A new method to mechanism kinematic chain isomorphism identification

doi:10.1016/S0094-114X(01)00084-2

Mechanism and Machine Theory

Volume 37, Issue 4, April 2002, Pages 411-417

https://doi.org/10.1016/S0094-114X(01)00084-2 Get rights and content

Abstract

A new method based on eigenvectors and eigenvalues is developed in this paper to identify isomorphism of mechanics kinematic chain. Kinematic chains are firstly represented by adjacent matrices. By comparing the eigenvalues and corresponding eigenvectors of adjacent matrices, the isomorphism of mechanism kinematics chain can easily be identified. The relationships in the mechanisms can also be obtained. A proof on this method is given in this paper. Some examples are provided to demonstrate the effectiveness of this method.

Introduction

Structural analysis and synthesis of mechanism is very important for the invention and innovation of mechanism. Isomorphism identification of mechanism kinematic chain is an essential step in kinematic mechanism synthesis. Undetected isomorphisms result in duplicate solutions and an unnecessary effort. Falsely identified isomorphism eliminates possible candidates for new mechanisms. Identifying isomorphism of kinematic chain by using characteristic polynomial method is a simple method [1], but the reliability of these methods was in question, as several counter-examples were found [2]. Some new approaches to these problems were also investigated, such as incident degree [3], [8], group theory [4], [5], adjacent-chain table [6], artificial neural network [7] and so on. However, most of these methods are complex and difficult to grasp and utilize. Therefore, in this paper, a new method to identify isomorphism of mechanism kinematic chain is presented by comparing the eigenvalue and eigenvector of adjacent matrix of kinematic chain. A simple proof to this method is given. This method is also used to obtain more information among mechanisms kinematic chains or in one mechanism kinematic chain.

Section snippets

Theorem to mechanism kinematic chain isomorphism identification and its proof

Suppose A, A^′ are adjacent matrices of mechanism kinematic chains; λ₁, λ₂,…,λ_n; λ₁^′, λ₂^′,…,λ_n^′ are eigenvalues of A, A^′, x₁, x₂,…,x_n; x₁^′, x₂^′,…,x_n^′, are eigenvectors of A, A^′. And the n independent eigenvectors compose nonsingular matrices X and X^′, respectively, with them as column vectors. $X= x_{1} x_{2} … x_{n},$ $X^{′} = x_{1}^{′} x_{2}^{′} … x_{n}^{′} .$ According to matrix theory [9], $X^{−1} AX= diag λ_{1} λ_{2} … λ_{n},$ $X^{′−1} A^{′} X^{′} = diag λ^{′}_{1} λ^{′}_{2} … λ^{′}_{n} .$

If the kinematic chains represented by A, A^′ are isomorphic, their eigenvalues can be modified to be in same

Examples of kinematic chains with 12 bars

There are three kinematic chains with 12 bars as shown in Fig. 1. They have the same characteristic polynomial coefficients and eigenvalues, so it makes the characteristic polynomials approach fail to work.

According to Fig. 1, the adjacent matrices for the three chains are shown below: $A= 000011000000000010001000000010000100000000100010111000000000100000110000000101001000000001000110010000100001001000010001000100010001000000001110; B= 01000110000010100001000001010000010000101000000000010100000110001$

Conclusion

In this paper, a new method to identify isomorphism is developed. By this method, the isomorphism of mechanism kinematic chain can easily be identified. In this method, the eigenvalues and eigenvectors of adjacent matrix of mechanism kinematics chain are used to analyze the mechanism structure. According to this method, we can find that the adjacent matrix is a map of mechanism kinematics chain, and the eigenvalues, eigenvectors and other characteristic values may reflect some nature and inner

References (9)

F.G. Kong et al.
An artificial neural network approach to mechanism kinematic chain isomorphism identification
Mechanism and Machine Theory
(1999)
H.S. Yan et al.
Linkage characteristic polynomials: definition, coefficients by inspection
ASME, Journal of Mechanical Design
(1981)
T.S. Mruthyunjaya
In quest of a reliable and efficient computational test for detection of isomorphism in kinematic chains
Mechanism and Machine Theory
(1987)
T.J. Jongsma
An efficient algorithm for finding optimum code under the condition of incident degree
Proceedings of Mechanical Conference
(1992)

There are more references available in the full text version of this article.

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A new method to mechanism kinematic chain isomorphism identification

Abstract

Introduction

Section snippets

Theorem to mechanism kinematic chain isomorphism identification and its proof

Examples of kinematic chains with 12 bars

Conclusion

Mechanism and Machine Theory

Linkage characteristic polynomials: definition, coefficients by inspection

ASME, Journal of Mechanical Design

In quest of a reliable and efficient computational test for detection of isomorphism in kinematic chains

Mechanism and Machine Theory

An efficient algorithm for finding optimum code under the condition of incident degree

Proceedings of Mechanical Conference