Four-bar Motion Research Articles

The Synthesis of Spherical Four-bars for Biomimetic Motion through Complete Solutions for Approximate Rigid Body Guidance

Abstract In this paper, we form a constrained optimization problem for spherical four-bar motion generation. Instead of using local optimization methods, all critical points are found using homotopy continuation solvers. The complete solution set provides a full view of the optimization landscape and gives the designer more freedom in selecting a mechanism. The motion generation problem admits 61 critical points, of which two must be selected for each four-bar mechanism. We sort solutions by objective value and perform a second order analysis to determine if the solution is a minimum, maximum, or saddle point. We apply our approximate synthesis technique to two applications: a hummingbird wing mechanism and a sea turtle flipper gait. Suitable mechanisms were selected from the respective solution sets and used to build physical prototypes.

On the application of spatial four-bar motion and axode generation for the design of a prosthetic canine knee joint

The novelty of this work is that for the first time, spatial four-bar linkage motion generation and axode generation models have been applied for the design of a joint to replicate natural canine knee motion. This canine knee design method incorporates spatial four-bar Revolute-Revolute-Spherical-Spherical linkage motion generation and axode generation models to ultimately produce the rolling cam surfaces needed to produce natural spatial canine knee motion. This work presents the latest findings from an ongoing study where a transfemoral prosthetic knee design method is being applied to produce concept prosthetic knee joints for canines that replicate natural spatial knee motion during treadmill gait. To date, this study has considered three canine subjects, ranging from ages 1.1 to 10 years and treadmill speeds ranging from 1.3 to 2.9 km/h.

Genetic Algorithm for Sensitivity Analysis of Automobile Hood Four-bar Mechanism Synthesis Using Motion Generation

In four-bar motion generation, linkage dimensions are calculated to achieve prescribed coupler positions. This work investigates the sensitivity of four-bar coupler motion sequences by analyzing position error margins and implementing a genetic algorithm (GA) for four-bar motion generation. As an application, the four-bar hood mechanism in the Plymouth Satellite mid-size automobile is considered. The results of the sensitivity analysis show that the mean average structural error between the prescribed and achieved hood positions is less than 0.015in for any quadratic analysis. In each demonstration, the proposed method consistently produced results that are scalable for 360° within a margin error of 0.06in.

A dual cam system for four-bar motion generation with adjustable length crank and follower links

A dual cam system for four-bar motion generation with adjustable length crank and follower links

Alternative Constraint Handling Technique for Four-Bar Linkage Motion Generation

This paper proposes an extension of a new concept for four-bar path generation and a new penalty constraint handling technique from our previous work to a motion generation problem. The proposed technique could neglect the timing constraint for problems without prescribed timing, which is found to be an efficient technique for a four-bar path generation problem. Later, the remaining constraints are solved with a new constraint handling technique that also gives high performance. The technique is one kind of penalty function techniques. The present work proposes to combine the previous two methods for solving the four-bar motion generation problems. The comparative study, optimisation of motion generation problems with a new and a traditional penalty techniqueare solved by using self-adaptive population size teaching-learning based optimization (SAP-TLBO), is presented. In this study, three new motion generation test problems are used to test the proposed technique. The results show that the new technique gives acceptable results and can be applied with the motion generation problems without prescribed timing.

Synthesis of multiple tasks of a planar six-bar mechanism by wavelet series

ABSTRACTThis paper introduces a method for the synthesis of the multiple tasks of a planar six-bar mechanism by wavelet series. Based on the theory of wavelet series, a mathematical model for assessing the output of multiple tasks of planar linkage mechanisms was generated. The multiple tasks that combine a function generation and a rigid body guidance will be presented throughout this paper. For function generation, a numerical atlas database, including 31,775 groups of basic dimensional types of a planar slider-crank mechanism was created based on the relationships of the wavelet feature parameters. For rigid body guidance synthesis, an output feature parameters database of a planar four-bar motion generation, including 178,810 groups of the basic dimensional types was created by analysing the relationships of the wavelet coefficients between the rigid body rotation angle and the coupler rotation angle of its corresponding basic dimensional types. The actual sizes and installation position parameters of the planar six-bar mechanism were calculated using the fuzzy identification method and theoretical formulas. Finally, an application is presented to illustrate the validity and practicality of the proposed theory.

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

Non-integer-period motion generation of a planar four-bar mechanism using wavelet series

In this paper, on the basis of the theory of the Haar wavelet series, the early proposed method, which is used to solve the problem of planar four-bar path synthesis, is extended to motion generation synthesis of a planar four-bar mechanism. By analysing the wavelet coefficients between the rigid-body rotation angle and the connecting rod rotation angle of its corresponding basic dimensional type, the internal relationships are determined. Based on this finding, an output feature parameters database of planar four-bar motion generation including 101,408 sets of the basic dimensional type was established. Using the fuzzy identification method and theoretical formulas to compute the actual size and installing positions parameters of a planar four-bar mechanism, the synthesis of a planar four-bar motion generation mechanism can be achieved using the established numerical atlas, fuzzy identification method and theoretical formulas.

Developments in quantitative dimensional synthesis (1970–present): four-bar path and function generation

Dimensional synthesis is a type of inverse problem in linkage kinematics where the objective is to calculate the linkage dimensions required to achieve prescribed linkage output motion. It includes three distinct subcategories: motion generation, path generation and function generation. The authors’ first work surveyed developments in quantitative four-bar motion generation. This work concludes the presentation of developments in four-bar dimensional synthesis by surveying the developments made in quantitative four-bar path and function generation since 1970. While early path and function generation methods (like motion generation methods) were primarily qualitative, ongoing advancements in computing hardware and software continue to make quantitative motion generation more practical.

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.

Developments in quantitative dimensional synthesis (1970-present): four-bar motion generation

Dimensional synthesis is a type of inverse problem in linkage kinematics where the objective is to calculate the linkage dimensions required to achieve prescribed linkage output motion. Motion generation is a particular category of dimensional synthesis where the objective is to calculate the linkage dimensions required to achieve a group of prescribed link positions. In motion generation for a four-bar linkage, positions are prescribed for the coupler link. While early motion generation methods were primarily qualitative, ongoing advancements in computing hardware and software continue to make quantitative motion generation more practical. By providing overviews of works representative of developments in quantitative four-bar motion generation since 1970, this work is essentially an overview that spans over 40 years of developments in quantitative four-bar motion generation.

Optimal synthesis of four-bar motion generator linkages using circular proximity function

In this paper, a new objective function named circular proximity function (CPF) is proposed for the synthesis of four-bar motion generator linkages. This function is designed to measure the similarity of a group of points in Cartesian space to a circular curve. This function lowers the number of optimization variables to only two variables reducing the computational cost significantly. The optimization process has been carried out by the method of Differential Evolution (DE). Four case studies have been investigated in this study to demonstrate the advantages and disadvantages of the proposed method.

Spherical four-bar motion generation and axode generation in cam mechanism design

Spherical four-bar motion generation and axode generation in cam mechanism design

On Adjustable Planar Four-Bar Motion Generation With Order, Branch and Circuit Defect Rectification

This work is an incremental extension of adjustable planar four-bar kinematic synthesis theory to consider not only synthesis, but also the elimination of the defects inherent in synthesis. A nonlinear equation system for moving pivot-adjustable planar four-bar motion generation that includes constraints for order defect, branch defect and circuit defect elimination is presented in this work. In the objective function of the equation system, the error between the prescribed and achieved precision positions is minimized. The equation system includes inequality constraints to eliminate order defects and branch defects. The equation system also includes a complete planar four-bar displacement model to eliminate circuit defects.

Performance Maps for a Bio-Inspired Robotic Condylar Hinge Joint

This paper presents performance charts that map the design space of a bio-inspired robotic condylar hinge joint. The joint mimics the design of the human knee joint by copying the condylar surfaces of the femur and tibia and by copying the four-bar motion of the cruciate ligaments. Four aspects of performance are modeled: peak mechanical advantage, RMS (root mean square) mechanical advantage, RMS sliding ratio, and range of movement. The performance of the joint is dependent on the shape of the condylar surfaces and the geometry of the four-bar mechanism. The design space for the condylar hinge joint is large because the four-bar mechanism has a very large number of possible configurations. Also, it is not intuitive what values of design parameters give the best design. Performance graphs are presented that cover over 12,000 different geometries of the four-bar mechanism. The maps are presented on three-dimensional graphs that help designers visualize the limits of performance of the joint and visualize tradeoffs between individual aspects of performance. The maps show that each aspect of performance of the joint is very sensitive to the geometry of the four-bar mechanism. The trends in performance can be understood by analyzing the kinematics of the four-bar mechanism and the shape of the condylar surfaces.

On cam system design to replicate spatial four-bar mechanism coupler motion

The coupler motion of a planar four-bar mechanism can be reproduced by rotating the mechanism's moving centrode over its fixed centrode. For spatial four-bar mechanisms, coupler motion can be replicated by moving the mechanism's moving axode over its fixed axode in a screw motion – a combination of simultaneous rotations and translations. This study presents a model to calculate the fixed and moving axodes of the revolute–revolute–spherical–spherical (RRSS) mechanism – one of the most basic spatial four-bar mechanisms. The axodes are useful in producing the contact surfaces for a cam system to replicate RRSS coupler motion.

Planar Four-Bar Motion and Path Generation with Order and Branching Conditions

This work presents an optimization model for planar four-bar motion and path generation. This model includes the four-bar linkage kinematic model by Suh and Radcliffe(13) as an objective function. This kinematic model allows for the direct minimization of the error between the prescribed and achieved coupler parameters while synthesizing the crank and follower links in a closed loop. Both order and branching conditions are included among the constraint equations.

A General Static Torque Constraint for Spatial Four-Bar Motion Generation With a Coupler Load

A general static torque constraint for spatial four-bar mechanisms with coupler forces is formulated in this work. With this constraint, the user can synthesize spatial four-bar motion generators that also support static coupler loads. This constraint is demonstrated in the synthesis of RRSS, SSRC, and 4R spherical motion generators to approximate prescribed coupler poses and do so within maximum specified driver torques for given coupler forces.

On the synthesis of spatial revolute—revolute—spherical—cylindrical motion generators with prescribed coupler loads

This work extends the concepts and methodologies of Shen et al. for the synthesis of spherical four-bar motion generators to the synthesis of spatial revolute—revolute—spherical—cylindrical (RRSC) motion generators with applied coupler loads. A constraint that includes coupler loads and driver static torques for the RRSC mechanism is formulated using the principle of virtual work. This constraint is combined with the conventional constraints for the R—R and C—S dyads to form a non-linear optimization problem from which RRSC mechanism solutions are calculated that approximate prescribed coupler poses and satisfy prescribed driver static torques for given coupler loads. The formulated RRSC coupler load and driver static torque constraint are also incorporated in a simplified RRSC motion generation model to form a simplified non-linear optimization problem for calculating RRSC mechanism solutions. This work demonstrates both the conventional and simplified non-linear optimization problems.

AN EXTENSION OF AN ALGORITHM FOR PLANAR FOUR-BAR PATH GENERATION WITH OPTIMIZATION

This work is an extension of the authors’ published work on a planar four-bar motion generation search algorithm with Grashof, transmission angle and linkage perimeter conditions [1], This latest work considers planar four-bar path generation with a coupler point load, crank static torque, crank transverse deflection and follower buckling in a modified search algorithm. As demonstrated in the example, a conventional methodology used in kinematic path generation has been expanded to consider static loading, elastic deflection and buckling in path generation. These factors must be considered in mechanical design, but are not the focus in traditional kinematic synthesis.