Synthesis Of Four-bar Mechanism Research Articles

Computer-aided design tool for typical flexible mechanisms synthesis by means of evolutionary algorithms

AbstractAccurate prediction for mechanisms’ dynamic responses has always been a challenging task for designers. For modeling easiness purposes, mechanisms’ synthesis and optimization have been mostly limited to rigid systems, making consequently the designer unable to vow that the manufactured mechanism satisfies the target responses. To address this limitation, flexible mechanism synthesis is aimed in this work. Two benchmark mechanisms being the core of myriad mechanical devices are of scope, mainly, the flexible slider-crank and the four-bar. In addition to the mechanism dimensions, materials properties have been embedded in the synthesis problem. Two responses are of interest for the slider-crank mechanism, the slider velocity, and the midpoint axial displacement for the flexible connecting rod. Whereas five responses have been compiled for the four-bar mechanism synthesis. A comparative analysis of seven optimization techniques to solve the synthesis problem for both mechanisms has been performed. Subsequently, an executable computer-aided design tool for mechanisms synthesis has been developed under MATLAB®. Numerical outcomes emphasize the limits of a single-response-based synthesis for a flexible mechanism. It has been proven that combining different responses alleviates possible error and fulfill high-accuracy requirement.

An Invariant Representation of Coupler Curves Using a Variational AutoEncoder: Application to Path Synthesis of Four-Bar Mechanisms

Abstract This paper focuses on the representation and synthesis of coupler curves of planar mechanisms using a deep neural network. While the path synthesis of planar mechanisms is not a new problem, the effective representation of coupler curves in the context of neural networks has not been fully explored. This study compares four commonly used features or representations of four-bar coupler curves: Fourier descriptors, wavelets, point coordinates, and images. The results demonstrate that these diverse representations can be unified using a generative AI framework called variational autoencoder (VAE). This study shows that a VAE can provide a standalone representation of a coupler curve, regardless of the input representation, and that the compact latent dimensions of the VAE can be used to describe coupler curves of four-bar linkages. Additionally, a new approach that utilizes a VAE in conjunction with a fully connected neural network to generate dimensional parameters of four-bar linkage mechanisms is proposed. This research presents a novel opportunity for the automated conceptual design of mechanisms for robots and machines.

Application of the path-repairing technique and virus optimization algorithm for the dimensional synthesis of four-bar mechanisms

This paper considers the synthesis of the four-bar mechanism. It is treated here as an optimization problem, in which an objective function is defined. To solve this problem, a metaheuristic called the virus optimization algorithm is employed. Furthermore, a new path-repairing technique recently published by Sleesongsom and Bureerat is applied instead of the very common technique related to the application of a penalty function. This makes the search by means of the metaheuristic more efficient. Furthermore, the obtained results are very accurate.

Dimensional synthesis of four-bar mechanisms using Rao algorithms and their variants

This paper presents the dimensional synthesis of four-bar mechanisms for path generation. The synthesis of the four-bar mechanisms is proposed using nine advanced optimization algorithms, namely the Rao-1 algorithm, Rao-2 algorithm, Rao-3 algorithm, self-adaptive multi-population (SAMP) Rao-1 algorithm, SAMP Rao-2 algorithm, SAMP Rao-3 algorithm, quasi-oppositional-based (QO) Rao-1 algorithm, QO Rao-2 algorithm, and QO Rao-3 algorithm. Six different cases of synthesis of a four-bar mechanism are considered. The brief literature review based on the synthesis of mechanisms is also presented. The designs of mechanisms obtained using Rao algorithms and their variants are compared with the designs acquired using other advanced optimization algorithms, such as genetic algorithm (GA), particle swarm optimization (PSO) algorithm, differential evolution (DE), imperialist competitive algorithm (ICA), hybrid GA-DE algorithm, simulated annealing (SA), Malaga university mechanism synthesis algorithm (MUMSA), IMMa optimization algorithm (IOA), hybrid Taguchi random coordinate search algorithm (HTRCA), hybrid GA and fuzzy logic (GA–FL) algorithm, ant-gradient method (AGM), teaching–learning-based optimization algorithm (TLBO), and hybrid Lagrange interpolation differential evolution algorithm (HLIDE). The comparison of results reveals the efficacy of Rao algorithms and their variants for synthesizing the four-bar mechanism.

Optimum synthesis of four-bar mechanisms considering clearance-induced uncertainty

This paper presents a multi-objective optimization model for the motion synthesis of four-bar mechanisms considering the effects of revolute joint clearances. A geometric method is proposed to compute the maximum clearances-induced positioning and orientation errors. The proposed error modeling method is validated using a simple example, and the results reveal its advantages of high accuracy and fast computation compared with the widely used existing methods. A multi-objective optimization model is developed to design the four-bar mechanism with the best accuracy. The positioning and orientation errors are modeled separately and equally. Moreover, three constraints are included in the optimization to ensure that the synthesized mechanism is a compact crank-rocker or double-crank mechanism free from branch and order defects. A study case of synthesizing a four-bar mechanism passing through 6 motion points is investigated to illustrate the proposed method. Finally, conclusions and a brief discussion are given.

A New Study on Optimization of Four-Bar Mechanisms Based on a Hybrid-Combined Differential Evolution and Jaya Algorithm

In mechanism design with symmetrical or asymmetrical motions, obtaining high precision of the input path given by working requirements of mechanisms can be a challenge for dimensional optimization. This study proposed a novel hybrid-combined differential evolution (DE) and Jaya algorithm for the dimensional synthesis of four-bar mechanisms with symmetrical motions, called HCDJ. The suggested algorithm uses modified initialization, a hybrid-combined mutation between the classical DE and Jaya algorithm, and the elitist selection. The modified initialization allows generating initial individuals, which are satisfied with Grashof’s condition and consequential constraints. In the hybrid-combined mutation, three differential 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, the elitist selection is also applied in HCDJ to find the best solutions for the next generation. To validate the feasibility of HCDJ, the numerical examples of the symmetrical motion of four-bar mechanisms are investigated. From the results, the proposed algorithm can provide accurate optimal solutions that are better than the original DE and Jaya methods, and its solutions are even better than those of many other algorithms that are available in the literature.

An open-source tool for path synthesis of four-bar mechanisms

Four-bar mechanisms are widespread in industrial and quotidian applications, but their design requires deep knowledge and experience in Mechanisms and Machine Theory. In this area, path synthesis is a common problem. It consists in calculating the length of the bars defining a mechanism so that a particular point of its structure follows the desired trajectory. This paper presents SALAR Mechanism Synthesizer, a new open-source software package that aims to be an easy-to-use yet powerful tool for addressing this kind of problem. This tool offers four different optimizers from the literature, one of them specially designed for mechanism synthesis. It also includes a new method for expressing and comparing paths, and hence, to define the optimization criterion. This method, referred to as normalized shape-descriptor vector(s) (NSDV(s)) and presented in this paper, allows us to handle paths without being affected by translation, rotation, and scale changes. The software package has been used for addressing six well-known problems from the literature. According to the results achieved, it arises as a valuable tool in terms of accuracy, convenience, and availability.

Optimum Synthesis of Four-Bar Mechanism by Using Relative Angle Method: A Comparative Performance Study

In this paper, the dimensional synthesis of the four-bar mechanism for path generation is formulated using the relative angle motion analysis and the link geometry parameterization with Cartesian coordinates. The Optimum Dimensional Synthesis using Relative Angles and the Cartesian space link Parameterization (ODSRA+CP) is stated as an optimization problem, and the solution is given by the differential evolution variant DE/best/1/bin. This study investigates the behavior and performance of such formulation and performs a comparative empirical study with the well-known synthesis method based on vector-loop equation motion analysis where different modifications in the metaheuristic algorithms are established in the literature to improve the obtained solution. Five study cases of dimensional synthesis for path generation with and without prescribed timing are solved and analyzed. The empirical results show that the way of stating the optimization problem in the ODSRA+CP significantly improves the search process for finding promising solutions in the optimizer without requiring algorithm modifications. Therefore, it is confirmed that the optimizer search process in the optimal synthesis of mechanisms is not the only way of improving the obtained solutions, but also the optimization problem formulation has a significant influence on the search for better solutions.

Path and function synthesis of multi-bar mechanisms using beetle antennae search algorithm

Beetle antennae search algorithm (BAS) is based on the searching behavior of longhorn beetles. This newly proposed metaheuristic algorithm is used in the path and function synthesis of multi-bar mechanism. The optimization results are compared with that of other metaheuristic algorithms like genetic algorithm (GA), particle swarm optimization (PSO) and differential evolution (DE) et al. While BAS uses only one group of initial parameters to search the best result, the convergence and efficiency are tested with eight case studies results. Newly added parameter q increases the possibility to find better results during iterations and deals with a higher number of independent parameters efficiently. Revised BAS exhibits good performance both in path synthesis of four-bar mechanism with or without prescribed timing, while optimized parameters increases from 6 to 34, and extends to path and function combined synthesis of Stephenson III six-bar double dwell mechanism with 14 parameters.

A Stackelberg game theoretic multi-objective synthesis of four-bar mechanisms

In this article, a Stackelberg game theoretic approach is used for multi-objective optimal synthesis of four-bar mechanisms for path generation. Two objective functions, namely, tracking error (TE)...

Hybrid Lagrange interpolation differential evolution algorithm for path synthesis

A hybrid Lagrange interpolation differential evolution algorithm (HLIDE) is proposed in this study for path synthesis of four-bar mechanisms. HLIDE combines two algorithms of differential evolution (DE) with Lagrange interpolation local search (LILS). LILS performs a local search in the neighborhood of the best individual to enhance the local exploitation capability of DE. In addition, an adaptive local search strategy is presented to further improve the efficiency of LILS. This technique can adaptively apply LILS based on the performance of LILS and DE in previous generations. To evaluate the efficiency and accuracy of HLIDE, five cases of path synthesis involving four-bar mechanisms were tested. Moreover, three well-known evolutionary algorithms, i.e., particle swarm optimization, teaching-learning-based optimization, and differential evolution, were implemented and compared in the five cases. Previous solutions for the same path generation problems by different evolutionary algorithms are summarized and compared in this study. Experimental results showed that HLIDE has significantly better performance in solving the synthesis problems of mechanisms when compared to other algorithms.

Comparative study of four-bar hyperbolic function generation mechanism with four and five accuracy points

Kinematic mechanisms are synthesized for a task. Function generation provides precise displacement at output links that obeys a given functional relations. This article describes the synthesis of four-bar mechanism for the hyperbolic function generation with four and five accuracy point, which is further optimized using least square method. This research is concerned with development of mathematical formulation based on Freudenstein–Chebyshev approximation theory, extended to four- and five-point synthesis function generation problem. The objective function is analyzed for the structural error between the generated function and the desired function. Resulting nonlinear equations are converted into set of linear equations applying the compatibility conditions and are solved using Gauss elimination method. The formulation is proposed for five position synthesis for algebraic and trigonometric function generation problem. Associated structural error is estimated. Comparison of estimated error through the formulation is carried out with the reported errors through graphical method. The error for hyperbolic function is estimated. Attempt is made to minimize the error through simple of least square technique. The results obtained are compared with Freudenstein–Chebyshev approximation method. Three hyperbolic functions, namely sinh(x), cosh(x) and tanh(x), are used to demonstrate the effectiveness of the proposed synthesis method.

A Fourier Descriptor-Based Approach to Design Space Decomposition for Planar Motion Approximation

In an earlier work, we have combined a curve fitting scheme with a type of shape descriptor, Fourier descriptor (FD), to develop a unified method to the synthesis of planar four-bar linkages for generation of both open and closed paths. In this paper, we aim to extend the approach to the synthesis of planar four-bar linkages for motion generation in an FD-based motion fitting scheme. Using FDs, a given motion is represented by two finite harmonic series, one for translational component of the motion and the other for rotational component. It is shown that there is a simple linear relationship between harmonic content of the rotational component and that of the translational component for a planar four-bar coupler motion. Furthermore, it is shown that the rotational component of the given motion identifies a subset of design parameters of a four-bar linkage including link ratios, while the translational component determines the rest of the design parameters such as locations of the fixed pivots. This leads naturally to a decomposed design space for four-bar mechanism synthesis for approximate motion generation.

Path synthesis of four-bar mechanisms using synergy of polynomial neural network and Stackelberg game theory

ABSTRACTIn this article, a novel approach based on game theory is presented for multi-objective optimal synthesis of four-bar mechanisms. The multi-objective optimization problem is modelled as a Stackelberg game. The more important objective function, tracking error, is considered as the leader, and the other objective function, deviation of the transmission angle from 90° (TA), is considered as the follower. In a new approach, a group method of data handling (GMDH)-type neural network is also utilized to construct an approximate model for the rational reaction set (RRS) of the follower. Using the proposed game-theoretic approach, the multi-objective optimal synthesis of a four-bar mechanism is then cast into a single-objective optimal synthesis using the leader variables and the obtained RRS of the follower. The superiority of using the synergy game-theoretic method of Stackelberg with a GMDH-type neural network is demonstrated for two case studies on the synthesis of four-bar mechanisms.

Two-Swim Operators in the Modified Bacterial Foraging Algorithm for the Optimal Synthesis of Four-Bar Mechanisms.

This paper presents two-swim operators to be added to the chemotaxis process of the modified bacterial foraging optimization algorithm to solve three instances of the synthesis of four-bar planar mechanisms. One swim favors exploration while the second one promotes fine movements in the neighborhood of each bacterium. The combined effect of the new operators looks to increase the production of better solutions during the search. As a consequence, the ability of the algorithm to escape from local optimum solutions is enhanced. The algorithm is tested through four experiments and its results are compared against two BFOA-based algorithms and also against a differential evolution algorithm designed for mechanical design problems. The overall results indicate that the proposed algorithm outperforms other BFOA-based approaches and finds highly competitive mechanisms, with a single set of parameter values and with less evaluations in the first synthesis problem, with respect to those mechanisms obtained by the differential evolution algorithm, which needed a parameter fine-tuning process for each optimization problem.

Optimal synthesis of four-bar mechanism for function generation with five accuracy points

In this paper, optimized kinematic synthesis of four-bar mechanism for function generation with five accuracy points is presented. Formulation of equations for the four-bar mechanism for function generations with five accuracy points are being performed using Freudenstein’s equation. As in the Freudenstein’s equation method approach, the desired output motions are achieved only at accuracy points, and at all other points, some structural error is present that does not follow any predictable scheme. Therefore, least square optimization technique has been used here for minimizing the structural error. Functions like log10(X), e(X) and tan(X) are used to verify the given method. The results obtained are compared with the results obtained using four accuracy points.

Optimum synthesis of path generating four-bar mechanisms using differential evolution and a modified error function

This paper proposes a new error function that can be used for optimum synthesis of path generating mechanisms. The proposed error function is formulated in a way that makes it suitable for use in applications when the primary requirement is that the synthesized path should match closely with the prescribed path. The proposed error function is used in the optimum synthesis of four-bar mechanisms. The method of differential evolution is used to carry out the optimization. Results for three path synthesis problems are presented. Results obtained by using the proposed error function are compared with those obtained by using the commonly used cumulative squares of Euclidean distance error function, on the basis of the values of transverse and longitudinal errors between the prescribed and synthesized paths. The usefulness of the proposed error function is established on the basis of the comparative results.

Pareto optimal synthesis of four-bar mechanisms for path generation

Abstract In this paper, hybrid multi-objective genetic algorithms (GAs) are used for Pareto optimum synthesis of four-bar linkages considering the minimization of two objective functions simultaneously. The important conflicting objective functions that have been considered in this work are tracking error (TE) and transmission angle’s deviation from 90° (TA). The obtained Pareto fronts demonstrate that trade-offs between these two objectives can be recognized so that a designer can optimally compromise for the selection of a desired four-bar linkage. Moreover, the obtained optimum points are also compared with those given in the literature in terms of both tracking error (TE) and transmission angle error (TA).

A New Approach to Shape Optimization for Closed Path Synthesis of Planar Mechanisms

A method for the synthesis of four-bar mechanisms to generate closed paths through shape optimization is herein introduced. The objective function is not based on Fourier descriptors, but rather on the cyclic angular deviation (CAD) vector associated with a set of desired points on the curve. A simple method is introduced to account for the starting point shift between the desired CAD and the generated one. Following shape optimization, a simple mathematical approach is devised to properly scale, rotate, and translate the mechanism to the desired configuration. Two case studies are presented to demonstrate the effectiveness and robustness of the proposed method.

Synthesis of four-bar mechanisms by Freudenstein–Chebyshev

A new dimensional synthesis method is described in the paper. The position function of the four-bar mechanism is presented by the Freudenstein’s equation and it is minimized by the Chebyshev’s best approximation theory. The target function is used as an exactly satisfied equation and the Freudenstein’s equation is considered as a Chebyshev’s polynomial. In some cases the method provides possibilities to find simple solutions of the synthesis tasks. These solutions are known as the best Chebyshev’s approach of the weighted target function. This approach is near the best one for the original target function. A numerical example given here explains the essence of the method.