Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 1, pp. 43-55, Warsaw 2018
DOI: 10.15632/jtam-pl.56.1.43

REDUCED NUMBER OF DESIGN PARAMETERS IN OPTIMUM PATH

SYNTHESIS WITH TIMING OF FOUR-BAR LINKAGE

Jacek Buśkiewicz

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: jacek.buskiewicz@put.poznan.pl

The paper presents themethod for the optimal synthesis of four-linkmechanism generating
open/closed paths with time prescription. Although the method is suitable for both closed
and open paths, it enables decreasing the number of design parameters describing dimen-
sions, orientation and position of a path generator. Compared to the methods presented in
the references, this is a one-phase synthesis method; although the number of design para-
meters is reduced, the method does not require affine transformations to be performed on
the synthesisedmechanism. The effectiveness of themethod is discussed based on examples
of three paths, with two taken from the literature.

Keywords: path synthesis, evolutionary algorithm, four-bar linkage

1. Introduction

Recent literature reveals a great variety of problems related to the path synthesis in which di-
mensionsof amechanismare sought, thepoint ofwhich traces adesiredpath.Thisdiversification
may be classified with respect to:

1. Methods of solutions: analytical, computer (deterministic, heuristic).

2. Formulation of the problem: closed- or open-path synthesis, path synthesiswith orwithout
timing, one- or two-phase path synthesis.

3. Type ofmethod: direct or indirect synthesismethod (based on an atlas of coupler curves).

Owing to great mathematical complexity (Erdman et al., 2001), computer methods are ma-
inly developed. The prevalent techniques are based on the genetic/evolutionary algorithms and
other heuristic methods patterned upon biological systems and social behaviours (Avilés et al.,
2010; Bulatović et al., 2013, 2016; Buśkiewicz, 2009, 2015; Buśkiewicz et al., 2010; Cabrera et
al., 2007, 2011; Ebrahimi and Payvandy, 2015; Gogate and Matekar, 2012; Kafash and Nahvi,
2015; Kunjur andKrishnamurty, 1997; Lin, 2010;Matekar andGogate, 2012; Nadal et al., 2015;
Penunuri et al., 2011; Shiakolas et al. 2002, 2005). Frequently, the methods are devoted to a
single specific path-synthesis task: synthesis of open-path generators, synthesis of closed-path
generators, path synthesis with and without timing (time prescription). In the path synthesis
with timing, each position of the coupler point corresponds to a prescribed angular position
of the active link. In many cases, this correspondence is of less significance. In the open-path
synthesis, a part of the coupler trajectory is prescribed.
The two-phase synthesis (Buśkiewicz et al., 2009;Buśkiewicz, 2010; Lin, 2014, 2015;McGarva

and Mullineux, 1993; Nadal et al., 2015; Sanchez Marin and Gonzalez, 2004; Smaili and Diab,
2007; Sunet al., 2015;UllahandKota, 1997) for thedirect synthesismethodconsists of two steps.
At first the shape synthesis is carried out followed by scale-rotation-translation transformations.
Thefirst step reduces the numberof design variables.This number is importantwhena synthesis
method is based on deterministic-probabilistic algorithms for optimal parameter determination.



44 J. Buśkiewicz

Every additional parameter increases the dimension of the solution space andmay increase the
computational cost. In general, the effectiveness of amethod depends essentially on the number
of design parameters, by which the objective function is expressed.

This paper deals with the direct synthesis method dedicated for an open/closed-path syn-
thesis with timing. Themathematical foundations of the usage of the Fourier descriptors (FDs)
as shape signatures (descriptors) of a closed curvewere formulated by Zahn andRoskies (1972).
Although a prevailing number of these methods may be used for closed paths, the wavelet
transform has been used to describe open paths by Sun et al. (2015). Smaili and Diab (2007)
presented the two-phase synthesis method based on the so-called cyclic angular deviation vec-
tor. Buśkiewicz et al. (2009) proposed a two-phase synthesis method in which the shape was
described by means of the curvature-based Fourier descriptors. The centroid, the direction of
the major principal axis and the perimeter of a curve enabled translation, rotation and scaling
of the mechanism to the desired configuration. Buśkiewicz (2010) also proposed a two-phase
synthesis method using the function of the distance of the curve from its centroid, to describe
the curve shape in terms of its normalised Fourier coefficients. Sanchez Marin and Gonzalez
(2004) utilised geometric properties to describe the path (without addressing harmonic series
theories) and to reduce the number of optimised parameters to five in the open-path synthesis.

The methods using decomposition of the path into normalised parameters, invariant with
respect to affine transformation, alsobelongs to the so-called indirect synthesismethod (Chuand
Sun,2010;Galán-Maŕın et al., 2009;Hoeltzel andChieng, 1990;McGarva, 1994;Mullineux, 2011;
Sun andChu, 2009; Vasiliu andYannou, 2001; Yu et al., 2007, 2012). The second phase consists
of searching through a computerised atlas database of normalised descriptors which are linked
to dimensions of themechanisms generating closed paths. Nonetheless, most techniques are one-
phase syntheses which either introduce new error functions or develop algorithms evaluating
an objective function (Acharyya andMandal, 2009; Avilés et al., 2010; Bulatović andDordević,
2009;Bulatović et al., 2013, 2016;Cabreraet al., 2002, 2007, 2011;EbrahimiandPayvandy, 2015;
Gogate andMatekar, 2012; Khorshidi et al., 2011;Kinzel et al., 2006;Kunjur andKrishnamurty,
1997; Lio 1997; Lio et al., 2000; Lin, 2010; Lin and Hsiao, 2017; Matekar and Gogate, 2012;
Penunuri et al., 2011, Sancibrian et al., 2004; Schmiedeler et al., 2014; Shiakolas et al., 2002,
2005; Smaili et al., 2005). The number of design parameters is as large as necessary to define
the problem. In the optimum synthesis with timing the design variables are not onlymechanism
dimensions and orientation but also angular positions of the input link corresponding with the
prescribed coupler points.

The number of techniques for open/closed-path synthesis with timing, defined by the de-
creased number of design parameters, is rather small (Nadal et al., 2015; Kafash and Nahvi,
2015). The paper presents themethod for the optimal synthesis of a four-link planarmechanism
generating open paths with time prescription, enabling a decrease in the number of design pa-
rameters describing dimensions, orientation and position of the path generator. Compared with
themethods presented in the references, this is a one-phase synthesis method; i.e. although the
number of design parameters is reduced, the method does not require affine transformations to
be performed on the synthesised mechanism. The paper is a continuation of the work (Buśkie-
wicz, 2015) dealing with the one-phase path synthesis without timing, defined by means of a
reduced number of design parameters.

2. Method description

2.1. Mathematical foundations

The method for path synthesis with timing is aimed at minimising the number of design
variables being optimised using a heuristic algorithm. The coupler curve is defined bymeans of
the discrete number of points. When the input link rotates by a given angle, the coupler point



Reduced number of design parameters in optimum path synthesis... 45

passes between two neighbouring curve points. The geometric scheme of the four-bar linkage is
shown in Fig. 1a.
The design variables optimised bymeans of the algorithm are (Fig. 1b):

• coordinates of the input link pivot: xO1, yO1,

• input link length: l1, and couplerAB length: l2,

• angle between armsAD andAB of the coupler: θ4,

• angular position of the input link θ10 corresponding with the first coupler point.

Six parameters define the four-bar linkage instead of the 10 required in the classical approach
to path synthesis with timing (l1, l2, l3, l4, l5,xO1,yO1,θ4,γ,θ10).
The input data are given as set {(xi,yi,θ1i), i=1, . . . ,n}, where (xi,yi) are the coordinates

of thepoints tobedrawnbythe couplerpointD for the input linkangularposition θ′1i = θ1i+θ10.
Itmeans that passing from the point i to the point i+1 is accompanied by rotation of the input
link by the angle θ1i+1−θ1i. The angle γ is determined in the phase of the computations and is
not known in the initial phase. This is why the angular position of the input link θ1 is measured
from the horizontal line. Let us assume that we have a set of arbitrary values of the design
variables defining a candidate for the optimummechanism. One has to evaluate how accurately
this mechanism approximates the given performance.

Fig. 1. (a) General geometric scheme of the four-bar linkage, (b) design variables (underlined) and
variables determined using design variables

The algorithm for determination of the remaining dimensions of themechanism is as follows.
Having certain design variables xO1, yO1, l1 and θ10, the positions of the jointA on the input

link for i=1, . . . ,n are computed

xAi =xO1+ l1cos(θ1i+θ10) yAi = yO1+ l1 sin(θ1i+θ10) (2.1)

Subsequently, the lengths are computed

|AiDi|=
√

(xAi−xi)
2+(yAi−yi)

2 (2.2)

as well as the trigonometric functions between the section AiDi and the horizontal line

cosαi =
xi−xAi
|AiDi|

sinαi =
yi−yAi
|AiDi|

(2.3)



46 J. Buśkiewicz

The optimum length of the arm |AD| minimises, in the sense of the least square method, the
sum of the squared deviations of the prescribed point D positions from the points generated
by dyad O1AD, defined by xO1, yO1, l1, θ10 and l5 for the prescribed angular positions of the
linkO1A

E1 =
n
∑

i=1

[(xAi+ l5cosαi−xi)
2+(yAi+ l5 sinαi−yi)

2] (2.4)

As l5 is not known, differentiating E1

∂E1

∂l5
=2

n
∑

i=1

[(xAi+ l5cosαi−xi)cosαi+(yAi+ l5 sinαi−yi)sinαi] = 0 (2.5)

one can obtain

l5 =
1

n

n
∑

i=1

[(xAi−xi)cosαi+(yAi−yi)sinαi] (2.6)

which equals the average value of |AiDi| for all i.
Then the desired design variables minimise the deviations of |AiDi| from l5. The next set of

design variables are the angle between the coupler arms AD and AB−θ4 and the coupler AB
length – l2. Having these values, one can compute the position of the jointB connecting the co-
upler and the output link.Using the complex number notation,wehave the subsequentpositions
of the link AB corresponding with the positions of the linkAD

AiBi =(xBi−xAi)+ j(yBi−yAi)= l2
AiDi

|AiDi|
ej(−θ4)

= l2(cosαi+jsinαi)[cos(−θ4)+jsin(−θ4)]

(2.7)

Hence, the coordinates

xBi =xAi+ l2cos(αi−θ4) yBi = yAi+ l2 sin(αi−θ4) (2.8)

are obtained by clockwise rotating the unit vector AiDi by the angle θ4 andmultiplicating the
result by l2. It is expected that the position of the jointB can be approximated by a circular arc.
The radius of the arc that best fits into these points equals the length of the output linkO2B,
and the centre of the arc becomes the pivot of this link. The parameters of this circular arc
may be computed using the method described in (Buśkiewicz, 2015). Let l3i denote the radius
of the circle with the centre at (xO2i,yO2i) passing through three points:Bi,Bi′ andBi′′, where:
i=1, . . . ,n, i′ = i+2, i′′ = i′+2, and the number point greater than n is diminished by n. The
centre coordinates of the i-th circle are

xO2i =
b1a22−b2a12
a11a22−a12a21

yO2i =
b2a11− b1a21
a11a22−a12a21

(2.9)

where

a1 =2(xBi−xBi′) a12 =2(yBi−yBi′)

a21 =2(xBi−xBi′′) a22 =2(yBi−yBi′′)

b1 =x
2
Bi+y

2
Bi−x

2
Bi′ −y

2
Bi′ b2 =x

2
Bi+y

2
Bi−x

2
Bi′′ −y

2
Bi′′

Then, the coordinates of the ground pin are

xO2 =
1

n

n
∑

i=1

xO2i yO2 =
1

n

n
∑

i=1

yO2i (2.10)



Reduced number of design parameters in optimum path synthesis... 47

Similarly, the average radius is taken as the length of the output link

l3 =
1

n

n
∑

i=1

l3i =
1

n

n
∑

i=1

|O2Bi| (2.11)

The final step is to compute the length of the immovable link, and cosine and sine of the angle γ

l4 = |O1O2| cosγ =
xO2−xO1
l4

sinγ =
yO2−yO1
l4

(2.12)

An alternative method for measuring the similarity of the set of points to the circular curve
has recently been presented by Kafash and Nahvi (2015). The Circular Proximity Function is
defined as

CPF =
n
∑

i=1

[(xBi−xO2)
2+(yBi−yO2)

2−R2] (2.13)

where the average value of the squared distances between the points B and the centre of the
hypothetical circle is as follows:

R=
1

n

n
∑

i=1

[(xBi−xO2)
2+(yBi−yO2)

2] (2.14)

The optimum circle centre coordinates minimises theCPF.

2.2. Objective function and Euclidian error

The four-bar linkage realises exactly the desired motion when the sum of deviations of the
arm length l5 from real distances |AiDi| and deviations of the length of the output link from
computed real distances |O2Bi| (themeasure of the deviation of the path of the jointB from an
ideal circular arc)

δ=
max |AiDi|−min |AiDi|

l5
+
max |O2Bi|−min |O2Bi|

l3
(2.15)

equals 0. Then δ is the minimised objective function and, simultaneously, it is the indirect
measure of the inaccuracy of the method. This error bears no information about the Euclidian
deviationbetweengivenpoints andpoints tracedby themechanismobtained.For thedetermined
dimensions of the four-bar linkage, the coordinates of the coupler points for the given angular
positions of the input link are computed. For this purpose, the angular position of the coupler
and the output linkO2B

θ21,2 =2arctan
−KE ±

√

K2E −4KDKF

2KD
θ31,2 =2arctan

−KB ±
√

K2B −4KAKC

2KA
(2.16)

are determined, where

KA =K3−K1+(1−K2)cos(θ1−γ) KB =−2sin(θ1−γ)

KC =K1+K3− (1+K2)cos(θ1−γ) KD =K5−K1+(1+K4)cos(θ1−γ)

KE =−2sin(θ1−γ) KF =K1+K5+(K4−1)cos(θ1−γ) K1 =
l4

l1

K2 =
l4

l3
K3 =

l21− l
2
2 + l

2
3+ l

2
4

2l1l3
K4 =

l4

l2
K5 =

−l21− l
2
2 + l

2
3 − l

2
4

2l1l2



48 J. Buśkiewicz

Each solution corresponds to the two configurations of the four-bar linkage in which the
mechanism can be assembled for the pre-set dimensions and the angular position of link O1A.
At this stage, the mechanism whose configuration has been generated is not known, therefore
the errors for both signs at the square roots in Eqs. (2.16) are computed and themore accurate
solution is taken into account.
Then the position of the pointD at the instant when the angular position of the input link

measured from the horizontal line equals to θ1i+θ10 is determined from the equations

xDi =xO1+ l1cos(θ1i+θ10)+ l5cos[θ2(θ1i+θ10−γ)+θ4+γ]

yDi = yO1+ l1 sin(θ1i+θ10)+ l5 sin[θ2(θ1i+θ10−γ)+θ4+γ]
(2.17)

The angular position of the crankwith respect to the axisO1O2 equals θ1i+θ10−γ. Therefore,
this angle is taken to compute θ2 fromEq. (2.16). In the references, the absolute Euclidian error
between the points given and generated is defined as follows

E =
n
∑

i=1

[(xDi−xi)
2+(yDi−yi)

2] (2.18)

2.3. Optimisation technique

The evolutionary algorithm (EA) (Goldberg, 1989) minimises the objective function (Eq.
(2.15)). The technique is described in depth in (Buśkiewicz, 2010, 2015). Preliminary numerical
simulations were performed to establish the parameters controlling the EA:
• The size of the population Nmax =80 (case I),Nmax =50 (cases II and III).

• The number of individuals to be crossed over in each generation lcr =30.

• The number of randomly generated individuals introduced to each population in the place
of the worst fitted ones lrnd =25.

• A feature of the new individual (N) is inherited from the parent (I) with the probability
pb =0.75 (from parent (II) with the probability 0.25).

• The initial value of the mutation coefficient λ=0.05 (the value is gradually decreased to
0.005), the initial disturbance coefficient µ=0.001 is gradually decreased to 0.00001).

• The minimum value of the error function (objective function) Emin depends on the case
being solved.

• The maximum number of iterations executed (generations) in the case of not achieving
the prescribedminimum value of the error functionLg =10000.

3. Numerical solutions

To prove the effectiveness of themethod, three examples are solved and discussed. Two of them
are taken from references and comparative analysis is carried out.
For all cases presented the points are gathered in sets in the forms as follows: tab =

{(xi,yi,θ1i), i=1, . . . ,n}. The coordinates are non-dimensional, the angles are in radians.

Case 1

The first case was presented by Acharyya and Mandal (2009). The authors of this paper
considered the synthesis with timing defined by data given in tab1

tab1 =
{

(0,0,0.5236),(1.9098,5.878,1.0472),(6.9098,9.511,1.5708),(13.09,9.511,2.0944),

(18.09,5.878,2.618),(20,0,3.1416)
}



Reduced number of design parameters in optimum path synthesis... 49

The input geometric constraints were:

• the range of the coordinates of the crank pivot: −50¬xO1, yO1 ¬ 50,

• the lengths of the crank: 5¬ l1 ¬ 50, and of the coupler: 5¬ l2 ¬ 50.

The computational time was 00:01:14 with 1000 iterations (generations in EA – 80 000 evalu-
ations of the objective function) performed. The evaluations of chosen parameters are shown in
Fig. 2a. The progress of the absolute Euclidian error is presented in Fig. 2b.

Fig. 2. (a) Evaluation of the crank length and crank pivot coordinates. (b) Evaluation of
the Euclidian distance

It is visible from Figs. 2a,b that the areas surrounding the found values of parame-
ters are localised in the first 200 iterations. The best solution of the problem with in-
stalling parameters for the first position (Fig. 3) is: l1 = 11.1149, l2 = 42.6226, θ4 =
−0.1368rad, γ = 6.2819rad, θ10 = 3.29rad, l3 = 11.9381, l5 = 52.4112, l4 = 43.30492,
O1(−43.3598,−0.0067), O2(−0.0549,−0.0609), D(−0.143478,0.271792), A(−52.058,−6.92637),
B(−11.032,4.63172) and θ1 = θ10+θ11(= tab11,3)= 3.8136rad (the angle is measured form the
horizontal line).

Fig. 3. Case 1: Synthesised four-bar linkage with paths given (continuous line) and generated

The method error δ = 2.767, and the absolute Euclidian error E = 2.10037. Three EAs
namelyGeneticAlgorithm(GA),Particle SwarmOptimization (PSO)andDifferential Evolution
(DE) were applied separately in (Acharyya and Mandal, 2009). The best solution obtained
using DE is presented in Table 1. The error reported by the authors is 2.349649, whereas the
error computed for the corresponding mechanism dimensions is 5.52074 – this difference was
also observed by Ebrahimi and Payvandy (2015). In all addressed methods 100000 or 200000
evaluations of the objective function were performed, whereas in the presented paper there
were 80000. The four-bar linkages generating this path were also determined in (Ebrahimi and
Payvandy, 2015) using: imperialist competitive algorithm(ICA)andparallel simulated annealing



50 J. Buśkiewicz

(SA). The best solution is also presented in Table 1. Ebrahimi and Payvandy also solved the
problem of the path synthesis withworkspace limits, which is not here considered. The ICAand
EAdiffer from each other in terms of the structure and, therefore, it is difficult to compare their
computational costs of the evaluations of the objective functions.

Table 1.Path 1: comparison of results

Acharyya and Ebrahimi and
Mandal (2009) Payvandy (2015) Proposed

DE ICA

l1 5 5 11.1149

l2 5.905345 7.08248 42.6226

l3 50 48.05733 11.9381

l4 50 50 43.30492

l5 18.819312 21.3969 52.4112

θ4 0 0.6956 −0.1368

xO1 14.373772 11.88034 −43.3598

y01 −12.444295 −16.08766 −0.0067

γ 0.463633 6.2831853 6.2819

θ10 – – 3.29

E 5.52074 2.5998 2.10037

Case 2

Kunjur and Krishnamurty (1997) presented the synthesis with timing of the path given by
18 points:

tab2 =
{

(0.5,1.1,0.3491),(0.4,1.1,0.6981),(0.3,1.1,1.0472),(0.2,1.0,1.3963),

(0.1,0.9,1.7453),(0.05,0.75,2.0944),(0.02,0.6,2.4435),(0,0.5,2.7925),(0,0.4,3.1416),

(0.03,0.3,3.4907),(0.1,0.25,3.8397),(0.15,0.2,4.1888),(0.2,0.3,4.5379),

(0.3,0.4,4.8869),(0.4,0.5,5.2360),(0.5,0.7,5.5851),(0.6,0.9,5.9341),(0.6,1,6.2832)
}

The range for the coordinates of the crank pivot is: −50 ¬ xO1, yO1 ¬ 50. The lengths
of the crank and coupler l1, l2 cannot exceed 50. The computational time is 00:00:36. Within
this time, 200 iterations have been performed – the number of evaluations of the objective
function is 10000. The optimum solution is expressed as follows: l1 = 0.4102, l2 = 1.2166,
θ4 =0.8342853rad, γ =0.1110rad, l3 =1.1230, l5 =0.7859, l4 =1.5395, θ10 =0.8721rad.
These parameters are completed with installing parameters for the first position

(Fig. 4): O1(0.363,−0.0874), O2(1.8931,0.0831), D(0.516114,1.08379), A(0.503501,0.297987),
B(1.41786,1.10058), θ1 = θ10+θ11(= tab21,3)= 1.2212rad.
The absolute Euclidian errorE =0.0185453, and the method error δ=0.1821. Taking into

account the solutions obtained bymeans of the EAs (Kunjur andKrishnamurty, 1997; Cabrera
et al. 2011; Nadal et al. 2015), the method presented in this paper gives the best solution. The
number of evaluations of the objective functions in these methods is 5000. In order to provide
higher repeatability, when 34 in 100 algorithm executions give the solution with errorE < 0.02,
twice as many evaluations of the objective function in the presentedmethod are taken. In order
to obtainmore accurate solutions, Lin andHsiao (2017) employed the Cuckoo Search (CS) and
teaching-learning-based optimization (TLBO) algorithms (200001 evaluations of the objective
function – twice as many evaluations resulted only in the smaller values of the mean error



Reduced number of design parameters in optimum path synthesis... 51

Fig. 4. Case 2: Synthesised four-bar linkage with paths given (continuous line) and generated

and standard deviation). TheModified Krill Herd algorithm applied by Bulatović et al. (2016)
produced the same degree of accuracy. The number of the objective function evaluations in the
MKHalgorithm is the product of the number of iterations and the number of repetitions – both
parameters are equal to 50. Comparedwith theEAs, the computational cost of one evaluation of
the objective function seems to be higher.This issue, however, was not discussed in (Bulatović et
al., 2016) since theMKH, as was the case with the ICA, differred from the EA in performance.
All the results are summarised in Table 2.

Table 2.Path 2: comparison of the results

Kunjur and Cabrera Nadal Bulatović Lin and
Krishnamurty et al. et al. et al. Hsiao Proposed
(1997) (2011) (2015) (2016) (2017)

l1 0.274853 0.297057 0.239834 0.4218 0.4238752 0.4102

l2 1.180253 3.913095 0.941660 0.87821 0.9142488 1.2166

l3 2.138209 0.849372 3.164512 0.58013 0.5989170 1.1230

l4 1.879660 4.453772 2.462696 1.00429 1.0539434 1.5395

l5 0.91561 2.65198 0.4834 0.5234 0.54481 0.7859

θ4 3.5680854 2.464734 1.5638233 0.8147729 0.822721 0.8342853

xO1 1.132062 −1.309243 0.569026 0.26886 0.2676546 0.363

yO1 0.663433 2.806964 0.350557 0.17715 0.1546514 −0.0874

γ 4.354224 2.7387359 4.788536 0.29294 0.2848225 0.1110

θ10 2.558625 4.853543 2.47266 0.88595 0.8915568 0.8721

E 0.043 0.0196 0.113595 0.00911 0.0090289 0.0185453

Case 3

Theset tab3presents the straight linediscretisedby6pointswith thecrankpositionsassigned
to

tab3 =
{

(5.0000,1.000,0.3491),(4.0000,1.000,0.6981),(3.0000,1.000,1.0472),

(2.0000,1.000,1.3963),(1.0000,1.000,1.7453),(0.0000,1.000,2.0944)
}

The input geometric constraints define the range for the crankpivot coordinates:−25¬xO1,
yO1 ¬ 25, and the maximum permissible lengths of the crank and coupler equal to 15. The



52 J. Buśkiewicz

computational time is 00:00:47 with 1000 iterations performed for Nmax = 50. The geometric
parameters of the optimummechanism are as follows: l1 =1.7930, l2 =6.6176, θ4 =0.4732rad,
γ=2.6679rad, θ10 =4.5633rad, l3 =6.6175, l5 =11.7914, l4 =3.475.

The absoluteEuclidian errorE =0.0000173, andmethod error δ=0.0038. Taking into acco-
unt the absolute error, one can state that the solution is very accurate. The installing parameters
for thefirstposition (Fig. 5) are:O1(5.5219,−9.0021),O2(2.4294,−7.4167),D(4.99683,0.99906),
A(5.87813,−10.7594), B(8.44532,−4.65988). θ1 = θ10+θ11(= tab31,3)= 4.9124rad.

Fig. 5. Case 3: Synthesised four-bar linkage with paths given (continuous line) and generated

4. Conclusion

Asmentioned in the introduction, twomain branches have been developed recently in the field
of mechanism synthesis. The first one deals with the enhancement of probabilistic determini-
stic algorithms. The other focuses on the minimisation of the number of design variables. It
is difficult to find papers on the reduction of the number of design parameters in the case of
open/closed-path synthesis with timing in the one-phase path synthesis. The presentedmethod
tries to overcome these difficulties and this is the distinctive feature of the method. Free paths
are considered to investigate the effectiveness of themethod. Although the algorithm employed
in this paper utilizes the classical EA, the results of the synthesis are comparable to the best of
those presented in the literature. The analysis performed is encouraging for further studies in
developing themethods decreasing the number of design parameters, whichmay be an alterna-
tive to more complex, frequently hybrid, new optimisation algorithms. The method, although
presented on the example of a four-bar linkage, can be applied in synthesis of other four-link
planar mechanisms: crank-slider mechanism andmechanismwith slotted links using techniques
described in (Buśkiewicz, 2015).

One can formulate some general observations on the way the results obtained by many
authors are compared. First of all, it seems reasonable to take into account somemeasure which
reflects the ratio of sizes of the mechanism and the synthesised path. Frequently, the working
space occupied by the optimal mechanismwhen tracing the path is much larger than the space
enclosed by the path alone. Such a solution is frequently useless for practical implementation.



Reduced number of design parameters in optimum path synthesis... 53

Moreover, in the case of an open-path synthesis, the general tendency is that a better solution
is characterised by a smaller ratio of the length of the synthesised part to the total path length.
This means that the path is generated for a very small angle of rotation of the input link, and
the greater part of themechanismmotion is deadmotion. Therefore, the comparison of different
methods based on the values of the objective functions should only be concluded very carefully.
Such an approach to mechanism synthesis, which consists of reducing the number of de-

sign parameters, imposes distinction between the problems with and without timing. In other
methods, when the problem without timing is considered, the orientations of the input link
corresponding with the coupler point synthesis are treated as additional design parameters.

Acknowledgment

This research was funded by Polish Ministry of Science and Higher Education (the grant DS

02/21/DSPB/3493).

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Manuscript received May 9, 2017; accepted for print July 17, 2017