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 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.

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 Computer Implementation of the Position Solution of Planar Linkages Using Vector Loop Decomposition

1998 ◽  
Author(s):  
Clint A. Kahler ◽  
J. Keith Nisbett ◽  
Clement R. Goodin
Keyword(s):  
Closed Form ◽  
Computer Software ◽  
Computer Implementation ◽  
Software Application ◽  
Prismatic Joints ◽  
Position Solution ◽  
Planar Linkages ◽  
Form Approach ◽  
Independent Loops

Abstract A general closed-form approach to the solution of loop equations of planar n-bar linkages is presented. Each loop of a set of canonical independent loops is decomposed to a set of vectors. Several common combinations of revolute and prismatic joints are defined. By evaluating the types of joints at each end of a vector, the magnitude and direction of the vector are determined to be known constants or unknown variables. This leads to an identification of the number of unknowns and the distribution of unknowns in the loop. This identification allows the unknowns to be found by matching the situation to one of the unique, closed-form cases for a solvable loop. A computer software application has been developed and is analyzed for efficiency.

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.

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 Ro-R-R-CP SPATIAL MECHANISMS WITH VECTOR ALGEBRAIC METHOD

1999 ◽  
Vol 23 (1A) ◽  
pp. 95-112
Author(s):  
C.M. Wong ◽  
K.C. Chan ◽  
Y.B. Zhou
Keyword(s):  
Closed Form ◽  
Algebraic Method ◽  
Input Output ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Spatial Mechanisms ◽  
Kinematic Loop

This paper presents the displacement analysis of the three variants of a spatial kinematic loop containing 3R and 1CP joints using vector algebraic method. The closed-form input-output displacement equations of this mechanism are derived as forth-order polynomials. Analytical steps and expressions are laid out uniformly and simply.

Closed-Form Displacement Analysis of Five-Link R-C-R-C-P Spatial Mechanism

1973 ◽  
Vol 2 (4) ◽  
pp. 238-240
Author(s):  
R. V. Dukkipati
Keyword(s):  
Closed Form ◽  
Second Order ◽  
Input Output ◽  
Variable Parameters ◽  
Displacement Analysis ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Dual Number ◽  
Input And Output ◽  
Order Polynomial

Using (3 x 3) matrices with dual-number elements, closed-form displacement relationships are derived for a spatial five-link R-C-R-C-P mechanism. The input-output closed form displacement relationship is obtained as a second order polynomial in the output displacement. For each set of the input and output displacements obtained from the equation, all other variable parameters of the mechanism are uniquely determined. A numerical illustrative example is presented. The derived input-output relationship can be used to synthesize an R-C-R-C-P function generating mechanism for a maximum of 15 precision conditions.

Displacement Analysis of the R0-2R-2C Spatial Linkage Mechanisms

1994 ◽  
Author(s):  
Y. B. Zhou ◽  
R. O. Buchal ◽  
R. G. Fenton
Keyword(s):  
Closed Form ◽  
Algebraic Method ◽  
Input Output ◽  
Displacement Analysis ◽  
Output Displacement

Abstract Closed form input-output displacement equations are derived for the R0-2R-2C mechanisms using the vector algebraic method. As compared to previous works, the proposed method is characterized by its standardized analysis steps, compact expressions and simplicity.

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