Computational Intelligence and Neuroscience
Volume 2018 (2018), Article ID 5462563, 16 pages
https://doi.org/10.1155/2018/5462563

Research Article

Optimal Synthesis of Four-Bar Linkage Path Generation through Evolutionary Computation with a Novel Constraint Handling Technique

Suwin Sleesongsom¹ and Sujin Bureerat²

¹Department of Aeronautical Engineering, International Academy of Aviation Industry, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
²Sustainable and Infrastructure Research and Development Center, Department of Mechanical Engineering, Faculty of Engineering, Khon Kaen University, Khon Kaen 40002, Thailand

Correspondence should be addressed to Suwin Sleesongsom; suwin.se@kmitl.ac.th

Received 26 March 2018; Revised 20 May 2018; Accepted 19 September 2018; Published 1 November 2018

Academic Editor: Saeid Sanei

Copyright © 2018 Suwin Sleesongsom and Sujin Bureerat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents a novel constraint handling technique for optimum path generation of four-bar linkages using evolutionary algorithms (EAs). Usually, the design problem is assigned to minimize the error between desired and obtained coupler curves with penalty constraints. It is found that the currently used constraint handling technique is rather inefficient. In this work, we propose a new technique, termed a path repairing technique, to deal with the constraints for both input crank rotation and Grashof criterion. Three traditional path generation test problems are used to test the proposed technique. Metaheuristic algorithms, namely, artificial bee colony optimization (ABC), adaptive differential evolution with optional external archive (JADE), population-based incremental learning (PBIL), teaching-learning-based optimization (TLBO), real-code ant colony optimization (ACOR), a grey wolf optimizer (GWO), and a sine cosine algorithm (SCA), are applied for finding the optimum solutions. The results show that new technique is a superior constraint handling technique while TLBO is the best method for synthesizing four-bar linkages.

1. Introduction

Since the last decade, many researchers have tried to solve the optimization for path generation of four-bar linkages using metaheuristic (MH) algorithms. The objective of path generation problem is to find dimensions of a mechanism, which minimize the target path and the actual path of a point on the coupler link. Path synthesis is one type of kinematic syntheses of four-bar mechanisms [1–13] in which such syntheses are basically classified into two groups. The first category is called dimensional synthesis [1, 2, 4, 5, 7–10]. This synthesis type aimed to find significant link lengths to achieve desirable function, path, and motion generation. The second synthesis is called type synthesis [6, 11] where a designer initially specifies a predefined motion transmission and is supposed not initially to know the mechanism type. This method is analogous to topology design in structural optimization. Having finished synthesizing, a certain mechanism type is received. Position analysis of the four-bar mechanism can be categorized into two groups. The first one is a vector loop or loop closure equation, which is the most traditional method in kinematic analysis, and it is one of the most popular analyses for path synthesis [1, 2, 4, 5, 7–15]. This equation can be solved by using Freudenstein equation. The second analysis technique is a straight forward and a simple method for position analysis involving the use of trigonometric laws for triangles, e.g., the law of cosine [3, 16, 17], whereas the six-bar linkage for steering mechanism also uses the same technique [18]. This work proposes a new computing technique for four-bar linkage position analysis by employing the concept of drawing an arbitrary rectangle using two circles.

The mechanism synthesis can be converted into optimization problem and be solved by using optimizers, where both nongradient- and gradient-based algorithms have been solved this problem. Recently, a nongradient-based optimizer, e.g., evolutionary algorithms (EAs) or metaheuristics (MHs), is a more popular selection in solving such optimization problems. It has been found that the advantages of using MHs are robustness, simplicity of use, and independence of function derivatives; however, they unavoidably lack convergence speed and consistency. At present, many algorithms in this group have been developed, which are expected to enhance in both convergence speed and consistency. Some of the most frequently used MHs for path synthesis are differential evolution (DE) [2, 3, 5, 8, 9, 11, 13], genetic algorithms (GA) [5, 13], particle swarm optimization (PSO) [5, 13], and an imperialist competitive algorithm (ICA) [13], etc. The use of gradient-based method, on the other hand, is somewhat questionable to deal with global optimization and nonsmooth constraints in the path synthesis. Nevertheless, if those aforementioned factors can be alleviated, the advantages of the gradient-based method are better convergence rate and consistency. In the literature, many researchers have combined MHs and a gradient-based optimizer for solving many kinds of real world problems, which is called a hybrid algorithm. Especially for part synthesis, the hybrid optimizers are introduced as the ant gradient [1], hybrid GA [4], and hybrid GA with sequential quadratic programming (SQP) [12]. The hybridization between two or more MHs was also studied such as hybrid GA-DE [7]. Furthermore, the path synthesis is extended to multiobjective optimization, which was solved by using a multiobjective genetic algorithm (MOGA) [10]. From the review literature, it was found that some MHs have been used for solving this task except the work by Sleesongsom and Bureerat [17]; therefore, one of the objectives of this paper is to present the comparative performance of a number of currently used MHs. Those algorithms include the artificial bee colony optimization (ABC), adaptive differential evolution with optional external archive (JADE), population-based incremental learning (PBIL), teaching-learning-based optimization (TLBO), the real-code ant colony optimization (ACOR), a grey wolf optimizer (GWO), a Jaya algorithm (Jaya), and a sine cosine algorithm (SCA).

The path generation is a mechanism synthesis to make a point on a coupler link move along the target path; thus, the objective function is the minimization of the sum square error between the target path and the actual path [5]. The design problem is a constrained optimization problem that comprises two constraints. The first constraint is set for the shortest link in the mechanism to be able to rotate with a complete revolution (crank) in either direction (clockwise or counter clockwise). The second constraint is assigned such that the four link lengths satisfy the Grashof criterion which results in a crank-rocker. From previous work, a simple exterior penalty function technique had used to deal with these constraints [1–5, 7, 8, 10–13]. The new technique proposed by [17] to neglect the first constraint from the optimization problem, which found new technique, provided the better result than the traditional exterior penalty technique. Additionally, the technique had improved in the result, but it increased in time consuming. From the present study can be concluded that the constraint handling technique is an inefficient technique, which needs an improvement [9, 13, 14]. Phukaokaew et al. [14] studied the number of unsuccessful runs from performing MHs for 30 runs, where the path synthesis optimization problems employ the penalty function (PF) technique. This means there is no guarantee that using this technique can promote the good results. The reason is that using the penalty technique leads to an overly narrow feasible region. As a consequence, MHs, which mostly have slow convergence rates, struggle to reach an optimum. Ebrahimi and Payvandy proposed the way to improve the constraint handling technique, which was still based on a penalty function, and they believe the proposed method can enhance the search performance [13].

This paper focuses on two aspects of investigation. Firstly, a new constraint handling technique for path synthesis of a four-bar linkage using MHs is proposed. The method is based on repairing illegitimate design solutions to become feasible solutions. The second investigation is to study the performance comparison of a number of established MHs for solving four-bar linkage path synthesis with the new constraint handling technique, where both convergence rate and consistency of the methods are measured.

The rest of this paper is organized as follows. Section 2 proposes an alternative position analysis of a four-bar linkage. The optimization problem and the constraint handling technique are detailed in Section 3. The numerical experiments are given in Section 4, while the design results are discussed in Section 5. The discussions and conclusions of the study are finally drawn in Sections 6 and 7, respectively.

2. Position Analysis of Four-Bar Linkages

The kinematic diagram of a four-bar linkage is shown in Figure 1. The four-bar linkage is the simplest and most commonly used linkage in many engineering applications. It is composed of a kinematic chain of four binary links connected with four revolute joints (denoted by capital letters) with one link being assigned as a frame. Applications for this mechanism are a window wiper, a door closing mechanism, rock crushers, etc. [9]. Based on the Gruebler equation for planar mechanisms, the mobility or degree-of-freedom of the mechanism is one; thus, it is a constrained mechanism fully operated by one actuator. The path generation for a four-bar linkage is a dimension-based design of the four-bar linkage lengths () and other parameters, which makes the trace point () on the coupler link follow the desire path ().

Figure 1: Four-bar linkage in a global coordinate system.

Let the coordinates of the joints O₂ and O₄ be () and (), respectively, so the coordinates of points B and O₄ can be computed as

where the angular positions and are positive counter-clockwise. The positions of points and are the intersection points of two circles as illustrated in Figure 1, which is calculated by a vector approach. First, let B and O₄ be the centres of the circles with radii and , respectively, and let be the distance between B and O₄. Then, generate , the unit vector from B to O₄, and , the unit vector perpendicular to . Given that is the vector from B to one of the intersection points, the angle between and denoted by is solved using the law of cosines

The intersection points can then be obtained as

The coupler curve is formed when the crank link rotates. From Figure 1, the coupler point coordinates () in the global coordinate can be expressed as

where is a unit vector in the direction from B to C and is a unit vector perpendicular to . The distances and can be computed using the given dimensions of , PB, and PC. Also, they can be set as design variables for optimal path synthesis. These vector forms of position analysis are used for four-bar linkage synthesis in this paper.

3. Optimization Problem and Constraint Handling

3.1. Optimization Problem

The path synthesis problem is converted to an optimization problem with an objective function expressed as the sum of square errors between the distances of and their corresponding . A set of the input angles () is assigned as design variables if the prescribed timing is not given. There are two major constraints. The first constraint is set in such a way that the generated mechanism type is a crank-rocker. This leads to two constraints: (i) sum of the shortest and longest links of linkage must be less than the sum of two remaining links and (ii) the shortest link must be an input link where one of its nearby links is set as a frame. The constraint (i) is denoted with Equation (7), while the constraint (ii) is denoted with Equation (6). In cases where the prescribed timing is not promoted, the second constraint is that the input values of must be in either ascending or descending order. The constraint (ii) is added to the design problem, which is denoted by , where . The optimization problem can then be written as

subject to

where , is the number of points on prescribed or target curve, and are input link angles. and are lower and upper bounds of a design vector , respectively.

A function evaluation is carried out for the optimization problem (5) by implementing the position analysis detailed in Section 2. To enable the position analysis, the constraints (6)–(8) must be first satisfied. In the previous studies, if the conditions are not met, the objective function will be modified by adding to it a great penalty value. However, in this work, the proposed path repairing algorithm (PRA) is assigned to repair all design solutions to always be feasible before performing position analysis of a four-bar linkage.

3.2. Constraint Handling

Normally, a traditional exterior penalty function technique has been used with the constrained optimization problem (5) [1–5, 7, 8, 10–13], but it was found to be inefficient [9, 13, 14, 17]. Using such a technique is questionable when prespecifying a penalty parameter. If the parameter is too small, the resulting optimum solution may be infeasible, but if it is too large, MH may not be able to find the optimum. This leads us to propose a new strategy to deal with the constraints called the path repairing technique. The technique can be separated into two parts as repair of the input link angle constraint (8) and the Grashof criterion constraint Equations (6) and (7).

3.2.1. Input Link Angle Constraint

For an optimization without prescribed timing, a repairing technique for this phase is shown in Algorithm 1 where the input variables are . Once it is found that the values of are not in either ascending or descending orders, the design variables in the part of will be repaired. Firstly, uniform random number for is generated. The lower bound is set to be 0.0001 in order to avoid repeated values of in cases that some random numbers become zeros. Those random values are then scaled in step 2 so that the sum of them does not exceed . The first angular position of an input link is , its original value. Then, the next value of for is accumulatively added to the next input angle until the last value is obtained as . Then, the new set of input angles is in an ascending direction before returning to the position analysis of a four-bar linkage. This concept was successfully used in a sprayed plate fin heat sink design [19]. The obtained sequence of input angles always obeys the constraint (8).

Input variables: infeasible

Output variables: feasible

(1)
Generate a uniform random number () whose values are in the range of [0.0001, 1]

(2)
Set , for .

(3)
For

(4)
Generate

(5)
End

Algorithm 1: A repairing technique for the prescribed timing constraint.

The vector of design variables for path synthesis of a particular four-bar linkage is . Variables are the link lengths of the linkage, and are the angular positions of link , also known as timings of the crank, while other variables are shown in Figure 1. The legitimated set of the timings must obey the condition . During an optimization process, if the set of timings is decoded as, for example, = 0.5, = 0.45, = 0.67, = 1.35, = 4.50, and = 2.10, they violate constraint (8). Then, Algorithm 1 is activated to repair these values. Five (for 6 timings) uniform random numbers are generated, for example, , , , , and . The values of are then scaled down according to step 2 in Algorithm 1 to meet the condition leading to

The output modified values of are then computed as

As a result, by using Algorithm 1, the timings are always feasible. The difference between and will never exceed .

3.2.2. Grashof’s Criterion Constraint

With the same reasons as for the previous constraint, the dimensions of {} must obey the conditions (6) and (7) so that the resulting mechanism is usable. Let the bound constraints of {} in Equation (9) be separately defined as

Then, the relation can be found:

The Grashof criterion repairing technique is shown in Algorithm 2. Firstly, four uniform random numbers {} in the range of [0.0001, 1] are generated. The lower bound is set to be 0.0001 for the same reason as Algorithm 1. Then the auxiliary variables for are computed as

With this computation, it is concluded that is their minimum and is their maximum. These four values fulfil the Grashof criterion if is an input crank. Condition (7) also holds if

Input: infeasible {} and bounds:

Output: feasible {}

(1)
Generate uniform random numbers {} whose values are in the range of [0.0001, 1]

(2)
If , swop their positions

(3)
Else if , set

(4)
Assign values

(5)
If , compute

(6)
Compute for

(7)
Assign and assign as the lengths of , respectively

Algorithm 2: Repairing Grashof criterion constraint.

Therefore, the computational steps 3-4 in Algorithm 2 are applied. Then, the values of {} are all scaled down so that . The values of can then be computed as

Then, set . The values of are assigned to be the lengths of , respectively. With such a computing scheme, the values of {} returned from activating Algorithm 2 will always be feasible. In the search process of MH, when a function evaluation is revoked, feasibility of a design solution will be checked. If it is infeasible, Algorithms 1 and 2 will be used to repair or alter the values of and in and send them back to the main process of a metaheuristic. That means all design solutions in a population of MH are always feasible.

Given that, for example, , , , and , the Grashof criterion is not fulfilled since . These values will then be regenerated by using Algorithm 2. From step 1, given that the values of are randomly generated as , , , and , then the auxiliary variables are computed based on (14) as

Since is greater than 1.00, the values of the auxiliary variables are modified as

Thus, the new link length values are obtained using (16):

which results in a crank-rocker four-bar linkage. In cases where the generated value of is greater than that of , their values are swapped. If they are equal, Step 3 in Algorithm 2 is activated ensuring that a crack-rocker is obtained after the repairing process.

4. Numerical Experiment

For evaluating the performance of the proposed path repairing technique, three path synthesis test problems of a four-bar linkage are used, whereas the optimizers are 7 established MHs. To validate the new approach, it will be compared with the exterior penalty function technique, which is traditionally applied in previous work. The path generation problems are detailed as [5, 14]:

Case 1. Path generation without prescribed timing

Design variables are

Target points are .

Limits of the design variables are as follows:

Case 2. Path generation with prescribed timing

Design variables are

Target points are .

Limits of the design variables are

Case 3. Path generation without prescribed timing

Design variables are

Target points are

Limits of the design variables are

The first synthesis problem has a straight line target path without a prescribed timing. The second test problem has a circular prescribed path with prescribed timing while the third test problem has an elliptic path without prescribed timing. The optimizers used to tackle the test problems are 7 well-known and newly developed metaheuristics. Their optimization parameter settings are given below. The population size nP = 100 is used for all algorithms, unless otherwise specified. It should be noted that the terms and variable definitions are from their original sources.

(1)Artificial bee colony algorithm (ABC) [20]: the number of food sources for employed bees is set to be nP/2. A trial counter to discard a food source is 100.(2)Real-code ant colony optimization (ACOR) [21]: the parameter settings are q = 0.2 and ξ = 1.(3)Teaching-learning-based optimization (TLBO) [22]: parameter settings are not required.(4)Adaptive differential evolution with optional external archive (JADE) [23]: The parameters are self-adapted during an optimization process.(5)Population-based incremental learning (PBIL) [24]. The learning rate, mutation shift, and mutation rate are set as 0.5, 0.7, and 0.2, respectively.(6)Grey wolf optimizer (GWO) [25]: the GWO has only two main parameters to be adjusted, a and C, where a is decreased from 2 to 0 and C is randomly generated in the range of [0, 2].(7)Sine cosine algorithm (SCA) [26]. The parameter decreases linearly from a = 2 to 0.

It should be noted that the parameter settings used in this paper are either the default values provided or suggested by their authors. Also, they have been used in some previous studies with acceptable results [17, 27–30].

The total number of iterations or generations for each optimization run is set to be 500. Any MH that uses a different population size will be terminated with the total number of function evaluations as 100 × 500. Each MH runs 30 times, so that to measure its convergence rate and consistency. It should be noted that most works in the literature tended to ignore examining the search consistency of MHs. This can be carried out by running MHs many times. In this paper, each MH is run 30 times for each optimization problem. The means of the objective function values obtained from the various MHs are used as their performance indicator. Also, the comparison based on nonparametric statistical Friedman test [17, 31] is employed. Thus, this study would be a proper baseline for the topic of using MHs for four-bar linkage path synthesis in the future.

In conclusion, the path repairing technique is proposed to increase the performance in solving four-bar linkage path synthesis. To evaluate the performance of the proposed technique, three path synthesis test problems and 7 established MHs are used to study. To validate the new constraint handling approach, it is compared with the results that obtained from using the classical exterior penalty function technique, the most popularly used technique for path synthesis. Moreover, statistical results including mean, minimum, maximum, and standard deviation are also reported.

5. Design Results

The optimization of path generation of four-bar mechanisms is to find the optimized link lengths and some other parameters, which minimize the objective function. Three case studies with and without prescribed timing are considered for performance testing of the optimizers and the new constraint handling technique. The mean values of objective functions are used as the main performance indicator, where the lower mean value is more reliable optimizer. The mean objective function value determines MH reliability as MH with lower mean value is likely to be more successful in solving the problem, even with only one optimization run.

The results of Case 1 obtained from the seven optimizers with the novel path repairing technique and the penalty technique are shown in Tables 1 and 2 respectively. In the tables, the number of successful runs (a run that obtains a feasible solution), the mean objective function values from 30 optimization runs (Mean), the worse result (Max), the best result (Min), the standard deviation (Std), and the best linkage that gives the minimum objective function value of each algorithm are given. Each parameter computes based on the objective function defined as the sum of squares of the distances between the desired points, and the actual points, while the term “Error” in the tables is an average distance error of the best found solution computed as

which has been used as a performance indicator in some previously published papers. is the number of points on the prescribed or target curve. From the results, it is seen that the proposed path repairing technique is far superior to the exterior penalty technique used in the previous studies [1–5, 7, 8, 10–13]. Based on the mean values of the objective function, all seven optimizer performances are greatly improved when implementing the path repairing algorithm as summarized in Table 3. All the optimizers can find the feasible solutions for all 30 runs. The optimizer that gives the best results is TLBO for both the mean value () and the best result (). The second and third best optimizers are ABC and GWO, respectively. The worst performer in this case, according to Mean, is SCA. In Table 2, only ABC can search for feasible solutions for all 30 runs. Based on the mean objective values, the performance of all optimizers deteriorates compared to the results from using PRA in Table 1. The best in cases of using the exterior penalty function technique is ABC, while the second best is GWO as it can search for feasible solutions for 29 from 30 runs with lower mean objective value compared to JADE. Figures 2 and 3 show the path traced by the coupler point of the best solution and the kinematic diagram of best linkage, respectively. It is found that TLBO with the novel design repairing technique gives the best result (Error = 0.0122) better than previous work [5, 14] (Error = 0.1227 (DE), and 0.0166 (DE), respectively). This design result cannot be compared with the previous work [15] due to the number of evaluation of the present work that is lower than ten times when compared with the reference. The best actual path as shown in Figure 3 is closer to the target path than in previous work [5, 14] and comparable with [15]. The four-bar linkage obtained from the best solution as shown in Figure 3 can completely rotate in an ascending direction.

Table 1: Comparative results for Case 1 with a novel path repairing technique.

Case 1: path generation without prescribed timing

Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

39.7861
52.4762
53.7995
57.9042
59.4760
33.4305
28.3704

10.1505
14.3110
9.4303
8.5359
9.1050
9.8739
5.8563

50.4447
48.0989
44.1070
20.5846
51.8860
38.5048
60.0000

35.2568
47.5363
46.5599
55.5223
20.4784
53.3193
38.7556

45.1511
30.2155
47.8722
5.0306
−6.9474
37.5632
60.0000

7.2885
−14.8707
32.8201
35.6648
−21.9732
−10.3487
−60.0000

50.7782
11.7094
60.0000
47.0842
−8.8106
−5.2992
−60.0000

−0.9366
−3.1269
−8.8920
12.5554
28.8944
59.6847
60.0000

82.2709
51.1387
45.4292
355.3772
96.7425
234.8353
0.0000

288.6774
313.0937
352.6949
9.4470
276.1848
334.3267
0.0000

18.7094
338.5807
24.3156
23.8648
298.9037
5.9786
71.2728

39.4220
359.8582
29.9612
37.4934
337.6217
31.4398
86.9490

57.0069
18.5113
56.1299
51.6572
358.4935
61.3596
103.9052

75.0134
44.8517
73.4859
67.9991
33.9200
90.2248
130.0967

103.2820
85.9854
136.6204
91.0354
57.5750
139.0187
166.4231

Min
0.6374
1.8255
6.7539
0.0010
49.1961
0.3271
42.5282

Mean
4.1297
7.5933
20.0681
0.3705
73.1582
8.5501
144.5432

Max
11.9015
17.0022
47.5797
2.7424
101.7039
94.4204
391.5969

Std
2.7684
3.4092
10.1640
0.6667
16.1132
17.0012
94.5604

Error
0.2380
0.4502
0.8232
0.0122
2.6807
0.2107
2.6387

Success
30
30
30
30
30
30
30

Success = number of successful runs.

Table 2: Comparative results for Case 1 with a penalty technique.

Case 1: path generation without prescribed timing

Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

38.2010
31.9116
34.6475
59.9108
22.0645
43.6675
60.0000

8.6064
7.4478
6.4582
29.9172
5.0000
5.3499
5.0000

22.4589
43.6552
21.6196
59.9967
60.0000
26.5380
47.3444

33.2769
31.8355
23.7955
54.2963
60.0000
22.5663
60.0000

20.0224
9.4994
30.1437
59.7723
4.2836
−17.2375
60.0000

24.8915
25.3461
−25.6646
11.4222
11.0367
43.5804
−60.0000

−5.1270
−0.4792
51.7915
41.5439
32.8024
59.9713
−60.0000

52.6358
38.6191
54.5888
−2.4092
29.3709
10.5553
−0.6986

212.3407
224.9538
211.7426
47.8784
15.3431
16.7149
0.0000

307.2217
241.9357
211.4543
304.0545
159.2096
342.4006
21.6624

357.3637
226.6521
243.5720
347.0862
246.5489
15.3455
201.7648

26.8761
204.4762
287.1713
359.9988
253.5892
37.5406
75.3665

52.9043
174.1217
311.7283
9.0531
299.5365
60.6278
3.1728

80.7316
141.3107
333.2289
16.0694
0.0000
91.7984
0.0000

110.8412
131.9133
58.1996
22.0740
111.4815
134.0130
360.0000

Min
0.9797
23.1751
23.1950
0.2399
523.5085
0.8853
358.3843

Mean
23.4148
297.8795
133.3815
175.9177
1005.078
147.9292
452.8526

Max
78.2443
1151.615
247.9302
1106.170
1541.802
399.6278
562.5967

Std
18.2255
242.1490
74.2606
226.5618
236.8720
120.4374
102.9596

Error
0.3545
1.6529
1.7811
0.1808
7.9631
0.3532
5.9932

Success
30
29
6
26
22
29
3

Success = number of successful runs.

Table 3: Comparative Min and Mean for Cases 1–3 with a novel path repairing technique and a penalty technique.

Case number
With
Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

1
PRA
Min
0.6374
1.8255
6.7539
0.0010
49.1961
0.3271
42.5282

Mean
4.1297
7.5933
20.0681
0.3705
73.1582
8.5501
144.5432

Success
30
30
30
30
30
30
30

Penalty technique
Min
0.9797
23.1751
23.1950
0.2399
523.5085
0.8853
358.3843

Mean
23.4148
297.8795
133.3815
175.9177
1005.078
147.9292
452.8526

Success
30
29
6
26
22
29
3

2
PRA
Min
7.3696
0.7614
2.7790
0.7614
3.6197
1.0934
22.7680

Mean
31.4436
13.3175
24.3873
9.2918
31.1379
27.5451
147.5295

Success
30
30
30
30
30
30
30

Penalty technique
Min
6.1512
8.5978
2.2458
0.7614
24.4416
2.6452
43.5787

Mean
58.0879
35.9131
20.6274
45.8361
98.6229
39.3821
340.6984

Success
30
30
21
29
30
30
30

3
PRA
Min
1.1331
2.2551
41.9444
0.0192
156.6075
26.8977
306.5633

Mean
7.7983
9.2387
116.5084
2.5706
310.5264
70.7824
568.2001

Success
30
30
30
30
30
30
30

Penalty technique
Min
1.0714
1212.697
42.6040
352.3451
0.0000
251.2720
0.0000

Mean
38.2367
1212.697
402.5957
576.6454
0.0000
493.2413
0.0000

Success
29
1
4
10
0
15
0

Figure 2: Best coupler curve obtained in Case 1.

Figure 3: Best mechanism obtained in Case 1.

For Case 2 with the number of target points at 6, the results obtained from using the various MHs with PRA and the penalty function technique are shown in Tables 4 and 5, respectively. The coupler curves and the best linkages obtained from TLBO are shown in Figures 4 and 5, respectively. In this case, all MHs are significantly improved by using PRA based on the mean objective function values. They can find feasible solutions for all runs with the given predefined number of function evaluations. Note that PBIL using PRA has higher mean value compared to that using the penalty function technique, but this is computed from only 21 successful runs when using the penalty function technique. Based on using PRA in Table 4, the best method is TLBO while the second and third best algorithms are JADE and PBIL, respectively. The worst optimizer is SCA according to the mean value. In Table 5, based on using the penalty function technique, the top three performers are JADE, GWO, and ABC in that order. In this design case, TLBO produces the best solutions for both constraint handling techniques. For this case, it is proved that using the second repairing technique (Algorithm 2) is better than the penalty function technique. The best optimizer is TLBO as with the first case which gives the best error result as 0.3073, while the results from previous work [5, 13, 14] are error = 2.3496 (DE), 2.5998 (ICA), and 1.6063 (JADE), respectively. The best actual path as shown in Figure 4 is closer with the target path than the previous work [5, 13, 14]. The four-bar linkage obtained from the best solution as shown in Figure 5 can completely rotate in an ascending direction in accordance with the prescribed timing and the circular target path.

Table 4: Comparative results of Case 2 with a novel path repairing technique.

Case 2: path generation with prescribed timing

Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

50.0000
47.3318
37.3034
47.3319
47.0787
48.7592
23.7320

8.7953
8.9594
9.7011
8.9594
9.5109
8.8102
6.4302

17.5326
26.1415
27.5790
26.1415
23.8293
23.8043
43.5310

50.0000
50.0000
40.1382
50.0000
44.9649
45.6184
37.4296

35.9767
43.5295
38.4166
43.5296
47.1950
49.8852
15.6308

−7.7228
−27.9916
−31.4443
−27.9914
−14.3055
−13.8744
−50.0000

16.0681
16.8224
16.2277
16.8224
17.8426
16.9234
14.9035

−34.3113
−50.0000
−48.1851
−50.0000
−47.7051
−49.9768
−50.0000

31.3196
48.3625
59.5519
48.3624
45.2184
44.2373
107.4695

Min
7.3696
0.7614
2.7790
0.7614
3.6197
1.0934
22.7680

Mean
31.4436
13.3175
24.3873
9.2918
31.1379
27.5451
147.5295

Max
52.3994
26.1423
92.0479
17.8459
107.1751
192.4375
660.2919

Std
13.9207
7.9546
18.4786
8.1308
23.2712
36.4676
141.0674

Error
1.0356
0.3073
0.6318
0.3073
0.6749
0.3723
1.6759

Success
30
30
30
30
30
30
30

Success = number of successful runs.

Table 5: Comparative result of Case 2 with a penalty technique.

Case 2: path generation with prescribed timing

Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

40.7603
32.5637
50.0000
47.3284
47.6986
49.0512
25.7137

5.8249
7.2738
8.9238
8.9594
6.4661
7.9356
5.0000

9.9768
22.0128
23.9506
26.1434
21.0563
21.2936
25.7783

38.6534
31.9534
46.8513
50.0000
48.2501
41.6373
38.5477

28.0556
42.1632
48.6935
43.5248
46.0280
50.0000
15.3979

−5.7200
−28.2724
−16.8126
−27.9988
−23.7354
3.4330
−50.0000

12.9887
12.8441
17.1186
16.8220
12.5560
16.2388
12.3447

−24.4168
−48.8226
−49.3931
−50.0000
−50.0000
−47.9755
−50.0000

34.5068
57.2910
46.9004
48.3661
41.6402
36.6935
67.0180

Min
6.1512
8.5978
2.2458
0.7614
24.4416
2.6452
43.5787

Mean
58.0879
35.9131
20.6274
45.8361
98.6229
39.3821
340.6984

Max
168.5279
97.6451
86.4201
609.1712
173.0658
108.0179
1064.047

Std
44.3671
17.0311
20.9987
139.0780
39.9827
32.7100
265.7480

Error
0.8997
1.0665
0.5467
0.3073
1.9211
0.6451
2.5063

Success
30
30
21
29
30
30
30

Success = number of successful runs.

Figure 4: Best coupler curve obtained in Case 2.

Figure 5: Best mechanism obtained in Case 2.

In Case 3 with the prescribed curve with 10 points, the results are obtained by those algorithms with PRA, and the penalty function techniques are shown in Tables 6 and 7, respectively. The coupler curve and the best linkage are shown in Figures 6 and 7, respectively. From the results, it is found that TLBO gives the best results for both the mean objective function value and the best result when using PRA. Similarly to the previous two cases, all MHs are greatly improved when using PRA. All the methods can find feasible solutions for all runs with using PRA. The top three performers in cases of using PRA are TLBO, ABC, and JADE in that order while the worst method is SCA based on the mean objective values. In the results of using penalty function technique, ACOR, and SCA cannot find a feasible solution. The best method when using penalty function technique is ABC while the performance of others cannot be evaluated as they can search for feasible solutions for a few optimization runs. The best result from TLBO gives an average distance error of 0.0324 while the previous work [5, 14] had error = 1.9523 (DE), and error = 0.1641 (DE), respectively. It means that the best actual path as shown in Figure 6 is closer with the target path than the previous work [5, 14]. The linkage obtained from the best solution as shown in Figure 7 can completely rotate in an ascending direction in accordance with the elliptic target path.

Table 6: Comparative results of Case 3 with a novel path repairing technique.

Case 3: path generation without prescribed timing

Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

80.0000
79.9162
69.0571
42.0053
66.5453
43.5326
80.0000

8.1263
9.7027
10.0219
8.0876
10.6498
19.6630
5.1563

80.0000
79.6300
80.0000
28.2660
65.3047
66.2180
64.8837

51.8439
22.1050
66.3977
24.1099
62.9137
42.3820
61.7959

−19.3276
13.4224
−6.0994
−4.4860
6.0790
25.1410
−40.4117

−11.0023
−10.1904
−27.8102
−4.7935
−6.9436
−25.6124
−80.0000

−1.2988
25.0942
−13.1694
11.1765
2.5885
−16.7352
25.4964

29.1004
3.1796
26.4926
3.5870
7.9850
8.3961
−80.0000

53.9222
176.9942
12.1922
203.0714
0.0000
10.8144
162.9594

349.6393
0.0377
8.8375
359.1656
52.8186
75.4675
47.9605

35.6886
31.3880
45.3792
38.0948
76.7817
88.6364
77.3485

77.1187
73.5207
79.6200
76.7126
111.7879
107.0112
78.7932

115.8808
117.4221
91.7790
115.5382
139.0174
129.7889
102.9839

153.8239
160.1615
140.7557
154.7689
168.2661
155.2651
132.9758

193.8898
201.3488
196.8064
196.1914
196.3339
189.1377
158.4142

234.8222
238.3213
225.1631
236.9290
236.0848
228.5280
178.2059

275.8371
275.8487
272.5570
277.6224
269.5633
255.6323
216.7998

311.5613
312.9753
298.1521
319.1629
298.2293
276.9471
227.5118

348.9137
355.2115
360.0000
359.1397
322.3910
290.1490
239.6187

Min
1.1331
2.2551
41.9444
0.0192
156.6075
26.8977
306.5633

Mean
7.7983
9.2387
116.5084
2.5706
310.5264
70.7824
568.2001

Max
30.3123
25.7054
306.4299
28.5182
551.2737
234.2163
1144.216

Std
7.1036
4.9496
62.3238
6.4004
83.5219
43.5185
153.6016

Error
0.2925
0.4315
1.8480
0.0324
3.5846
1.4398
5.2041

Success
30
30
30
30
30
30
30

Success = number of successful runs.

Table 7: Comparative results of Case 3 with a penalty technique.

Case 3: path generation without prescribed timing

Parameters
ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

68.1732
19.9901
40.6029
17.7554
0.0000
59.7863
0.0000

9.4618
5.8357
8.1608
5.0231
0.0000
5.0000
0.0000

80.0000
43.1677
50.0664
32.9105
0.0000
74.9475
0.0000

80.0000
50.3385
34.8760
31.2567
0.0000
30.6040
0.0000

−1.1813
18.7411
−0.3738
5.8641
0.0000
15.4730
0.0000

−13.1623
−25.0130
26.8058
24.6039
0.0000
80.0000
0.0000

−1.1627
41.0708
26.7918
10.3944
0.0000
30.7488
0.0000

16.3687
12.8362
30.3681
−16.0968
0.0000
−70.6740
0.0000

360.0000
134.8235
99.1181
309.4384
0.0000
4.4956
0.0000

0.0000
172.6179
32.5643
351.2755
0.0000
360.0000
0.0000

37.4238
325.2499
72.4952
52.7076
0.0000
34.6517
0.0000

75.2680
337.0868
88.5868
94.6252
0.0000
102.8443
0.0000

114.8952
20.4030
112.0993
143.0734
0.0000
135.9833
0.0000

152.8726
182.4764
173.3556
190.6071
0.0000
164.5501
0.0000

191.5270
331.0880
201.7909
317.9020
0.0000
186.7582
0.0000

232.1421
82.2786
258.3224
129.4480
0.0000
188.6709
0.0000

274.8734
134.4962
280.6872
300.9699
0.0000
8.0308
0.0000

314.3132
241.5786
318.6172
335.0357
0.0000
19.9826
0.0000

0.0000
303.1435
43.3601
357.2376
0.0000
51.8922
0.0000

Min
1.0714
1212.697
42.6040
352.3451
0.0000
251.2720
0.0000

Mean
38.2367
1212.697
402.5957
576.6454
0.0000
493.2413
0.0000

Max
110.6796
1212.697
781.4347
990.3743
0.0000
1901.836
0.0000

Std
25.2906
0.0000
398.2755
210.1756
0.0000
402.5266
0.0000

Error
0.3074
10.1061
1.9909
4.9788
0.0000
4.6902
0.0000

Success
29
1
4
10
0
15
0

Success = number of successful runs.

Figure 6: Best coupler curve obtained in Case 3.

Figure 7: Best mechanism obtained in Case 3.

Table 3 gives a summary of the comparative performance of the various metaheuristics for solving the four-bar linkage path synthesis using the new constraint handling technique and the classical exterior penalty function technique. It shows that the results from using PRA are totally superior to those obtained from using the penalty function.

It is shown that TLBO with PRA is superior to the other MHs using the penalty technique. In this study, the Friedman test and the Tukey–Kramer test are used for comparing the results. Based on the Friedman test, TLBO ranks 1st whereas the second best is JADE at value (0.02) < α (0.05) as shown in Table 8. It can be summarized that TLBO is the best performer for solving the four-bar path synthesis problems cases 1–3. Based on the Tukey–Kramer test, the mean column ranks of TLBO are significantly different from SCA.

Table 8: Average ranking and value of performance index of MHs achieved by Friedman test.

Average ranking of each algorithm Friedman
value

ABC
JADE
PBIL
TLBO
ACOR
GWO
SCA

3.3333 (3)
2.6667 (2)
4.3333 (5)
1 (1)
5.6667 (6)
4 (4)
7 (7)
0.02

The mean values obtained from using all optimizers of the test problems with PRA and the penalty function technique are shown in Table 3. In the statistical study, the results from ACOR and SCA are discarded because they cannot find any feasible solution in Case 3 (without using PRA). This table shows that MHs using the PRA approach give better mean than their counterparts that employ the penalty function technique for all test problems. The of average results of Friedman test are given in Table 9 which shows that MHs with the PRA technique significantly outperforms those using the penalty function technique at value (0.00016) < α (0.05).

Table 9: Performance comparison of each case study with and without a novel path repairing technique for all algorithms.

value
Average ranking of each technique Friedman

With PRA
Without PRA

0.00016
1.0556 (1)
1.9444 (2)

Figures 8–10 illustrate the search histories of the best runs of TLBO for Case 1, Case 2, and Case 3, respectively. The search history compares the best runs obtained from using the penalty function technique and the path repairing algorithm. It can be seen that even though TLBO using PRA started with the worse solution (Case 3), the path repairing technique still guided the optimizer to converge considerably faster than when using the penalty function technique. All three figures show the superiority of the proposed path repairing technique for four-bar linkage path synthesis.

Figure 8: Search history of the best result obtained in Case 1.

Figure 9: Search history of the best result obtained in Case 2.

Figure 10: Search history of the best result obtained in Case 3.

6. Discussion

Further discussion is provided in order to investigate the behavior of the proposed constraint handling technique PRA and why it is efficient when used with TLBO. Figure 11 shows the search history of the best runs of TLBO for the case 1 problem using PRA and the penalty function (PF) technique. The figure displays the number of iterations versus the number of infeasible solutions for both runs. The illustration also separates the number of infeasible solutions due to the timing constraint and those caused by the link length constraints. It can be seen that, with the use of PRA, infeasible solutions due to the link length constraints vanished after approximately 50 iterations. The same conclusion is applied in cases of using PF. By using PRA, the numbers of infeasible solutions due to the timing constraint disappear after around 85 iterations while the number of infeasible solutions due to the timing constraint when using PF cannot be suppressed throughout the search process. From this particular comparison, it is shown that the proposed PRA is efficient for dealing with both link length and timing constraints while the penalty function technique failed to solve the timing constraint problem.

Figure 11: Repairing and penalty function histories of Case 1 for 200 iterations.

Figure 12 shows the search history (objective function versus iterations) of the best runs of the three design cases from using TLBO in combination with PRA. In this figure, the best objective function values obtained from the teaching and learning phases of TLBO are plotted separately. It can be seen from all three path synthesis problems that the best results produced by the learning phase are equal to or better than those obtained from the teaching phase. This implies that the reproduction using the learning operator of TLBO works well with the proposed PRA. In TLBO, the teaching phase is used for exploitation while the learning phase emphasizes more on exploration. That means the proposed technique tends to be efficient with an exploration-based reproduction operator. As a result, further development of TLBO for path synthesis of a four-bar linkage can adapt from the original version by adding a self-adaptive strategy for population sizing.

Figure 12: Teacher phase and student phase function evaluation history of Cases 1–3 for 500 iterations.

7. Conclusions

This paper presents a technique to find the optimum parameters of a four-bar linkage for path synthesis using metaheuristics including ABC, JADE, PBIL, TLBO, ACOR, GWO, and SCA. The new technique, called a path repairing technique, is proposed to handle the synthesis constraints effectively. The comparative results of the three case studies show that the new path repairing technique is superior to the penalty function technique traditionally used in four-bar linkage path syntheses in the previous studies. The comparative performance of the metaheuristics shows that the TLBO with PRA is the most efficient method based on both convergence rate and consistency. The results in this work can consider as the baseline for developing an efficient optimizer for four-bar linkage path syntheses in the future. Any optimizer should be tested by running it many times to solve the synthesis problems, and the mean value of an objective function should be used as a performance indicator.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors are grateful for the financial support provided by the King Mongkut’s Institute of Technology Ladkrabang, the Thailand Research Fund (RTA6180010), and the Postdoctoral Program from Research Affairs, Graduate School, Khon Kaen University (58225).

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