An Innovative Approach to Detect Isomorphism in Planar and Geared Kinematic Chains Using Graph Theory

Kamesh, Vinjamuri Venkata; Rao, Kuchibhotla Mallikarjuna; Rao, Annambhotla Balaji Srinivasa

doi:10.1115/1.4037628

Cited by 28 publications

(7 citation statements)

References 61 publications

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“…Sunkari and Schmidt (2006) introduced the eigenvalue and eigenvector of the adjacency matrix as the criterion into the isomorphism identification method. Moreover, Venkata Kamesh et al (2017) detected the isomorphism of linkage and geared KCs using the concept of net distance in graph theory; counterexamples have been found for 10link KCs in this paper. By considering the simplicity of the method, scholars, such as Shin and Krishnamurty (1992), Tang and Liu (1993), and Rai andPunjabi (2018, 2019), have conducted extensive research on code-based methods.…”

Section: Introductionmentioning

confidence: 56%

“…There are many methods for isomorphism discrimination, which can also be used for similarity recognition, such as the eigenvalue, eigenvector, characteristic polynomial (Uicker and Raicu, 1975;Mruthyunjaya, 1984;Mruthyunjaya and Balasubramanian, 1987;Yan and Hall, 1982;He et al, 2005;Cubillo and Wan, 2005;Sunkari and Schmidt, 2006), Hamming number-based (Rao and Varada Raju, 1991;Rao, 1993, 2008;Dharanipragada and Chintada, 2016;Sun et al, 2017), code-based (Ambekar and Agrawal, 1987;Shin and Krishnamurty, 1992;Tang and Liu, 1993;Rai andPunjabi, 2018, 2019), and distance-based or path-based (Yadav et al, 1996;Venkata Kamesh et al, 2017) methods. For example, Uicker and Raicu (1975) applied characteristic polynomials of adjacent matrices to isomorphism identification.…”

Section: Introductionmentioning

confidence: 99%

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Novel loop tree for the similarity recognition of kinematic chains

et al. 2022

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Abstract. The similarity recognition of kinematic chains (KCs) is helpful for improving the efficiency of configuration synthesis, which has been paid more and more attention in recent years. The existing recognition methods are divided into the definition method and feature constant method. Among them, the definition method is difficult to adopt in practice because of its long operation time, especially when the number of similar vertices in KCs is large. In this paper, the new concepts of a loop tree (LT) and a loop tree matrix (LTM) have been proposed, which improve the efficiency of similarity recognition. This method is applied on the complete structure of the following: 8-link with 1 DOF (degree of freedom), 9-link with 2 DOF, 10-link with 1 DOF, 12-link with 1 DOF, 13-link with 2 DOF, 14-link with 3 DOF, 15-link with 4 DOF planar single-joint KCs, and contracted graphs with up to six independent loops. All results are verified by the definition method to prove the good applicability, reliability, and efficiency of the proposed method. Simultaneously, the application case of the similarity recognition in a mechanism creation is given to provide a reference for an innovative design.

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Section: Introductionmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 99%

Novel loop tree for the similarity recognition of kinematic chains

et al. 2022

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“…Max (Min) code method is a new labeling algorithm for KC, in which the number of links increases, the size of binary sequence and later the computational complexity increases. The distance method proposed by Kamesh [25] and Sun [26] generated the distance matrix by extracting the distance information between links, then obtained the path code or distance sequence string which represented links' information based on the distance matrix, and arranged the path code or distance sequence string of links in ascending or descending order. The isomorphic result was obtained by comparing whether the path codes or distance sequences were the same.…”

Section: Introductionmentioning

confidence: 99%

“…11(11),12(12),13(13),14(14),15(15),16(1 6),17(17),18(18),19(19),20(20),21(21),22(22) ,23(23),24(24),25(25),26(26),27(27),28(28),2 9(29),30(30),31(31),32(32),33(33),34(34),35( 35),36(31),37(37),38(38),39(39),40(40) 1(19),2(20),3(9),4(4),5(10),6(37),7(32),8( 28),9(3),10(5),11(22),12(21),13(18),14(3 8),15(15),16(31),17(27),18(13),19(1),20( 2),21(12),22(11),23(33),24(35),25(29),26 (39),27(17),28(8),29(25),30(30),31(16),3 2(7),33(23),34(40),35(24),36(36),37(6),3 8(14),39(26),40(34) …”

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confidence: 99%

A new Isomorphism Identification Method for Kinematic Chain Based on Exchange and Compare of Lower-Triangle Matrix with Whole Information

Sun

Liu

et al. 2023

Preprint

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Isomorphism identification is an important step and costs major counting labor (more than 90%) in kinematic chain (KC) structure synthesis, many isomorphism identification methods have been proposed by researchers in past decades. The identification principle is simple and easy to be understood, the result has no objection, and the programming is simple and efficient, these are main criterions for isomorphism identification method. A new Matrix with Whole Information (MWI) has been proposed to describe the structure information of KC, and the principles of determining multiple joints and polygonal links are also proposed. Based on the principle that the Lower Triangular Matrix with Whole Information (LTMWI) can uniquely express the structure of KC, a new isomorphism identification method, based on the exchange and compare of LTMWI is proposed. This method is based on the law of exchange of links’ numbers and key points’ numbers, and two KCs are expressed by LTMWI, in which one LTMWI is selected as the datum matrix, the key points are arranged in different groups and non-zero elements of key points are moved forward, the definite and indefinite key points are obtained, under the assumption that all key points in the same positions have the mapping relationships one-to-one between exchange matrix and datum matrix, a finite number of exchange matrices are generated and compared with datum matrix, then isomorphism can be drawn. This method has the advantages such as simple principle, easy to program and only analysis and comparisons are needed in the identification process. The isomorphism identification results and the mapping relationships of links and even revolute pairs can all be obtained. The isomorphic analysis results of several KCs have shown the above characteristics of this method.

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“…This method is test for simple and multiple joints also in epicyclic gear trains for its efficiency. Kamesh et al [3] presented graph theory to detect isomorphism in planar and geared kinematic chains. Mruthyunjaya [4] started with a multiple jointed binary chain and transformed them gradually in stages until all the joints became simple joints.…”

Section: Introductionmentioning

confidence: 99%

A Parametric Approach to Detect Isomorphism and Inversion in the Planar Kinematic Chains

Dewangan

Shukla²

2021

Lecture Notes in Mechanical Engineering

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This work deals with the problem of detection of isomorphism and inversions among planar kinematic chains. To detect isomorphism and inversions a least distance matrix (LDM) is formed using the values on the basis of the parameter. Isomorphism is detected on the basis of chain string (CS) of least distance matrix (LDM). An inversion is generated by grounding a different links in the kinematic chain. Here, link string (LS) is generated for the different kinematic inversions of the chains so that we can detect kinematic inversions with the help of link string (LS).

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An Innovative Approach to Detect Isomorphism in Planar and Geared Kinematic Chains Using Graph Theory

Cited by 28 publications

References 61 publications

Novel loop tree for the similarity recognition of kinematic chains

Novel loop tree for the similarity recognition of kinematic chains

A new Isomorphism Identification Method for Kinematic Chain Based on Exchange and Compare of Lower-Triangle Matrix with Whole Information

A Parametric Approach to Detect Isomorphism and Inversion in the Planar Kinematic Chains

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