Four-bar Linkage 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.

A Matrix-Based Approach to Unified Synthesis of Planar Four-Bar Mechanisms for Motion Generation With Position, Velocity, and Acceleration Constraints

Abstract This paper introduces a novel matrix-based approach for the simultaneous type and dimensional synthesis of planar four-bar linkage mechanisms, accommodating various practical constraints, including position, velocity, acceleration, and joint placements. Traditional design processes segregate type synthesis, the determination of joint and link configurations, from dimensional synthesis, which involves specifying link sizes and pivot locations. This segregation often leads to complexities in addressing the complete design challenge. The novel methodology proposed in this paper departs from the conventional sequential design approach by concurrently evaluating type and dimensional parameters using a data-driven matrix formulation. The crux of the paper’s methodology involves formulating a singular design equation through a transformation matrix, parameterized by the Cartesian parameters of the mechanism’s dyads. This formulation linearly expresses a broad range of constraints, facilitating the identification of viable solutions through singular value decomposition and null space analysis. This integrated approach not only simplifies the synthesis process but also provides direct insights into the mechanism’s parameters, encompassing both type and dimensions, thereby obviating the need for further interpretative steps common to the use of quaternions and kinematic mapping. In essence, the paper presents two main contributions: the development of a unified design equation capable of encompassing a wide array of constraints within the mechanism synthesis process, and the introduction of an algorithm that effectively identifies all potential planar four-bar linkage mechanisms by accurately satisfying up to five constraints. This approach promises to enhance the design and optimization of mechanical systems by offering a more holistic and efficient pathway to mechanism synthesis.

Experimental Structural Design of a Novel Variable-Sweep Wing Based on a Four-Bar Planar Linkage

Experimental Structural Design of a Novel Variable-Sweep Wing Based on a Four-Bar Planar Linkage

Design and analysis of folding mechanism for automotive seats based on space maximization

To solve the problem of maximizing space utilization of car folding seats within limited spatial constraints, a design method for space-maximizing four-bar folding seat hinges is proposed. First, we will investigate the existing types of car folding seats, identify the mainstream configurations, and determine the optimal position of the four-bar linkage hinge point based on the constraints present within the car using theoretical analysis methods. Secondly, kinematic and dynamic models of the four-bar linkage mechanism are established, followed by the formulation of kinematic and dynamic equations. Finally, numerical methods are employed to solve these equations and obtain the kinematic and dynamic characteristics. The results indicate that the design method proposed in the article, which aims to maximize the space of the hinge four-bar folding seat, is efficient and feasible. This approach optimizes the utilization of interior space and enhances passenger comfort while accommodating the folding function of the seat within the existing vehicle environment. This method holds significant engineering implications for the design of folding seats in similar types of vehicles.

Trajectory tracking control of a four-bar linkage prosthetic knee with magnetorheological damper

In order to improve biomimicry, a four-bar linkage prosthetic knee with magnetorheological (MR) damper was developed in this paper. The principle and structure of four-bar linkage MR prosthetic knee was introduced and its kinetic model was established. A MR damper matching the proposed prosthetic knee was designed, fabricated and tested, and the forward and inverse mechanical model were conducted. In addition, to solve the problems of complex model, poor bionics and low trajectory tracking accuracy in the MR prosthetic knee, a PD-type iterative learning control (ILC) controller was developed. The simulated and experimental test results show that the positive maximum error of the knee joint swing trajectory based on PD-type ILC controller are 2.2°(speed of 0.5 m/s), 4.1°(speed of 1.0 m/s), and 1.9°(speed of 1.5 m/s), respectively, and the negative maximum error of that are −1.4°(speed of 0.5 m/s), −6.7°(speed of 1.0 m/s), and −5.5°(speed of 1.5 m/s), respectively, which effectively tracks the reference swing trajectory.

Design and Simulation of an Innovative Modular Four-Bar Linkage Robotic Arm: Analysis and Applications

In response to the issues of traditional manipulators, such as difficulty in controlling the terminal posture during motion, low control precision, and complex structure, a modular mechanical arm based on a parallel four-bar terminal posture maintenance mechanism has been designed to meet the special requirements of terminal posture maintenance. By analyzing the motion mechanism of the manipulator, the shape space of the manipulator, the terminal position space, and the relationships of forward and inverse kinematics have been derived. A simulation model of the linkage manipulator was established in the Robotic Toolbook for Matlab environment, and structural simulation analysis and kinematic equation verification were conducted. The end positioning and attitude control of different manipulators are simulated by Matlab programming. The working space of the manipulators is calculated by Monte Carlo method, and compared with that of traditional planar manipulators. The results show the validity of the kinematics equation and the rationality of the end positioning control scheme of the linkage manipulator. Finally, the experiment verifies that the attitude retention rate of the modular decoupling manipulator is 97.78% under high load, which verifies the reliability of this scheme. In addition, the manipulator can simplify the complexity of the mechanical structure and complete the control requirements of the terminal attitude under the premise of increasing the working space as much as possible, which lays a foundation for the design and optimization of the manipulator in the field of modular terminal attitude.

Deep generative model-based synthesis framework of four-bar linkage mechanisms with target conditions

Abstract This paper proposes a deep generative model-based framework for synthesizing four-bar linkage mechanisms that satisfy specified kinematic and quasi-static conditions. We define two objective functions for crank-rocker mechanisms using kinematic workspaces and geometric configurations. Our approach utilizes a conditional generative adversarial network (cGAN) modified for mechanism synthesis, which learns the relationship between mechanism requirements and linkage lengths. The results demonstrate that the proposed model successfully generates multiple distinct mechanisms meeting specific kinematic and quasi-static requirements. We compare our cGAN approach to traditional optimization methods and other deep learning-based generative models. Our method offers several advantages over traditional design approaches, enabling efficient generation of diverse yet feasible design candidates while exploring a large design space. By considering both kinematic and quasi-static requirements, the proposed model can produce more effective mechanisms for real-world applications. This makes it a promising tool for linkage mechanism design, offering designers a way to efficiently generate multiple viable design options that satisfy key performance criteria.

Eight-Bar Elbow Joint Exoskeleton Mechanism

This paper deals with the design and kinematic analysis of a novel mechanism for the elbow joint of an upper-limb exoskeleton, with the aim of helping operators, in terms of effort and physical resistance, in carrying out heavy operations. In particular, the proposed eight-bar elbow joint exoskeleton mechanism consists of a motorized Watt I six-bar linkage and a suitable RP dyad, which connects mechanically the external parts of the human arm with the corresponding forearm by hook and loop velcro, thus helping their closing relative motion for lifting objects during repetitive and heavy operations. This relative motion is not a pure rotation, and thus the upper part of the exoskeleton is fastened to the arm, while the lower part is not rigidly connected to the forearm but through a prismatic pair that allows both rotation and sliding along the forearm axis. Instead, the human arm is sketched by means of a crossed four-bar linkage, which coupler link is considered as attached to the glyph of the prismatic pair, which is fastened to the forearm. Therefore, the kinematic analysis of the whole ten-bar mechanism, which is obtained by joining the Watt I six-bar linkage and the RP dyad to the crossed four-bar linkage, is formulated to investigate the main kinematic performance and for design purposes. The proposed algorithm has given several numerical and graphical results. Finally, a double-parallelogram linkage, as in the particular case of the Watt I six-bar linkage, was considered in combination with the RP dyad and the crossed four-bar linkage by giving a first mechanical design and a 3D-printed prototype.

Analyzing Higher-Order Curvature of Four-Bar Linkages with Derivatives of Screws

Analyzing Higher-Order Curvature of Four-Bar Linkages with Derivatives of Screws

Continuously Variable Transmission and Stiffness Actuator Based on Actively Variable Four-Bar Linkage for Highly Dynamic Robot Systems

Continuously Variable Transmission and Stiffness Actuator Based on Actively Variable Four-Bar Linkage for Highly Dynamic Robot Systems

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.

Trajectory Synthesis and Linkage Design of Single-Degree-of-Freedom Finger Rehabilitation Device

For injured and after-stroke patients who temporarily lose their hand’s grasping abilities, assisting them in regaining their index finger mobility is very important in the rehabilitation process. In this paper, a finger rehabilitation device based on one degree-of-freedom (DOF) linkage mechanism is designed, aiming to lead the index finger through the flexion–extension trajectory during grasping tasks. Two types of one-DOF mechanisms, a four-bar linkage and a Watt-I six-bar linkage, are synthesized for the task trajectory. Various algorithms such as PSO, GA, and GA–BFGS are adopted and compared for the synthesis of these two types of mechanisms, among which the Watt-I six-bar linkage obtained with GA–BFGS shows the optimal performance in accuracy. Clinical biomechanical data are utilized to perform static analyses of the mechanisms, and the feasibility of the Watt-I six-bar linkage models is tested, compared, and demonstrated. Finally, the prototype of the six-bar linkage as well as a wearable exoskeleton finger rehabilitation device are designed to show how they are applied in the finger rehabilitation scenario.

Bi-level game theoretic approach for robust design: A case study of path-generating four-bar

This study addresses the bi-level multi-objective optimization problems (MOP) that raise in robust design and optimization of engineering systems through establishing a state-of-the-art game theoretic scenario. A novel leader-follower decentralized decision-making scenario is proposed, leveraging the synergy of game theory, Robust Design Optimization (RDO), Monte Carlo Simulation (MCS), and Artificial Intelligence (AI). The proposed algorithm can be employed for optimum robust Pareto design of a wide range of dynamical systems. In order to achieve a robust design, both the mean and variance of each objective function are considered as players in a multi-agent game setting. In this approach, both Stackelberg and cooperative games are utilized to model the behaviors of the players. Genetic Programming (GP) meta-models are employed to capture the Stackelberg protocol between two levels specifically for constructing the follower's rational reaction set (RRS). Additionally, the Nash bargaining function is -utilize to model the cooperative behaviors among players in each level. The proposed approach is applied and demonstrated through a case study involving multi-objective robust design of planar four-bar linkages. In this manner, four objective functions are assigned to four players within the system. Each player is responsible for optimizing a specific objective criterion, namely the mean of tracking error (TE), variance of tracking error, mean of transmission angle and variance of transmission angle (TA) of the linkage. As a result, the four-objective optimization problem of mechanism is transformed into a single-objective robust synthesis problem. The comparisons of the results show a significant enhancement in the robust behavior of the linkage, while ensuring that deterministic criteria such as quality of motion and precision are preserved.

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.

Design and kinematic analysis of a deployable mechanism for soft sheets grasping

In the automation process of the garment industry, a notable challenge arises from the presence of flexible materials characterized by irregular shapes, posing difficulties for robotic grasping. This article addresses this challenge by proposing a gripper manipulator centered around the 4 P-RRP quadrilateral cone folding mechanism as the core driving mode, with the angled four-bar linkage mechanism serving as the extension arm. Utilizing screw theory, the developability of the 4 P-RRP four-pyramid mechanism with one degree of freedom is verified, forming the basis for its adoption as the central driving mechanism. Furthermore, an in-depth analysis of the geometric properties of the robot’s extension arm leads to the derivation of an optimal folding ratio structure, which is seamlessly integrated into the robot design. Kinematic equations for the mechanism are formulated and solved using the D-H parameter method to obtain the kinematic solution. The investigation demonstrates that the manipulator utilizing the quadrilateral cone folding mechanism significantly enhances adaptability to irregular shapes. By incorporating this proposed mechanism, robots can efficiently manipulate flexible materials across various garment manufacturing applications.

Four-bar linkage reconfigurable robotic wheel: Design, kinematic analysis, and experimental validation for adaptive size modification

Abstract This article presents the development of a robot capable of modifying its size through a wheel reconfiguration strategy. The reconfigurable wheel design is based on a four-bar retractable mechanism that achieves variation of the effective radius of the wheel. A reconfiguration index is introduced based on the number of retractable mechanisms that predicts the radius of configuration according to the number of mechanisms implemented in the wheel. The kinematics of the retractable mechanism is studied to determine the theoretical reconfiguration radius during the transformation process, it is also evaluated numerically with the help of the GeoGebra software, and it is validated experimentally by image analysis using the Tracker software. The transformation process of the robot is investigated through an analysis of forces that consider the wheel in contact with the obstacle, the calculation of the wheel torque and the height of the obstacle to be overcome are presented. On the other hand, the experimental validation of the robot reconfiguration process is presented through the percentage of success shown by the robot to overcome obstacles of 50, 75, 100 and 125 mm. In addition, measurements of energy consumption during the transformation process are reported. Reconfigurable wheels, capable of adapting their size, offer innovative solutions to various challenges across different applications such as robotic exploration and search and rescue missions to industrial settings. Some key issues that these wheels can address include terrain adaptability enhancing a robot’s mobility over uneven surfaces, or obstacles; enhanced robotic design; cost-effective design; space efficiency; and versatility in applications.

Hybrid origami-linkage vaulted deployable structures: parametric development and form-finding

AbstractThe demand for adaptable structures is steadily increasing in contemporary architectural practice. Deployable structures, characterized by their flexibility and versatility, offer innovative solutions across demanding engineering and architectural contexts. Among the various types of deployable structures, two main categories emerge: lattice deployable structures and continuous surface deployable structures. This paper explores the creation of hybrid deployable vaulted structures by incorporating kinetic elements from both deployable structure categories. A combination of rigid thick origami surfaces and retractable structural frames composed of four-bar and scissor-hinged linkages is examined, while both direct and inverse design methodologies are explored. To facilitate the parameterization and kinematic study of the proposed structures, an algorithm is developed within the Rhino Grasshopper visual programming environment, which is then utilized for form-finding and design optimization of the generated deployable vaults. The present study aims to contribute to the evolution of lightweight, kinetic elements for versatile architectural applications. The proposed hybrid deployable structures are designed to adjust to evolving spatial and functional requirements while maintaining structural integrity.

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

An Improved Fourier-Based Method for Path Generation of Planar Four-Bar Linkages without Prescribed Timing

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 issues with time parameters assignment for path generation without prescribed time in previous studies. In this paper, an improved Fourier-based point-to-point combination method is presented, which generates more points by Fourier approximation and assigns the time parameters to the given points while allowing discarding solutions with order defects. This method relies on the Fourier coefficients, improving the accuracy of synthesis solutions, and simplifying the computational procedure. Time parameters are assigned directly to the given points, which avoids the complex calculations to find intersection points in the given path, eliminates combinations that would lead to solutions with order defects, and simplifies the assessment process of synthesis results. The parameters obtained by the point-to-point combination method can be used as the parameters of the input dyad, skipping the first set of design equations for faster calculation. Several examples are presented to demonstrate the advantages of the proposed synthesis method, which is easy-understanding, computationally efficient, and yields more accurate solutions than available synthesis methods.