Synthesis Of Four-bar Linkages Research Articles

On the solution of seven-precision-point path synthesis of planar four-bar linkages based on the solution region methodology

This paper makes further research on the application of the solution region synthesis methodology (SRSM) in path synthesis of planar four-bar linkages. Detailed process of establishing the solution region for seven precision-point path synthesis is illustrated. In order to obtain the solution region where one point corresponds to one linkage and the feasible solution region where linkages distributes different zones according to curve types, this paper compares four different solution region schemes and finally establishes the solution region with B1 as the reference. Compared with the eight-point case, the range of the solution region in this paper is larger and the statistics on the feasible solutions is more than twice of that in the case with eight points. Moreover, the feasible solutions obtained in this case have more curve types.

Fourier-Based Function Generation of Four-Bar Linkages With an Improved Sampling Points Adjustment and Sylvester's Dialytic Elimination Method

Abstract Four-bar linkages are critical fundamental elements of many mechanical systems, and their design synthesis is often mathematically complicated with iterative numerical solutions. Analytical methods based on Fourier coefficients can circumvent these difficulties but have difficulties with sampling points adjustment and solutions of the design equations in previous studies. In this paper, an improved Fourier-based analytical synthesis method is presented, which transforms the function generation synthesis of planar four-bar linkages into a problem of solving design equations. Calculation of the Fourier coefficients is discussed, including the discretization of the prescribed function and an improved sampling points adjustment method. It is shown that the Fourier coefficients can be computed efficiently and accurately by discretizing the prescribed function with a small number of sampling points. The proposed sampling adjustment method overcomes the difficulty of easily resulting in non-Grashof solutions by considering the complete period of the prescribed function. An improved Sylvester's dialytic elimination method is presented to solve design equations. The method reduces the computation time and avoids cumbersome procedures without generating additional invalid solutions. Several examples are presented to demonstrate the advantages of the proposed synthesis method, which is easy-understanding and efficient, and yields more accurate solutions than available synthesis methods.

A planar bisection strategy for multipoint path synthesis of four-bar linkages

In order to improve the global search ability of the optimization algorithm when solving the path synthesis problem, this paper proposes a synthesis method of a four-bar mechanism based on the bisection strategy of the Lévy-PSO evolutionary algorithm. First, a synthesis equation of the target point is established based on the closed-loop vector method. Use the bisection strategy to cut the hinge coordinate variation range to reduce the probability of premature maturity of the algorithm. The error algorithm of the circular proximity function is used to reduce the search dimension of the optimization algorithm and improve search efficiency. Use the improved particle swarm algorithm based on Lévy to improve global search capabilities. Four multi-point path synthesis problem examples are given to verify the effectiveness and accuracy of the method.

Complete Solutions for the Approximate Synthesis of Spherical Four-Bar Function Generators

Abstract Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. Instead of using local methods, we conduct global design searches by direct computation of all critical points. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, which allows for a full exploration of the design space, whereas local solvers can only find the minimum nearest to an initial guess. We form equality-constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. We consider the general case where all mechanism dimensions may vary and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. The discretization points are contained within the coefficients of the stationarity polynomials, so the algebraic structure of the synthesis equations remains invariant to the number of points. The results of our computational work were used to design a mechanism that coordinates the folding wings. We also use this method to parameterize mechanism dimensions for a family of hyperbolic curves.

Path synthesis of planar four-bar linkages for closed and open curves using elliptical Fourier descriptors

Path synthesis of planar four-bar linkages for closed and open curves using elliptical Fourier descriptors

Computing complete solution sets for approximate four-bar path synthesis

The approximate path synthesis of four-bar linkages has been framed and solved with many different optimization techniques. Here we present a polynomial objective that is invariant to the number of approximate design positions selected, and a solution technique capable of finding all minima. The invariance property caps compute time despite increasing the size of input task specification data. This is performed by collecting a variable amount of task data into an invariable number of polynomial coefficients, called moments, before numerical optimization begins. The minima are found by applying the method of random monodromy loops to the zero gradient polynomial system of the aforementioned objective. This procedure finds all critical points, including the local and global minimum, and provides an in-process estimate of the percentage of critical points found. We applied our methodology to four-bar path synthesis problems of various computational scales by altering dimensional pre-specifications. The most general case was estimated to have 1,820,238 ± 3810 critical points, while pre-specification of one or two ground pivots yielded 26,052 and 503 roots, respectively, as validated by a trace test. The results are applied to a variety of examples.

A Bi-Invariant Approach to Approximate Motion Synthesis of Planar Four-Bar Linkage

This paper presents a planar four-bar approximate motion synthesis technique that uses only pole locations. Synthesis for rigid-body guidance determines the linkage dimensions that guide a body in a desired manner. The desired motion is specified with task positions including a location and orientation angle. Approximation motion synthesis is necessary when an exact match to the task positions cannot be obtained. A linkage that achieves the task positions as closely as possible becomes desired. Structural error refers to the deviations between the task positions and the linkage’s generated positions. A challenge in approximate motion synthesis is that structural error involves metrics that include location and orientation. A best-fit solution is not evident because the structural error is based on an objective function that combines the location and orientation. Such solutions lack bi-invariance because a change in reference for the motion changes the values of the metric. This work uses only displacement poles, described solely by their coordinates, as they sufficiently characterize the relative task positions. The optimization seeks to minimize the distance between the poles of the task positions and the poles of the generated positions. The use of poles results in a bi-invariant statement of the problem.

A differential evolution algorithm combined with Lévy Flight for dimensional synthesis of four-bar linkage

The paper considers dimensional synthesis of a path generator four-bar linkage by applying the modified Differential Evolution (DE) algorithm. During the search of the space, the DE algorithm may skip the right solution and lead to premature convergence. If the step is reduced, the search space is reduced as well, so that convergence becomes slower. Search efficiency may be achieved by including Lévy Flight in the DE algorithm. Namely, when the DE algorithm identifies the space of best solution, then a Levý Flight step is used for local search in that region, which significantly increases the exploitation capability and allows obtaining a very efficient solution of the optimization problem. The algorithm was tested with three examples of dimensional synthesis of a path generator four-bar linkage. The proposed algorithm can be very efficiently applied to solving very complex optimization problems.

On the exact motion synthesis of planar four-bar-linkages with prismatic and revolute joints

This paper introduces a novel and simple approach that can be used in the exact motion synthesis of planar four-bar linkages with prismatic (P) and revolute (R) joints to up four given poses. Only basic knowledge of vectors is required and does not depend on either, advanced concepts of kinematic mapping or cumbersome graphic skills. As a result, the design equations are very simple, and they can provide up to a maximum of twenty solutions, which can be real or imaginary. Each pair of different real solutions allows the design of a mechanism that can accurately achieve the given poses. The details of the methodology are illustrated by considering several challenging examples found in the literature.

Finding straight line generators through the approximate synthesis of symmetric four-bar coupler curves

The approximate path synthesis of four-bar linkages with symmetric coupler curves is presented. This includes the formulation of a polynomial optimization problem, a characterization of the maximum number of critical points, a complete numerical solution by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired curve and finds several optimal four-bar linkages with a coupler trace that approximates it. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a structure that remains the same for any desired trace leading to a generalized result. Ground pivot locations are set as chosen parameters, and it is shown that the objective has a maximum of 73 critical points. The theoretical work is applied to the design of straight line paths. Parameter homotopy runs are executed for 1440 different choices of ground pivots for a thorough exploration. These computations found the expected linkages, namely, Watt, Evans, Roberts, Chebyshev, and other previously unreported linkages which are organized into a 2D atlas using the UMAP algorithm.

Synthesis of Four-Bar Linkages by Four Infinitely Close Relative Positions and Pressure Angle

A computer-applicable linear mathematical model has been developed to determine Burmester’s curves for infinitely close relative positions (cubic of stationary curvature), which indirectly uses Carter-Hall’s circle. By varying a free parameter and using elements of kinematic and analytical geometry, an incomparably simpler solution is achieved than that obtained by the third-degree equations of the Burmester's curves for stationary curvature. The mathematical model for the synthesis of four-bar linkages includes and a condition for the pressure angle, whereupon is uniquely defined the kinematic diagram of the mechanism. Of the pressure angle, the reactions of the forces in the kinematic pairs and the force sizing of the mechanism depend. The model would facilitate the engineers in the synthesis of four-bar linkages by generating a function approximating a given function in the vicinity of a given position, where the two functions have four infinitely close common points (3rd-order approximation). An example of the synthesis of a four-bar linkage illustrates the application of the model, which is linear - it includes only equations of straight lines written in Cartesian coordinates, which is why it is convenient for computer calculations.

A Dynamic Loading Mechanism for a Morphing Winglet

Abstract A loading mechanism for a variable cant angle morphing winglet is proposed in this paper. The mechanism provides a method of loading a morphing winglet with the simulated aerodynamic lift force by maintaining its application direction always orthogonal to the surface of the winglet throughout the range of motion. The loading mechanism is developed by type synthesis of four-bar linkages through enumeration of prismatic and revolute joints. The final design is a R-P-R-P four-bar linkage modified by merging the middle P-R joint together to form a planar joint named Shark-fin. The shape of Shark-fin is optimized to minimize its weight effect on the overall loading system. A half-scale prototype was built as a reference to address several implementation issues including the neutral loading condition. Afterwards, a full-size dynamic loading mechanism was successfully constructed and implemented to load test a full-size morphing winglet.

A generative model for path synthesis of four-bar linkages via uniform sampling dataset

Solving the problem of path synthesis for four-bar linkages via either analytical or numerical algorithms may entail issues such as mechanism defects and the need to guess at initial values. Recently, methods for solving such problems using neural network-based schemes show that these issues can be avoided. Despite the success in resolving the issues, there exist areas for further enhancement of the accuracy of the neural network-based scheme. In this work, a learning-based framework including preprocessing, data generation, and model training for the path synthesis of four-bar linkages is presented. The preprocessing starts by regenerating the target path with evenly distributed points along the path, followed by the normalization of the shape and feature extraction. For data generation, unsupervised learning, that is, K-means clustering, is employed to uniformly adjust the distribution of paths of different shapes in the dataset so that robustness of the model can be achieved. As for model training, models based on datasets of different classes of four-bar linkage as well as a classifier to determine the suitable generative model for the target path are constructed. Finally, several examples, including closed and open paths, are illustrated to verify the effectiveness of the framework.

Comparative Study on the Synthesis of Path-Generating Four-Bar Linkages Using Metaheuristic Optimization Algorithms

Four-bar linkages are one of the most widely used mechanisms in industries. This paper presents a comparative study on the accuracy and efficiency of the optimum synthesis of path-generating four-bar linkages using five metaheuristic optimization algorithms. The utilized metaheuristic methods included two swarm intelligence-based algorithms, i.e., particle swarm optimization and hybrid particle swarm optimization, and three evolutionary-based algorithms, i.e., differential evolution, ensemble of parameters and mutation strategies in differential evolution, and linearly ensemble of parameters and mutation strategies in differential evolution. The objective function to be minimized is the sum of squares of the distance between the generated points and the precision points of a coupler point. The optimal design of four-bar linkages must meet the Grashof’s criteria and exhibit sequential constraints that can prevent the occurrence of order defect. This study investigated five representative cases of the dimensional synthesis of four-bar path generators with and without prescribed timing and compared the optimal solutions of the utilized five metaheuristic methods to those of previously reported algorithms in literature. The improved metaheuristic methods exhibited superior optimal solution and enhanced reliability compared to the original methods. Moreover, three improved metaheuristic methods were not only easy implemented, but also more efficient for solving the optimal synthesis problems, particularly for high dimensional problems.

Optimal synthesis of four-bar linkages for path generation using the individual repairing method

Abstract. Constraints are one of the main factors that hinder the optimal synthesis of linkages to obtain the global optimum. At present, the most commonly used constraint handling method is called the penalty function (PF) method, but this technique has proven to be inefficient and unreliable in path synthesis. In this paper, a novel and effective constraint handling method called the individual repairing (IR) method is developed for the optimization model of four-bar mechanisms for path synthesis. The function of the IR method is to check the conformity of each individual to the Grashof condition, the sequence condition, and the variable range condition after population initialization and each iteration as well as to repair all unreasonable individuals according to the corresponding repairing technique to make them all satisfy the constraint conditions. In other words, the new constraint handling method establishes a transformation method so that each infeasible point in the search space uniquely corresponds to a feasible point. Based on the IR method, the optimization of path synthesis can be effectively carried out. Finally, the validity and reliability of the new constraint handling method are verified using three examples.

The Synthesis of Planar Four-Bar Linkage for Mixed Motion and Function Generation.

The synthesis of four-bar linkage has been extensively researched, but for a long time, the problem of motion generation, path generation, and function generation have been studied separately, and their integration has not drawn much attention. This paper presents a numerical synthesis procedure for four-bar linkage that combines motion generation and function generation. The procedure is divided into two categories which are named as dependent combination and independent combination. Five feasible cases for dependent combination and two feasible cases for independent combination are analyzed. For each of feasible combinations, fully constrained vector loop equations of four-bar linkage are formulated in a complex plane. We present numerical examples to illustrate the synthesis procedure and determine the defect-free four-bar linkages.

An algebraic input–output equation for planar RRRP and PRRP linkages

In this paper, the algebraic input–output (IO) equations for planar RRRP and PRRP linkages are derived by mapping the linkage displacement constraints into Study’s soma coordinates and then using tangent half-angle substitutions to transform the trigonometric into algebraic expressions. Both equations are found to be equivalent to the one that has already been derived for RRRR linkages, giving exciting new insight into kinematic analysis and synthesis of planar four-bar linkages. The algebraic properties of the IO curve equations yield information regarding the topology of the linkage, such as the sliding position limits of the prismatic joints and (or) the angle limits of the rotational joints. Additionally, the utility of the equations is successfully demonstrated with two approximate synthesis examples.

A fully analytical method for coupler-curve synthesis of planar four-bar linkages

Coupler-curve synthesis is a classical problem in linkage kinematics. A recent study shows that the solutions to the coupler-curve synthesis problems of four-bar linkages can be obtained based on a combination of analytical and graphical methods. In this work, a fully analytical method is developed to find the exact solutions of coupler-curve equations for the planar four-bar linkages. The developed method is based on a novel algebraic formulation of the system of coefficient equations, which yields a small system of four equations and four variables about the positions of pivoting joints. With any given algebraic coupler curve of a planar four-bar linkage, the position parameters of pivoting joints can be easily solved, while the other parameters can be calculated by the remaining equations. Two cases are provided to demonstrate and validate the fully analytical method.

Solution region synthesis methodology of planar eight-bar linkage for four specified positions of a 4R chain

The solution region methodology for solving the problem of four-bar linkage synthesis with four specified positions was extended to solve the problem of eight-bar linkage synthesis. The processes to build solution regions for synthesizing different types of eight-bar linkages are described, and the methods of building solution regions are divided into five types. First, the synthesis equation is derived, and the curve expressed by the synthesis equation is called the solution curve. Second, the process to build the spatial solution regions from the solution curves is detailed, and a new defect identification method is developed for building the spatial feasible solution region, which is a set of linkage solutions meeting four positions and excluding defects. Finally, linkage solutions that do not meet practical engineering requirements are eliminated from the spatial feasible solution region to obtain the useful spatial solution region. The examples demonstrate the feasibility of the proposed method. The proposed synthesis methodology is simple and easy to program, and provides reference for four specified position synthesis of other multi-bar linkages.

A frame-independent comparison metric for discrete motion sequences

To evaluate the kinematic performance of designed mechanisms, a statistical-variance-based metric is proposed in this article to measure the “distance” between two discrete motion sequences: the reference motion and the given task motion. It seeks to establish a metric that is independent of the choice of the fixed frame or moving frame. Quaternions are adopted to represent the rotational part of a spatial pose, and the variance of the set of relative displacements is computed to reflect the difference between two sequences. With this variance-based metric formulation, we show that the comparison results of two spatial discrete motions are not affected by the choice of frames. Both theoretical demonstration and computational example are presented to support this conclusion. In addition, since the deviation error between the task motion and the synthesized motion measured with this metric is independent of the location of frames, those corresponding parameters could be excluded from the optimization algorithm formulated with our frame-independent metric in kinematic synthesis of mechanisms, and the complexity of the algorithm are hereby reduced. An application of a four-bar linkage synthesis problem is presented to illustrate the advantage of the proposed metric.