Six and Seven Position Triad Synthesis Using Continuation Methods

1994 ◽  
Vol 116 (2) ◽  
pp. 660-665 ◽  
Author(s):  
T. Subbian ◽  
D. R. Flugrad
Keyword(s):  
Finite Number ◽  
Continuation Method ◽  
Synthesis Problem ◽  
Continuation Methods ◽  
Path Generation ◽  
Motion Generation ◽  
Number Of Solutions ◽  
Position Synthesis

A continuation method is used for the synthesis of triads for motion generation with prescribed timing applications. The procedure is applied to solve both six and seven position synthesis problems. Triad Burmester curves are generated for the six position synthesis problem and an eight-bar mechanism is designed to illustrate the procedure. For the seven position synthesis problem, a finite number of solutions are obtained. A geared five-bar, seven position path generation example is considered.

“MKCIRCLES”: An Interactive Computer Graphics Design Tool to Solve the Generalized Three-Precision-Position Synthesis Problem

1994 ◽  
Author(s):  
Pi-Ming Cheng ◽  
Raed N. Rizq ◽  
Arthur G. Erdman
Keyword(s):  
Computer Graphics ◽  
Design Tool ◽  
Synthesis Problem ◽  
Path Generation ◽  
Motion Generation ◽  
Interactive Computer Graphics ◽  
Linkage Design ◽  
Planar Regions ◽  
Angular Displacements ◽  
Position Synthesis

Abstract A new interactive computer graphics program (MKCIRCLES) has been developed to solve the following three-precision-position dyad synthesis tasks: (i) motion generation, (ii) path generation with prescribed timing and (iii) a new solution strategy for (a) motion generation for a user-specified range of the prescribed-timing angular displacements, and (b) path generation with prescribed timing for a user-specified range of the rotations of the floating link. The latter two cases address a problem that is encountered in linkage design; namely, the need to specify limits that certain variables may range through as opposed to specifying fixed values that constrain the design unnecessarily, thus increasing the total number of designs from which to choose. As a result of this new approach, two planar regions, representing all permissible dyad ground-pivot locations and all permissible dyad moving-pivot locations, are identified and plotted. The program uses the properties of the circle-point circles (K1-circles) and the center-point circles (M-circles) throughout the synthesis procedure. MKCIRCLES also allows the designer to define a region in which the ground-pivots are constrained to lie and determine the corresponding moving-pivot region (and vice versa). The program is shown to be a useful design tool and provides greater geometric and kinematic insight into the general three-precision-position synthesis problem.

A Complex Number Approach for Absolute Precision Position Synthesis for Three Precision Positions

1992 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Closed Form ◽  
Complex Number ◽  
Closed Form Solution ◽  
Form Solution ◽  
Path Generation ◽  
Motion Generation ◽  
The Absolute ◽  
Position Problem ◽  
Position Synthesis

Abstract Dyads can be synthesized by prescribing the precision point coordinates and the absolute planar orientations of one dyad vector at each of three precision positions. This differs from traditional complex number methods wherein the vector orientations are described relative to one another. Absolute precision position synthesis can be performed for both motion generation, and path generation with prescribed timing. The method presented uses vector loop equations and complex number notation to produce a closed form solution for the three absolute precision position problem. Absolute precision position synthesis is applicable to cases that require specific coupler geometries. The synthesis of flat-folding mechanisms is an example of one such application.

Complete Solutions to Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With M-Homogenization

1992 ◽  
Author(s):  
Anoop K. Dhingra ◽  
Jyun-Cheng Cheng ◽  
Dilip Kohli
Keyword(s):  
Group Theory ◽  
Matrix Method ◽  
Numerical Results ◽  
Path Generation ◽  
Link Function ◽  
Homotopy Methods ◽  
Motion Generation ◽  
The Matrix ◽  
Function Generators ◽  
Complete Solutions

Abstract This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

An Improved Harmony Search Algorithm for a Planar Parallel Robot Synthesis

2018 ◽  
Vol 35 ◽  
pp. 185-197
Author(s):  
Badreddine Aboulissane ◽  
Dikra El Haiek ◽  
Larbi El Bakkali
Keyword(s):  
Search Algorithm ◽  
Harmony Search ◽  
Parallel Robot ◽  
Harmony Search Algorithm ◽  
Path Generation ◽  
Motion Generation ◽  
Kinematic Synthesis ◽  
Design Variables ◽  
Joint Coordinates ◽  
Improved Harmony Search

The objective of kinematic synthesis is to determine the mechanism dimensions such as link lengths, positions or joint coordinates, in order to approximate its output parameters such as link positions, trajectory points, and displacement angles. Kinematic synthesis is classified into three categories: function generation, path generation, and motion generation. This paper is dedicated only to path generation. As the number of trajectory points increases, analytical methods are limited to obtain precisely mechanism solutions. In that case, numerical methods are more efficient to solve such problems. Our study proposes an improved heuristic algorithm applied to four-bar mechanism path-generation. The objective of this work is to find optimum dimensions of the mechanism and minimize the error between the generated trajectory and the desired one, taking into consideration constraints such as: Grashof condition, transmission angle, and design variables constraints. Finally, our results are compared with those found by other evolutionary algorithms in the literature.

Real Continuation Method and its Application in Kinematic Synthesis

2000 ◽  
Author(s):  
Liu Anxin ◽  
Yang Tingli
Keyword(s):  
High Efficiency ◽  
Continuation Method ◽  
Linear Equations ◽  
Path Generation ◽  
Kinematic Synthesis ◽  
Real Solutions ◽  
Non Linear ◽  
Four Bar Linkage ◽  
Non Linear Equations

Abstract Real continuation method for finding real solutions to non-linear equations is proposed. Synthesis of planar four-bar linkage for path generation with nine precision points is studied using this method. The proposed method has high efficiency and can best be used for solving synthesis problems.

Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad

2012 ◽  
Author(s):  
Chintien Huang ◽  
Chenning Hung ◽  
Kuenming Tien
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Quadratic Equation ◽  
Geometric Constraints ◽  
Numerical Continuation ◽  
Polynomial Equations ◽  
Quartic Polynomial ◽  
Solutions Of Equations ◽  
Position Synthesis

This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.

Five Position Triad Synthesis With Applications to Four- and Six-Bar Mechanisms

1993 ◽  
Vol 115 (2) ◽  
pp. 262-268 ◽  
Author(s):  
T. Subbian ◽  
D. R. Flugrad
Keyword(s):  
Continuation Method ◽  
Solution Procedure ◽  
Motion Generation

The displacement equations for trials with motion generation and prescribed timing capabilities are developed and cast in polynomial forms. These equations are solved for five precision points using a continuation method. A detailed description of the method used to generate the solution curves is provided in the paper. Two infinities of solutions can be obtained for the problem under consideration as we are solving four equations in six unknowns. The solution procedure discussed is applied to synthesize a four-bar function generating mechanisms and a six-bar mechanism.

The No-Three-In-Line Problem

1968 ◽  
Vol 11 (4) ◽  
pp. 527-531 ◽  
Author(s):  
Richard K. Guy ◽  
Patrick A. Kelly
Keyword(s):  
Finite Number ◽  
The Other ◽  
Probabilistic Argument ◽  
Number Of Solutions ◽  
Other Hand ◽  
Image Position

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

Finite Position Synthesis of 5-SS Platform Linkages Including Partially Specified Joint Locations

2017 ◽  
Author(s):  
Xin Ge ◽  
Anurag Purwar ◽  
Q. J. Ge
Keyword(s):  
Linear Structure ◽  
Motion Generation ◽  
Numerical Examples ◽  
Moving Points ◽  
Unified Algorithm ◽  
Moving Platform ◽  
Position Synthesis ◽  
Partial Specification ◽  
Generation Problem ◽  
Novel Algorithm

A 5-SS platform linkage generates a one-degree-of-freedom motion of a moving platform such that each of five moving points on the platform is constrained on a sphere, or in its degenerated case, on a plane. It has been well established a 5-SS platform linkage can be made to guide though seven positions exactly. This paper investigates the cases when the number of given positions are less than seven that allows for partial specification of locations of the moving points. A recently developed novel algorithm with linear structure in the design equations has been extended for the solution of the problem. The formulation of this expanded motion generation problem unifies the treatment of the input positions and constraints on the moving and fixed joints associated with the 5-SS platform linkage. Numerical examples are provided to show the effectiveness of the unified algorithm.

Kinematic Mapping Based Evaluation of Assembly Modes for Planar Four-Bar Synthesis

2005 ◽  
Author(s):  
Hans-Peter Schro¨cker ◽  
Manfred L. Husty ◽  
J. Michael McCarthy
Keyword(s):  
Image Space ◽  
Synthesis Problem ◽  
New Method ◽  
Kinematic Mapping ◽  
Position Synthesis ◽  
Four Bar Linkage

This paper presents a new method to determine if two task positions used to design a four-bar linkage lie on separate circuits of a coupler curve, known as a “branch defect.” The approach uses the image space of a kinematic mapping to provide a geometric environment for both the synthesis and analysis of four-bar linkages. In contrast to current methods of solution rectification, this approach guides the modification of the specified task positions, which means it can be used for the complete five position synthesis problem.

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