4R Mechanism Research Articles

An Analytical Method for the Forward Kinematics Analysis of 6-SPS Parallel Mechanisms

The displacement analysis of the general spatial 7R mechanism, which is an abstraction from the serial industrial robots, is regarded as one of the most difficult tasks in the kinematics analysis of spatial mechanisms. This problem is finally solved by Chinese scholars. There is dual relation between serial robots and parallel robots. The displacement analysis of 6-SPS parallel mechanisms is also a very difficult problem without substaintial progress made in the past ten years. A new method is presented to describe the rotation matrix. Through analyzing the coupling relationships between the moving platform position and attitude variables, eleven compatible algebraic equations used to solve this problem are obtained. Based on this, through eliminating the higher terms step by step, the problem of the forward kinematics analysis of 6-SPS parallel mechanisms can be finally expressed as a seventeen order algebraic equation, and maybe it is the best result of this kind of problem.

Transformation from Spatial RSSR Mechanism to Spherical 4R Mechanism

A method of transformation from spatial RSSR mechanism to spherical 4R mechanism for spatial driving mechanism design is presented. The necessary conditions of transformation from spatial RSSR mechanism to spherical 4R mechanism are also analyzed. Furthermore,it is proved theoretically that,spatial RSSR mechanism,by defined parameters,can be transformed into spherical 4R mechanism with axes intersection at one point. The prerequisite for each design parameter in transformation from spatial RSSR mechanism to spherical 4R mechanism,when spatial RSSR acts as a driving crank,is thus deduced,which lays a theoretical basis for the application of practical engineering. Based on the method of transformation from spatial RSSR mechanism to spherical 4R mechanism,theoretical calculation and test simulation,a spatial rapier-driving mechanism applicable for high-speed rapier looms is developed. Through this example,this method that spatial RSSR mechanism is replaced by spherical 4R mechanism is verified. The finding indicates that all the requirements of spatial RSSR mechanism can be met and the spherical 4R shows better performance,manufacturability and assemblability. Priority of spherical 4R mechanism is also validated in terms of structure,manufacturability,maintenance and kinematics and dynamics performance.

Dual quaternion-based inverse kinematics of the general spatial 7R mechanism

The theory of dual quaternion and its use in serial mechanisms are described in this paper. A closed-form solution to the inverse kinematic analysis of the general 7-link 7R mechanism is presented. Dixon's resultant is used and the input—output equation is expressed in the form of a 6×6 determinant equated to zero, and the formulae to determine other angular displacements are expressed in the closed form. Numerical example confirms these theoretical results. The whole process is very simple and easy to program, which supplies a new method for the real use of the 7R mechanism.

Analytical synthesis of function generating spherical four-bar mechanism for the five precision points

This paper presents an analytical method for synthesis of function generating spherical 4R mechanisms for the five precision points. For the design requirements an additional parameter, reference value of output angle, ψ 0, was added to angular link length parameters, α i ( i = 1, … , 4). In the dimensional synthesis procedure, a novel approach of polynomial approximation method was proposed to determine these five design parameters. Using this method, a set of five non-linear equations was easily transformed into a set of fifteen linear equations. Hence, the problem was reduced to the solution of a cubic polynomial equation. Moreover, a graphical method in a CAD environment is proposed to verify the solutions. ▪

Synthesis of a single-loop, overconstrained six revolute joint spatial mechanism for two-position cylindrical rigid body guidance

The synthesis and analysis of a single-loop, overconstrained spatial mechanism with six binary links and six revolute joints for the guidance of a cylindrical rigid body between two positions are presented. The geometric constraints that make the spatial 6R mechanism movable are first introduced. Four features of the geometrical constraints of this mechanism are summarized to demonstrate its mobility and used for dimensional synthesis. Steps for mechanism dimensional synthesis are developed based on the descriptive and analytical geometry. The mechanism analysis based on analytical geometry is also presented. One of the advantages of this spatial mechanism is that the toggle position can be integrated into the synthesis. This is especially attractive in the synthesis of the wheel retract and twist mechanism for aircraft landing gears and the automated grinding wheel or tool changer for machine tools. A detailed example is used to illustrate the mathematical models for mechanism synthesis and analysis.

Design of Spherical 4R Mechanisms: Function Generation for the Entire Motion Cycle

Spherical 4R mechanisms are studied for which the crank is relatively smaller than the remaining links. The theory of small-crank mechanisms is applied to obtain approximate descriptions for the follower angular displacement in terms of the input crank angle. The follower angle is presumed to comprise a mean and a perturbational motion. This results in an approximate expression in which the follower displacement is given as a linear combination of simple harmonic functions of the first and second harmonics of the crank angle. The approximate equations are utilized for synthesis of spherical 4R mechanisms for function generation. In contrast to the conventional design procedures, the use of the approximate equations allows the synthesis of spherical mechanisms in which a prescribed function is satisfied for the entire motion of the mechanism. In addition to design examples, sample error charts are provided to assist the designer in ascertaining feasible ranges for design and corresponding orders of error.

Experimental and Finite Element Modal Analyses of Planar Mechanisms With Three-Dimensional Geometries

This article presents the results of numerical and experimental modal analyses of a planar four-bar (4R) mechanism having three dimensional geometry due to the offsets between its links. An experimental mechanism is built and modern modal testing techniques are used. A five-node isoparametric finite element is developed and used to model the mass and flexibility of the links and joints, link offsets, and drive and load shafts. The results obtained from the finite element solution matched closely the experimental results, a testimony to the accuracy of the proposed element. The results indicated that link offsets have considerable influence on the dynamic characteristics of a mechanism system and should be considered in dynamic modeling of planar mechanisms.

Active Control of Elastodynamic Vibrations of a Four-Bar Mechanism System With a Smart Coupler Link Using Optimal Multivariable Control: Experimental Implementation

This paper presents an experimental investigation on active control of the elastodynamic response of a four-bar (4R) mechanism system using “smart” materials featuring piezoelectric sensor/actuator (S/A) pairs and multivariable optimal control. The experimental 4R mechanism is made such that its coupler link is flexible, its follower link is slightly less flexible and its crank is relatively rigid. Two thin plate-type piezoceramic S/A pairs are bonded to the flanks of the coupler link at the high strain locations corresponding to the first and second vibration modes. Based on the optimal multivariable control theory, a controller which consists of a linear quadratic regulator (LQR) and a Luenberger observer as a state estimator is designed and implemented. The results of the experimental investigation prove that in order to prevent high mode excitations, the controller design should be based on the modes representing vibrations of all components comprising the mechanism system rather than the modes corresponding to the link to which the S/A pairs are bonded. Response amplitude attenuation ratios up to 50 percent are achieved and high mode excitations are prevented.

Kinematic Analysis of General Spatial 7-Link 7R Mechanisms. 2nd Report. A Sensitivity Analysis and 7R Link Joints.

The sensitivities of the general spatial 7-link, 7R mechanism, namely, the ratios of the small changes in angular displacements of the links to the deviations of the kinematic constants, are estimated as the solutions to the simultaneous equations obtained from partially differentiating the necessary and sufficient equations for the purpose of displacement analysis. The partial differential coefficients that comprise the elements in the simultaneous equations are given in a recurrence formula. By using this procedure, the sensitivities of a constant-velocity joint as a special case of the general spatial 7-link, 7R mechanism are analyzed, and they are divided into four classes. Moreover, by using the noninfluencing or mutually compensating characteristic of sensitivities, asymmetric constant-velocity joints of the 7R linkage type are derived.

An enzymatic immunoassay for insulin-like growth factor-I (IGF-I)

In the mechanisms area, minimization of the magnitude of the acceleration of the center of mass (ACoM) implies shaking force balancing. This article shows an efficient optimum synthesis method for minimum acceleration of the center of mass of a spherical 4R mechanism by using dual functions as well as the counterweights balancing method. Once the dual function for ACoM has been formulated, one can minimize the shaking forces from a kinematic point of view. We present the synthesis of a spherical 4R mechanism for the case of a path generation task. The synthesis process involves the optimization of two objective functions; this multiobjective problem is solved by using the weighted sum method implemented in the evolutionary algorithm known as differential evolution. Our results show that the magnitude of the ACoM can be reduced by about 20 times, which for the presented synthesis, is traduced to a reduction of about 89% for the shaking forces.

Comparison of three immunoassay kits for serum thyroglobulin

In the mechanisms area, minimization of the magnitude of the acceleration of the center of mass (ACoM) implies shaking force balancing. This article shows an efficient optimum synthesis method for minimum acceleration of the center of mass of a spherical 4R mechanism by using dual functions as well as the counterweights balancing method. Once the dual function for ACoM has been formulated, one can minimize the shaking forces from a kinematic point of view. We present the synthesis of a spherical 4R mechanism for the case of a path generation task. The synthesis process involves the optimization of two objective functions; this multiobjective problem is solved by using the weighted sum method implemented in the evolutionary algorithm known as differential evolution. Our results show that the magnitude of the ACoM can be reduced by about 20 times, which for the presented synthesis, is traduced to a reduction of about 89% for the shaking forces.

Immunoenzymatic assay of insulin-like growth factor binding protein-3

In the mechanisms area, minimization of the magnitude of the acceleration of the center of mass (ACoM) implies shaking force balancing. This article shows an efficient optimum synthesis method for minimum acceleration of the center of mass of a spherical 4R mechanism by using dual functions as well as the counterweights balancing method. Once the dual function for ACoM has been formulated, one can minimize the shaking forces from a kinematic point of view. We present the synthesis of a spherical 4R mechanism for the case of a path generation task. The synthesis process involves the optimization of two objective functions; this multiobjective problem is solved by using the weighted sum method implemented in the evolutionary algorithm known as differential evolution. Our results show that the magnitude of the ACoM can be reduced by about 20 times, which for the presented synthesis, is traduced to a reduction of about 89% for the shaking forces.

Kinematic Analysis of General Spatial 7-Ling 7R Mechanisms. 1st Report. A Displacement Analysis Using Newton-Raphson's Method.

A closed-loop equation of the general spatial 7-link 7R mechanism is derived using the dual number quaternion algebra for the line vectors of seven links and seven axes of rotation. The necessary and sufficient components for the displacement analysis are chosen from among twelve components of the closed-loop equation in the form of line vectors, and they are expressed in a recurrence formula for the purpose of numerical calculation. The functional relationship between the input and output angular displacement can be traced by solving the set of six component equations using Newton-Raphson's method in the case where the driving link rotates or oscillates.

Kinematic Analysis and Dimensional Synthesis of Bennett 4R Mechanism

The displacement equation of the Bennett 4R mechanism is directly derived and the output angular velocity and acceleration are analyzed. The coupler curve, its higher order curvature analysis, and coupler-link motion are also provided. In practical applications, this mechanism is used for dimensional synthesis, function, path and motion generation in the optimization technique. Moreover, some numerical examples are provided to illustrate the design algorithm.

Mobility Analysis and Type Identification of Four-Link Mechanisms

An alternative derivation of the simple criteria, in terms of linkage parameters, demarking the zones of nonexistence of double-crank and crank-rocker RSSR mechanisms is presented. The existence of a crank is verified by polynomial discriminants of the limit position. Closed from solutions are obtained for type identification for 4R spherical, RCCC, RSSP, and planar 4R mechanisms by treating these as the limiting cases of an RSSR mechanism.

Synthesizing spatial 7R mechanism with 16-assembly configurations

Using complex numbers, a special case of a spatial 7R mechanism is discussed. The computer calculations show that the input-output equation of this mechanism is still 16th degree in tan-half-angle of the output angular displacement. By adjusting a group of dimensions of this mechanism, a general 7R mechanism with 16-assembly configurations is discovered.

Movable Spatial 6R Mechanisms With Three Adjacent Parallel Axes

Based on derived matrix loop equations and their algebraic manipulations, we obtain the necessary mobility criteria, and prove that there are only three types of movable spatial 6R mechanisms with three adjacent parallel axes. The first type is a general form of Sarrus 6R mechanism. The second type can be regarded as a combination of spherical and planar 4R mechanisms. The third type is a hybrid linkage, first mentioned by Waldron. All these three types are proper linkages confirmed by iteration and optimization methods.

MOVABLE SPATIAL 6R MECHANISMS WITH THREE ADJACENT CONCURRENT AXES

Based on derived matrix loop equations and their algebraic manipulations, we systematically Identify three types of movable spatial 6R mechanisms with three adjacent concurrent axes. The first type was not discovered algebraically. The second type is a general form of the Cardan mechanism. The third type is a plane-symmetry mechanism which was derived by Waldron.

Kinematic analysis of the general spatial 7R mechanism

The displacement. velocity and acceleration analysis of the general spatial 7R mechanism is discussed in this paper. Based on the method proposed in Ref. [2], an input-output algebra equation of the 16th degree in the tan-half-angle of the output angular displacement is derived. The derivation process and computation are considerably simple. A program written in AI language is used to derive the coefficients of displacement equations: therefore the amount of manual work is greatly decreased. The results are verified by a numerical example. The researches of this paper and Ref.[5] found a base for establishing an expert system of spatial mechanism analysis in the future.

A new vector theory for the analysis of spatial mechanisms

This paper develops a new vector theory for the analysis of spatial mechanisms on the basis of the vector analysis theory and dual-number algebra and the recursive notation presented by J. Duffy. The new vector theory includes the displacement, velocity and acceleration equations for the open kinematic chains and loop mechanisms. The displacement equations for the loop mechanisms are classified into four groups with two forms, namely the primary, secondary, projective and the fourth-group loop/half-angle equations, which have been applied to the derivation of the 16th degree polynomial input-output equations in the tan-half-angle of the output angular displacements for the general spatial 7-link 7R and 6R-P mechanisms, etc. In the analysis of the velocity and acceleration, the differential calculus are changed into vectorical algebraic operations and the unified equations are expressed in the vector forms which are linear in the velocity and acceleration variables.