A new method to mechanism kinematic chain isomorphism identification
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,…,λn; λ1′, λ2′,…,λn′ are eigenvalues of A, A′, x1, x2,…,xn; x1′, x2′,…,xn′, are eigenvectors of A, A′. And the n independent eigenvectors compose nonsingular matrices X and X′, respectively, with them as column vectors.According to matrix theory [9],
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:
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
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2022, Mechanism and Machine TheoryCitation Excerpt :The key to the necessary and sufficient condition of isomorphism proposed in this paper is the orthogonal matrix D makes F=TDPT a permutation matrix, and the final judgment basis must be consistent with FTAF = B. However, the judgment basis proposed by Sunkari and Schmidt [14]., Chang et al [22]., and He et al [25].
Structural synthesis towards intelligent design of plane mechanisms: Current status and future research trend
2022, Mechanism and Machine TheoryCitation Excerpt :This issue has aroused widespread concern among scholars. Various kinds of KC isomorphism detection methods have been successfully developed, including those methods based on characteristic polynomial [19–23], link or joint paths/distances [24–29], Hamming number [30], standard code [31–35], permutation groups [36], eigenvalues and eigenvectors [37–41], neural network or evolutionary [42–48], link or joint adjacency/incidence relationship [49–52], loops [53–57] and other methods [58–65]. If correct synthesis results can be derived when an isomorphism detection method is applied to the KC synthesis, this method can be verified as being reliable.
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2018, Mechanism and Machine TheoryCitation Excerpt :This idea behind this was that if only the shortest route was considered, effect of other links that contribute significantly in the transmission of motion from input to output link is completely neglected. Cubillo and Wan [22] discussed necessary and sufficient conditions of the eigenvalues and eigenvectors of adjacent matrices of isomorphic chains and also they revised the theory published by Chang et al. [19] about mechanism kinematic chain isomorphism using adjacent matrices. The main advantage of this method is its easy computer execution.