian Institute of Technology, Madras (1969). [9] SPANG H. A. A review of minimization techniques for re)n-linear functions. SIAM Rev. 4. 343-365 t 1962). [I01 Wl LDE D. J. Optimum Seekim,, Methods. Prentice Hall, New York (1964).
APPENDIX
--Y[l-t-n(n--~)](~'x):~-L--v[l-l-("t-~):~l(b"+~ax):~. ~ -~ - ' , c . , " Z~--x
[
a,-~,
( 1)J[
~ n-
x
-x
l+n
(')] -7 n
.,,
,
(a~,.r)'-'--c,
1
-- y [ l --{-(/z - 1 ):z ] (oJix) 2 h.~ " Z.,=y
a~
_
n--
_•_[. --y[l +(n--~:)'-'l(oJ.,y)e g~
Z., = x [a,_, + aJ._, ( n -- 1 ) ( 1
--x[l+n(n--l)](o~._,y)
__
"-' 1 _ x I l + ( n _ l ) - ' l ( e . _ , + ~ o . , _ y ) . , C,>
l
7.,.
271
1
--V "
n--
1+
\
b.,.+cotx)(co.,_y)--; gz
x/
j
h.,'