Isomorphism Identification Research Articles

The extension of two-Lipschitz operators

The paper deals with some further results concerning the class of two-Lipschitz operators. We prove first an isometric isomorphism identification of two-Lipschitz operators and Lipschitz operators. After defining and characterize the adjoint of two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from cartesian product of two complemented subspaces of a Banach space to the cartesian product of whole spaces. Also, we show that every two-Lipschitz functional defined on cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm, under some requirements on domaine spaces.

Derivation of high-contracted topology graphs for the type synthesis of complex closed robotic mechanisms with more mechanical advantages

AbstractChallenges/shortages of existing research: Both a topology graph TG and a contracted graph CG are the important tool for the type synthesis of mechanisms and have been studied. Let (b, t, q, p, h) be (binary, ternary, quaternary, pentagonal, hexagonal) link, respectively. Existing TG may include (b, t, q, p, h, …), and existing CG may include (t, q, p, h, …) and exclude b. Therefore, their derivation and isomorphism identification are quite complicated.The significance of this research: A novel high-contracted topology graph Gj/i including n (q, p, h, …) is proposed in this paper. Since Gj/i excludes (b, t), Gj/i must be more simple than complex TG and CG, and the derivation and isomorphism identification of various Gj/i must be more easily than that of the complex TG and CG.This paper focuses on the derivation and isomorphism identification of various Gj/i for deriving TG and type synthesis of the complex closed robotic mechanisms with more mechanical advantages and useful functions. First, the conceptions for deriving Gj/i and the relations between Gj/i and the basic links are explained, the relative rules are determined for representing Gj/i and reducing isomorphism of Gj/i using the character strings and the digit groups, and many different Gj/i are represented based on determined rules. Second, many different Gj/i (j = 0, …, 17) are constructed using a circle, several composite curves, and vertex points on the circle based on the character strings and the digit groups. Third, a condition for identifying isomorphism Gj/i is discovered and proved using the composite curves and their connections in Gj/i. Some isomorphism Gj/i are identified from constructed different Gj/i. Finally, different Gj/i (G7/7, G11/11) are illustrated to derive TGs, several complex closed robotic mechanisms are synthesized using derived TGs for creating robotic gripper, a long-stroke robotic pumping unit, and a tunnel robotic excavator.

The Equivalent Mechanical Model of Topological Graphs and the Isomorphism Identification of Kinematic Chains

Abstract In type synthesis of mechanisms, isomorphic identification of kinematic chains is a key issue, which has been studied for many years. In this paper, a new topological invariant, a character array of edge is constructed based on a novel model called an equivalent mechanical model. This invariant is used as a necessary condition for recognizing the edges of two topological graphs. On this basis, a method of constructing mapping by closed loop is proposed. The constructed mapping is used as the adequacy of identification so as to accurately identify the isomorphism of two graphs. This method is sufficient and necessary, which has been successfully tested in the 10-vertex, 12-vertex, 15-vertex, 28-vertex topological graphs. This method is limited to planar mechanisms that consist of revolute joints.

An efficient approach of graph isomorphism identification using loop theory and hopfield neural networks

An efficient approach of graph isomorphism identification using loop theory and hopfield neural networks

Identification of Isomorphism in Kinematic Chains by Using the Reduced Graph Matrix

Identification of Isomorphism in Kinematic Chains by Using the Reduced Graph Matrix

An isomorphism identification method of kinematic chain based on optimal arrangement and comparison of branch-chain matrix derived from dendrogram graph

Isomorphism identification is an essential step in the structure synthesis of Kinematic Chain (KC), and needs a large amount of analysis and calculation. To find an isomorphism identification method with simple rules, scientific feasibility and less analysis and calculation has always been a research hotspot of mechanism scholars. In this paper, the structure information of KC is described by dendrogram structure graph with multiple joints, and Branch-chain Matrix (BM) is separated from the dendrogram structure. The characteristics of BM are analyzed, and the concepts of intimacy between branch-chains and Repeatability Matrix (RM) corresponding to BM are proposed. Based on fact that both dendrogram graph and BM can uniquely determine the structural information of one KC, a new isomorphism identification method for KC, based on row optimal rearrangement and comparison of BM, is proposed. The operation steps are discussed in detail, and several cases are analyzed to show this method has advantages such as easy rules, small calculation of retrieval and comparison, and easy to be programmed.

Computer-aided synthesis of planar mechanisms with one multiple-joint

• An isomorphism identification algorithm is presented for multiple-joint mechanisms. • A method for synthesizing planar mechanisms with one multiple-joint is proposed. • The database of planar multiple-joint mechanisms with up to 16 links is created. In the innovative design for mechanism-based devices and robots, it is crucial to synthesize as many diverse kinematic structures of available mechanisms as possible. Despite the widespread applications of multiple-joint mechanisms in various mechanical systems, few contemporary approaches concentrated on the synthesis of multiple-joint mechanisms. Herein, a computer-aided method is presented to synthesize all feasible non-isomorphic mechanisms with one multiple-joint. First, tricolor topological graphs are introduced that denote the structures of multiple-joint mechanisms. Second, two vertex classification methods are proposed, based on the calculations for the distance and vertex degree, to reduce the number of times required for the isomorphism identification of the generated tricolor topological graphs. Third, an approach is presented to automatically eliminate isomorphic tricolor topological graphs whose fixed vertices exist in the same advance vertex group. Then, a computer-aided method is proposed for synthesizing the tricolor topological graphs of planar mechanisms with one multiple-joint. Finally, a database of synthesized multiple-joint mechanisms with up to 16 links and 11 degrees of freedom (DOFs) is created, and the synthesis results are compared and discussed.

Graph isomorphism identification based on link-assortment adjacency matrix

Graph isomorphism identification based on link-assortment adjacency matrix

A novel compound topological invariant for isomorphism detection of planar kinematic chains

Abstract. The isomorphism identification of the kinematic chain (KC) based on graph theory definition has no advantage in efficiency, especially when the number of links in the KCs is large. The topological characteristic constants for isomorphism identification, such as the power value sequence (PVS), least distance matrix sequence (LDMS), and loop number (LN), are proposed. The fourth PVS, the LDMS, and the LN are compared and arranged in descending order, to form a strong complementary chain, which can identify KCs of at least 15 links with 4 degrees of freedom (DOF). The method is applied to the complete atlas of the following: 8-link 1 DOF, 9-link 2 DOF, 10-link 1 DOF, 12-link 1 DOF, 13-link 2 DOF, and 15-link 4 DOF planar single-joint KCs, 6-link 1 DOF and 7-link 1 DOF planetary gear trains, 8-link 1 DOF planar multiple joint KCs, and contracted graphs with up to six independent loops. All results are in agreement with the reported ones in the literature. Thus, the proposed method possesses good versatility and has been verified as being reliable and efficient.

The singular value decomposition method of improved incidence matrix for isomorphism identification of epicyclic gear trains

Abstract. The epicyclic gear train (EGT) is an advanced gear transmission mechanism, which is widely applied in drive systems. It is of great significance to eliminate the same structure in the type synthesis of EGTs. In this paper, an isomorphism identification method of EGTs based on the singular value decomposition of improved incidence matrix is proposed. Firstly, the improved incidence matrix is used to describe the structure of EGTs. Then, the degree sequence of links and kinematic pairs can be extracted and used as a basic information for preliminary screening. The improved incidence matrix can effectively distinguish multiple joints. Next, the topological connection relationship between the links and the kinematic pairs is extracted by the singular value decomposition of the matrix, which is used as the final value for isomorphism identification of EGTs. Finally, the effectiveness of this method is verified through a lot of examples.

Representation of planar kinematic chains with multiple joints based on a modified graph and isomorphism identification

The representation and isomorphism identification of kinematic chains (KCs) with multiple joints are crucial issues to be solved in mechanism research. In this paper, a modified graph—planar face graph (PF graph), is proposed. As a key concept, “face” has been put forward to represent the case that a single, specific relationship is incident with multiple incidents. Some other basic concepts relating to PF graph are also explained in detail. PF graph is used to represent KCs with multiple joints and it reflects the uniqueness of KCs and realizes the one-to-one correspondence between KCs and adjacency and incident matrices. It also establishes relationships with other methods for representing KCs and it is proved to be suitable for complex mechanical systems. Moreover, the sufficient and necessary condition of isomorphism are derived, and two properties of permutation similarity are analysed. Based on them, the modified eigenvalue eigenvector method and the maximum path method are proposed, and they are integrated to form a new algorithm. In addition, the reliability and efficiency of the algorithm are proved.

The Equivalent Mechanical Model of Topological Graphs and the Isomorphism Identification of Kinematic Chains

The Equivalent Mechanical Model of Topological Graphs and the Isomorphism Identification of Kinematic Chains

A simple and efficient method for isomorphism identification of planar kinematic chains

Isomorphism identification plays an important role in structural design and innovative design. Based on the adjacency matrix and loop theory, a new method is proposed in this paper to identify the isomorphic kinematic chains. It enriches the application of loop-based theory for isomorphism identification. In the kinematic chain, links and joints are connected alternatively and every link corresponds to a fixed link degree. Due to the inherent characteristics, the labeled sequence of links can be random, which does not affect the result of isomorphism identification. By the programming software MATLAB, some examples with 6-, 8-, 10-, 11-, 12-link kinematic chains, and 15-vertex topological graphs are presented. Results show that the proposed method applies to topology graphs and kinematic chains with one or multiple joints. Compared with other methods, the proposed method is confirmed correctly. And there is no counterexample. It lays a solid foundation for structural synthesis in the future.

Isomorphism identification algorithm and database generation for planar 2-6 DOFs fractionated kinematic chains combined by two or three non-fractionated kinematic chains

Existing methods for synthesizing fractionated kinematic chains usually avoid generating isomorphic fractionated kinematic chains rather than identifying isomorphic chains. Consequently, no isomorphism identification algorithm for fractionated kinematic chains, especially those with complex structures, has been reported. This study, based on a new isomorphism identification algorithm, proposes a systematic method to synthesize all possible non-isomorphic fractionated kinematic chains with up to 16 links and up to 6 degrees of freedom (DOFs), which are combined by two or three non-fractionated kinematic chains. First, the structural constraint equations are presented to obtain all possible combinations of fractionated kinematic chains. Second, the two concepts of a split-vertex characteristic graph and a combined characteristic graph are developed to acquire the corresponding combined characteristic code, which is the unique representation of planar fractionated kinematic chains and is applied to identify the isomorphism problem. Then, all possible 2-6 DOFs non-isomorphic fractionated kinematic chains combined by two or three non-fractionated kinematic chains are synthesized. Moreover, the corresponding databases of the synthesized fractionated kinematic chains are established. Finally, the synthesis methods and results of this study and the ones in the existing literature are compared and discussed in detail.

Eliminating isomorphism identification method for synthesizing nonfractionated kinematic chains based on graph similarity

A novel method for eliminating isomorphism identification is proposed to improve synthesis efficiency and synthesize planar nonfractionated kinematic chain (KCs) automatically. This method is based on the vertex insertion of contracted graphs. First, similar edges of contracted graphs are divided into groups, in which the similar edges are found and their characteristic matrices are calculated. The edge types are divided based on whether or not isomerism occurs after vertices are inserted. Then, the vertices are inserted into contracted graphs according to the edge condition. In this process, all isomerism caused by the location and number of inserted vertices is reserved, and the property change of similar edges is checked. Lastly, the rigid subchains of remaining isomerism are distinguished. Contracted graphs with four independent loops and some of their inserted vertices are presented in appendix. A complete set of nonfractionated KCs with up to seven independent loops and three degrees of freedom is also provided. The veracity and efficiency of the method are confirmed by conducting a comparative analysis between this synthesis and other literature results.

Numbering method for the kinematic chain isomorphism recognition of a planar mechanism

The kinematic chain isomorphism identification of a planar mechanism is an important part of the innovative design of kinematic chain regeneration. Based on the structural characteristics of the kinematic chain of a planar mechanism, a new method, i.e. the numbering method, is proposed for isomorphism identification. Firstly, the links of a kinematic chain are numbered according to certain rules, and they are normalized. Then, the normalized contracted link adjacency matrix is used to represent the connection of the kinematic chain graph. Based on whether the contracted link adjacency matrix of two kinematic chains is the same, we can judge whether the two links are similar.

Synthesis method for planetary gear trains without using rotation graphs

The displacement graphs (d-graphs) are the final form of planetary gear trains (PGTs) synthesis. The existing configuration synthesis methods for the d-graphs of PGTs need rotation graphs (r-graphs) as the intermediate step and produce a large amount of isomorphic KCs. This paper proposes a method to obtain d-graphs directly from the geared graph. Meanwhile, a uniform isomorphism identification method suitable for parent graphs, geared graphs, and d-graphs is presented. Finally, a fully automatic method based on parent graphs is tested on synthesizing 1-DOF PGTs with up to nine links. Our synthesis results and those in the literature are comparatively analyzed, and the reasons for contradictory synthesis results are analyzed.

A new method for isomorphism identification of planetary gear trains

Abstract. Planetary gear trains (PGTs) are widely used in machinery such as vehicles, pulley blocks, wrist watches, machine tools, and robots. During the process of structural synthesis of PGTs using graph theory, isomorphism identification of graphs is an important and complicated problem. The reliability of the isomorphism detection method directly determines the accuracy of the synthesis result. In this paper, a novel isomorphism identification method for PGTs is proposed. First, a new weighted adjacent matrix is presented to describe the topological graph of PGTs, which has is unique in describing the structure of PGTs. Then, the weighted distance matrix is proposed and the sum of the matrix is obtained, which can determine whether the planetary gear trains is isomorphic or not. Eventually, the examples demonstrate that this new method can be accurately and effectively performed.

Mechanism isomorphism identification based on artificial fish swarm algorithm

The aim of this paper is to propose a practical solution for mechanism kinematic chain isomorphism identification – an artificial fish swarm algorithm. The artificial fish model of mechanism isomorphism identification is established, and behavioral way of the artificial fish is designed. According to isomorphism identification features of topological graph, the process of mechanism isomorphism identification based on artificial fish swarm algorithm is confirmed. The rationality and reliability of artificial fish swarm algorithm on the isomorphic identification of mechanism have been illustrated by a specific example, which provides a new method for intelligent CAD system design of mechanism. It builds a basis for future work in isomorphism identification of mechanism with high efficiency. Isomorphic identification of mechanism will contribute to rational qualitative analysis of mechanism design, perfection of irrationality can be done timely, which is the key factor for mechanical manufacturing. In this paper, we introduce the mechanism kinematic chain firstly, then optimization of artificial fish swarm algorithm is illustrated, and it is shown that how fish swarm algorithm is applied to mechanism kinematic chain. Finally, the feasibility and efficiency of the method are verified by the example of 10 bars, and the complex mechanism can be identified by the example of 14 bars and 18 bars.

Detecting similar piping design failures in shipbuilding based on sub-graph isomorphism identification

Detecting similar piping design failures in shipbuilding based on sub-graph isomorphism identification