Abstract
Based on the mathematic representation of loops of kinematic chains, this paper proposes the “⊕” operation of loops and its basic laws and establishes the basic theorem system of the loop algebra of kinematic chains. Then the basis loop set and its determination conditions, and the ways to obtain the crucial perimeter topological graph are presented. Furthermore, the characteristic perimeter topological graph and the characteristic adjacency matrix are also developed. The most important characteristic of this theory is that for a topological graph which is drawn or labeled in any way, both the resulting characteristic perimeter topological graph and the characteristic adjacency matrix obtained through this theory are unique, and each has one-to-one correspondence with its kinematic chain. This characteristic dramatically simplifies the isomorphism identification and establishes a theoretical basis for the numeralization of topological graphs, and paves the way for numeralization and computerization of the structural synthesis and mechanism design further. Finally, this paper also proposes a concise isomorphism identification method of kinematic chains based on the concept of characteristic adjacency matrix.
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References
Freudenstein F, Dobrjanskyj L. On a theory of type synthesis of mechanisms. In: Proceedings of 11th International Congress of Applied Mechanics. Berlin: Springer, 1964. 420–428
Lu K C. The Graph Theory and Applications (in Chinese). Beijing: Tsinghua University Press, 1984. 3–20
Yang T L. Basic Theory of Mechanical System—Structical·Kinematical·Dynamics (in Chinese). Beijing: Mechanical Press, 1995. 1–69
Cao W Q. The Analysis and Synthesis of Linkage Mechanisms (in Chinese). 2nd ed. Beijing: Science Press, 2002. 1–58
Yan H S, Hall A S. Linkage characteristic polynomials: definitions, coefficients by inspection. Trans ASME J Mech Design, 1981, 103(3): 578–589
Mruthyunjaya T S, Balasubramanzan H R. In quest of a reliable and efficient computational test for detection of isomorphism in kinematic chains. Mech Mach Theory, 1987, 22(2): 131–140
Rao A C, Varada R D. Application of the hamming number technique to detect isomorphism among kinematic chains and inversion. Mech Mach Theory, 1991, 26(1): 55–75
Luo Y F, Yang T L, Cao W Q. Identification of isomorphism for kinematic chains using vertex incident degree. In: The 7th National Conference on Mechanisms in Bei daihe (in Chinese), 1990. 16–20
Ambeker A G, Agrawal V P. Canonical numbering of kinematic chains and isomorphism problem: Min code. Mech Mach Theory, 1987, 22(5): 453–467
Luo Y F, Yang T L, Cao W Q. Identification of isomorphism for kinematic chains using incident degree and incident degree code. Chin J Mech Eng (in Chinese), 1991, 27(2): 44–50
Shin J K, Krishnamurty S. Development of a standard code for colored graphs and its application to kinematic chains. Trans ASME J Mech Design, 1994, 116(1): 189–196
Chu J K, Cao W Q. Identification of isomorphism among kinematic chains and inversion using link’s adjacent-chain-table. Mech Mach Theory, 1994, 9(1): 53–68
Liu C H, Yang T L. Using the topological structure code permutation group method of polyhedral solid for identifying kinematic chains. Mech Sci Tech (in Chinese), 1999, 27(2): 180–185
Feng C, Chen Y. Mechanism kinematics chain isomorphism identification based on genetic algorithms. Chin J Mech Eng (in Chinese), 2001, 37(10): 29–30
Kong F G, Zhou H J. A new method to mechanism kinematic chain isomorphism identification. Chin Mech Eng (in Chinese), 1997, 8(2): 30–33
Chang Z Y, Zhang C, Yang Y H, et al. A new method to mechanism kinematic chain isomorphism identification. Mech Mach Theory, 2002, 37(4): 411–417
Mruthyunjaya T S. Kinematic structure of mechanisms revisited. Mech Mach Theory, 2003, 38(4): 279–320
Rao A C, Prasad Raju Pathapati V V N R. Loop based detection of isomorphism among chains, inversions and type of freedom in multi-degree of freedom chains. ASME J Mech Design, 2000, 122(1): 31–42
Ding H F, Huang Z. A new topological description method of kinematic chain. In: The 14th National Conference on Mechanisms in Chongqing, 2004. 161–164
Ding H F, Huang Z. The systemic research on loop characteristics of kinematic chains and its applications. In: ASME Conference, MECH-84283, California, USA, 2005
Huang Z, Ding H F. Software for topological graphs sketching and the creation of characteristic representation codes of kinematic chains automatically. Authorized number: 2005SR04488
Ding H F, Huang Z. Software for creating atlas database of kinematic chains. Authorized number: 2005SR04487
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Supported by the National Natural Science Foundation of China (Grant No. 50575197)
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Huang, Z., Ding, H. Theory of loop algebra on multi-loop kinematic chains and its application. SCI CHINA SER E 50, 437–447 (2007). https://doi.org/10.1007/s11431-007-0057-6
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DOI: https://doi.org/10.1007/s11431-007-0057-6