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Optimum synthesis of four-bar function generating mechanism
Mechanism and Machine Theory, 1972, Vol. 7, pp. 387-398.
Pergamon Press.
Printed in Great Britain
Optimum Synthesis of Four-Bar Function Generating Mechanism N. I. Levitskii* Y. L. Sarkissyan.~
and G. S. Gekchian§
Received 12 October 1971 Abstract This paper considers the general problem of determining five parameters specifying a four-bar linkage which synthesizes a given function and at the same time satisfies some limiting conditions. Introduction OPTIMUM synthesis of function generating four-bar mechanisms is represented here as a nonlinear programming problem. We note briefly the main reasoning involved in formulating the synthesis problem. The method developed by the authors is based on the well-known method of replacing the direct expression of deviation from the given law by a generalized polynomial called a weighted difference. Such substitution is inadmissible, however, when the transmission angle is close to 0 ° or 180°. Thus, to apply the weighted approximation method, it is necessary to restrict the magnitude of the transmission angle within certain predetermined limits. In design work, the following order of linkage mechanism synthesis is generally accepted. 1. Type synthesis or the choice of type of mechanism. 2. Determination of the parameters of the chosen type so that the desired function (path) or the positions can be approximated. 3. The checking of calculations in connection with the supplementary conditions of synthesis including: (a) the condition of crank existence; (b) the requirement of favorable transmission angles; (c) the requirement of desirable ratios of link lengths. Certain disadvantages are inherent in this multistep design procedure. It is commonly known that the second stage of the procedure contains the main volume of computations, since it usually involves the determination of an approximating polynomial by general approximation methods. Checking these results often reveals the impossibility of the practical realization of the obtained mechanism; then the only way out is to modify both the general and additional conditions of synthesis, which at times appears to be inadmissible. This process may need to be repeated over and over; it brings about immense waste of computer time and may lead to no desirable results at all. Therefore, *Prof. Dr., Institutefor the Studyof Machines,Moscow,USSR. ~:Docent, PolytechnicalInstitute,Yerevan, USSR. §Eng., PolytechnicalInstitute,Yerevan, USSR.
387
388
it becomes essential to unite the last two stages in designing and to solve the problem as a whole. Since the necessary conditions for permissibility of weighted approximation as well as the additional design constraints are expressed in the form of inequalities, the classical methods of differential calculus aimed at finding an extremum for a function of several variables appears to be unfit and one has to resort to mathematical programming methods. The transition to minimization methods may be regarded as completely justified even in many cases of synthesis without constraints in view of the fact that the solution by other methods has proved to be very cumbersome [2].
The Method of Weighted Approximation The four-bar mechanism shown in Fig. l(a) can be used to approximately generate the given function
(1)
t~ = F ( ¢ )
within a required range/~00, ~m/. A four-bar linkage produces in general some function t~m = Fro(C)
(2)
defined by means of three relative lengths a, b, c and the angles a,/3 specifying the position of reference lines I, II (Fig. 1). For the mechanism to synthesize the given function, one should choose combinations of the parameters so that the deviation A = ~bm - - d~ = Fro(a, b, c, ~x,/3, ~ ) - - F ( ¢ )
will differ from zero on the approximation segment as little as possible.
Ca)
c ~...
IB
¢ rr
(b)
C,t
Figure 1.
~
c
(3)
389 But the complicated irrational form of A when (3) is expressed in terms of the mechanism proportions does not allow us to use this formula for the evaluation of the sought-for parameters. Therefore, we make use of some equivalent but rather simple expression characterizing the structural error (deviation), which is obtained as follows: If we place an additional slider along the direction of the rod BC, we obtain a five-link mechanism with two-degrees of freedom, and so the links AB and CD can be put into motion independently in accordance with the now arbitrary functions ¢(t), 6(t), thus fulfilling the prescribed function ~ = F(¢) exactly. Meanwhile, the distance BC will vary as a definite function of mechanism dimensions and the angle ~o. The projections of BC follow from the projection of the contour ABCD on the axes x and y: brcos 3 = c cos (tk+/3) + 1 - a cos ( ~ + a),] brsin 8 = c sin (qs+/3) - a sin ( ¢ + a).
(4)
Squaring and summing the equations (4) we obtain: b ~ = a ~-+ c" + 1 - 2 a c
cos (qs- ~ +/3 - a) - 2a cos (~ + a) + 2c cos (qs+/3).
(5) The displacement of the slider measured from the point C determined by the constant magnitude b, can be expressed as follows: Ab=b-b
(6)
r.
Imagine the mechanism proportions selected so that the displacement Ab deviates rather negligibly from zero within the interval under consideration and consequently, the variable distance br remains close to the constant magnitude b. Then it is not difficult to be convinced that fixing the size BC at the value b yields a four-bar mechanism which produces approximately the given function ~b= F(~). Since the sum b + br differs slightly from the constant magnitude 2b throughout the interval of approximation, by rewriting the expression (6) we have: Ab = b - be = b" - b7 _-_ A___~o b + br 2b'
(7)
where the difference in the squares, Aq, is the weighted difference while q = b + b t = 2 b is the parametrical weight (it depends on the parameters). A~ in response to the formula (5) is represented in the following form: A,7 = 2 a c
cos (~b-- ~ + fl -- a) + 2a cos (~p+ a)
--2c cos
( q s + /3) + b z - - a 2 - c 2 - - 1.
(8)
Furthermore, from expression (7), we note that the minimization problems for the displacement Ab and, subsequently, for the deviation Aqs may be substituted by the equivalent minimization problem for weighted difference Aq introducing, beforehand, certain limiting conditions set out below. One can derive the dependence between the quantities Aqs and Aq taking advantage of a small displacement diagram constructed on the basis of the following: The small displacements of mechanism points may be considered to be proportional to their M M T . Vol. 7, No. 4 - 8
390
velocities. Supposing that, in addition to A D , the driving link AB is fixed, we plot to an arbitrary scale the velocity diagram for the resulting one-degree-of-freedom mechanism. In Fig. lb, point P represents ground, PC represents the velocity of the slider joint center C, while CC~ is the relative velocity of the point C~ on the connecting rod with respect to the joint center C. As a result of the proportionality between the velocities and the small displacements, we have: Ab
CC~ - -= sin r~ ~c PC
(9)
or
``b = c ',tO- sin "0.
(10)
In accordance with the equality (10) the formula (7) assumes the following form: AtO = 2bc sin "O'
(I 1)
By projecting the contour A B C D on the direction perpendicular to C D we obtain an expression by means of which one can reduce formula (1 1) to the following form: '``q AtO = 2c [sin (tO+/3)-- a sin (tO--~ + / 3 - - a)]"
(12)
T h e use of Aq to characterize the deviation from a given function is impermissible when the values o f ~ + a are nearly 0 ° and 180°, as follows from formula (12). Thus, the minimization of the weighted difference Aq is in line with the minimization of the deviation AtOif the following inequalities are observed: "~min ~ ~per.,
~rnax
~< 180°--v/p~r.
(13)
which constrain the extremum values of transmission angle within limits defined by the permissible magnitude v/perT h e general method of least-square weighted approximations introduced in this paper is applicable to a wide range of different planar and spatial synthesis problems. Although this paper only treats the example o f function generation synthesis, these methods can also be applied to path synthesis, coplanar body guidance synthesis, and analogous spatial problems.
Choice of Objective Function We express the weighted difference Aq determined by formula (8) in the form of a generalized polynomial, supposing that tO= F(~), we have: Aq = 2 [Pofo (~) + P,f~ (¢) + " " " + Psf~ (¢) + P6f6 (¢) ],
(14)
where f0(9) = cos ~, fl(~o) = sin ~,
P0 = a cos a, PL = -- a sin a,
A(~o) = cos
P,_, = - c c o s / 3 ,
f~(~o) = f4(~) = fs(~o) = fn(~o) =
to,
sin tO, 1, cos ( t o - ~ ) , sin (tO- ~),
P3 = c sin r , P4 = ( b ' - a ' - ' - c ~ ' -
(15)
1)/2,
and the coefficients P5, P6 are subject to the coupling equations:
391
Ps=-(PoP2+PIP3),
(16)
Ps = P I P z - P s P 3 .
(17)
From this form of the weighted difference, the coefficients Po, P~, P.~, P3, P4 can be determined. The form of the weighted approximation method developed in the previous paragraphs enables us to apply various objective functions. The first of them is formed by summing up the squares of weighted difference evaluated at chosen points within the approximation interval. Having recourse to expression (14), allows this sum to be written in the following quadratic form: S = CooPoz + • • " +
C66P69-+ 2Co, PoP1 + " • + 2CosPoP6 + 2Cv,_PtP2 + • • •
.d~2 C 16P IP6 + 2 C23P2 P3 + " " " + 2 C2eP2 P6 + 2 C34 P3 P4 +" • • + 2 C3sP3 P6 "k-.2C45PJ,P5 + 2 C46P4Ps + 2 C~6P5 P6,
( 18)
where Ck, = ,=o ~" f~(~')f/(~')
{ kl = O, 1,2 . . . . 6
Yk = ~ F(~)fk(cpi)
O, 1,2,
i=O
6.
(19)
Alternatively, the sum of the squares of the deviation Aqj could serve as a criterion for optimization
S =
l_~ ~q2(t~i)
(20)
4b2c ~' sin" "0
Similar expressions for the weighted difference Aq and the objective functions (as defined in (18) and (20)) may be easily derived for different types of spatial and planar linkages. It should be pointed out that the functions (18) and (20) can be utilized only when seeking quadratic or interpolative approximations. Finally, as an objective function, one can take also the expression of A~b given by formula (12). In this case, the optimization problem is reduced to one of finding a minimax vector in the space of soughtfor parameters, i.e. finding that combination of the five parameters which yields the minimum value for the largest one (in absolute magnitude) among all the values of the deviation Aqj evaluated at the chosen points i = 0, 1. . . . m within the approximation range. We express this: Aqf,t. = min max Aq,i, a,b,c,a,l$,w.
where w is the multitude of possible deviation values. It can be seen that for large values of m the approximations obtained in the foregoing manner are the best. Let us carry out an efficiency comparison of the objective functions (12), (18), (20), supposing the movement to the minimum is realized by means of a simple search. Let some arbitrary values of sought-for proportions a, b, c, a,/3 be picked and the corresponding values of Po, P1, P2, Pz, P4 determined. The objective function (18) is rep-
392 resented as a quadratic form with the numerical coefficients C,~ given by formula (19). In order to compare it with the other versions, we will consider it be computed only once. The objective function (20) cannot be reduced to a quadratic form because the term sin ~ 0, which includes both unknown dimensions and input angle ¢ (the function to = F(~) is prescribed) forms part of the denominator. It is also computed once for the each combination of unknowns, but the volume of necessary computations will be nearly m times greater even though the sum (18) contains the same number of items. Finally the objective function (12) should also be computed m times to determine the maximum value of the deviation tO(for each chosen combination of P~). The analysis produced above shows convincingly the great advantages of the objective function (18) in comparison with the expressions (12), (20) even for a blind search. This advantage is emphasized still more when the desired minimum is found by means of algorithmic search methods since the objective function (12) becomes unusable in this case, while the application of expression (20) is restricted practically in view of the complexity of its partial derivatives. Unlike functions (12) and (10), when minimizing the mean square sum (18) the inequality limitations (13) should be imposed. This would seem to detract from the merits of this objective function, but this fact appears to be unimportant in practice because one generally has to meet these same requirements on permissible transmission angles. Thus, in the optimum synthesis of function-generating four-bar linkages, it is most advisable to evaluate the sought-for parameters from the condition that the mean square summation differs least from zero. When executing the minimization procedure certain limitations, characterizing supplementary conditions of synthesis and the weighted approximation permissibility condition, are to be imposed. These constraints are expressed: (a) in the inequality form gi(Po, P,, P._,. . . . P,,) >1 0,
i = 0, 1. . . . K
(21)
v=0,1,..q
(22)
(b) in the equality form Fv(Po, P~,P2 . . . . P,,)=0.
The coupling equations (16), (17) are also to be reckoned with equality-restrictions. The problem under consideration requires the search for an optimum point P* (in n size space) among all points P satisfying the conditions (21) and (22) at which the minimum of the objective function is attained, i.e. S ( P * ) = min S ( P ) , PEfl
where ~ is the domain defined by constraints (21) and (22).
The General Inequalities for Optimum Synthesis of Function Generating FourBar Linkages Having manufacturing and design considerations in mind and also the installation of the mechanism, we require that the ratio of any two link lengths not exceed a given number k. Since it is impossible to predict before synthesis which link has the greatest or smallest length, all possible combinations of the relative lengths are to be considered.
393
T h e stated condition yields the following inequalities:
a~k,
a~,
a~kc,
a~,
a~kb,
a~,
1
b~k,
1
b~,
1
c~, l
c~k,
i
b~kc, b~c.
(23)
1
F u r t h e r m o r e , in o r d e r to a v o i d large force r e a c t i o n s in the m e c h a n i s m ' s joints and large loads on the links, the following r e q u i r e m e n t is t a k e n into a c c o u n t : the transmission angle m u s t not o v e r s t e p the limits d e t e r m i n e d by the m i n i m u m and m a x i m u m permissible values. T h i s r e q u i r e m e n t m a y be written as (24)
'~min ~> "r~per., T~max ~ "OtPer..
F r o m Fig. 2, triangles B ' C ' D and B " C " D yield r e s p e c t i v e l y : b'-' + c z - 1 + 2a - a ~ (25)
C O S "Omin =
2bc b"-+ c"- -- 1 -- 2a -- a 2 C O S T]max "~"
(26)
2bc M a k i n g use o f the inequalities (24), we obtain:
2bc cos "Oper. ~> b2 +
1 + 2a -- a 2,
(27)
2bc cos "O'per. ~< b2 + c2 -- 1 -- 2a -- a 2.
(28)
C
0
B"
F i g u r e 2.
C2 --
C I
394
It can be readily seen that the link AB may perform a complete rotation if it passes through the positions AB' and AB", corresponding to the extremum values of transmission angle, i.e. the inequalities (27), (28) are sufficient conditions for the existence of crank AB. According to Grashof's theorem the shortest link in the crank-androcker mechanism is the crank while in the drag-link mechanism it is the fixed link. By virtue of the above remark the last nine (last three columns) inequalities in (23) are meaningless if (27) and (28) are observed. Thus, when doing an optimum synthesis the constraints (27), (28) will be imposed and in addition the following:
a >i kc, a>~kb,
(29)
a < a<
If it is necessary to obtain a drag-link mechanism, the inequalities (27), (28) should be satisfied and also: a-<~.
l
1, b~<~.
I < b,
1, c~<~
l
(30)
Finally, if it is required to provide complete rotation of the driving link AB, but the total swing angle of output link may take any arbitrary value, the following restrictions are imposed besides (27), (28): a>~k,
a<~ 1,
a >1 kc,
a ~c,
a>~kb,
a<~b,
1
(31)
1
in view of the fact that either the crank AB or the fixed link AD may have the least length. Utilizing the relations (15), one can express the sought-for parameters in the form of the following functions of Pi: a = k/PoZ+Pt-',
c
=
X/Pe2q--P32, a = a r c t g
(
--Po]
(32) 13 = arctg --
, b = X/PoZ + Pl 2 "4-P2' + P.~'-'+ 2P~
Using the expressions (32), the inequalities (27)-(30) can be reduced respectively
395
to the forms: o
,,
.)
,)
,,
.7
X/(Po- + P~" + P2- + P3- + 2P4)(P~- + P:V) cos ~per. -P,_'--P,~"--P4+
X/-~+P(
" >1 O,
(33)
-- X/(P0'-' + PI"- + Be 2 + e3" + 2 P4)( P22 + P3") c o s "r'/'per. +P~"+P3"+P4+
V~-+P('
>~ O.
(34)
Po"- + Pt 2 - k2(P". "-+ P3") ~ O,
P . , " + P 3 2 + 1 + 2 P 4 ~> 0,
Po"+P12-k
P2"- + P3"- -- Po 2 - Pt 2 ~> 0,
"- ~ O,
( 1 - k")(Po"- + Pt 2) - k"-(P..,"-+ P:,"-+ 1 + 2P4) t> 0, 1
1 - Po" - Pf'- >~ O.
- - - - P o " - - - P t ' - ' ~ O, k"
Po"- + P['- - 1 1> 0,
1 ---" k"-
p.,2 + p3 2 __ 1 >1 0,
P22--
P3 '' >>-
0,
1
- - - - Po"- -- P ('- -- P.,.2 -- P:~2 - 1 - 2 P 4 >t 0, k"
(35)
(36)
Po 2 + PI" + P2"- + P3 2 + 2P4 ~ 0.
T h e sets of inequalities (33), (34), (35) and (33), (34), (36) are for the design of crank-rocker and drag-link mechanisms respectively. Minimization Procedure T o minimize the mean square sum (1 8) with or without the limiting conditions, the gradient, steepest descent and random search methods have been tested. In accordance with the gradient method, the gradient of the objective function is defined for every point of the search and the next step is taken in a direction opposite to the gradient direction. Consequently, the coordinates of the new point are determined by the formula P,,+l=Pim-hi,
i=0,1 .... n
where hi = K o ( A S / A P 3 m are the increments proportionate to the partial derivatives of the mean square sum S at the point m. T h e search by steepest descent method begins similarly, by specifying at first the gradient direction for the initial point, and then the m o v e m e n t to the minimum is along the line coinciding with the direction opposite to the gradient and so on till the point which yields the least value of the objective function is attained. In the random search algorithm applied here, a step is taken from the starting point in a random direction and the function value is calculated there. If this function value at the new point is greater than the previous one, a reverse is executed, i.e. a movement in the opposite direction with the same step size from the starting point. If the new attempt fails to lead to any satisfactory result, a return to the initial point is undertaken and a new random direction is selected. Using a self-adjusting algorithm makes it possible to correct, in the course of the optimization, the current magnitude of the special coefficient, K0, controlling the step size so as to increase search efficiency (an adaptive property).
396
To start the search when we have the limiting constraints we express at first all the inequalities in the form: g(Po, P~, P._,. . . . P,,) >1 O, then utilize an auxiliary objective function represented in the following form: = S+EMj,
where M~ = gi for violated constraints and M~ = 0 for constraints satisfied at the point under consideration. The sought-for solution is found by means of zigzag movements along the boundary of the permissible domain. Making use of the above algorithms, and expressions (2), (14)-(19), (33), (34), (35), a digital computer program (for computer "Hrazdan-2") was written to determine the design parameters for four-bar function generating mechanisms.
/
"~b
145°59'53 "
232"0,5'31"
./ -t I--
Figure 3(a).
260°01'12"
/
229 °05'23''
". Co x.
Bo
Figure 3(b).
(b)
/i Cm
397
Examples Let us consider two examples to demonstrate the efficacy of the method developed above for both cases of synthesis: (a) without constraints, (b) with constraints. I. We seek to generate by means of a four-bar linkage, the function ¢ = (g/1000)¢ ~ within the range 0 ~< 9 ~< 100°. The search for a minimum of the objective function (18) with the coefficients Ckt defined by (19) leads to several solutions corresponding to different local minimums. The coordinates of two local minimum points are given below: P0 I-1.298451 1I-0.423279
P1 P2 P3 -0.875430 0.355518 -0.456155 2.396711 1.117555 -1.600419
P4 S 0-879098 0-000047 0.297389 0-00083
Then the dimensions of the corresponding mechanisms are computed from formulas (32), and the maximum deviation value is found making use of relation (17). a
I II
c
b
ct
/3
Al/tma x
1.566006 0-578812 2-354799 --0-593191 0.908822 --0-00432044 2.433669 1.714831 1-126138 1.396061 0-860668 --0-00194989
The mechanisms are drawn in Figs. 3a and 3b. They have similar proportions to the linkages obtained in [ 1] by the interpolative method. II. As the second example, let us undertake the task of evaluating the five parameters for the optimum design of a crank-and-rocker mechanism which could represent the function y = Iogx in the interval x = 1 to x = 10, when the values of v/per, ~'perand k are given. The exploration of the corresponding surface determined by the quadratic form (18) indicates again that several local minimums are available. For instance, utilizing as a Bm
(a)
(b)
/i \
A/ ~
. "~.
o
\\ .." Figure 4(a)
Figure 4(b),
398
starting point the values: P,, = 0 . 5 , P., = P., = P:~ : P4 = - 0 . 5 , the following solution was obtained by random search method: Po = 0-591182, P~ = - 0 - 3 1 2 9 4 6 , P., = 0-464068, P:~ : - 0 - 3 2 0 4 1 0 , P4 = - - 0 . 3 0 1 6 5 1 , S = 0-000018, yielding a mechanism with the proportions: a---0-668910, b = 1.077564, c = 0"563047, c~ = 27°52'55 ''. /3 = 55°22 ' 14" and AO~.~ = 0-003 (Fig. 4a). This m e c h a n i s m corresponding to one of the partial minimums for objective function (l 8) differs considerably from the minimum determined in [3] by use of Lagrange's method. Further, the search by the gradient method e m p l o y e d the constraints (33), (34), (35). The computations, with the value of k (the permissible ratio of the link-lengths) being fixed, revealed a significant sensitivity of the minimum value of the objective function to variations of permissible transmission angles vh,~, ~'po~. Thus, taking ~p~. = 30 °, v/'po~.= 150 ° we s u c c e e d e d in minimizing the objective function only to S = 0-0245, then with a 10 ° change to v/~,~. = 20 °, v/'p~. = 160 ° we obtained the value S = 0-00458. Finally assuming ~ = 10 °, ~'~r. = 170 °, the optimum point P* was found defined by the parameters: P0-= 0.714695, P~ = - 0 - 1 2 4 5 1 4 , P~ = - 0 . 3 1 5 7 3 9 , Pa = - 0 - 6 7 1 4 8 2 , P4 ~ -- 0 " 5 5 0 0 0 9 , yielding S = 0 . 0 0 0 5 4 6 . The dimensions of the resulting m e c h a n i s m obtained from formulas (32) are as follows: a = 0.725468, c = 0.742684, b = 0-988358, c~ = 9052 ' 18", /3 = - 6 4 0 4 8 ' 5 0 ''. This mechanism (Fig. 4b) satisfies all the limitations imposed and yields a sufficiently accurate function generator.
References [l] A R T O B O L E V S K I I I., L E V I T S K l l N. I. and C H E R K U D I N O V S. A., Synthesis o f Plane Mechanisms. Fismatgis (1959). [2] S A R K I S S Y A N Y. L. and G E K C H I A N G. S. O n the optimum design of four-bar function gen, ..-ating linkage, Machinovedenie, No. 3, 25-31 (1969). [3] L E V I T S K I I N. I. and S A R K I S I A N Y. L. O n the special properties of Lagrange's multipliers in the least-square synthesis of m e c h a n i s m s , J. Mechanisms 3, 3 - I 0 (1968). OnTUMa.~t~HU~ cltnTea nepe~aTOqltoro qeTblp~x3BeHsxKa H. I,t[..r]eBHTCgK.~, tO. f[. CapKHC,qH, ]7. f[. F~K~i~iH P e 3 ~ o M e - B BBO,~[HO~ ~aCT~t CTaTbH nOKaBblBaeTcR tle~ecoo6pa3HOCTb npe,~CTaB~leHH~I 3aJlaqH flpOCKTHpOBaHH~ NCpC~RTO"~HOI'O ',,ICTblpeX3BeHHHKa B BH~C 3a~aqH HeJ'IHHe~HOrO rlpoI'paMMHpOBaHKR. }~a~Ie¢ I'IOB,pO6HO orlRCblBaCTCR MCTOJ1 B3BelJ.ICHHOFO KBa~paTl4'.IeCKOrO rlpH(~.qH)KeH1431, IIpc~YlO;.KeHHblI~[ H. H. .TIcBI~TCKH2v[. ,~[OKa3blBaCTCR, ~'ITO BMCCTO CJIO>KHOr'O l,[ppauHoaa.r'lbHoro Bblpa~KeHRR (3) OTKJ]OHeHHM OT :3aJ1aH.aofi qbyHKllHH I'IpR CKHTC3e MO;,KHO MHHHMH3HpOBaTb ]IOCTaTOqHO npocToe abtpaY,
Pergamon Press.
Printed in Great Britain
Optimum Synthesis of Four-Bar Function Generating Mechanism N. I. Levitskii* Y. L. Sarkissyan.~
and G. S. Gekchian§
Received 12 October 1971 Abstract This paper considers the general problem of determining five parameters specifying a four-bar linkage which synthesizes a given function and at the same time satisfies some limiting conditions. Introduction OPTIMUM synthesis of function generating four-bar mechanisms is represented here as a nonlinear programming problem. We note briefly the main reasoning involved in formulating the synthesis problem. The method developed by the authors is based on the well-known method of replacing the direct expression of deviation from the given law by a generalized polynomial called a weighted difference. Such substitution is inadmissible, however, when the transmission angle is close to 0 ° or 180°. Thus, to apply the weighted approximation method, it is necessary to restrict the magnitude of the transmission angle within certain predetermined limits. In design work, the following order of linkage mechanism synthesis is generally accepted. 1. Type synthesis or the choice of type of mechanism. 2. Determination of the parameters of the chosen type so that the desired function (path) or the positions can be approximated. 3. The checking of calculations in connection with the supplementary conditions of synthesis including: (a) the condition of crank existence; (b) the requirement of favorable transmission angles; (c) the requirement of desirable ratios of link lengths. Certain disadvantages are inherent in this multistep design procedure. It is commonly known that the second stage of the procedure contains the main volume of computations, since it usually involves the determination of an approximating polynomial by general approximation methods. Checking these results often reveals the impossibility of the practical realization of the obtained mechanism; then the only way out is to modify both the general and additional conditions of synthesis, which at times appears to be inadmissible. This process may need to be repeated over and over; it brings about immense waste of computer time and may lead to no desirable results at all. Therefore, *Prof. Dr., Institutefor the Studyof Machines,Moscow,USSR. ~:Docent, PolytechnicalInstitute,Yerevan, USSR. §Eng., PolytechnicalInstitute,Yerevan, USSR.
387
388
it becomes essential to unite the last two stages in designing and to solve the problem as a whole. Since the necessary conditions for permissibility of weighted approximation as well as the additional design constraints are expressed in the form of inequalities, the classical methods of differential calculus aimed at finding an extremum for a function of several variables appears to be unfit and one has to resort to mathematical programming methods. The transition to minimization methods may be regarded as completely justified even in many cases of synthesis without constraints in view of the fact that the solution by other methods has proved to be very cumbersome [2].
The Method of Weighted Approximation The four-bar mechanism shown in Fig. l(a) can be used to approximately generate the given function
(1)
t~ = F ( ¢ )
within a required range/~00, ~m/. A four-bar linkage produces in general some function t~m = Fro(C)
(2)
defined by means of three relative lengths a, b, c and the angles a,/3 specifying the position of reference lines I, II (Fig. 1). For the mechanism to synthesize the given function, one should choose combinations of the parameters so that the deviation A = ~bm - - d~ = Fro(a, b, c, ~x,/3, ~ ) - - F ( ¢ )
will differ from zero on the approximation segment as little as possible.
Ca)
c ~...
IB
¢ rr
(b)
C,t
Figure 1.
~
c
(3)
389 But the complicated irrational form of A when (3) is expressed in terms of the mechanism proportions does not allow us to use this formula for the evaluation of the sought-for parameters. Therefore, we make use of some equivalent but rather simple expression characterizing the structural error (deviation), which is obtained as follows: If we place an additional slider along the direction of the rod BC, we obtain a five-link mechanism with two-degrees of freedom, and so the links AB and CD can be put into motion independently in accordance with the now arbitrary functions ¢(t), 6(t), thus fulfilling the prescribed function ~ = F(¢) exactly. Meanwhile, the distance BC will vary as a definite function of mechanism dimensions and the angle ~o. The projections of BC follow from the projection of the contour ABCD on the axes x and y: brcos 3 = c cos (tk+/3) + 1 - a cos ( ~ + a),] brsin 8 = c sin (qs+/3) - a sin ( ¢ + a).
(4)
Squaring and summing the equations (4) we obtain: b ~ = a ~-+ c" + 1 - 2 a c
cos (qs- ~ +/3 - a) - 2a cos (~ + a) + 2c cos (qs+/3).
(5) The displacement of the slider measured from the point C determined by the constant magnitude b, can be expressed as follows: Ab=b-b
(6)
r.
Imagine the mechanism proportions selected so that the displacement Ab deviates rather negligibly from zero within the interval under consideration and consequently, the variable distance br remains close to the constant magnitude b. Then it is not difficult to be convinced that fixing the size BC at the value b yields a four-bar mechanism which produces approximately the given function ~b= F(~). Since the sum b + br differs slightly from the constant magnitude 2b throughout the interval of approximation, by rewriting the expression (6) we have: Ab = b - be = b" - b7 _-_ A___~o b + br 2b'
(7)
where the difference in the squares, Aq, is the weighted difference while q = b + b t = 2 b is the parametrical weight (it depends on the parameters). A~ in response to the formula (5) is represented in the following form: A,7 = 2 a c
cos (~b-- ~ + fl -- a) + 2a cos (~p+ a)
--2c cos
( q s + /3) + b z - - a 2 - c 2 - - 1.
(8)
Furthermore, from expression (7), we note that the minimization problems for the displacement Ab and, subsequently, for the deviation Aqs may be substituted by the equivalent minimization problem for weighted difference Aq introducing, beforehand, certain limiting conditions set out below. One can derive the dependence between the quantities Aqs and Aq taking advantage of a small displacement diagram constructed on the basis of the following: The small displacements of mechanism points may be considered to be proportional to their M M T . Vol. 7, No. 4 - 8
390
velocities. Supposing that, in addition to A D , the driving link AB is fixed, we plot to an arbitrary scale the velocity diagram for the resulting one-degree-of-freedom mechanism. In Fig. lb, point P represents ground, PC represents the velocity of the slider joint center C, while CC~ is the relative velocity of the point C~ on the connecting rod with respect to the joint center C. As a result of the proportionality between the velocities and the small displacements, we have: Ab
CC~ - -= sin r~ ~c PC
(9)
or
``b = c ',tO- sin "0.
(10)
In accordance with the equality (10) the formula (7) assumes the following form: AtO = 2bc sin "O'
(I 1)
By projecting the contour A B C D on the direction perpendicular to C D we obtain an expression by means of which one can reduce formula (1 1) to the following form: '``q AtO = 2c [sin (tO+/3)-- a sin (tO--~ + / 3 - - a)]"
(12)
T h e use of Aq to characterize the deviation from a given function is impermissible when the values o f ~ + a are nearly 0 ° and 180°, as follows from formula (12). Thus, the minimization of the weighted difference Aq is in line with the minimization of the deviation AtOif the following inequalities are observed: "~min ~ ~per.,
~rnax
~< 180°--v/p~r.
(13)
which constrain the extremum values of transmission angle within limits defined by the permissible magnitude v/perT h e general method of least-square weighted approximations introduced in this paper is applicable to a wide range of different planar and spatial synthesis problems. Although this paper only treats the example o f function generation synthesis, these methods can also be applied to path synthesis, coplanar body guidance synthesis, and analogous spatial problems.
Choice of Objective Function We express the weighted difference Aq determined by formula (8) in the form of a generalized polynomial, supposing that tO= F(~), we have: Aq = 2 [Pofo (~) + P,f~ (¢) + " " " + Psf~ (¢) + P6f6 (¢) ],
(14)
where f0(9) = cos ~, fl(~o) = sin ~,
P0 = a cos a, PL = -- a sin a,
A(~o) = cos
P,_, = - c c o s / 3 ,
f~(~o) = f4(~) = fs(~o) = fn(~o) =
to,
sin tO, 1, cos ( t o - ~ ) , sin (tO- ~),
P3 = c sin r , P4 = ( b ' - a ' - ' - c ~ ' -
(15)
1)/2,
and the coefficients P5, P6 are subject to the coupling equations:
391
Ps=-(PoP2+PIP3),
(16)
Ps = P I P z - P s P 3 .
(17)
From this form of the weighted difference, the coefficients Po, P~, P.~, P3, P4 can be determined. The form of the weighted approximation method developed in the previous paragraphs enables us to apply various objective functions. The first of them is formed by summing up the squares of weighted difference evaluated at chosen points within the approximation interval. Having recourse to expression (14), allows this sum to be written in the following quadratic form: S = CooPoz + • • " +
C66P69-+ 2Co, PoP1 + " • + 2CosPoP6 + 2Cv,_PtP2 + • • •
.d~2 C 16P IP6 + 2 C23P2 P3 + " " " + 2 C2eP2 P6 + 2 C34 P3 P4 +" • • + 2 C3sP3 P6 "k-.2C45PJ,P5 + 2 C46P4Ps + 2 C~6P5 P6,
( 18)
where Ck, = ,=o ~" f~(~')f/(~')
{ kl = O, 1,2 . . . . 6
Yk = ~ F(~)fk(cpi)
O, 1,2,
i=O
6.
(19)
Alternatively, the sum of the squares of the deviation Aqj could serve as a criterion for optimization
S =
l_~ ~q2(t~i)
(20)
4b2c ~' sin" "0
Similar expressions for the weighted difference Aq and the objective functions (as defined in (18) and (20)) may be easily derived for different types of spatial and planar linkages. It should be pointed out that the functions (18) and (20) can be utilized only when seeking quadratic or interpolative approximations. Finally, as an objective function, one can take also the expression of A~b given by formula (12). In this case, the optimization problem is reduced to one of finding a minimax vector in the space of soughtfor parameters, i.e. finding that combination of the five parameters which yields the minimum value for the largest one (in absolute magnitude) among all the values of the deviation Aqj evaluated at the chosen points i = 0, 1. . . . m within the approximation range. We express this: Aqf,t. = min max Aq,i, a,b,c,a,l$,w.
where w is the multitude of possible deviation values. It can be seen that for large values of m the approximations obtained in the foregoing manner are the best. Let us carry out an efficiency comparison of the objective functions (12), (18), (20), supposing the movement to the minimum is realized by means of a simple search. Let some arbitrary values of sought-for proportions a, b, c, a,/3 be picked and the corresponding values of Po, P1, P2, Pz, P4 determined. The objective function (18) is rep-
392 resented as a quadratic form with the numerical coefficients C,~ given by formula (19). In order to compare it with the other versions, we will consider it be computed only once. The objective function (20) cannot be reduced to a quadratic form because the term sin ~ 0, which includes both unknown dimensions and input angle ¢ (the function to = F(~) is prescribed) forms part of the denominator. It is also computed once for the each combination of unknowns, but the volume of necessary computations will be nearly m times greater even though the sum (18) contains the same number of items. Finally the objective function (12) should also be computed m times to determine the maximum value of the deviation tO(for each chosen combination of P~). The analysis produced above shows convincingly the great advantages of the objective function (18) in comparison with the expressions (12), (20) even for a blind search. This advantage is emphasized still more when the desired minimum is found by means of algorithmic search methods since the objective function (12) becomes unusable in this case, while the application of expression (20) is restricted practically in view of the complexity of its partial derivatives. Unlike functions (12) and (10), when minimizing the mean square sum (18) the inequality limitations (13) should be imposed. This would seem to detract from the merits of this objective function, but this fact appears to be unimportant in practice because one generally has to meet these same requirements on permissible transmission angles. Thus, in the optimum synthesis of function-generating four-bar linkages, it is most advisable to evaluate the sought-for parameters from the condition that the mean square summation differs least from zero. When executing the minimization procedure certain limitations, characterizing supplementary conditions of synthesis and the weighted approximation permissibility condition, are to be imposed. These constraints are expressed: (a) in the inequality form gi(Po, P,, P._,. . . . P,,) >1 0,
i = 0, 1. . . . K
(21)
v=0,1,..q
(22)
(b) in the equality form Fv(Po, P~,P2 . . . . P,,)=0.
The coupling equations (16), (17) are also to be reckoned with equality-restrictions. The problem under consideration requires the search for an optimum point P* (in n size space) among all points P satisfying the conditions (21) and (22) at which the minimum of the objective function is attained, i.e. S ( P * ) = min S ( P ) , PEfl
where ~ is the domain defined by constraints (21) and (22).
The General Inequalities for Optimum Synthesis of Function Generating FourBar Linkages Having manufacturing and design considerations in mind and also the installation of the mechanism, we require that the ratio of any two link lengths not exceed a given number k. Since it is impossible to predict before synthesis which link has the greatest or smallest length, all possible combinations of the relative lengths are to be considered.
393
T h e stated condition yields the following inequalities:
a~k,
a~,
a~kc,
a~,
a~kb,
a~,
1
b~k,
1
b~,
1
c~, l
c~k,
i
b~kc, b~c.
(23)
1
F u r t h e r m o r e , in o r d e r to a v o i d large force r e a c t i o n s in the m e c h a n i s m ' s joints and large loads on the links, the following r e q u i r e m e n t is t a k e n into a c c o u n t : the transmission angle m u s t not o v e r s t e p the limits d e t e r m i n e d by the m i n i m u m and m a x i m u m permissible values. T h i s r e q u i r e m e n t m a y be written as (24)
'~min ~> "r~per., T~max ~ "OtPer..
F r o m Fig. 2, triangles B ' C ' D and B " C " D yield r e s p e c t i v e l y : b'-' + c z - 1 + 2a - a ~ (25)
C O S "Omin =
2bc b"-+ c"- -- 1 -- 2a -- a 2 C O S T]max "~"
(26)
2bc M a k i n g use o f the inequalities (24), we obtain:
2bc cos "Oper. ~> b2 +
1 + 2a -- a 2,
(27)
2bc cos "O'per. ~< b2 + c2 -- 1 -- 2a -- a 2.
(28)
C
0
B"
F i g u r e 2.
C2 --
C I
394
It can be readily seen that the link AB may perform a complete rotation if it passes through the positions AB' and AB", corresponding to the extremum values of transmission angle, i.e. the inequalities (27), (28) are sufficient conditions for the existence of crank AB. According to Grashof's theorem the shortest link in the crank-androcker mechanism is the crank while in the drag-link mechanism it is the fixed link. By virtue of the above remark the last nine (last three columns) inequalities in (23) are meaningless if (27) and (28) are observed. Thus, when doing an optimum synthesis the constraints (27), (28) will be imposed and in addition the following:
a >i kc, a>~kb,
(29)
a < a<
If it is necessary to obtain a drag-link mechanism, the inequalities (27), (28) should be satisfied and also: a-<~.
l
1, b~<~.
I < b,
1, c~<~
l
(30)
Finally, if it is required to provide complete rotation of the driving link AB, but the total swing angle of output link may take any arbitrary value, the following restrictions are imposed besides (27), (28): a>~k,
a<~ 1,
a >1 kc,
a ~c,
a>~kb,
a<~b,
1
(31)
1
in view of the fact that either the crank AB or the fixed link AD may have the least length. Utilizing the relations (15), one can express the sought-for parameters in the form of the following functions of Pi: a = k/PoZ+Pt-',
c
=
X/Pe2q--P32, a = a r c t g
(
--Po]
(32) 13 = arctg --
, b = X/PoZ + Pl 2 "4-P2' + P.~'-'+ 2P~
Using the expressions (32), the inequalities (27)-(30) can be reduced respectively
395
to the forms: o
,,
.)
,)
,,
.7
X/(Po- + P~" + P2- + P3- + 2P4)(P~- + P:V) cos ~per. -P,_'--P,~"--P4+
X/-~+P(
" >1 O,
(33)
-- X/(P0'-' + PI"- + Be 2 + e3" + 2 P4)( P22 + P3") c o s "r'/'per. +P~"+P3"+P4+
V~-+P('
>~ O.
(34)
Po"- + Pt 2 - k2(P". "-+ P3") ~ O,
P . , " + P 3 2 + 1 + 2 P 4 ~> 0,
Po"+P12-k
P2"- + P3"- -- Po 2 - Pt 2 ~> 0,
"- ~ O,
( 1 - k")(Po"- + Pt 2) - k"-(P..,"-+ P:,"-+ 1 + 2P4) t> 0, 1
1 - Po" - Pf'- >~ O.
- - - - P o " - - - P t ' - ' ~ O, k"
Po"- + P['- - 1 1> 0,
1 ---" k"-
p.,2 + p3 2 __ 1 >1 0,
P22--
P3 '' >>-
0,
1
- - - - Po"- -- P ('- -- P.,.2 -- P:~2 - 1 - 2 P 4 >t 0, k"
(35)
(36)
Po 2 + PI" + P2"- + P3 2 + 2P4 ~ 0.
T h e sets of inequalities (33), (34), (35) and (33), (34), (36) are for the design of crank-rocker and drag-link mechanisms respectively. Minimization Procedure T o minimize the mean square sum (1 8) with or without the limiting conditions, the gradient, steepest descent and random search methods have been tested. In accordance with the gradient method, the gradient of the objective function is defined for every point of the search and the next step is taken in a direction opposite to the gradient direction. Consequently, the coordinates of the new point are determined by the formula P,,+l=Pim-hi,
i=0,1 .... n
where hi = K o ( A S / A P 3 m are the increments proportionate to the partial derivatives of the mean square sum S at the point m. T h e search by steepest descent method begins similarly, by specifying at first the gradient direction for the initial point, and then the m o v e m e n t to the minimum is along the line coinciding with the direction opposite to the gradient and so on till the point which yields the least value of the objective function is attained. In the random search algorithm applied here, a step is taken from the starting point in a random direction and the function value is calculated there. If this function value at the new point is greater than the previous one, a reverse is executed, i.e. a movement in the opposite direction with the same step size from the starting point. If the new attempt fails to lead to any satisfactory result, a return to the initial point is undertaken and a new random direction is selected. Using a self-adjusting algorithm makes it possible to correct, in the course of the optimization, the current magnitude of the special coefficient, K0, controlling the step size so as to increase search efficiency (an adaptive property).
396
To start the search when we have the limiting constraints we express at first all the inequalities in the form: g(Po, P~, P._,. . . . P,,) >1 O, then utilize an auxiliary objective function represented in the following form: = S+EMj,
where M~ = gi for violated constraints and M~ = 0 for constraints satisfied at the point under consideration. The sought-for solution is found by means of zigzag movements along the boundary of the permissible domain. Making use of the above algorithms, and expressions (2), (14)-(19), (33), (34), (35), a digital computer program (for computer "Hrazdan-2") was written to determine the design parameters for four-bar function generating mechanisms.
/
"~b
145°59'53 "
232"0,5'31"
./ -t I--
Figure 3(a).
260°01'12"
/
229 °05'23''
". Co x.
Bo
Figure 3(b).
(b)
/i Cm
397
Examples Let us consider two examples to demonstrate the efficacy of the method developed above for both cases of synthesis: (a) without constraints, (b) with constraints. I. We seek to generate by means of a four-bar linkage, the function ¢ = (g/1000)¢ ~ within the range 0 ~< 9 ~< 100°. The search for a minimum of the objective function (18) with the coefficients Ckt defined by (19) leads to several solutions corresponding to different local minimums. The coordinates of two local minimum points are given below: P0 I-1.298451 1I-0.423279
P1 P2 P3 -0.875430 0.355518 -0.456155 2.396711 1.117555 -1.600419
P4 S 0-879098 0-000047 0.297389 0-00083
Then the dimensions of the corresponding mechanisms are computed from formulas (32), and the maximum deviation value is found making use of relation (17). a
I II
c
b
ct
/3
Al/tma x
1.566006 0-578812 2-354799 --0-593191 0.908822 --0-00432044 2.433669 1.714831 1-126138 1.396061 0-860668 --0-00194989
The mechanisms are drawn in Figs. 3a and 3b. They have similar proportions to the linkages obtained in [ 1] by the interpolative method. II. As the second example, let us undertake the task of evaluating the five parameters for the optimum design of a crank-and-rocker mechanism which could represent the function y = Iogx in the interval x = 1 to x = 10, when the values of v/per, ~'perand k are given. The exploration of the corresponding surface determined by the quadratic form (18) indicates again that several local minimums are available. For instance, utilizing as a Bm
(a)
(b)
/i \
A/ ~
. "~.
o
\\ .." Figure 4(a)
Figure 4(b),
398
starting point the values: P,, = 0 . 5 , P., = P., = P:~ : P4 = - 0 . 5 , the following solution was obtained by random search method: Po = 0-591182, P~ = - 0 - 3 1 2 9 4 6 , P., = 0-464068, P:~ : - 0 - 3 2 0 4 1 0 , P4 = - - 0 . 3 0 1 6 5 1 , S = 0-000018, yielding a mechanism with the proportions: a---0-668910, b = 1.077564, c = 0"563047, c~ = 27°52'55 ''. /3 = 55°22 ' 14" and AO~.~ = 0-003 (Fig. 4a). This m e c h a n i s m corresponding to one of the partial minimums for objective function (l 8) differs considerably from the minimum determined in [3] by use of Lagrange's method. Further, the search by the gradient method e m p l o y e d the constraints (33), (34), (35). The computations, with the value of k (the permissible ratio of the link-lengths) being fixed, revealed a significant sensitivity of the minimum value of the objective function to variations of permissible transmission angles vh,~, ~'po~. Thus, taking ~p~. = 30 °, v/'po~.= 150 ° we s u c c e e d e d in minimizing the objective function only to S = 0-0245, then with a 10 ° change to v/~,~. = 20 °, v/'p~. = 160 ° we obtained the value S = 0-00458. Finally assuming ~ = 10 °, ~'~r. = 170 °, the optimum point P* was found defined by the parameters: P0-= 0.714695, P~ = - 0 - 1 2 4 5 1 4 , P~ = - 0 . 3 1 5 7 3 9 , Pa = - 0 - 6 7 1 4 8 2 , P4 ~ -- 0 " 5 5 0 0 0 9 , yielding S = 0 . 0 0 0 5 4 6 . The dimensions of the resulting m e c h a n i s m obtained from formulas (32) are as follows: a = 0.725468, c = 0.742684, b = 0-988358, c~ = 9052 ' 18", /3 = - 6 4 0 4 8 ' 5 0 ''. This mechanism (Fig. 4b) satisfies all the limitations imposed and yields a sufficiently accurate function generator.
References [l] A R T O B O L E V S K I I I., L E V I T S K l l N. I. and C H E R K U D I N O V S. A., Synthesis o f Plane Mechanisms. Fismatgis (1959). [2] S A R K I S S Y A N Y. L. and G E K C H I A N G. S. O n the optimum design of four-bar function gen, ..-ating linkage, Machinovedenie, No. 3, 25-31 (1969). [3] L E V I T S K I I N. I. and S A R K I S I A N Y. L. O n the special properties of Lagrange's multipliers in the least-square synthesis of m e c h a n i s m s , J. Mechanisms 3, 3 - I 0 (1968). OnTUMa.~t~HU~ cltnTea nepe~aTOqltoro qeTblp~x3BeHsxKa H. I,t[..r]eBHTCgK.~, tO. f[. CapKHC,qH, ]7. f[. F~K~i~iH P e 3 ~ o M e - B BBO,~[HO~ ~aCT~t CTaTbH nOKaBblBaeTcR tle~ecoo6pa3HOCTb npe,~CTaB~leHH~I 3aJlaqH flpOCKTHpOBaHH~ NCpC~RTO"~HOI'O ',,ICTblpeX3BeHHHKa B BH~C 3a~aqH HeJ'IHHe~HOrO rlpoI'paMMHpOBaHKR. }~a~Ie¢ I'IOB,pO6HO orlRCblBaCTCR MCTOJ1 B3BelJ.ICHHOFO KBa~paTl4'.IeCKOrO rlpH(~.qH)KeH1431, IIpc~YlO;.KeHHblI~[ H. H. .TIcBI~TCKH2v[. ,~[OKa3blBaCTCR, ~'ITO BMCCTO CJIO>KHOr'O l,[ppauHoaa.r'lbHoro Bblpa~KeHRR (3) OTKJ]OHeHHM OT :3aJ1aH.aofi qbyHKllHH I'IpR CKHTC3e MO;,KHO MHHHMH3HpOBaTb ]IOCTaTOqHO npocToe abtpaY,