A Method for Recognizing Topological Symmetry of Kinematic Chain
-
摘要: 运动链拓扑对称性需要识别以减少机械装置新构型的备选方案,提高机械结构设计的效率。本文基于连杆邻接矩阵提出了一种新的运动链拓扑对称性识别方法。首先以连杆邻接矩阵为已知量导出汉明矩阵,计算汉明矩阵的平方阵与连杆邻接矩阵的立方阵,其次将上述平方阵与立方阵求积得积矩阵且将此积矩阵各行中的元素按降序排列得积矩阵行序列,最后将序列中各元素与拓扑因子相乘并求和得出运动链的拓扑对称性识别码(Topological symmetry recognition code,TSRC)。若同一拓扑图中两顶点的对称性识别码相同,则它们是对称的,否则是非对称的。通过单铰、复铰运动链和行星轮系的实例证明此方法的有效性,结果表明:此方法简单、高效且便于计算机编程。Abstract: To improve the efficiency of the mechanical structural design, it is necessary to identify the topological symmetry of a kinematic chain to reduce the options of the configuration of the new mechanical device. The Hamming matrix is derived from the link adjacency matrix. The cubic matrix of the adjacency matrix and the square matrix of the Hamming matrix are calculated. The square matrix and the cubic matrix are computed to obtain the product matrix. The elements in each row of the product matrix are arranged in descending order to obtain their row sequences. Finally, topological symmetry recognition code (TSRC) of the kinematic chain is obtained by summing the product of the elements in the above sequences with the topological factor. If the symmetry recognition codes of two vertices in a topological graph are the same, they are symmetric; otherwise they are not. The effectiveness of this method is proved by some examples of a single joint kinematic chain, multi-joint kinematic chain and planetary gear train. The results show that this method is simple, efficient and easy for computer programming.
-
Key words:
- kinematic chains /
- topological symmetry /
- link adjacency matrix /
- Hamming matrix train
-
图 6 单铰运动链实例[14]
表 1 图1各构件的TSRC值
构件号 1 2 3 4 5 6 TSRC 10524 10036 10036 10524 10036 10036 表 2 图6a)各构件的TSRC值
组别 TSRC 构件 组别 TSRC 构件 1 118544 2,10 5 144806 4 2 120996 7 6 145134 8 3 126096 6 7 147240 3,9 4 133930 5 8 182290 1 表 3 图6b)各构件的TSRC值
组别 TSRC 构件 组别 TSRC 构件 1 114150 8 6 130834 3 2 122272 2 7 140924 4 3 124384 7 8 142052 9 4 126408 10 9 144084 5 5 129846 6 10 169560 1 表 4 图6a)各构件的连杆邻接串
构件
标号连杆邻接串 1 34, 000001311301, (32, 000001212301)×3 2 50, 000014310001, 40, 000002402101 3 40, 000012203101, 40, 000002402101, 32, 000001212301 4 (40, 000002402101)×2, 36, 000000252001 5 40, 000012203101, 34, 000001311301 6 50, 000014310001, 36, 000000252001 7 50, 000014310001, 40, 000002402101 8 (40, 000002402101)×2, 32, 000001212301 9 40, 000012203101, 40, 000002402101, 32, 000001212301 10 50, 000014310001, 40, 000002402101 表 5 图7a)各顶点的TSRC值
组别 TSRC 顶点 组别 TSRC 顶点 1 8584724 4 6 9211224 10 2 8877346 7 7 9895040 5 3 8888906 6 8 10181800 3 4 8918560 2 9 1398484 8 5 9208086 9 10 1398692 1 表 6 图7b)各顶点的TSRC值
组别 TSRC 顶点 组别 TSRC 顶点 1 694288 6 4 887856 5,7 2 752736 4,8 ~ 10 5 1190688 1,3 3 812240 2 表 7 图7a)各顶点的CPVS值
组别 SPVS 顶点 组别 SPVS 顶点 1 2826268400000 8 6 26984440000 9,10 2 2826268222000 1 7 26654420000 7 3 281087542000 3 8 7754442000 4 4 28774441000 5 9 7654440000 6 5 261084441000 2 表 8 图8a)各顶点的TSRC值
组别 TSRC 顶点 组别 TSRC 顶点 1 3354030 6,7 3 4216814 3,5 2 4015348 2,4 4 5113004 1 表 9 图8b)各顶点的TSRC值
组别 TSRC 顶点 组别 TSRC 顶点 1 1226220 6 4 1583910 4 2 1360842 3 5 1598334 5 3 1467846 2 6 1768554 1 -
[1] 颜鸿森. 机械装置的创造性设计[M]. 姚燕安, 王玉新, 郭可谦, 译. 北京: 机械工业出版社, 2002YAN H S. Creative design of mechanical devices[M]. YAO Y A, WANG Y X, GUO K Q, trans. Beijing: China Machine Press, 2002 (in Chinese) [2] 邓涛, 周豪, 唐鹏. 并联混合动力传动系统结构创新设计研究[J]. 汽车工程, 2018, 40(9): 997-1004DENG T, ZHOU H, TANG P. A research on the structural Innovative design of parallel hybrid powertrain system[J]. Automotive Engineering, 2018, 40(9): 997-1004 (in Chinese) [3] YAN H S, CHIU Y T, et al. On the number synthesis of kinematic chains[J]. Mechanism and Machine Theory, 2015, 89: 128-144 doi: 10.1016/j.mechmachtheory.2014.08.012 [4] RIZVI S S H, HASAN A, KHAN R A. An efficient algorithm for distinct inversions and isomorphism detection in kinematic chains[J]. Perspectives in Science, 2016, 8: 251-253 doi: 10.1016/j.pisc.2016.03.022 [5] TUTTLE E R, PETERSON S W, TITUS J E. Further applications of group theory to the enumeration and structural analysis of basic kinematic chains[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1989, 111(4): 494-497 doi: 10.1115/1.3259027 [6] TUTTLE E R, PETERSON S W, TITUS J E. Enumeration of basic kinematic chains using the theory of finite groups[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1989, 111(4): 498-503 doi: 10.1115/1.3259028 [7] YAN H S, HWANG Y W. The specialization of mechanisms[J]. Mechanism and Machine Theory, 1991, 26(6): 541-551 doi: 10.1016/0094-114X(91)90037-5 [8] 褚金奎, 张然, 邹炎火. 平面机构拓扑结构理论及在机构创新设计中的应用[M]. 北京: 科学出版社, 2017CHU J K, ZHANG R, ZOU Y H. Topological structure theory of planar mechanism and its application in innovative design of mechanism[M]. Beijing: Science Press, 2017 (in Chinese) [9] WANG Y X, YAN H S. Computerized rules-based regeneration method for conceptual design of mechanisms[J]. Mechanism and Machine Theory, 2002, 37(9): 833-849 doi: 10.1016/S0094-114X(02)00036-8 [10] 王玉新, 蹇军, 颜鸿森. 构件相似性判定的关系码方法[J]. 天津大学学报(自然科学与工程技术版), 2000, 33(3): 290-293WANG Y X, JIAN J, YAN H S. Distinguishing the similarity of links by relation code method[J]. Journal of Tianjin University (Science and Technology), 2000, 33(3): 290-293 (in Chinese) [11] 宋黎, 杨坚. 用邻接矩阵判断含复铰平面运动链同构和拓扑对称的新方法[J]. 机械科学与技术, 2005, 24(8): 1005-1008 doi: 10.3321/j.issn:1003-8728.2005.08.035SONG L, YANG J. A new method based on adjacent matrix for identifying isomorphism and topological symmetry in planar kinematic chain with multiple joints[J]. Mechanical Science and Technology for Aerospace Engineering, 2005, 24(8): 1005-1008 (in Chinese) doi: 10.3321/j.issn:1003-8728.2005.08.035 [12] 张锦丽. 基于图谱库的平面4和6自由度可分离运动链自动结构综合[D]. 秦皇岛: 燕山大学, 2014ZHANG J L. Automatic structure synthesis of fractionated kinematic chains based on atlas database[D]. Qinhuangdao: Yanshan University, 2014 (in Chinese) [13] SUN L, CUI R J, YE Z Z, et al. Similarity recognition and isomorphism identification of planar kinematic chains[J]. Mechanism and Machine Theory, 2020, 145: 103678 doi: 10.1016/j.mechmachtheory.2019.103678 [14] RAO A C, RAJU D V. Application of the hamming number technique to detect isomorphism among kinematic chains and inversions[J]. Mechanism and Machine Theory, 1991, 26(1): 55-75 doi: 10.1016/0094-114X(91)90022-V [15] 丁华锋, 黄真. 平面机构统一拓扑描述模型的建立及同构判别[J]. 机械工程学报, 2009, 45(3): 99-103 doi: 10.3901/JME.2009.03.099DING H F, HUANG Z. Uniform topological representation model of planar mechanisms and isomorphism identification[J]. Chinese Journal of Mechanical Engineering, 2009, 45(3): 99-103 (in Chinese) doi: 10.3901/JME.2009.03.099 [16] YANG W J, DING H F, ZI B, et al. New graph representation for planetary gear trains[J]. Journal of Mechanical Design, 2018, 140(1): 012303 doi: 10.1115/1.4038303 [17] 徐俊明. 图论及其应用[M]. 合肥: 中国科学技术大学出版社, 2010XU J M. Graph theory and its application[M]. Hefei: University of Science and Technology of China Press, 2010 (in Chinese) [18] DHARANIPRAGADA V, CHINTADA M. Split hamming string as an isomorphism test for one degree-of-freedom planar simple-jointed kinematic chains containing sliders[J]. Journal of Mechanical Design, 2016, 138(8): 082301 doi: 10.1115/1.4033611 [19] KUO C H, SHIH C J. Computational identification of link adjacency and joint incidence in kinematic chains and mechanisms[J]. Journal of Mechanical Design, 2008, 130(8): 084501 doi: 10.1115/1.2936931