1-s2.0-S0094114X08000281-main

Available online at www.sciencedirect.com Mechanism

Mechanism and Machine Theory 44 (2009) 235–246

www.elsevier.com/locate/mechmt

andMachine Theory

On the optimum synthesis of a four-bar linkage usingdifferential evolution and method of variable controlled

deviations

Radovan R. Bulatovic a,*, Stevan R. Dordevic b

a Faculty of Mechanical Engineering, University of Kragujevac, Kraljevo, Serbiab Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia

Received 28 November 2006; received in revised form 28 January 2008; accepted 3 February 2008Available online 14 March 2008

Abstract

The synthesis of a four-bar linkage in which the coupler point performs approximately rectilinear motion is presented inthis paper. The Grashof four-bar linkage whose geometries allow minimum deviations from the problem given for differentparts of the crank cycle has been chosen. The motion of the mechanism crank point located in the prescribed environmentof the given point on the observed segment is followed within the prescribed values of allowed deviation. Allowed devia-tions change during the optimisation process from the given maximum values to the given minimum ones. Very high accu-racy for motion along a straight line at a large number of given points has been achieved by using the method of variablecontrolled deviations and by applying differential evolution algorithm (DE). The mechanism obtained is not symmetricaland the ratios of lengths between individual members of the mechanism are not normalised.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Optimum; Synthesis; Four-bar linkage; Rectilinear motion; DE algorithm; Variable controlled deviations

1. Introduction

A lot of working processes require motion of a point of the working member along different paths whichcan be rectilinear or have a very complex shape. Regardless the path shape, it is difficult to design a mechanismwhich will exactly achieve desired motion, so there should be an orientation toward the mechanisms by meansof which the task can be approximately realised. Imprecisions in a mechanism arising due to mistakes in man-ufacture and mounting are multiplied by the number of members of the mechanism. Therefore, there is a ten-dency in practice to use lever mechanisms with as few number of members as possible and as simple structureas possible. Such mechanisms are a four-bar linkage, a slider crank mechanism, an inverted slider crank mech-anism, etc.

0094-114X/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmachtheory.2008.02.001

* Corresponding author. Tel.: +381 36 383 377; fax: +381 36 383 269.E-mail address: [email protected] (R.R. Bulatovic).

mailto:[email protected]

236 R.R. Bulatovic, S.R. Dordevic / Mechanism and Machine Theory 44 (2009) 235–246

Accomplishment of exact rectilinear motion is practically impossible for a four-bar linkage. There arenumerous mechanisms which approximately achieve rectilinear motion on a segment. They are mostly mech-anisms in which the coupler curve is symmetrical, and the point on the working part (coupler) whose motion isobserved lies on the direction normal to the direction of the support and it coincides with its centre line, whichis at the same time the centre line of the coupler curve. Such mechanisms are the Chebychev linkage (the crankand the rocker have the same lengths) and the Roberts linkage (the crank, the rocker and the coupler have thesame lengths). In the Hoeken’s four-bar linkage, the coupler point also traces rectilinear motion, the couplercurve is symmetrical, and the centre line of the coupler curve passes through the point of rocker support. Thislinkage satisfies the Grashof conditions, the crank is the shortest member and during a working cycle itdescribes a complete circle. The coupler and the rocker have the same lengths. The Watt’s mechanism foraccomplishing approximately rectilinear motion is also known, with the coupler being the shortest memberand with the coupler and the rocker having the same lengths.

Norton [1,2] describes a four-bar linkage whose geometries deliver a minimum error in either position orvelocity deviation over various portions of the crank cycle when the crank is the driving member and has con-stant angular velocity. Deviations are less than 1% in precision points and less than a few percent for the con-stant velocity for a part of the cycle bigger than 50%. Paper [1] analyses the difference between the originalHoeken’s mechanism and the family of Hoeken’s mechanisms similar to the Bunduwongse and Ting (BT)mechanism [3]. BT is a symmetrical mechanism in which the coupler is the shortest member, and the couplerpoint lies in the middle of the coupler. It has normalized dimensions 1, 4, 3, 4 (the coupler is the member withthe unit length, the crank and the rocker are of the length 4, and the support of the length 3). In the Hoeken’smechanism, the coupler point lies on the direction of the coupler but on the opposite side, and its dimensionsare also normalized with 1, 2.5, 2.5, 2 (the unit member is the crank, the coupler and the rocker have thelengths 2.5, and the support has the length 2). The mentioned paper analyses and defines the Hoeken’s mech-anism with different ratios of lengths, but in such a way that it achieves highly accurate rectilinear motion aswell as the coupler point motion with the velocity which is very close to the angular velocity. It gives a table ofoptimisation results presenting, for different working angles of the crank and different ratios of the lengths ofthe given mechanism, the percentage of the cycle as well as deviations from rectilinear motion and deviationsfrom constant velocity. Thus, for example, during optimisation of rectilinear motion, for the working angle of80�, which makes 22.2% of the working cycle, the deviation from rectilinear motion is 0.001%, and the devi-ation from constant velocity is 6.27%. Also, during the optimisation of conditions of constant velocity for thesame working angle, the deviation from constant velocity is 0.34%, and the deviation from rectilinear motionis 0.503%. In the first case, the ratio between dimensions is 1, 3.738, 3.738, 2.825, and in the second case it is 1,2.463, 2.463, 1.975.

Dordevic [4] defined the method of controlled deviations as the method for approximate synthesis by whichthe mechanism functions would meet technological requirements on the observed segment. The requirementsof the method fully support the requirements of a technological process. The boundary functions are definedthrough the given (desired) function which corresponds to an idealised process and through allowed deviationswhich represent constant values and do not change during an optimisation process. A successful synthesisrequires determination of space around the given functions in which the functions of the mechanism haveto be found.

Papers [5,6] present the Genetic Algorithm (GA) as the main technique in the synthesis of mechanisms. Theprocedure of GA described in the paper Kunjur and Krishanamury [6] is not directly applied to the synthesisof mechanisms because there are highly non-linear constraints of the optimisation problem. Certain modifica-tions in the main GA in relation to the constraints and avoidance of early convergence in the solution weremade. The paper presents a formulation of GA which, with constraints, leads the mechanism synthesis pro-cedure into the global minimum region.

Cabrera et al. [5] also applied the GA method for the synthesis of the path of planar mechanisms. Theobjective function has two parts. The first part is calculation of the position error between the given pointsand the points within the reach of the resulting mechanism. The second part of the objective function refersto deduced constraints which are prescribed by the mechanism. Three cases of synthesis of a four-bar linkagewith very fast convergence of the objective function in the vicinity of the optimal solution were considered, theerror is very small and reaches the approximate value zero in the first 100 generations.

R.R. Bulatovic, S.R. Dordevic / Mechanism and Machine Theory 44 (2009) 235–246 237

Storn and Price [7] successfully applied the DE algorithm during optimisation of certain well-known non-linear, non-differentiable and non-convex functions. Papers (cf. [8–11]) and Ref. [12] give a detailed descriptionof the DE algorithm as well as its application to various optimisation problems.

Shiakolas et al. [13] performed synthesis of a six-bar linkage with dwell, combining DE and the geometriccentroid of precision positions technique, where dwell is defined by timing relative to the motion of the inputmember (crank). The coupler curve contains 18 precision points with two circular arcs. Dependence at preci-sion points on circular arcs in relation to the input crank angle is given.

This paper aims at realisation of synthesis of a non-symmetrical four-bar linkage where the optimisationmay result on a symmetrical mechanism, thus accomplishing high accuracy rectilinear motion of a point onthe working member of the mechanism.

2. Analysis of a four-bar linkage

A four-bar linkage, as a typical representative of planar linkages, is considered. The relevant parametersdefining the mechanism geometry are presented in Fig. 1. In addition to these parameters, further analysis usesthe following parameters:

xp – the initial position of the point M of the coupler on the path (which corresponds to the initial angle u0),xk – the final position of the point M of the coupler on the path (which corresponds to the final angle uk),u1 = uk � u0 – the working angle of the crank which corresponds to the given path.

The position of the point B is the defined by the expressions:

xB ¼ xA þ a � cos u; ð1ÞyB ¼ yA þ a � sin u: ð2Þ

The distance EB ¼ s is defined by the expression

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxE � a � cos uÞ2 þ ðyE � a � sin uÞ2

qð3Þ

and with the positive direction of the Ox-axis it creates the following angle:

O

EA

ϕθ

B

M

x

C

y A x

y E

a

d

b

s

c

r

x

x k

x

y

A E

p

D

Fig. 1. Geometry of a four-bar linkage.


h ¼ arctgyE � a � sin uxE � a � cos u

� �; �p 6 h 6 p: ð4Þ

The coordinates of the point D are determined by the expressions

xD ¼ xA þ xE þ c � cosðh� t � cÞ; ð5ÞyD ¼ yA þ yE þ c � sinðh� t � cÞ; ð6Þ

where t is the coefficient whose value is

t = 1, for the case when the point D is above the line segment s (Fig. 1), andt = �1, for the case when the point D is below the line segment s (crossed mechanism),

and where c, the angle created between the rocker and the line segment s, is defined by the expression

c ¼ arccoss2 þ c2 � b2

2 � s � c

� �; 0 6 c 6 p: ð7Þ

The angle w created between the coupler BD and the positive part of the Ox-axis is determined by theexpression

w ¼ arctgyD � yB

xD � xB

� �: ð8Þ

Finally, the position of the point M of the coupler, the point moving along the desired path, is given by thefollowing equations:

xM ¼ xA þ a � cos uþ d � cos wþ r � cosp2þ w

� �; ð9Þ

yM ¼ yA þ a � sin uþ d � sin wþ r � sinp2þ w

� �: ð10Þ

3. Optimum synthesis of a four-bar linkage and tools employed

3.1. Formulation of the optimisation problem

The paper considers the following optimisation problem:Find optimum dimensions of the mechanism for the desired rectilinear path traced by the point M (Fig. 1)

of the coupler of the four-bar linkage so that the objective function has the minimum value.Thus defined optimisation problem can be given the following general mathematical formulation:

minimise f ðXÞ; ð11Þsubject to gjðXÞ 6 0; j ¼ 1; . . . ; ng; ð12Þ

f(X) is the objective function, gj(X) 6 0 represent the constraints defined by the search space, ng is the totalnumber of constraints.

X = {x1, . . . ,xD}T represents the design vector consisting of D design variables. The design variables are thevalues which should be defined during the optimisation procedure. Each design variable is defined by its lowerand upper boundaries. For the case of the four-bar linkage (Fig. 1), the design vector is X = (a,b,c,d, r,xA,yA,xE,yE,u0,u1)T.

By introducing the constraints in the objective function, Eqs. (11) and (12) can be transformed into the fol-lowing form:

Minimise F ðXÞ ¼ f ðXÞ þ PðX ; kjÞ; ð13Þ

where P(X,kj) are the penalty functions which can be presented in the following way:


PðX ; kjÞ ¼Xng

j¼1

kj � ðmaxð0; gjðXÞÞÞ2

" #: ð14Þ

When the solution is found outside the region considered, then the current parameters of the solution aresquared and multiplied by large positive numbers (penalty factors) kj, and then added to the numerical valuesof the objective function.

3.2. DE algorithm

The DE algorithm is briefly described here, and the control parameters of the algorithm are also dealt with.A detailed description of the DE algorithm can be seen in Refs. [7–12].

DE is a simple, but still strong evolutionary algorithm for realisation of the global minimum in numerousreal optimisation problems. The DE algorithm, similarly to GA, has the following control parameters: thepopulation size NP, the crossover constant CR and the mutation constant F. Coding of chromosomes withreal numbers, i.e. presentation of chromosomes as vectors of real values, is used in numerical applicationsof DE in optimisation processes.

Generation of the initial population is performed stochastically. At that, the population size NP iscommonly ten times bigger than the number of design variables. At the beginning, each design variable is arandom value which is found within the defined upper and lower boundaries. While defining the bound-aries, attention should be paid that the values of design variables are not out of the range which is reallyacceptable.

The mutation constant in DE is a real parameter, which controls the increase of difference between twoindividuals in the search space. The difference between two randomly chosen vectors defines the magnitudeand direction of mutation. When the difference is added to a randomly chosen vector, it becomes a mutantvector. The basic idea of DE is that mutation is self-adaptive in the search space and the current population.At the beginning of the optimisation process, the magnitude of mutation is large because the vectors in thepopulation are far away from the search space. When the process starts to converge, the magnitude of muta-tion starts to decrease. The self-adaptive mutation in DE leads the solution of the optimisation process towardthe global minimum.

For taking the best values for CR, there are certain basic rules defined in [12]. High values are effective forall problems, but they are not always the fastest. The problems with heavy interaction between design vari-ables generally require a high CR. But, if interaction between design variables is lower, a lower CR can beused, which results in obtaining a satisfactory solution with a smaller number of iterations (faster solution).The interaction between design variables in this paper is high so that a high CR, i.e. CR = 0.8 is used in allexamples. Also, on the basis of recommendations in [12], the values of other control parameters in all exam-ples are: NP = 110 and F = 0.6.

3.3. Method of variable controlled deviations

For the purpose of achieving ideal operation of a mechanism within a device, it is necessary that at a certainsection of the path the observed point should move according to the precisely defined law. The desired law ofmotion in a general case can be represented by the shape function and the position function.

In a large number of cases, the mechanism will perform its operation successfully even when the workingpart of its path does not coincide with the ideal one. It is enough that the real path is in the ‘‘controlled” spacearound the ideal path where the allowed deviations are prescribed in advance.

For the four-bar linkage in this paper, the motion of the coupler point on a segment is defined by expres-sions (9) and (10), within which all design variables figure either directly or indirectly.

Solution of the problem is reduced to forming of the objective function in which there are also deviationsbigger than the allowed ones. Minimisation of the objective function results in bringing the mechanism pathwithin the allowed deviations around the given path. As the values of real deviations do not figure in the syn-thesis process, it can be concluded that, if there is one solution, then there is an infinite number of indefinite


similar solutions. The following relations give the projections of differences between actual and alloweddeviations:

ni ¼jyMðiÞ � ydðiÞj � dyðiÞ if jyMðiÞ � ydðiÞj � dyðiÞ > 0;

0 otherwise;

�ð15Þ

gi ¼jxMðiÞ � xdðiÞj � dxðiÞ if jxMðiÞ � xdðiÞj � dxðiÞ > 0;

0 otherwise:

�ð16Þ

i ¼ 1; 2; . . . ;N ;

where N is the number of given points on the path, xM and yM are the real coordinates of the point M definedby Eqs. (9) and (10), xd and yd are the coordinates of the point M, i.e. the coordinates lying on the given path,dx and dy are the allowed deviations of the position of the point M from the given position.

Without reducing the generality of consideration, it is taken that the given path is the segment of the lengthL on the straight line (l), which is defined by yd = c = const. The coordinate xd is defined by the expressions

xd1 ¼ xp for u ¼ u0;

xdi ¼ xp þ Lði� 1� xxÞ=ðN � 1Þ for u0 < u < u0 þ u1;

xdN ¼ xk ¼ xp þ L for u ¼ u0 þ u1;

i ¼ 2; 3; . . . ;N � 1 and xx 2 ð0; 1Þ:

ð17Þ

By introducing the variable xx whose value during optimisation is randomly changed in the interval (0,1), thewhole path is found within the required controlled space.

The controlled deviations dx and dy change during the optimisation process. At the beginning of the syn-thesis process, the boundaries of allowed deviations are set considerably out of those that should be accom-plished and they are marked as maximum values in Fig. 2. The synthesis process is activated and as soon as theobjective function reaches the zero value, i.e. when all points of the generated path are in the prescribed envi-ronment of the given path, the allowed deviations decrease, and the synthesis process continues. Variable con-trolled deviations both for the shape function and the position function are thus introduced in theoptimisation process, directly in the DE algorithm. The rule for decrease of allowed deviations during the opti-misation process is described by the following equations:

dy ¼ dy � c1

dx ¼ dx

�; when dy > dymid and dx > dxmin;

dy ¼ dy

dx ¼ dx� c2

�; when dy 6 dymid and dy P dymin and dx > dxmin;

dy ¼ dy � c3

dx ¼ dx

�; when dy 6 dymid and dy > dymin and dx 6 dxmin:

ð18Þ

M(x d(i), (i))y

dxmin

M(xM(i), M(i))y

dxmax

dym

in

dym

id

dym

ax

x

y

desired path

d

Fig. 2. Decrease of controlled deviations.


The values c1, c2 and c3 are positive numbers given at the beginning of the optimisation process. It is desirablefor c3 to be significantly smaller than the remaining two constants because in that case the path approaches thegiven path more safely and the local minima are avoided. The value dymid represents a mid-value of the con-trolled deviation dy.

Fig. 3 presents the given functions in the field of allowed deviations. The shaded parts represent undesireddeviations which should be minimised.

3.4. Objective function

On the basis of Eq. (13), the objective function is defined by the following expression:

f ða; b; c; d; r; xA; yA; xE; yE;u0;u1Þ ¼ k �XN

i¼1

½n2i þ g2

i � þXm

j¼1

½k1 � g21j þ k2 � g2

2j� þ k3 � g23; ð19Þ

where N is the number of given points representing the rectilinear portion, and k, k1, k2 and k3 are large num-bers influencing the importance of values multiplied by them.

The first member in the given expression for the objective function represents the sum of squares of devi-ations of the current path out of the controlled space around the given path, while the second and the thirdmembers refer to the rest of the path and Grashof conditions, respectively.

The constraints are defined in the following manner:

(1) Scope of initial values of design variables. In the DE algorithm, the initial values of design variables aredefined by their upper and lower boundaries, i.e. a broader range of values which can be taken as theinitial solutions in the relevant area is given. Ref. [12] proposes the constraints directly in the DE algo-rithm, which allow that the values of design variables remain within the mentioned boundaries duringthe whole optimisation process. This has also been applied in this paper so that the constraints whichunable negative values of certain design variables have not been introduced in the objective function.Namely, it is enough to take positive numbers for the lower and upper boundaries of those variableswhose values must not be negative. This refers to the lengths of the crank, the coupler and the rocker.

(2) The rest of the path. In the Eq. (19), m is the number of given points representing the rest of the path, andg1 and g2 are the constraints which refer to the global shape of the rest of the path:

g1j ¼ðxðjÞ � x rÞ � ðxðjÞ � x lÞ for xðjÞ � x r > 0 or xðjÞ � x l < 0;

0 for xðjÞ � x r < 0 and xðjÞ � x l > 0;

�ð20Þ

g2j ¼ðyðjÞ � y upÞ � ðyðjÞ � y lowÞ for yðjÞ � y up > 0 or yðjÞ � y low < 0;

0 for yðjÞ � y up < 0 and yðjÞ � y low > 0;

�ð21Þ

L

x

ξ

ξ

η

η

ϕ

ϕ1

Fig. 3. (a) The shape function; (b) the position function.


where x_r is the right-hand boundary, x_l is the left-hand boundary, y_low is the lower boundary and y_up isthe upper boundary of the rectangular area, and x(j), y(j) are the real coordinates of position of a point of thecoupler, which lie outside the rectilinear part of the path.It happens that the optimisation process results in ‘‘stretching” of the rest of the path or in its undesired shape.That is the reason why a rectangular area around the rectilinear portion of the path in which the rest of thepath should lie is given. The constraint of the whole path by the rectangular area has shown to be morefavourable than the constraint of the working angle because different values of working angles as well as amuch more acceptable path can be obtained.

(3) Grashof conditions. The sum of lengths of the shortest and the longest member of a four-bar linkage mustbe smaller than the sum of lengths of the remaining two members. The constraint referring to Grashofconditions in Eq. (19) is g3:

TableInitial

LowerUpper

TableConsta

c1 (mm2

x_l (m�270

g3 ¼lmax þ lmin � lmid1 � lmid2 for lmax þ lmin � lmid1 � lmid2 > 0;

0 otherwise;

�ð22Þ

where lmax = max(a,b,c, f), lmin = min(a,b,c, f), f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

E þ y2E

p, lmid1 and lmid2 are the lengths of the remaining

two members.

4. Results

The described method has been applied for the synthesis of a four-bar linkage which accomplishes rectilin-ear motion. The method has been used in 10 examples in which the desired path is a segment of the straightline parallel to the Ox-axis. The beginning of the segment coincides with the coordinate beginning, and itslength is 400 mm. The number of precision points has been varied for different cases. The smallest numberof observed precision points is 16, and the largest one is 80. All of them are distributed along the given segmentof the straight line and in each iteration their positions are changed stochastically. Since the DE method hasbeen applied, and it requires the initial boundaries for design variables, the same values have been used for allexamples and they are presented in Table 1.

The constants used in the optimisation process are presented in Table 2. The maximum number of itera-tions in all examples is 5000.

Tables 3 and 4 show the final results of the optimisation processes. The presentation of 10 obtained solu-tions (mechanisms) is given, as well as the number of precision points, the number of iterations, the minimumdeviations of the shape function and the position function. The values of deviations of the shape function, theposition function and the working angle are also given in percentage.

The use of a large number of precision points has aimed at as precise accomplishment of rectilinear portionof the path as possible. Synthesis has also been performed for different values of minimum deviations of theshape function and the position function.

1values of design variables

a (mm) b (mm) c (mm) d (mm) r (mm) xA (mm) yA (mm) xE (mm) yE (mm) u0 (rad) u1 (rad)

boundary 0 0 0 �900 �900 �900 �900 �900 �900 �3 �3boundary 900 900 900 900 900 900 900 900 900 3 3

2nts used in the optimisation process

) c2 (mm) c3 (mm) dymax (mm) dxmax (mm) dymid (mm) m e (error)2 0.1 dymin + 10 dxmin + 10 dymin + 2 5 0

m) x_r (mm) y_low (mm) y_up (mm) k k1 k2 k3

670 �300 300 106 108 108 106

Table 3The obtained mechanisms with deviations of the shape function and the position function in (%), as well as the values of the working anglealso in (%)

Example 1 Example 2 Example 3 Example 4 Example 5

a (mm) 125.978916 96.705406 139.591303 135.497672 129.412911b (mm) 411.399256 371.127732 604.312170 467.980994 468.645202c (mm) 541.496506 328.692471 893.584293 480.876733 449.268731d (mm) 592.409710 775.161111 681.718428 901.629932 978.557388r (mm) �128.981898 �281.463296 �236.511665 �83.625963 8.220348xA (mm) �41.142091 4.050515 �65.002319 �112.894205 �160.207318yA (mm) �479.253399 �727.970387 �581.438747 �767.149999 �844.281041xE (mm) 302.434924 177.225533 393.056215 336.678285 351.114253yE (mm) �108.279994 124.917147 �243.074691 14.002536 22.629577u0 (rad) �1.835903 �1.535475 �1.930653 �2.039952 �2.194976u1 (rad) �2.472382 �2.199662 �2.157540 �1.898473 �1.852450N 16 32 48 64 80Number of iterations 567 744 758 834 2080dymin (mm) 0.05 0.04 0.03 0.02 0.01dxmin (mm) 7 3 3 3 3dy (%) 0.025 0.02 0.015 0.01 0.005dx (%) 3.5 1.5 1.5 1.5 1.5u1 (%) 39.34918 35.00871 34.33832 30.21514 29.48266

Table 4The obtained mechanisms with deviations of the shape function and the position function in (%), as well as the values of the working anglealso in (%)

Example 6 Example 7 Example 8 Example 9 Example 10

a (mm) 176.819373 177.504353 199.385132 189.437426 203.611519b (mm) 619.410129 625.399922 733.773373 683.792342 717.963041c (mm) 747.520042 753.337923 1079.669236 766.947339 878.658085d (mm) 1006.536306 1018.699233 994.863267 1207.147526 1174.481687r (mm) �143.022337 �142.953350 �183.812366 �118.031209 �111.341095xA (mm) �201.388994 �200.485390 �213.072039 �265.279043 �279.296622yA (mm) �834.444226 �845.523541 �805.770289 �1015.939001 �966.625118xE (mm) 460.200949 464.503441 557.097831 507.155936 553.408409yE (mm) �102.977265 �102.817834 �336.628634 �59.485926 �156.426044u0 (rad) �2.325397 �2.310155 �2.365534 �2.417108 �2.464141u1 (rad) �1.591044 �1.579201 �1.521278 �1.379626 �1.380132N 80 80 80 80 80Number of iterations 1195 1241 1274 1286 1399dymin (mm) 0.005 0.004 0.003 0.002 0.002dxmin (mm) 3 3 3 3 3dy (%) 0.0025 0.002 0.0015 0.001 0.001dx (%) 1.5 1.5 1.5 1.5 1.5u1 (%) 25.32251 25.13376 24.21189 21.95743 21.96548


According to Tables 3 and 4, it is obvious that in certain examples some values of the design variables areout of the boundaries defined in Table 1. Retention of the design variables within the given initial boundaries,for small values of allowed deviations of the shape function and the position function, has made the synthesisprocess unrealisable. Therefore, constraints up to some values of allowed deviations of the shape function andthe position function have been used, all until the entrance of the mechanism into the zone of solution, andthen those constraints have been excluded and the optimisation process has continued. Exclusion of the con-straints at the very beginning would result in an unfavourable solution, in the sense of disproportion of themembers of the mechanism or negative lengths of the members.

Figs. 4–7 present the final mechanisms from Example 1 and Example 10, as well as their paths. The mech-anism from the first example has the biggest minimum deviation of the shape function 0.05 mm and the

–600 –400 –200 0 200 400 600 800 1000

–1000

–800

–600

–400

–200

0

200

400

Fig. 4. Mechanism from Example 1.

–300 –200 –100 0 100 200 300 400 500 600 700–500

–400

–300

–200

–100

0

100

200

300

400

500

Fig. 5. Path of the mechanism from Example 1. The number of precision points is 16, and the maximum deviation from the straight line is0.025%.


position function 7 mm, with the smallest number of precision points 16. In contrast to this one, the mech-anism from the tenth example has the deviation of the shape function of 0.002 mm, or, in percentage0.001%, and the deviation of the position function whose value is 3 mm or 1.5%. The number of precisionpoints for this mechanism is 80. These are the two mechanisms representing all mechanisms shown in Tables3 and 4.

–600 –400 –200 0 200 400 600 800 1000

–1000

–800

–600

–400

–200

0

200

400

Fig. 6. Mechanism from Example 10.

–300 –200 –100 0 100 200 300 400 500 600 700–500

–400

–300

–200

–100

0

100

200

300

400

500

Fig. 7. Path of the mechanism from Example 10. The number of precision points is 80, and the maximum deviation from the straight lineis 0.001%.


5. Conclusion

This paper described the process of optimal synthesis of a four-bar linkage by the method of variable con-trolled deviations with the application of the DE algorithm. The method of variable controlled deviations issuitable because deviation of motion of the working member point in relation to the projected path can befollowed at any moment. In the largest number of cases, when variable controlled deviations are not applied


to the synthesis process, and when the projected deviations are considerably small, realisation of the synthesisof a four-bar linkage becomes practically impossible. Namely, if constant controlled deviations are taken inthe optimisation process, the value of the objective function stagnates and convergence is impossible. How-ever, the application of variable controlled deviations for a relatively small number of iterations results inthe solutions in which the objective function decreases until the zero value. Deviations are controlled by pre-scribing their allowed values within which motion of the coupler point is followed. The method is illustratedby the example of synthesis of a four-bar linkage where the point on the working member moves along therectilinear segment the length of which is 400 mm. For the given example, this paper presents the results ofsynthesis in which a very high accuracy was realised. Namely, a mechanism in which the deviation of the shapefunction is 0.001%, and the deviation of the position function is 1.5%, was obtained. These values correspondto the working angle of 79.07� or 1.380132 rad, i.e. 21.96548% of the coupler motion cycle. Further research inthis field will be directed to elaboration of the ways of application of this method for the case of an arbitraryshape of the path which should be realised by a point on the working member of the four-bar linkage duringits motion.

References

[1] R.L. Norton, In search of the ‘‘perfect” straight line and constant velocity too – design charts for optimum straight-line crank-rockerfour bar linkages with near constant velocity, in: Proc. 6th Applied Mechanisms and Robotics Conference, Cincinnati, OH, 1999.

[2] R.L. Norton, Design of Machinery an Introduction to the Synthesis and Analysis of Mechanisms and Machines, third ed., McGraw-Hill Inc., New York, 2004.

[3] R. Bunduwongse, K.L. Ting, Synthesis of straight line mechanism with fifth order burmester points at inflection pole andtransmission angle control, in: Second Applied Mechanisms and Robotics Conference, Cincinnati OH, II.A.3-1, 1991.

[4] S.R. Dordevic, A New Approach to Technological Requirements of Motion of Mechanisms in Their Synthesis, DoctoralDissertation, Faculty of Mechanical Engineering, University of Belgrade, Belgrade, 1988 (in Serbian).

[5] J.A. Cabrera, A. Simon, M. Prado, Optimal synthesis mechanisms with genetic algorithms, Mechanism and Machine Theory 37 (10)(2002) 1165–1177.

[6] A. Kunjur, S. Krishanamury, Genetic algorithms in mechanical synthesis, Journal for Applied Mechanisms and Robotics 4 (2) (1997)18–24.

[7] R. Storn, K. Price, Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces, Journal ofGlobal Optimization 11 (4) (1997) 341–359.

[8] K. Price, R. Storn, A simple evolution strategy for fast optimization, DR. Dobb’s Journal 264 (1997) 18–24.[9] R. Storn, On the usage differential evolution for function optimization. <http://http.icsi.berkley.edu/~storm/litera.html>.

[10] J. Lampinen, A Bibliography of Differential Evolution Algorithm, Lappeenranta University of Technology, Finland, 2001.[11] S. Kukkonen, J. Lampinen, Comparison of generalized differential evolution to other multi-objective evolutionary algorithms, 2004.

<http://www.imamod.ru/~serge/arc/conf/ECCOMAS_2004/ECCOMAS_V2/proceeding/pdf/716.pdf> (Online).[12] K.V. Price, R.M. Storn, J.A. Lampinen, Differential Evolution – A Practical Approach to Global Optimization, Springer-Verlag,

Berlin Heidelberg, 2005.[13] P.S. Shiakolas, D. Koladiya, J. Kebrle, On the optimum synthesis of six-bar linkages using differential evolution and the geometric

centroid of precision positions technique, Mechanism and Machine Theory 40 (3) (2005) 319–335.

http://http.icsi.berkley.edu/~storm/litera.html

http://www.imamod.ru/~serge/arc/conf/ECCOMAS_2004/ECCOMAS_V2/proceeding/pdf/716.pdf

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