Computational Intelligence and Neuroscience

Volume 2018, Article ID 5462563, 16 pages

https://doi.org/10.1155/2018/5462563

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

^{1}Department of Aeronautical Engineering, International Academy of Aviation Industry, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand^{2}Sustainable 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; ht.ca.ltimk@es.niwus

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 ().