Polynomial Solutions to the Displacement Analysis of 10-Link 1-DOF Mechanisms

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
A. N. Almadi ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Solution Process ◽  
Input Output ◽  
Analysis Problem ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Polynomial Solutions ◽  
Elimination Procedure ◽  
Half Angle ◽  
Univariate Polynomial

Abstract This paper presents closed-form polynomial solutions to the displacement analysis problem of planar 10-link mechanisms with 1 degree-of-freedom (DOF). Using the successive elimination procedure presented herein, the input-output (I/O) polynomials as well as the number of assembly configurations for five mechanisms resulting from two 10-link kinematic chains are presented. It is shown that the displacement analysis problems for all five mechanisms can be reduced to a univariate polynomial devoid of any extraneous roots. This univariate polynomial corresponds to the I/O polynomial of the mechanism. In addition, one of the examples also illustrates how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are also presented.

A Framework for Closed-Form Displacement Analysis of 10-Link 1-DOF Mechanisms

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
A. N. Almadi ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Input Output ◽  
Analysis Problem ◽  
Loop Closure ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Solvable In Closed Form ◽  
Univariate Polynomial ◽  
Form Approach ◽  
Independent Loops

Abstract This paper presents a closed-form approach, based on the theory of resultants, to the displacement analysis problem of planar 10-link 1-DOF mechanisms. Since each 10-link mechanism has 4 independent loops, its displacement analysis problem can be written as a system of 4 reduced loop-closure equations in 4 unknowns. This system of 4 reduced loop closure equations, for all non-trivial mechanisms resulting from 230 10-link kinematic chains, can be classified into 22 distinct structures. Using the successive and repeated elimination procedures presented herein, it is shown how each of these structures can be reduced into a univariate polynomial devoid of any extraneous roots. This univariate polynomial corresponds to the input-output (I/O) polynomial of the mechanism. Based on the results presented, it can be seen that the displacement analysis problem for all 10-link 1-DOF mechanisms is completely solvable, in closed-form, devoid of any extraneous roots.

Closed-Form Displacement Analysis of SDOF 8 Link Mechanisms

1996 ◽  
Author(s):  
Abdulaziz N. Almadi ◽  
Anoop K. Dhingra ◽  
Dilip Kohli
Keyword(s):  
Closed Form ◽  
Analysis Problem ◽  
Algebraic Form ◽  
Single Degree Of Freedom ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Closed Form Solutions ◽  
Single Degree ◽  
Half Angle

Abstract This paper presents closed-form solutions to the displacement analysis problem of planar 8-link mechanisms with a single degree of freedom (SDOF). The degrees of I/O polynomials as well as the number of possible assembly configurations for all 71 8-link mechanisms resulting from 16 8-link kinematic chains are presented. Three numerical examples illustrating the applicability of the successive elimination procedure to the displacement analysis of 8-link mechanisms are presented. The first example deals with the determination of I/O polynomial for an 8-link mechanism containing no four-bar loops. The second and third examples, address in detail, some of the problems associated with the conversion of transcendental loop-closure equations into an algebraic form using tangent half-angle substitutions. These examples illustrate how extraneous roots can get introduced during the displacement analysis of mechanisms, and how one can derive an I/O polynomial devoid of the extraneous roots. Extensions of the proposed approach to the displacement analysis of SDOF spherical 8-link mechanisms is also presented.

A Framework for Closed-Form Displacement Analysis of Planar Mechanisms

1999 ◽  
Vol 121 (3) ◽  
pp. 392-401 ◽  
Author(s):  
A. N. Almadi ◽  
A. K. Dhingra ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Solution Process ◽  
Computational Procedure ◽  
Comprehensive Treatment ◽  
Planar Mechanisms ◽  
Algebraic Form ◽  
Displacement Analysis ◽  
Successive Elimination ◽  
Half Angle ◽  
Univariate Polynomial

This paper presents a closed-form approach, based on the theory of resultants, to the displacement analysis problem of planar n-link mechanisms. The successive elimination procedure presented herein generalizes the Sylvester’s dialytic eliminant for the case when p equations (p ≥ 3) are to be solved in p unknowns. Conditions under which the method of successive elimination can be used to reduce p equations (in p unknowns) into a univariate polynomial, devoid of extraneous roots, are presented. This univariate polynomial corresponds to the I/O polynomial of the mechanism. A comprehensive treatment is also presented on some of the problems associated with the conversion of transcendental loop-closure equations, into an algebraic form, using tangent half-angle substitutions. It is shown how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are presented. The computational procedure is illustrated through the displacement analysis of a 10-link 1-DOF mechanism with 4 independent loops.

Closed-Form Displacement Analysis of 9-Link, 2-DOF and 10-Link, 3-DOF Mechanisms

1996 ◽  
Author(s):  
Abdulaziz N. Almadi ◽  
Anoop K. Dhingra ◽  
Dilip Kohli
Keyword(s):  
Closed Form ◽  
Degrees Of Freedom ◽  
Computational Procedure ◽  
Companion Paper ◽  
Input Output ◽  
Analysis Problem ◽  
Numerical Examples ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Solvable In Closed Form

Abstract This paper addresses the closed-form displacement analysis problem of all mechanisms which can be derived from 9-link kinematic chains with 2-DOF, and 10-link kinematic chains with 3-DOF. The successive elimination procedure developed in the companion paper is used to solve the resulting displacement analysis problems. The input-output polynomial degrees as well as the number of assembly configurations for all mechanisms resulting from 40 9-link kinematic chains, and 74 10-link kinematic chains with non-fractionated degrees of freedom (DOF) are given. The computational procedure is illustrated through two numerical examples. The displacement analysis problem for all mechanisms resulting from these chains is completely solvable, in closed-form, devoid of any spurious roots.

A Framework for Closed-Form Displacement Analysis of Planar Mechanisms

1996 ◽  
Author(s):  
Abdulaziz N. Almadi ◽  
Anoop K. Dhingra ◽  
Dilip Kohli
Keyword(s):  
Closed Form ◽  
Solution Process ◽  
Computational Procedure ◽  
Comprehensive Treatment ◽  
Planar Mechanisms ◽  
Algebraic Form ◽  
Displacement Analysis ◽  
Successive Elimination ◽  
Half Angle ◽  
Univariate Polynomial

Abstract This paper presents a closed-form approach, based on theory of resultants, to the displacement analysis problem for n-link planar mechanisms. The proposed approach, called the method of successive elimination, generalizes Sylvester’s dialitic eliminant to the case when m equations (m ≥ 3) are to be solved in m unknowns. Conditions under which the method of successive elimination can be used to reduce m equations (in m unknowns) into a univariate polynomial, devoid of spurious roots, are presented. This univariate polynomial corresponds to the 1/0 polynomial of the mechanism. A comprehensive treatment is also presented on some of the problems associated with the conversion of transcendental loop-closure equations into an algebraic form using tangent-half-angle substitutions. It is shown how trigonometric manipulations in conjunction with tangent-half-angle substitutions can lead to extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these spurious roots are presented. The computational procedure is illustrated through the displacement analysis of a 10-link SDOF mechanism which has 4 independent loops.

Displacement Analysis of Spatial Five-Link Mechanisms Using (3×3) Matrices With Dual-Number Elements

1969 ◽  
Vol 91 (1) ◽  
pp. 152-156 ◽  
Author(s):  
A. T. Yang
Keyword(s):  
Closed Form ◽  
Algebraic Equation ◽  
Input Output ◽  
Fourth Degree ◽  
Displacement Analysis ◽  
Dual Number ◽  
Closure Equation

A closure equation, in terms of matrices with dual-number elements, for spatial five-link mechanisms, is presented in this paper. From the equation, a set of displacement equations for a RCRCR mechanism with general proportions is obtained; the input-output relationship is expressed as a fourth-degree algebraic equation and formulas to determine other linkage variables are expressed in closed form.

On Trigonometric Formulations of Polynomial Equations

2006 ◽  
Author(s):  
Harvey Lipkin
Keyword(s):  
Real Axis ◽  
Projective Transformation ◽  
Polynomial Equations ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Real Roots ◽  
Entire Real Axis ◽  
Half Angle ◽  
Sine Polynomials ◽  
Circular Points

The displacement analysis of open and closed kinematic chains is based on polynomial equations whose variables are functions of relative joint displacements. The objective of this paper is to investigate new and interesting properties of the transformations between the canonical cosine-sine polynomials and the even degree tan-half angle polynomials associated with displacement kinematics. Using a homogeneous coordinate formulation, it is shown that the coefficients of the polynomials are linearly related by a projective transformation whose elements can be defined recursively. The canonical cosine-sine polynomial is then transformed to a cosine or a sine polynomial which can be rooted by usual techniques. However, all real roots are bracketed between −1 and +1 which can have numerical advantages over a corresponding tan-half angle polynomial for which the entire real axis must be searched. It is also demonstrated how polynomial solutions corresponding to circular points at infinity in the tan-half angle, which are typically introduced as extraneous roots via algebraic elimination, may be easily factored out by the transformation to the cosine-sine formulation.

Forward Displacement Analyses of the 4-4 Stewart Platforms

1992 ◽  
Vol 114 (3) ◽  
pp. 444-450 ◽  
Author(s):  
W. Lin ◽  
M. Griffis ◽  
J. Duffy
Keyword(s):  
Closed Form ◽  
Stewart Platform ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Half Angle ◽  
Stewart Platforms ◽  
Forward Displacement Analysis

A forward displacement analysis in closed-form is performed for each case of a class of Stewart Platform mechanisms. This class of mechanisms, which are classified into three cases, are called the “4-4 Stewart Platforms,” where each of the mechanisms has the distinguishing feature of six legs meeting either singly or pair-wise at four points in the top and base platforms. (This paper only addresses those 4-4 Platforms where both the top and base platforms are planar.) For each case, a polynomial is derived in the square of a tan-half-angle that measures the angle between two planar faces of a polyhedron embedded within the mechanism. The degrees of the polynomials for the first, second, and third cases are, respectively, eight, four, and twelve. All the solutions obtained from the forward displacement analyses for the three cases are verified numerically using a reverse displacement analysis.

Displacement Analysis of Spatial Six-Link, 5R-C Mechanisms: A General Solution

1985 ◽  
Vol 107 (3) ◽  
pp. 353-357 ◽  
Author(s):  
Xu Li Ju ◽  
J. Duffy
Keyword(s):  
General Solution ◽  
Angular Displacement ◽  
Problem Formulation ◽  
Input Output ◽  
Displacement Analysis ◽  
Numerical Example ◽  
Single Operation ◽  
Half Angle ◽  
Output Equation

Four angular displacement equations are derived for the spatial 5R-C hexagon from which an input-output equation of 16th degree in the tan-half-angle of the output angular displacement for each of the RCRRRR, RRCRRR mechanisms and the yet unsolved RRRRRC2 mechanism can be obtained by the elimination of two unwanted variables in a single operation. This novel problem formulation is a general solution for all 5R-C mechanisms. Results are verified by a numerical example.

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