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 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.

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.

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 Closed-Form Approach to Coupler-Curves of Multi-Loop Mechanisms

1998 ◽  
Author(s):  
A. K. Dhingra ◽  
A. N. Almadi ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Computational Procedure ◽  
Multivariate System ◽  
Algebraic Form ◽  
Numerical Examples ◽  
Solution Approach ◽  
Half Angle ◽  
Curve Equation ◽  
Form Approach

Abstract This paper presents a closed-form approach, based on the theory of resultants, for deriving the coupler curve equation of 16 8-link mechanisms. The solution approach entails successive elimination of problem unknowns to reduce a multivariate system of 8 equations into a single bivariate equation. This bivariate equation is the coupler curve of the mechanism under consideration. Three theorems which summarize key coupler curve characteristics are presented. The computational procedure is illustrated through two numerical examples. These examples address in detail some of the problems associated with the conversion of transcendental loop equations into an algebraic form using tangent half-angle substitutions. An extension of the proposed approach to the determination of degrees of input-output (I/O) polynomials and coupler curves for general n-link mechanisms is also presented.

A Closed-Form Approach to Coupler-Curves of Multi-Loop Mechanisms

1999 ◽  
Vol 122 (4) ◽  
pp. 464-471 ◽  
Author(s):  
A. K. Dhingra ◽  
A. N. Almadi ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Computational Procedure ◽  
Multivariate System ◽  
Algebraic Form ◽  
Numerical Examples ◽  
Solution Approach ◽  
Half Angle ◽  
Curve Equation ◽  
Form Approach

This paper presents a closed-form approach, based on the theory of resultants, for deriving the coupler curve equation of 16 8-link mechanisms. The solution approach entails successive elimination of problem unknowns to reduce a multivariate system of 8 equations in 9 unknowns into a single bivariate equation. This bivariate equation is the coupler curve equation of the mechanism under consideration. Three theorems, which summarize key coupler curve characteristics, are outlined. The computational procedure is illustrated through two numerical examples. The first example addresses in detail some of the problems associated with the conversion of transcendental loop equations into an algebraic form using tangent half-angle substitutions. An extension of the proposed approach to the determination of degrees of input-output (I/O) polynomials and coupler curves for a general n-link mechanism is also presented. [S1050-0472(00)01104-1]

Displacement Analysis of Spatial Seven-Link 5R-2P Mechanisms

1977 ◽  
Vol 99 (3) ◽  
pp. 692-701 ◽  
Author(s):  
J. Duffy
Keyword(s):  
Significant Advance ◽  
Input Output ◽  
Spherical Torus ◽  
Numerical Examples ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Spatial Mechanisms ◽  
Special Cases

Input-output displacement equations of eighth degree are derived for general spatial seven-link (RPPRRRR, RRRPPRR), (RPRRRPR, RPRRPRR), and (RPRPRRR) mechanisms. The results are verified by numerical examples. The solutions of these mechanisms constitute a significant advance in the theory of analysis of spatial mechanisms. They contain as special cases the solutions for spatial seven-link 4R-3P slider-crank mechanisms, the solutions for all five-link 3R-2C and six-link 4R-P-C mechanisms that have appeared in the literature, together with the solutions for a multitude of solved and unsolved mechanisms containing spherical, torus, and plane kinematic pairs.

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.

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.

Displacement Analysis of a Six-DOF Parallel Manipulator With 3R3P and 4R2P Chains

1998 ◽  
Author(s):  
R. Kamra ◽  
D. Kohli ◽  
A. K. Dhingra
Keyword(s):  
Parallel Manipulator ◽  
Solution Procedure ◽  
Analysis Problem ◽  
Forward Displacement ◽  
Numerical Examples ◽  
Displacement Analysis ◽  
Tangent Method ◽  
Six Dof ◽  
Six Degree Of Freedom ◽  
Forward Displacement Analysis

Abstract This paper addresses the forward displacement analysis of a six degree of freedom platform manipulator which is actuated by three different configurations involving six chains with six joints in each chain. The displacement analysis problem involves finding all possible positions and orientations of the platform along with all the joint variables in response to the inputs supplied. The forward displacement analysis problem is solved using the “Suppressed Tangent method.” The proposed solution procedure is illustrated through three numerical examples. The first example deals with forward displacement analysis of a platform manipulator actuated by six 3R3P chains whereas the second and third examples deal with a manipulator actuated by five 3R3P and one 4R2P, and four 3R3P and two 4R2P chains respectively.

A Displacement Analysis of Spatial Six-Link 4-R-P-C Mechanisms—Part 2: Derivation of Input-Output Displacement Equation for RCRRPR Mechanism

1974 ◽  
Vol 96 (3) ◽  
pp. 713-717 ◽  
Author(s):  
J. Duffy ◽  
J. Rooney
Keyword(s):  
Angular Displacement ◽  
Input Output ◽  
Numerical Examples ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Displacement Equation

The input-output displacement equation is expressed as a degree eight polynomial in the half-tangent of the output angular displacement. The equation can be used to generate input-output functions of spatial five-link RCRCR and RCRRC mechanisms. The results are illustrated by numerical examples.

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