Synthesis Of Planar Four-bar Linkages Research Articles

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.

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

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.

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.

MINLP models for dimensional synthesis of planar four-bar linkages

Most works on mechanism synthesis consider models that employ trigonometric functions and rely on derivative-free optimisation, such as bio-inspired and hybrid heuristics. Derivative-free strategies have the advantage of being applied directly to explore the design space. However, such strategies require tuning to achieve satisfactory results and do not provide certificates of solution quality. To this end, this work presents two alternative models: the first one consists of a trigonometric function-based model; while the second relies solely on natural coordinates, in which the nonlinear terms are bilinear. Both models result in mixed-integer nonlinear programming (MINLP) formulations. Mechanism constraints are conveniently expressed in this formulation, including Grashof's condition, prescribed timing, and increasing and decreasing angle sequence. In standard numerical cases of four-bar linkage path synthesis, global and local MINLP solvers yielded provably optimal designs in some cases and in others they outperformed works from the literature.

MINLP models for dimensional synthesis of planar four-bar linkages

Most works on mechanism synthesis consider models that employ trigonometric functions and rely on derivative-free optimisation, such as bio-inspired and hybrid heuristics. Derivative-free strategies have the advantage of being applied directly to explore the design space. However, such strategies require tuning to achieve satisfactory results and do not provide certificates of solution quality. To this end, this work presents two alternative models: the first one consists of a trigonometric function-based model; while the second relies solely on natural coordinates, in which the nonlinear terms are bilinear. Both models result in mixed-integer nonlinear programming (MINLP) formulations. Mechanism constraints are conveniently expressed in this formulation, including Grashof's condition, prescribed timing, and increasing and decreasing angle sequence. In standard numerical cases of four-bar linkage path synthesis, global and local MINLP solvers yielded provably optimal designs in some cases and in others they outperformed works from the literature.

A Motion Synthesis Approach to Solving Alt-Burmester Problem by Exploiting Fourier Descriptor Relationship Between Path and Orientation Data

This paper presents a generalized framework to solve m-pose, n-path-points mixed synthesis problems, known as the Alt-Burmester problems, using a task-driven motion synthesis approach. We aim to unify the path and motion synthesis problems into an approximate mixed synthesis framework. Fourier descriptors are used to establish a closed-form relationship between the path and orientation data. This relationship is then exploited to formulate mixed synthesis problems into pure motion synthesis ones. We use an efficient algebraic fitting based motion synthesis algorithm to enable simultaneous type and dimensional synthesis of planar four-bar linkages.

A Task-Driven Approach to Optimal Synthesis of Planar Four-Bar Linkages for Extended Burmester Problem

The classic Burmester problem is concerned with computing dimensions of planar four-bar linkages consisting of all revolute joints for five-pose problems. We define extended Burmester problem as the one where all types of planar four-bars consisting of dyads of type RR, PR, RP, or PP (R: revolute, P: prismatic) and their dimensions need to be computed for n-geometric constraints, where a geometric constraint is an algebraically expressed constraint on the pose, pivots, or something equivalent. In addition, we extend it to linear, nonlinear, exact, and approximate constraints. This extension also includes the problems when there is no solution to the classic Burmester problem, but designers would still like to design a four-bar that may come closest to capturing their intent. Machine designers often grapple with such problems while designing linkage systems where the constraints are of different varieties and usually imprecise. In this paper, we present (1) a unified approach for solving the extended Burmester problem by showing that all linear and nonlinear constraints can be handled in a unified way without resorting to special cases, (2) in the event of no or unsatisfactory solutions to the synthesis problem, certain constraints can be relaxed, and (3) such constraints can be approximately satisfied by minimizing the algebraic fitting error using Lagrange multiplier method. We present a new algorithm, which solves new problems including optimal approximate synthesis of Burmester problem with no exact solutions.

A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

This paper studies the problem of planar four-bar motion generation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the image space of planar displacements, we obtain a class of quadrics, called generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using singular value decomposition (SVD). The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

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.

A novel analytical method for function generation synthesis of planar four-bar linkages

In this paper, a novel analytical approach is presented to solve function generation synthesis problems for planar four-bar linkages with no limitations on the number of precision points. Firstly, the output rotation-angle function is formulated according to the Fourier series components of the input angle, and is substituted into a vector loop equation. Then, the relationship between the design parameters of the linkage and the Fourier coefficients of the output function is obtained. On the basis of this, the new design equations are established. Finally, a cubic polynomial equation is obtained by dialytic elimination. Three non-trivial solutions for the new design equations are ultimately obtained analytically. Three examples are provided to verify the effectiveness of the proposed technique. As a result, a general formula is derived for calculation of the design parameters of a planar four-bar linkage using the Fourier coefficients of the prescribed function.

On the Extension of a Fourier Descriptor Based Method for Planar Four-Bar Linkage Synthesis for Generation of Open and Closed Paths

In an earlier work, we have presented an efficient method for synthesizing crank-rocker mechanisms that are capable of generating perceptually simple and smooth low harmonic closed curves. In this paper, we seek to extend this approach to the synthesis of planar four-bar linkages for the generation of open curves. Instead of using Fourier transform that requires a function to be defined over the entire period, we combine finite Fourier series in a curve fitting scheme for the approximation of periodic as well as nonperiodic paths. This yields a general method for planar four-bar path generation that is applicable to both closed and open paths.

A COMPREHENSIVE SOLUTION OF THE CLASSIC BURMESTER PROBLEM

The classic Burmester problem aims at finding the geometric parameters of a planar four-bar linkage whose coupler link attains a prescribed set of finitely separated poses. The solution proposed is claimed to be comprehensive because it (a) includes all four types of dyads—RR (revolute-revolute), PR (prismatic-revolute), RP and PP—and (b) gives due consideration to the numerics behind the solution. A PR dyad is treated as a RR dyad with its fixed joint centre at infinity, similar interpretations applying to RP and PP dyads. The paper includes the synthesis of planar four-bar linkages in its full generality, that of dyads with P joints being given the utmost attention. Finally, the underlying numerics receives the attention seldom found in the literature on the subject, our main concern being numerical robustness.

Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming

This paper presents an original approach to the kinematic synthesis of planar mechanisms for finitely separated positions. The technique, referred to here as geometric constraint programming, uses the sketching mode of commercial parametric computer-aided design software to create kinematic diagrams. The elements of these diagrams are parametrically related so that when a parameter is changed, the design is modified automatically. Geometric constraints are imposed graphically through a well-designed user interface, and numerical solvers integrated into the software solve the relevant systems of equations without the user explicitly formulating those equations. This allows robust algorithms for the kinematic synthesis of a wide variety of mechanisms to be “programmed” in a straightforward, intuitive manner. The results provided by geometric constraint programming exhibit the accuracy and repeatability achieved with analytical synthesis techniques, while simultaneously providing the geometric insight developed with graphical synthesis techniques. The key advantages of geometric constraint programming are that it is applicable to a broad range of kinematic synthesis problems, user friendly, and highly accessible. To demonstrate the utility of the technique, this paper applies geometric constraint programming to three examples of the kinematic synthesis of planar four-bar linkages: Motion generation for five finitely separated positions, path generation for nine finitely separated precision points, and function generation for four finitely separated positions.

Approximate Synthesis of Spatial Linkages

Freudenstein’s approximate synthesis of planar four-bar linkages is generalized to spatial linkages, and the conditions under which this generalization is applicable are expressed. Three cases of synthesis of spatial linkages to generate functions of one variable between nonparallel axes are considered in detail: (a) The spherical four-bar linkage; (b) a variation of the four-bar linkage in which two turning pairs are replaced by ball-and-socket joints, a linkage which may be designed to generate arbitrary functions with up to seven accuracy points; and (c) a second variation of the four-bar linkage where three turning pairs are replaced by cylinder pairs, a linkage capable of being designed to generate a variable-pitch helical motion with three accuracy points.