Kinematic Chain Isomorphism Research Articles

Detection of Isomorphism of Kinematic Chains Using two DOF Using CSJV Matrix

Detection of Isomorphism of Kinematic Chains Using two DOF Using CSJV Matrix

A new structural code for detecting isomorphism of kinematic chains with simple, multiple and gear pairs

A new structural code for detecting isomorphism of kinematic chains with simple, multiple and gear pairs

A new structural code for detecting isomorphism of kinematic chains with simple, multiple and gear pairs

A new structural code for detecting isomorphism of kinematic chains with simple, multiple and gear pairs

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

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

Identification of Cospectral Simple Jointed Kinematic Chains using Characteristic Polynomial of Degree Matrix and Structural Matrix

Different methods to address the isomorphism in kinematic chains existed earlier and is well reported. In all the methods, link-adjacency matrix (LAM) represents the structural information of a given kinematic chain. Having gathered a large collection of co-spectral chains belonging to a large number of chain categories possessing (containing up to 20 links and up to 13 degrees of freedom), the present work aims at addressing the isomorphism by developing and implementing a methodology for isomorphism based on characteristic coefficients of the degree matrix and structural matrix representations of a specified Simple Jointed Kinematic Chain (SJKC) being computed and comparing. While the above two tests proved to be successful in most cases, two instances of cospectral cases were discovered wherein the above polynomial-based tests failed to detect dissimilarity in the chain structures. These cases are reported in the paper.

Computer-aided digitization method for structural synthesis of planar non-fractioned kinematic chains with pure revolute joints

Structural synthesis plays an important role in the conceptual design of mechanisms. Based on the link assortment array, links are divided into binary links and multi-vertex links. From the relation of binary links and multi-vertex links, a new form of the adjacency matrix is introduced to illustrate planar kinematic chain. It enriches the representative form of the adjacency matrix, which can provide a new direction for rigid-chain detection and structural synthesis. Without auxiliary graph models, based on the link-path adjacency matrix, characteristic code, and Sudoku Crosswords, a computer-aided digitization method is proposed to synthesize the whole family of planar non-fractioned kinematic chains. Compared with most existing methods of structural synthesis, isomorphic kinematic chains and some rigid chains are identified and deleted in the generation of link-path adjacency matrix, which can improve the efficiency of structural synthesis. By synthesizing planar non-fractioned kinematic chains with up to 10 links and 5 degrees of freedom, results confirm that the proposed method is available.

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.

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.

Automatic Generation of N-Bar Planar Linkages Containing Sliders

In this work, a generalized algorithm is introduced to generate all alternatives of planar N-bar kinematic chains (KCs) with simple joints containing sliders. A simple graphical technique is introduced to enumerate all available N-bar chains with prismatic (P) joints. Then, a new topological Loop Code (TLC) is presented to detect isomorphic chains during the enumeration process in addition to detecting rejected KCs. A visual C++ code is developed for automatic enumeration and detection of rejected KCs and isomorphic KCs. Examples of 6, 8, and 10-bar KCs are presented to illustrate algorithm procedures. As a result, 21, 16, and 1350 KCs have P-joints for Stephenson, Watt, and 8-bar chains, respectively. Also, 308 KCs are obtained for a 10-bar KC with up to 3 sliders.

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.

Topological invariant and definition method for detecting isomorphism in planar kinematic chains

Isomorphism identification is an essential step in mechanism configuration synthesis. Although various detection methods have been proposed, some of them can only effectively identify kinematic chains (KCs) within 10 links or complex programs that are needed to identify multilink KCs. In this study, a new isomorphism identification method is proposed based on the distance concept of graphs and the graph theory definition of isomorphism. In addition to two complex 21- and 28-link planar simple-joint KCs (PSKCs), the proposed algorithm is tested on the complete atlas of 8-link 1-DOF, 9-link 2-DOF, 10-link 1-DOF, 12-link 1-DOF, and 13-link 2-DOF PSKCs. The algorithm is also tested on 6-link 1-DOF and 7-link 1-DOF planetary gear trains (PGTs) to detect isomorphism. All results are in agreement with those of the existing literature. The method is fully automated via a computer program and has been verified to be reliable and efficient.

An Application of Modified Path Matrix Approach for Detection of Isomorphism Among Epicyclic Gear Trains

The identification of isomorphism in epicyclic gear trains has been found a lot of attention by researchers for the last few years. Various methods have been suggested by different authors for the detection of isomorphism in planer kinematic chains and epicyclic gear trains (EGTs), but everyone has found some difficulties to address new issues. In this paper, a modified path matrix approach was presented in order to compare all the distinct geared kinematic mechanisms. A new method based on the matrix approach and corresponding train values is required to identify isomorphism among epicyclic gear trains and their mechanisms. The proposed method was examined on the basis of various examples from four-link, five-link, six-link, and eight-link one-degree-of-freedom EGTs and six-link two-degree-of-freedom EGTs. All the examples have been found satisfactory results with existing literature.

A distinct matrix representation of the planar kinematic chains and isomorphism recognition

Structural synthesis of kinematic chains has been an indispensable area of the mechanism-design problem. The duplication may occur while developing kinematic chains. Therefore, an isomorphic test is required to eliminate duplication. For this purpose, the numbers of methods are proposed during recent years. However, most of the methods are complex and difficult to understand, and fulfil the only primary condition, but not the secondary conditions for isomorphism detection. In the present work, a new method is introduced to detect isomorphism in planar kinematic chains (KCs) fulfilling both primary and secondary conditions. First, KC’s are topologically transformed into skeleton diagrams, and then skeleton matrices [S] and identification strings [IS] are formulated consequently. In order to detect isomorphism, the IS is considered as an invariant string of a KC which in turn, enables the detection of isomorphism between the KCs. The proposed method accurately recognizes isomorphism up to 12 links KCs with no counter examples found in the literature. Three examples with one degree of freedom having 10 links 12 joints, 10 links 13 joints and 12 links three degree of freedom systems are introduced to reveal the reliability and strength of the proposed method.

A novel algorithm for the isomorphism detection of various kinematic chains using topological index

For mechanism design, the isomorphic kinematic chains (KCs) should be deleted to improve design efficiency. At present, most methods of isomorphism detection involve complex concepts and comparison of intermediate parameters. Moreover, when the number of components increases, it is complex and difficult to discriminate a large number of KCs in short time. However, the molecular topological index is proposed to identify isomer in organic chemistry, which shows excellent abilities for isomer recognition. In some respects, topological graph of kinematic chain is similar to chemical molecular model. Therefore, this idea is adopted to obtain extended adjacency identification index (EAID) which fits for isomorphism detection of KCs, namely KC-EAID. If two topological graphs have same KC-EAID value, then they are isomorphic. Otherwise, they are not. Three kinds of kinematic chains, including simple joints KCs, multiple joints KCs and gear-link KCs are given to verify effectiveness of this index in isomorphism detection. For KC-EAID index, it requires a few known quantities and no comparison of intermediate parameters. The algorithm only includes the calculation of matrices, which saves operation time. Therefore, KC-EAID can be used as a powerful tool for isomorphism detection in number synthesis of KCs.

Isomorphic identification for kinematic chains using variable high-order adjacency link values

Isomorphism identification is a crucial issue in the type synthesis of kinematic chains (KCs). To date, various topological characteristics have been used to detect isomorphism, but the manner of the correspondence between two isomorphic KCs has seldom been investigated. In this paper, a method using variable high-order adjacency link values to identify isomorphisms in KCs is proposed. First, the definition of improved high-order adjacency link values, which are used to describe the characteristics of KCs, is introduced in detail. The variable high-order adjacency link values are then calculated repeatedly through reassignment procedure according to the repetitions of their elements for one KC. Finally, isomorphisms are identified and all manner of the correspondence are detected by comparing two high-order adjacency link strings from two KCs. The proposed method is tested on 8-link, 15-link, and 28-link KCs, and all results demonstrate its feasibility and efficiency.

Isomorphism Detection of Planar Kinematic Chains With Multiple Joints Using Information Theory

Isomorphism (structural similarity) of kinematic chains (KCs) of mechanisms is an important issue in the structural synthesis, which must be identified to avoid the duplicate structures. Duplication causes incorrect family size, i.e., distinct KCs with a given number of links (n) and degree of freedom (dof). Besides simple joints kinematic chains (SJKCs), multiple joints kinematic chains (MJKCs) are also widely used because of their compact size and the methods dealing with such KCs are few. The proposed method deals with two different structural invariants, i.e., primary structural invariants (provide only the necessary condition of isomorphism), such as link connectivity number (LCN) of all the links, link connectivity number of chain (CCN), joint connectivity number (JCN) of all the joints, and joint connectivity number of chain (JCNC), and secondary structural invariants (provide the sufficient condition of isomorphism), such as power transmission (P) and transmission efficiency (Te). Primary structural invariants are calculated using a new link–link connectivity matrix (LLCM), whereas secondary structural invariants are calculated using the concept of entropy of information theory. The method has been successfully tested for 10 and 11 links MJKCs (illustrative examples taken in the paper) and for the families of 18 MJKCs with 8 links, 2 MJs, 1-dof, and 3 independent loops; 22 MJKCs with 8 links, 1 MJ, 1-dof, and 3 independent loops; and 83 MJKCs with 9 links, 1 MJ, 2-dof, and 3 independent loops.

A new algorithm of links labelling for the isomorphism detection of various kinematic chains using binary code

Isomorphism of kinematic chains (KCs) has always been a critical issue for the researchers dealing with structural synthesis. Consequently, many researchers of repute presented various methods during the last eight decades for the KCs with either simple and/or multiple joints. Binary code is one of such various methods, but the major problem lies in the algorithm of links labelling, which becomes cumbersome, in particular, for large KCs. The paper presents a simple algorithm of links labelling used to find out a binary sequence which, in turn, provides a maximum binary code (chain invariant). The algorithm is tested for six, seven, eight, nine, ten, eleven, twelve and fifteen links with simple joints, seven and eight links KCs with multiple joints and finally, the Epicyclic gear trains (EGTs) with four, five and six links for its efficiency and reliability. The results are in full agreement with the references taken for the purpose. The paper discusses, in a unique way, the decoding of the binary codes of different KCs also.

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

Detection of isomorphism in planar and geared kinematic chains (GKCs) is an interesting area since many years. Enumeration of planar and geared kinematic chains becomes easy only when isomorphism problem is resolved effectively. Many researchers proposed algorithms based on topological characteristics or some coding which need lot of computations and comparisons. In this paper, a novel and simple algorithm is proposed based on graph theory by which elimination of isomorphic chains can be done very easily without any tedious calculations or comparisons. A new concept “Net distance” is proposed based on the graph theory to be a quantitative measure to assess isomorphism in planar kinematic chains (PKCs) as well as GKCs. The proposed algorithm is applied on nine-link two-degrees-of-freedom (DOF) distinct kinematic chains completely and the results are presented. Algorithm is tested on examples from eight-link 1-DOF, ten-link 1-DOF, 12-link 1-DOF, and 15link 4-DOF PKCs. The algorithm is also tested on four-, six-link 1-DOF GKCs to detect isomorphism. All the results are in agreement with the existing literature.

Gradient method for identification of isomorphism of planar kinematic chains

Detection of isomorphism among large family of kinematic chains is always a challenge and numerous methods are also proposed with their inherent strength and weaknesses. Gradient concept is presented for identification of isomorphism and inversions of kinematic chains. In the present work, based on gradient analogy, gradient matrices are proposed to distinctly denote the structure of each kinematic chain. Further, First gradient chain string and Second gradient chain string two invariants from the matrix in the form of strings are identified to be used for detection of isomorphism and similarly First gradient link string and Second gradient link string and First gradient chain string of inversion are formed for detection of inversions. The proposed novel method is easy, exhaustive and reliable. It is applied successfully for detection of isomorphism and distinct inversions to all known families of kinematic chains i.e. single degree of freedom, multi degree of freedom, and detection of isomorphism in graphs.

An Improved Genetic Algorithm Approach on Mechanism Kinematic Structure Enumeration with Intelligent Manufacturing

During the process of mechanism kinematic structure enumeration, isomorphism identification of graphs is an important and complicated problem. The problem is known to be a NP-complete problem. In this paper, according to the mechanism kinematic chain isomorphism identification criteria, a highly efficient hybrid genetic algorithm model is proposed for isomorphism identification. The model method is coupled with genetic algorithm, optimal choice, and optimal crossover operation. It shows a quick convergence rate of the late operation and can avoid convergence to local optimum. Simulation results show that the hybrid algorithm is more rapid and effective compared with simple genetic algorithm and the improved neural network algorithm.