1. Introduction
Symmetrical motion mechanisms in which the forward and return strokes are the same movement have been applied in many mechanical systems. Four-bar mechanisms with symmetrical and asymmetrical motions are commonly used in mechanical devices such as sewing machines, round balers, and suspension systems of automobiles [
1,
2]. Achieving higher precision design of four-bar mechanisms, whose symmetrical motions satisfy the input path of a point in the coupling link, is still a challenge for the kinematic dimensional calculation. It results from the highly nonlinear objective function with many constraints.
Regarding methods for designing four-bar mechanisms, Zhang [
3] proposed the graphical method, and Freudenstein [
1] has been used in analysis to compute the kinematic dimensions of linkages. However, this method resulted in low precision and waste time [
4]. In recent decades, random search algorithms such as differential evolution [
2,
5,
6,
7,
8], genetic algorithm [
4,
9], simulated annealing algorithm [
10], and particle swarm optimization [
2,
11,
12] have been used to solve kinematic dimensions of the mechanisms to increase accuracy in mechanism design.
The synthesis of four-bar mechanisms has been mentioned in recent years due to their wide applications in mechanical systems. Fernandez et al. [
13] proposed the determination of kinematic dimension in order to minimize the objective function based on the dimensional constraint equation of mechanisms. The design of a four-bar mechanism used for the shadow robot is presented in [
14]. Ramon et al. [
15] applied the combination of difference evolution and local search algorithms for synthesis planer mechanisms, in which the four-bar mechanism is one of the examples in this work. These works, however, have limited the application cases of four-bar mechanisms. The teaching-learning based optimization algorithm has been used to determine the kinematic dimension of four-bar mechanisms [
16,
17]. In these researches, the signed timing has not been considered. Varedi-Koulaei [
18] synthesized four-bar mechanisms using graphical and analytical methods.
With random search algorithms, the Jaya algorithm is also a newly proposed approach [
19]. This method has been applied for solving numerous optimization problems [
20,
21]. In order to obtain an optimal solution by using the Jaya algorithm, a high computational cost is essentially required. Thus, to improve its performance, the Jaya algorithm has been combined with other algorithms [
5,
22,
23]. Up until now, the application of the Jaya algorithm in designing four-bar mechanisms is still limited [
24].
One of the most well-known random search algorithms is DE algorithm which can be found in [
25]. DE algorithm has been used to find the optimal solution for a lot of problems [
26,
27,
28]. It is similar to the other random search, as an optimal solution is found by using the DE algorithm, which also requires a considerable computation cost. For this reason, several modifications of DE were proposed [
29,
30,
31,
32,
33,
34,
35]. The combination of DE and other algorithms such as GA [
36,
37], PSO [
38,
39], and fireworks algorithm (FA) [
40,
41] was proposed. Furthermore, some additional modifications of DE were also proposed for path synthesis of four-bar mechanisms such as Cabrera [
6], Ortiz [
7], Lin [
8]. Concretely, Cabrera [
6] proposed a modified crossover to change the values of genes in the mutant and target vectors to achieve better values of objective functions for the next generation. Based on the work of Cabrera, Ortiz [
7] suggested a tuning technique for the control parameters of F, CP, MP, and range to avoid the multiple executions of the algorithm until their proper values are found. Furthermore, Lin [
7] proposed a new combined mutant operator of DE. In the combined mutation, DE/best/1, DE/current to best/1, and DE/rand/1 are used in the first
ranking parents, middle-ranking parents (ranked between
), and other inferior parents, respectively.
From the discussions mentioned above, the investigation of a hybrid algorithm between DE and Jaya for improving the optimal dimensional synthesis of the four-bar mechanism has not yet been reported. Thus, this study proposes a novel hybrid-combined differential evolution and Jaya algorithm (called HCDJ) to fill the above research gap. The current hybrid algorithm combines DE and Jaya to improve the accuracy of optimal solutions. By combining the mutation operator DE with a Jaya operator, the global exploration ability of HCDJ is significantly enhanced, and thus the solution accuracy is also improved. Concretely, in the modified mutation stage, three groups of mutations are used in the first
, middle, and remaining populations, respectively. DE/best/1 and DE/best/2, DE/current to best/1 and Jaya operator, and DE/rand/1, and DE/rand/2 belong to the first, second, and third groups, respectively. Additionally, the refined initialization stage is applied to choose individuals that are satisfied with Grashof’s condition and consequential constraints. Finally, the elitist selection technique is used in the selection phase to determine the best solutions for the next generation. To prove the effectiveness and robustness of the HCDJ algorithm in terms of accuracy, five numerical examples for the dimensional synthesis of four-bar mechanisms with symmetrical motions are performed, and outcomes obtained by the proposed methodology are compared with those of some available algorithms in the literature. The rest of this article is arranged as follows.
Section 2 presents the optimization problem of four-bar mechanisms. In
Section 3, a brief review of the classical DE and Jaya is presented firstly, and then a perspective scheme of the proposed HCDJ is discussed. In
Section 4, five commonly examined numerical examples for the dimensional synthesis of the four-bar mechanisms are performed to validate the effectiveness and robustness of HCDJ. Then, results obtained in five cases are discussed in
Section 5. Finally, some conclusions are provided in
Section 6.
2. Dimensional Synthesis of the Four-Bar Mechanisms
In this study, we focus on the dimensional synthesis of the four-bar mechanisms and the optimization problem, which is used to determine the kinematic dimensions of linkages and positions of pin-joints denoted by
and
(see
Figure 1) by giving the input path of the coupler point C in link 3.
and
are set as target points indicated by the input point and the designed point, respectively, i.e.,
and
. Therefore, the objective function can be written as follows [
4]:
where
N is the number of points in the path of the coupler, and
is preventative for the design variables’ vector characterized by kinematic dimensions and positions of the linkages and can be expressed as
where
,
,
, and
denote the kinematic dimensions of links 1, 2, 3, and 4, respectively;
a,
b,
,
, and
are shown in
Figure 1;
,
,
,
are preventative for the angle positions of link 2 at input positions of coupler C, respectively.
In Equation (
1), the position of the coupler C in the reference frame
can be computed based on the closure loop as found in [
5]. It can be calculated as
The angle positions of links 3, i.e.,
can be computed as
where
and
J are calculated as the following equations
in which,
The position of the point C in the global coordinate
, as shown in
Figure 1, can be simply computed as
It should be noted that the design variable is in the range of with its upper bound of and lower bound of .
In addition to the constraint on the design variable, there are also two more constraints, namely the Grashof’s condition and the sequence of input angles. These constraints can be displayed as follows.
The Grashof’s condition allows the mechanism to have an entire ration link that is connected with the frame (link 1); this condition is denoted by
. If Grashof’s condition is true,
, in contrast
. Mathematically, this constraint could be given as
in which,
;
s and
l denote the shortest and longest lengths, respectively;
p and
q are other lengths.
The constraint for the sequence of input angles is that the angle-position values of link 2 are in sequence; this condition is denoted by
. If this condition of input angles is true,
, in contrast
. Mathematically, this constraint is expressed as
in which,
is equal to
;
is the value of
in its
position; Z is the number of input angles and
is the remainder of the quotient of
.
These conditions need to be put into the objective function; thus, the objective function in Equation (
1) can be rewritten as
where
is the penalty constant.
4. Numerical Examples
This section presents several examples in design kinematic dimensions of four-bar mechanisms with symmetrical motions, which have been investigated in [
2,
4,
7,
8,
44] by using GA, DE, and PSO, respectively. Thus, results obtained by HCDJ are compared with those of GA, DE, and PSO in such algorithms. The four-bar mechanisms with symmetrical motions must satisfy the Grashof’s condition, as shown in
Section 2 in order to make the symmetrical motion of the mechanisms. In Cases 1 and 5, the problem is the path generation without prescribed timing. In contrast, in Cases 2, 3, and 4, the problems are known by the input of the coupler point. In this work, Jaya, DE, and HCDJ algorithms are applied for finding the optimum solutions, as shown in
Figure 1, and the optimal results of these algorithms are compared with the other algorithms that have been used in the previous studies. The population sizes (NP) in Cases 1 and 5 are equal to 100, and the population sizes are equal to 50 in both Cases 2, 3, and 4. The maximum iterations in Cases 1, 4, and 5 are equal to 1000 and are equal to 100 in Cases 2 and 3. By investigations, the values of
in Equation (
10) are chosen as
in Cases 1 and 5,
in Cases 2 and 3 and
in Case 4. Since Jaya, DE, and HCDJ are random-optimization algorithms, each different run provides different optimal solutions. To tackle this problem, DE, Jaya, and HCDJ used 50 independent times. Subsequently, the minimal errors of Jaya, DE, and HCDJ are provided and compared with other algorithms in the literature. In addition, the standard deviation and mean values of the minimal errors in 50 runs of Jaya, DE, and HCDJ are also reported. For validations of the obtained solutions, the synthesized mechanisms are illustrated in GeoGebra classic 5.
4.1. Case 1
In this case, the input data of the coupler point are presented in [
2,
4,
7]. The design variables’ vector, input path and boundaries for design parameters are respectively provided in Equation (
18) to Equation (
20).
The design parameters are presented by a vector as follows:
The input data of the coupler pointer is given as follows:
The boundary conditions of the design parameters can be expressed as follows:
4.1.1. Effects of the Parameters of and in HCDJ on the Optimal Solutions
This section investigates the effects of the parameters of
and
in HCDJ on the optimal solutions. Firstly, the mutant factor (
) used in HCDJ is investigated by considering the following five cases: 0.4, 0.5, 0.6, 0.7, and the range
. In this examination, the values of
,
and
are equal to 0.3,
and
, respectively. Obtained results are shown in
Table 1. It can be seen that when F is equal to 0.7, the HCDJ algorithm yields the best optimal solutions compared to other cases of
. Next, the seven different cases of
are investigated in which the values of
,
and
are equal to
,
and
, respectively. Obtained results are shown in
Table 2. It can be seen that when
is equal to 0.2, the HCDJ algorithm yields the best optimal solutions compared to other cases of
. From the obtained results in
Table 1 and
Table 2, the suitable values of
and
are set to 0.7 and 0.2, respectively, and are recommended for the HCDJ.
4.1.2. Effects of the Parameters of and in HCDJ on the Optimal Solutions
Next, the effects of the parameters of
and
in HCDJ on the optimal solutions are studied. In this examination, the values of
F and
are set to 0.7, and 0.2, respectively. The values of
and
used in HCDJ are investigated by considering 16 different cases. The obtained results are shown in
Table 3 and
Table 4. It can be seen that when
and
are equal to
and
, the HCDJ algorithm yields best optimal solutions compared to other cases of
and
. From the obtained results in
Table 3 and
Table 4, the suitable values of
and
are chosen as
and
, respectively, in HCDJ operators and are recommended for the HCDJ.
4.1.3. Comparison Performances of HCDJ with Other Available Methods in the Literature
Table 5 provides the optimal results obtained by HCDJ, and other approaches. It can be seen that HCDJ gives the best optimal solutions in all algorithms,
for HCDJ. Additionally,
Figure 5 shows the best path traced by the coupler in Case 1 by using HCDJ and there is a very good traced path of the point C. Additionally, the convergence rates of the HCDJ, DE, and Jaya are shown in
Figure 6. The HCDJ reaches the optimal solutions faster than the DE and the Jaya.
4.2. Case 2
In this case, path generation with a prescribed timing of four-bar mechanism is performed, which is also investigated in [
2], the inputs for the optimization problem are six coupler points and these points belong to a semi-circular arc. Thus, the design variables’ vector is defined as follows:
The six input coupler points are chosen as follows:
The boundary conditions of the design parameters can be expressed as follows:
Table 6 shows the optimal solutions obtained by HCDJ and other algorithms. It can see that HCDJ gives the best optimal solutions in all algorithms,
for HCDJ. Additionally,
Figure 7 shows the best path traced by the coupler in Case 2 with using HCDJ and there is a very good traced path of the coupler (point C in link 3). Furthermore, the convergence speed of the HCDJ, DE, and Jaya are also illustrated in
Figure 8. The HCDJ algorithm reaches the optimal solutions much faster than the DE and the Jaya.
4.3. Case 3
For the third case, the coupler point traces a close loop path generation in which 18 coupler points are included and prescribed timing is required. This problem was first presented in [
44]. Thus, the vector of design variable is defined as follows:
The 18 desired coupler points are chosen as follows:
The boundary conditions of the design parameters can be expressed as follows:
Table 7 shows the optimal solutions obtained by HCDJ and other algorithms. It can be seen that HCDJ gives the best optimal solutions in all algorithms,
for HCDJ. Additionally,
Figure 9 shows the best path traced by the coupler in Case 3 using HCDJ, and there is a very good traced path of the coupler (point C in link 3). Furthermore, the convergence speed of the HCDJ, DE, and Jaya are also illustrated in
Figure 10. The HCDJ reaches the optimal solutions much faster than the DE and the Jaya.
4.4. Case 4
In the Case 4, a path generation problem with prescribed timing is performed. Six coupler points in a vertical straight line are used as inputs. Then, the vector of design variables is given as follows
The 18 desired coupler points are selected as follows:
Lower and upper boundary for design variables is taken in an interval as the following equation which is given as follows:
Table 8 shows the optimal solutions obtained by HCDJ and other algorithms. It can be seen that HCDJ gives the best optimal solutions in all algorithms,
for HCDJ. Additionally,
Figure 11 shows the best path traced by the coupler in Case 4 using HCDJ and there is a very good traced path of the coupler (point C in link 3). Furthermore, the convergence speed of HCDJ, DE, and Jaya is also illustrated in
Figure 12. The HCDJ algorithm reaches the optimal solutions much faster than the DE and Jaya algorithms.
4.5. Case 5
In the fifth case, an elliptical path generation problem without prescribed timing is investigated. The path consists of 10 target points. The elliptical path with a major axis of 20 units and a minor one of 16 units is considered. The center’s coordinate is at (10, 10) and the major axis is kept horizontal.
The vector of design variables is given as follows:
The desired coupler points are chosen as follows:
The boundary conditions of the design parameters can be expressed as follows:
Table 9 shows the optimal solutions obtained by HCDJ and other algorithms. It can be seen that HCDJ and DE in [
8] give the best optimal solutions in all algorithms,
for HCDJ and
for DE [
8]. Additionally,
Figure 13 shows the best path traced of the coupler in Case 5 using HCDJ, and there is a very good traced path of the coupler (point C in link 3). Furthermore, the convergence speed of the HCDJ, DE, and Jaya are also illustrated in
Figure 14. The HCDJ algorithm reaches the optimal solutions much faster than the DE and the Jaya algorithms.
6. Conclusions
This study proposed a newly hybrid-combined algorithm, called HCDJ, as a combination of the classical DE and Jaya algorithms for the optimally dimensional design of four-bar mechanisms with symmetrical motions. The combined algorithm has a good global search to improve the optimal solution quality by using modified initialization, a hybrid-combined mutation between the classical DE and Jaya algorithm, and the elitist selection. The modified initialization generates initial individuals that are satisfied with Grashof’s condition and consequential constraints. In the hybrid-combined mutation, three different groups of mutations are combined. DE/best/1 and DE/best/2, DE/current to best/1 and Jaya operator, and DE/rand/1, and DE/rand/2 belong to the first, second, and third groups, respectively. In the second group, DE/current to best/1 is hybrid with the Jaya operator. Additionally, in the selection stage, the best candidates are produced for the next generation by using the elitist selection technique. Five numerical examples, including two path generations with prescribed timing and three without prescribed timing, are performed to find the optimal designs of the four-bar mechanisms. The obtained solutions of HCDJ are compared with those of the original DE, Jaya, and other algorithms existing in the literature. The optimal results using the HCDJ algorithm have indicated that it can achieve better performances in terms of the solution accuracy than the original DE and Jaya, even in many other algorithms. Accordingly, the proposed HCDJ algorithm is expected to apply not only to symmetrical motion mechanisms, but also asymmetrical motions of mechanisms and various engineering problems.