A joint–joint matrix representation of planar kinematic chains with multiple joints and isomorphism identification

Abstract

The synthesis of the kinematic chain needs to obtain the information of the kinematic chain accurately and comprehensively. Isomorphism identification is an essential step in kinematic chain synthesis. In this article, a novel isomorphism determination method of planar kinematic chains with multiple joints based on joint-joint matrix description was proposed. First, a joint-joint matrix is presented to describe the kinematic chain, which can uniquely represent the kinematic chain structure. Then links and joints information were extracted from the matrix. And the link code and joint code were introduced to represent the link attributes and joint attributes, respectively. Furthermore, the standardization rules of joint-joint matrix are proposed. Isomorphism of kinematic chain is identified by comparing links, joints, and matrices. And the relationship between the links and the joints corresponding to the isomorphic kinematic chain is determined. Finally, the examples demonstrate that the method is novel and efficient.

Similar Vertices and Isomorphism Detection for Planar Kinematic Chains Based on Ameliorated Multi-Order Adjacent Vertex Assignment Sequence

“…Step 4: There are two corresponding relations by comparing S 7 a,1 with S 7 b,1,1 and S 7 b,1,2 : b a M 1 = {28, 29, 30, 9, 8, 1, 2, 3, 4, 5, 6, 7, 14, 16, 17, 18, 19, 20, 21, 26, 25, 24, 23, 27, 15, 13, 11, 12, 10, 22}, b a M 2 = {28, 29,30,9,8,1,2,3,4,5,6,7,14,16,17,18,19,20,25,26,21,22,23,27,15,13,11,12, 10, 24}.…”

“…By using this method, the sufficiency of isomorphism detection was obtained based on vertex correspondence. Other isomorphism detection methods, such as partition matching algorithm [13], hybrid immune algorithm [14], optimized circuit simulation [15][16][17], connectivity number and entropy [18], binary code [19], information theory [20], joint matrix [21,22], loop theory [23][24][25][26], high-order adjacency link values [27], are noticeable too. Although all the abovementioned synthesis methods can identify the isomorphism, they cannot accurately find the similar vertices that are beneficial for mechanism selection and isomorphism detection.…”

Structural synthesis of plane kinematic chain inversions without detecting isomorphism

“…Ding and Huang (2007) improved the perimeter-loop-graph-based method to detect isomorphism. Sun et al (2018) applied the joint-joint matrix to represent planar kinematic chains with multiple joints and studied the isomorphism detection of kinematic chains by comparing links, joints and matrices.…”

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

“…Rai and Panjabi [5] proposed work with different structural invariants, i.e., primary structural invariants for isomorphism identification and secondary structural invariant provides power transmission and transmission efficiency. Sun et al [6] joint-joint matrix proposed to describe the kinematic chain, which can uniquely represent the kinematic chain structure. The links and joints information were extracted from joint-joint matrix and the link code and joint code were use to represent the link and joint attributes isomorphism detection was done by comparing links, joints and matrices.…”