Spherical Four-bar Mechanism Research Articles

Synthesis of Coupler Curves of Spherical Four-bar Mechanism Through Fast Fourier Transform

The geometrical model of spherical four-bar mechanism is proposed,the harmonic component of the coupler curves is analysed by using the Fourier series theory.The relationship between the harmonic component of the coupler curves and the harmonic component of the rotation-angle function and the dimensional type of spherical four-bar mechanism is discovered.Based on the basic dimensional type of spherical four-bar mechanism,a numerical atlas database of harmonic characteristic parameters of the spatial four-bar mechanism comprising more than 6 millions of basic dimensional types is build up.Mathematic equations for spherical four-bar generator on an arbitrarily arranged position are set up by using Fourier theory.The formulas which can compute the position for coupler point,real size and installing dimensions of the spherical four-bar mechanism are deduced.Based on those theoretical formulas and the numerical atlas database of the harmonic characteristic parameters,making the spherical four-bar mechanism path generation is possible.Finally,an example is given to prove the feasibility of the method.

Computer-Aided Modeling and Manufacturing of Spherical Mechanisms via a Novel Web Tool

In this paper, we present a novel web-based computer-aided modeling and manufacturing software tool for spherical mechanisms. Our purpose is to facilitate the analysis, dynamic simulation, and manufacture of one degree of freedom spherical four-bar mechanisms. First, a brief review of some of the current computer-aided design software for spherical four-bar mechanisms is presented. These software packages provide the three-dimensional visualization and computational capabilities necessary to synthesize and analyze spherical four-bar mechanisms. However, to date, no readily available and effective tools exist to aid in the modeling and manufacture of spherical mechanisms. Next, the kinematics of spherical four-bar mechanisms are reviewed as they pertain to their geometric modeling and manufacture. Finally, we present our web-based implementation of a computer-aided modeling, simulation, and manufacturing methodology for spherical four-bar mechanisms called SFBDESIGNER (for spherical four-bar designer). SFBDESIGNER facilitates the design, dynamic simulation, prototyping, and manufacture of spherical four-bar mechanisms.

Computer-aided manufacturing of spherical mechanisms

In this paper a computer-aided manufacturing methodology for spherical four-bar mechanisms is presented. First, the kinematics of spherical mechanisms are reviewed as they pertain to their manufacture. This is followed by a brief review of some of the current computer-aided design (CAD) software for spherical four-bar mechanisms, e.g., S phinx , S phinx PC, I sis , and O siris . These software packages provide the three-dimensional visualization and computational capabilities necessary to design spherical four-bar mechanisms. However, to date no tools exist to aid in the manufacture of spherical mechanisms. Finally, we present our implementation of this CAM methodology for spherical four-bar mechanisms called S phinx CAM. S phinx CAM, when used with the CAD tools mentioned above, facilitates the design, visualization, prototyping and manufacture of spherical four-bar mechanisms.

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. ▪

IN MEMORIAM: Dr. An Tzu Yang (1923–2003)

IN MEMORIAM: Dr. An Tzu Yang (1923–2003)

VEMECS: A virtual reality interface for spherical mechanism design

Mechanisms are one of the fundamental elements used by engineers in the design of machines. The design or synthesis of these elements has been studied for hundreds of years. More recently, the focus of kinematic synthesis research has been in reformatting the graphical synthesis methods into an analytical form compatible with computer processing. One of the thrusts of this research concentrates on the human/computer interface (HCI) between the user and the computer design software. The work presented in this paper addresses this issue of developing a new HCI for mechanism design based on virtual reality techniques. Current computer-aided mechanism design can be seen in the areas of analysis, topological, and dimensional synthesis (Erdman 1995). In each of these areas, the trend is toward developing user interfaces that are more compatible with the cognitive and perceptual nature of the designer. Mechanism synthesis computer programs like KINSYN (Rubel et al . 1977), LINCAGES-4 (Erdman and Riley 1981) and SPHINX (Larochelle et al . 1993) utilize the traditional HCI of monitor, keyboard and mouse. More recently, virtual reality (VR) is being examined as a natural human interface for mechanism design. Osborn (1994) developed the first virtual environment for the synthesis of spherical four-bar mechanisms, SphereVR, which used the Newton-Raphson iterative approach to solve the design equations. This paper takes a slightly different approach by combining the reliable and robust solution algorithms of SPHINX1.0 with VR technology. VR is used for all interaction, manipulation, and navigation throughout the design process, while SPHINX1.0 computational routines are used for computing the solution mechanisms. Natural and intuitive skills of the designer are used through reliance on a head tracked three-dimensional display and three-dimensional interaction. The result is a program called VEMECS (Virtual Environment MEChanism Synthesis). This paper outlines the organization and operation of VEMECS, and concludes with a discussion of the lessons learned in development and implementation of this approach.

Synthesis of Spherical Four-Bar Mechanisms for Two or Three Prescribed Coupler-Curve Cusps

This paper presents a new approach for the synthesis of spherical four-bar mechanisms with two or three prescribed coupler curve cusps. According to the configuration of a spherical four-bar mechanism whose coupler point coincides with the instant center at two separated positions, two kinds of degenerated curves for the corresponding spherical Burmester curves are obtained. For each degenerated curve, a synthesis procedure is proposed for the dimensional synthesis of spherical four-bar mechanisms to trace a coupler curve with two prescribed cusps. The possible types of spherical four-bar mechanisms with two-cusp coupler curves are discussed. Furthermore, special spherical double-rockers are also presented for tracing a symmetric coupler curve with two cusps, tracing a doubly symmetric coupler curve with two cusps or tracing a symmetric coupler curve with three cusps.

Branch and circuit defect elimination in spherical four-bar linkages

This paper presents algebraic-geometrical methods to eliminate branch and circuit defects in the synthesis of spherical four-bar mechanisms. Regions containing potentially circuit defective mechanism (PCDM) designs have been identified. Exclusion of the PCDM region then becomes a sufficient condition for the elimination of the circuit defect. For branch defect elimination, the approach of Gupta and Tinubu ( Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641–647 [1]) has been extended for spherical four-bar mechanisms. Furthermore, regions which yield designs with fully rotatable input-links have been identified. Examples have been given for three and four precision point function generators.

Complete Solution of the Five-Position Synthesis for Spherical Four-Bar Mechanisms

Complete Solution of the Five-Position Synthesis for Spherical Four-Bar Mechanisms

Classification of all self-motions of the original Stewart-Gough platform

The ‘original’ Stewart-Gough (S-G) platform is a parallel 6-6 mechanism with both bases isosceles triangles cut at vertices. All six points of the upper base are connected by telescopic legs with six points of the lower base by spherical joints. If the length of all six legs remains constant then the S-G platform is stiff (a structure), i.e. it has no movability in general. Exceptions from this rule are called self-motions of the S-G platform and they can appear only in very special circumstances. They are known as Borel-Bricard motions, because they have been studied by Borel and Bricard at the beginning of this century [Borel E., Mémoire sur les déplacements à trajectoires sphériques. Mémoires présentés par divers savants, Paris, 1908, 33(2), 1–128. Bricard R., Mémoire sur les déplacements à trajectoires sphériques. Journal de l'École Polytechnique, 1906, 11(2), 1–96 1,2. Some classes of Borel-Bricard motions are known, but their classification is far from complete. In this paper we have determined all self-motions of the ‘original’ S-G platform. As a byproduct new types of Borel-Bricard motions have been discovered.During this task one has to use a computer for three different purposes. At first algebraic equations of self-motions have to be determined. For this purpose we have used Maple V on a workstation. The use of a workstation is necessary, because we had to handle expressions having up to 20,000 terms. After equations of the self-motion have been derived one has to visualize the result, which means to plot trajectories of points of the platform and corresponding positions of the upper base. Here we have two possibilities. If the motion can be parameterized, we can plot trajectories by a suitable graphical system. If the motion cannot be parameterized (this can happen because it is algebraic of order up to 8), we have to compute numerically some locations of the upper base of the platform and then to interpolate them. We did not go tot he last step because this would be beyond the scope of the paper. As a result we have shown that self-motions of the ‘original’ S-G platform can be translatory motions, pure rotations, generalized screw motions with fixed axis, spherical four-bar mechanisms or more general space motions. One of the special cases of the general self-motion is a space motion with three circular trajectories, which is not spherical. This motion yields a new spatial mechanism, which is a spatial analogy of the well known planar or spherical four-bar. It seems that Bricard already knew about the existence of such a motion.

Singularities of Spherical Four-Bar Motion

The link curve of a spherical four-bar mechanism can be separated into two branches. The condition that the double points of the curve must satisfy is given. The double points may be the double points of either branch, or the intersection points of the two branches.

Cross ratio in sphere geometry and its application to mechanism design

In this paper a cross ratio theory applied in sphere geometry and its application on the spherical mechanism is presented. By expanding the cross ratio concept in plane geometry, cross ratio is proved to be equally valid in sphere geometry. Then, an alternative form of spherical cross ratio is proposed. An application of spherical cross ratio for the design of spherical four-bar mechanisms is presented.

Rotatability Considerations for Spherical Four-Bar Linkages With Applications to Robot Wrist Design

The following result for spherical four-bar mechanisms is proved: Let all links of a spherical four-bar linkage be in the range [0, π]. Consider two linkages—one primary (p) and another secondary (s) according to the condition αip + αfp < αis + αfs where αs = π − αp. Then the input link is fully rotatable iff αip + αfp < αc + α0, |αip − αfp| > |αc − α0| and αip + αfp + αc + α0 < 2π. An alternate form of this result also is developed. Applications of these results in determining full or partial rotatability of grippers in spherical robot wrists are discussed.

Minimization of Inertia-Induced Forces in Spherical Four-Bar Mechanisms. Part 1: The General Spherical Four-Bar Linkage

Computer-aided design procedures have been developed for the optimum mass distribution of the links of high-speed spherical four-bar linkages. The analysis, which includes a quadratic-programming technique, allows an optimum trade-off between shaking forces, shaking moments, bearing reactions, and input-torque fluctuation. The results are illustrated in the case of a Hooke joint and a wobble-plate linkage.

Minimization of Inertia-Induced Forces in Spherical Four-Bar Mechanisms. Part 2: Wobble-Plate Engines

In Part 2 of this paper, computer-aided design procedures have been developed for the optimum mass distribution of high-speed wobble-plate engines.

Approximate Synthesis of Spherical Four-Bar Mechanisms : 2nd Report, A Method for Calculating Co-ordinates of Burmester Points

Approximate Synthesis of Spherical Four-Bar Mechanisms : 2nd Report, A Method for Calculating Co-ordinates of Burmester Points

Approximate Synthesis of Spherical Four-Bar Mechanisms : 2nd Report, A Method for Calculating Co-ordinates of the Burmester Points

In this report, firstly a relationship between five parameters of a two-link mechanism with two degrees of freedom and three position variables of an arc fixed on its second link is derived in order to establish a method for calculating co-ordinates of the Burmester points with respect to five finitely separated spherical positions. Secondly it is shown that by using this relationship as a basic equation co-ordinates of the Burmester points can be determined by means of a simple procedure that consists in solving a set of five linear equations and a sixths order equation. The analytical method established in this report with the numerical results proves that the number of the Burmester points in spherical motion is generally zero, two, four, or six.

Konjugierte Stellungen bei sphäerischen Viergelenkgetrieben und die zugehoerigen speziellen Koppelkurven

Conjugate positions of spherical four-bar mechanisms are corresponding to planar mechanisms those having the same output-angle ‘β’ and consequently two conjugate positions for every possible value of ‘β’. The geometric relations for conjugate positions of spherical four-bar mechanisms are derived in this paper. Furthermore, it is demonstrated how to gain spherical coupler curves having two or three cusps by kinematic inversion of special conjugate positions.

On the synthesis of the spherical four-bar

The synthesis of the spherical four-bar mechanism for angular co-ordinations of the cranks with five precision points is reduced to the solution of a quintic. The spherical slider-crank is also considered.

Static Force and Torque Analysis Using 3 × 3 Screw Matrix, and Transmission Criteria for Space Mechanisms

The method of determining the static force and torque distributions in space mechanisms by use of the 3 × 3 screw matrix is presented. Transmissivities in favor of rotation and translation, and angles of transmission in space mechanisms are defined. Transmissivities are used to show why some space mechanisms have no dead center positions. In the process, the dual equilibrium equations, one for each link, are written in dual vector form, then they are solved simultaneously for the dual force components. Explicit expressions for forces and torques in the RCCC and RSSR space mechanisms are obtained. Torque distribution in the spherical four-bar mechanism is reduced from the results for the RCCC space mechanism. The numerical example considers a symmetrical crank-rocker RSSR space mechanism, where the output torque is due to the load inertia. Variations in the input torque, output torque, mechanical advantage, and transmission angle for one cycle are given for both geometric inversions.