Coupler Point Research Articles

Dimensional synthesis of motion generation of a planar four-bar mechanism

A motion generation method for a planar four-bar mechanism without prescribed timing is proposed in this article. A characteristic of coupler points is found: for the coupler points of a four-bar mechanism in a standard installation position rotated by the corresponding input angle clockwise around the origin of the Cartesian coordinate system xOy, the generated points lie on the feature coupler circles. Next, based on this, the design process is divided into three steps. In the first step, the relative input angles, the relative coupler angles and five geometric parameters of the desired four-bar mechanism are optimized. In the second step, the basic dimensional types and initial input angle are determined, and then, the input angles are obtained. In the third step, the real size of the desired linkage mechanism is calculated. Because the dimension of the design variables in each step is exceptionally small, the motion synthesis method can yield design solutions with high precision in a short time. Six comparison examples are presented to demonstrate the efficacy of the proposed method. In addition, the method is used to design a robot for lower limb rehabilitation for application to the human foot.

Using instantaneous invariants of the curvature theory in motion generation synthesis of 4R mechanisms with higher-order coupler point kinematics

This paper aims at exploiting the unique characteristics of instantaneous invariants and pole locations in synthesizing motion generation four bar (4R) planar mechanisms with higher-order coupler point kinematics (i.e. specifying coupler point velocity and/or acceleration in addition to its position and coupler orientation). The proposed methodology targets to ease the complexity of such design problems by reducing the number of non-linear governing equations that are solved to generate the desired dyads of the final 4R mechanism design. The new approach integrates graphical and analytical techniques to synthesize 4R mechanisms with prescribed coupler point position, velocity, and acceleration. Two case studies are presented; the first presents the synthesis of a motion generation 4R with 3 prescribed coupler point positions and 2 prescribed coupler point velocities; the second presents the synthesis of a motion generation 4R with 2 prescribed coupler point positions, 2 prescribed coupler point velocities, and 1 prescribed coupler point acceleration. When compared to classical kinematic-based approaches, the proposed method prominently reduces the size and complexity of the set of governing non-linear equations.

Dimensional synthesis of motion generation in a spherical four-bar mechanism

A new objective function for optimizing spherical, four-bar mechanisms for motion generation is presented. Rigid-body poses of a spherical four-bar mechanism in a standard installation position are investigated, and a normalization processing method is proposed. After processing the coupler points of the mechanism, the feature points generated are located on the circles. The formation principle of the feature coupler circles is analysed, and the internal relationship between the centre angle of the adjacent feature points on the feature coupler circles and the coupler angle of the spherical four-bar mechanism is determined. An input angle determination method is proposed for the spherical four-bar mechanism in a general installation position. An objective function is then established to optimize the basic dimensional types, relative input angles, and installation position parameters of the desired spherical four-bar mechanism. The proposed method is applicable to both prescribed and unprescribed timing problems. Notably, the dimension of the optimization variables is only eight, which is exceptionally low for multiple position motion generation without prescribed timing; therefore, the global optimal solution is more easily obtained. The optimization process is carried out by the genetic algorithm. The feasibility and effectiveness of our proposed method are demonstrated by examples.

Neural Network Based Control of Four-Bar Mechanism with Variable Input Velocity

For control applications, the angular velocity of the drive crank of a four-bar mechanism is traditionally assumed to be constant. In this paper, we propose control of variable velocity of the drive crank to obtain the desired output motions for the coupler point. To estimate the reference trajectory for the crank velocity, a neural network is trained with data from the kinematic model. The control law is designed from feedback linearization of the tracking error dynamics and a Proportional–Integral–Derivative (PID) controller. The applicability of the proposed scheme is validated through simulations for three variable speed profiles, obtaining excellent results from the system.

Higher order kinematic formulas and its numerical computation employing dual numbers

In the analysis of robots and mechanisms, the computation of velocities and accelerations is frequently needed. In other cases, higher-order kinematic formulas such as the jerk and the jounce/snap—the third and fourth-order time derivative of the position vector, respectively—are also needed. While procedures for computing velocity and accelerations can be found elsewhere in the literature, the computations of the jerk and especially of the jounce/snap are not so common. In this study, a novel formulation to compute all kinematic quantities up to the fourth order, based on the dual number is presented. In dual number formalism, kinematic analysis equations are computed automatically once the position vector is constructed. This provides a concise and efficient method to conduct kinematic analysis. To prove the validity of the implemented formulation, we compute the velocity, acceleration, jerk, and jounce/snap for the end effector of the RC robot manipulator. In addition, we compute the jerk and the jounce/snap for the coupler point of the spherical 4 R mechanism.

Nature-Inspired Intelligent Synthesis of a Spherical Mechanism for Passive Ankle Rehabilitation Using Differential Evolution

The ankle rehabilitation in certain injuries requires passive movements to aid in the prompt recovery of ankle movement. In the last years, parallel ankle rehabilitation robots with multiple degrees of freedom have been the most studied for providing such movements in a controlled way. Nevertheless, the high cost does not make it viable for home healthcare. Then, this paper presents an optimization approach where a spherical mechanism of one-degree-of-freedom is proposed as a low-cost ankle rehabilitation device to provide the passive rehabilitation exercise for plantar flexion/dorsiflexion and adduction/abduction ankle movements. The approach is formulated as a mono-objective constraint optimization problem where the relative motion angle of the mechanism, the Grashof criterion, the force transmission, and the rehabilitation routine are included. The link lengths of the mechanism parameterized in Cartesian coordinates are found by the two most representative differential evolution variants. The statistical analysis of optimizers indicates that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$DE/rand/1/bin$ </tex-math></inline-formula> finds, on average, more promising solutions through algorithm executions than the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$DE/best/1/bin$ </tex-math></inline-formula> . The numerical simulation results and the motion simulation of the CAD model illustrate the obtained ankle rehabilitation mechanism, indicating that the percentage error between the desired rehabilitation path and the curve generated by the coupler point of the mechanism is in the interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[{0.036,0.437}]\%$ </tex-math></inline-formula> . Manufacturing the ankle rehabilitation mechanism with a 3D printer validates the optimization approach and verifies the resulting mechanism.

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.

Path Analysis for Hybrid Rigid–Flexible Mechanisms

Hybrid rigid–flexible mechanisms are a type of compliant mechanism that combines rigid and flexible elements, being that their mobility is due to rigid-body joints and the relative flexibility of bendable rods. Two of the modeling methods of flexible rods are the Cosserat rod model and its simplification, the Kirchhoff rod model. Both of them present a system of differential equations that must be solved in conjunction with the boundary constraints of the rod, leading to a boundary value problem (BVP). In this work, two methods to solve this BVP are applied to analyze the influence of external loads in the movement of hybrid compliant mechanisms. First, a shooting method (SM) is used to integrate directly the shape of the flexible rod and the forces that appear in it. Then, an integration with elliptic integrals (EI) is carried out to solve the workspace of the compliant element, considering its buckling mode. Applying both methods, an algorithm that obtains the locus of all possible trajectories of the mechanism’s coupler point, and detects the buckling mode change, is developed. This algorithm also allows calculating all possible circuits of the mechanism. Thus, the performance of this method within the path analysis of mechanisms is demonstrated.

A General Method for Constructing Planar Cognate Mechanisms

Abstract Cognate linkages are mechanisms that share the same motion, a property that can be useful in mechanical design. This article treats planar curve cognates, that is, planar mechanisms with rotational joints whose coupler points draw the same curve, as well as coupler cognates and timed curve cognates. The purpose of this article is to develop a straightforward method based solely on kinematic equations to construct cognates. The approach computes cognates that arise from permuting link rotations and is shown to reproduce all of the known results for cognates of four-bar and six-bar linkages. This approach is then used to construct a cognate of an eight-bar and a ten-bar linkage.

Measuring curvature of trajectory traced by coupler of an optimal four-link spherical mechanism

This research paper presents design of a gripper finger mechanism and a new curvature measurement method of fingertip trajectory. Firstly, the conditions of a grasp provided by gripper, which was designed from a spherical mechanism, was explained. Therefore, trajectory length of coupler point was modeled as objective function which is maximized. Ball-Burmester theory was used to form constraints which can provide a spherical curve with curvature close to zero. A plausible dimension range of mechanism, transmission angle, instantaneous invariants were arranged as constraint function. A meta-heuristic algorithm previously developed by one of authors was used to search the best instantaneous invariants based on the type of mechanism. The optimal solution was prototyped, and three-fingered gripper assembled. Secondly, an endoscopic camera peculiarly was used to measure curvature of the trajectory traced by coupler point of an optimal four-link spherical mechanism which was supposed to compose a finger of a robotic gripper. The coupler of the mechanism was expected to trace approximately a great circle arc to provide a grasp. Therefore, the technique to measure curvature was based on image processing and geodetic curvature theory. A calibrated endoscopic camera was used for acquisition of video record above the prototype which was centered in the center of the camera screen. The curvature of coupler point of optimal spherical mechanism design and prototype were compared. According to performed regression analysis, results of the measurement were %97 reliable for the useful part of the trajectory.

Mechanism analysis of a main landing gear of transporting aircraft: A design learning perspective

The landing gear lifetime is estimated based on the safe-life design concept and its design requires thorough calculations. The individual parts and sub-assemblies must be sensibly designed to function safely and efficiently for the specified design lifetime without failure. In a design process of a main landing gear system, the kinematics/mechanisms analysis shall be first carried out as a primary task for feasibility of the gear retraction/extension with the satisfactory link paths. This paper presents the analytical kinematics analysis of a complex mechanism involving multiple four-bar linkages with multiple coupler points of the main landing gear (MLG) of CN-235 aircraft. The kinematics analyses were performed in order to figure out the overall link positions of the main landing gear mechanism and make scenarios of a deformed component from its designed geometry for possibilities of a lock failed position.

Comprehensive approach for the dimensional synthesis of a four-bar linkage based on path assessment and reformulating the error function

The optimal dimensional synthesis methodology proposed in this article does not follow the more common approaches of recent years, such as heuristic methods, but aims to perfect one of the earlier and efficient approaches—gradient methods. The four-bar linkage in its general form is applied, analyzing crank and rocker input, ensuring the nonexistence of circuit and order defects. The procedure for obtaining the optimal mechanism is based on a characterization of the circuits and branches of the path traced by the coupler point, establishing the corresponding actuation mode based on a series of indices that avoid the use of penalty functions. The proposed optimization methodology defines a new error function that includes an additional characteristic, the slope of the tangent to each point of the path. Including the tangent enables better fitting to the desired curve without sacrificing the possibility of calculating the gradient analytically, and obtaining, in general, better results than by defining additional points. Although this paper focuses primarily on the dimensional synthesis of path generation, the bases of the procedure have been also applied to rigid body motion, as shown in the illustrative examples.

Appropriate synthesis of the four-bar linkage

The uncertainties arising from the fabrication and operation of mechanisms, specifically the four-bar linkage, significantly affect the expected performance of the mechanism. To accommodate uncertainties during the dimensional synthesis of the four-bar linkage, the desired coupler curve response is described by any number of precision point elements and trajectory elements, where each element is specified with an allowable error. A design description which accounts for bounded uncertainties is termed an appropriate design. An appropriate synthesis method is developed to synthesize the entire set of appropriate design solutions corresponding to the desired response. This method is able to completely explore the continuous design space and each synthesized appropriate design guarantees that the resulting four-bar linkage has a corresponding coupler point which lies inside each precision point response element, and a corresponding set of continuous coupler points which remain inside the trajectory response elements from start to finish.

Optimal Synthesis of six bar Mechanism using Particle Swarm Optimization

This paper presents the optimal synthesis of a six-bar Watt-II mechanism with rotational joints in which a tracing point on the coupler traces a desired path. The objective of the work is to synthesize a six-bar mechanism whose coupler point generates the path as close to the given path as possible. The tracing path is the combination of two circular arcs with eighteen precision points with prescribed timing. A mathematical model is formed by two vector loop closure equations, and design constrained are imposed to improve the result quality. ‘Particle Swarm Optimization’ (PSO), an evolutionary technique, is used to obtain the optimal solution of the given six bar mechanism at first time, which has been used by various authors in case of four bar mechanism. Final design parameters and convergence rate of the optimal solutions are presented.

Optimal Design and Redundancy Resolution of a Novel Robotic Two-Fingered Exoskeleton

This article deals with the optimal design and redundancy resolution of a seven DOF robotic two-fingered hand exoskeleton intended for rehabilitation purposes. The exoskeleton is designed to track the human digit motion accurately. As the human digit joint cannot be modeled by single revolute joint due to instantaneously varying center of rotation, a 4-bar mechanism is employed to model each phalanx of a finger. Optimal 4-bar linkages are designed by minimizing the error between the Cartesian trajectories of the human phalanx and those of the coupler points in respective 4-bar mechanisms. It is demonstrated that the designed 4-bar based exoskeleton can track the human digit motion accurately. Performance analysis of the developed device has been carried out by resolving its kinematic redundancy during a fine object translation motion by instantaneously optimizing manipulability measure.

Design Improvement of Four Bar Linkages using Different Soft-Computing Methods

Modeling of optimized flexible four bar mechanism considering joint clearances are developed at first. The dynamic analysis such as Moment, Torque, Energy and structural analysis such as Deformation, different mode of Stress are done by considering load in coupler point of optimized flexible four bar linkage. The deformation and stress are compared by considering different materials. Lagrange's equation is used to define the direction of the joint clearance. Differential evolution algorithm is used to optimize the length of links which reduce the gap between the desirable and actual path.

Synthesis Theory and Optimum Design of Four-bar Linkage with Given Angle Parameters

Abstract. In this paper, a synthesis method is proposed for the 5-point-contact four-bar linkage that approximates a straight line with given angle parameters. The given parameters were the angles and the location of the Ball point. Synthesis equations were derived for a general Ball–Burmester point case, the Ball–Burmester point at an inflection pole, and the Ball point that coincided with two Burmester points, resulting in three respective groups of bar linkages. Next, taking Ball–Burmester point as the coupler point, two out of the three bar-linkage combinations were used to generate three four-bar mechanisms that shared the same portion of a rectilinear trajectory. Computation examples were presented, and nine cognate straight-line mechanisms were obtained based on the Roberts-Chebyshev theory. Considering that the given parameters were angles which was arbitrarily chosen, with the other two serving as the horizontal and vertical axes, so the solution region graphs of the solutions for three mechanism configurations were plotted. Based on these graphs, the distribution of the mechanism attributes was obtained with high efficiency. By imposing constraints, the optimum mechanism solution was straightforwardly identified by the designers. For the angular parameters prescribed in this paper, the solutions for three straight-line mechanism configurations were obtained, along with nine cognate straight-line mechanisms that shared the same portion of the rectilinear trajectory. All the fixed pivot installation locations and motion performances differed, thus providing multiple solutions to the trajectory of the synthesis of mechanisms.

Appropriate analysis of the four-bar linkage

Uncertainties are inherent in the fabrication and operation of mechanisms. In terms of the four-bar linkage, uncertainties in the geometric parameters result in non-exact solutions for the coupler point. A design description which accounts for bounded uncertainties is termed an appropriate design. Appropriate analysis routines are analysis routines for guaranteeing that specific requirements will be satisfied, which is applicable to mechanisms described by an appropriate design. These routines rely on interval analysis to provide reliable results and are able to handle the uncertainties present in a mechanism. An appropriate analysis routine which is able to determine the worst-case bounds of the coupler curve resulting from uncertainties is proposed. As well, routines are introduced for evaluating the classification and assembly of mechanisms with uncertainties. Examples are presented to demonstrate the coupler curves of four-bar linkages described by an appropriate design which corresponds to each linkage classification, including folding linkages.

Planar Linkage Synthesis for Mixed Motion, Path, and Function Generation Using Poles and Rotation Angles1

The kinematic synthesis of planar linkage mechanisms has traditionally been broken into the categories of motion, path, and function generation. Each of these categories of problems has been solved separately. Many problems in engineering practice require some combination of these problem types. For example, a problem requiring coupler points and/or poses in addition to specific input and/or output link angles that correspond to those positions. A limited amount of published work has addressed some specific underconstrained combinations of these problems. This paper presents a general graphical method for the synthesis of a four bar linkage to satisfy any combination of these exact synthesis problems that is not overconstrained. The approach is to consider the constraints imposed by the target positions on the linkage through the poles and rotation angles. These pole and rotation angle constraints (PRCs) are necessary and sufficient conditions to meet the target positions. After the constraints are made, free choices which may remain can be explored by simply dragging a fixed pivot, a moving pivot, or a pole in the plane. The designer can thus investigate the family of available solutions before making the selection of free choices to satisfy other criteria. The fully constrained combinations for a four bar linkage are given and sample problems are solved for several of them.

Defect-free optimal synthesis of crank-rocker linkage using nature-inspired optimization algorithms

This paper proposes the synthesis of a human knee exoskeleton using the reduced number of necessary and sufficient constraints. The proposed human knee exoskeleton consists of a defect-free crank-rocker mechanism capable of accurately moving its coupler point along the prescribed trajectory. For synthesizing the crank-rocker mechanism based on proposed reduced number of constraints, an optimization problem is formulated to synthesize the mechanism. An algorithm based on teaching-learning-based-optimization (TLBO) is presented to solve this highly nonlinear optimization problem. The optimization minimizes the error between generated and prescribed trajectory and simultaneously avoids any defect in the synthesized mechanism. The penalty method is used to manage all the constraints. Besides a realistic nontrivial example of human knee flexion/extension, a straight line trajectory is also considered to demonstrate the effectiveness of the refinement scheme in the optimal syntheses of planar crank-rocker linkage free from all defects. The optimization problem is solved using well-established nature-inspired algorithms such as genetic algorithm (GA), particle swarm algorithm (PSO), and teaching-learning-based-optimization (TLBO) algorithm. It is found that TLBO is computationally more efficient than other algorithms used here. Additionally, the proposed human knee exoskeleton model is experimentally validated for one gait cycle.