Isomorphism Identification Research Articles

Computer-aided structure decomposition theory of kinematic chains and its applications

Abstract Rigid sub-chain detection and isomorphism identification are two of the most difficult problems in the computer-aided structure synthesis of kinematic chains. This paper aims to find an efficient solution which can be fulfilled within the computer itself to the problem of rigid sub-chain detection of planar kinematic chains, especially for complex planar mechanisms. Based on the array representation of the loops of kinematic chains, this paper first introduces two operations of loops, the “Θ” and “⊕” operations. Then a new theory of structure decomposition of kinematic chains is proposed on the basis of the concept of the independent loop set. After that a new method grounded on the theory is proposed for rigid sub-chain detection. Finally, the optimized algorithm for structure decomposition is presented and the corresponding program for rigid sub-chain detection is developed as well.

An improved artificial immune algorithm for mechanism kinematic chain isomorphism identification

An improved clonal selection artificial immune algorithm is presented to identify the isomorphism of the mechanism kinematic chains. The principles of isomorphism identification are based on the graph theory that adjacency matrices of two isomorphic graphs can interconvert. The clonal selection artificial immune algorithm is improved by adding a process named saving and updating operator after the process of refreshing. The high efficiency and robustness of this improved algorithm is proved by comparing it with simple clonal selection algorithm. The simulation results show that the improved algorithm can identify isomorphic kinematic chains successfully and rapidly.

Topological expression mode approach of Satellite Gear Mechanism for intelligent CAD

To achieve computer aided designs for mechanism in intelligent CAD, one method is to apply graph theory. During the process of enumeration of kinematic structures using graph theory, the topological expression mode of mechanisms is an important and complicated problem. A graph model of satellite gear mechanism is presented based on investigation of the topological characteristics of the kinematic structure of satellite gear mechanism with graph theory. In addition, some measures for the identification of isomorphism of the graph mode are discussed. The work will be a reliable topological expression mode for intelligent CAD of satellite gear mechanism.

Theory of loop algebra on multi-loop kinematic chains and its application

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.

Improving Neural Networks for Mechanism Kinematic Chain Isomorphism Identification

Detection of isomorphism among kinematic chains is essential in mechanical design, but difficult and computationally expensive. It has been shown that both traditional methods and previously presented neural networks still have a lot to be desired in aspects such as simplifying procedure of identification and adapting automatic computation. Therefore, a new algorithm based on a competitive Hopfield network is developed for automatic computation in the kinematic chain isomorphism problem. The neural approach provides directly interpretable solutions and does not demand tuning of parameters. We have tested the algorithm by solving problems reported in the recent mechanical literature. Simulation results show the effectiveness of the network that rapidly identifies isomorphic kinematic chains.

A new theory for the topological structure analysis of kinematic chains and its applications

Abstract Based on the array representation of loops in topological graphs of kinematic chains, this paper proposes two basic loop operations, “Θ” and “⊕”, for the first time. The existent conditions and properties of “⊕” operation are researched and four laws about the operation are presented. Furthermore, after the important concepts of the independent loop set and its selection theorem are proposed, the loop relationship of kinematic chains is revealed; thus an original theory of loop analysis is established. Finally, some applications are given under the basic theory above, such as the isomorphism identification, the detection of rigid sub-chains, and the freedom type analysis of kinematic chains.

Isomorphism identification for epicyclic gear mechanism based on mapping property and ant algorithm

To achieve computer-aided design for mechanism, one method that can be applied is graph theory. During the process of kinematic structure enumeration using graph theory, isomorphism identification of graphs is an important and complicated problem. The problem is known to be NP-complete problem. It is very important to improve the isomorphism identification efficiency and reliability. The objective of this research work is to use ant algorithm to improve the isomorphism identification efficiency and reliability for epicyclic gear mechanism. Based on mapping property between graphs, the isomorphism identification problem can be changed into a matrix operation problem. Two graphs must be isomorphism if the two matrixes of the two graphs can be exactly by exchanging their certain rows and columns, whereas show the two graphs must be non-isomorphic. A mixed model was developed for isomorphism identification by integrating the ant algorithm and mapping property between graphs. The gratifying results can be achieved while parameters are selected in appropriate situation by using the model. Finally, a computer program has been developed in Visual C++. Some examples confirm the validity of the model. The work will be a reliable isomorphism identification algorithm for intelligent CAD of epicyclic gear mechanism.

The Establishment of the Canonical Perimeter Topological Graph of Kinematic Chains and Isomorphism Identification

Some new concepts, such as the perimeter loop, the maximum perimeter degree-sequence, and the perimeter topological graph, are first presented in this paper, and the method for obtaining the perimeter loop is also involved. Then, based on the perimeter topological graph and some rules for relabeling its vertices canonically, a one-to-one descriptive method, the canonical adjacency matrix set of kinematic chains, is proposed. Another very important characteristic of the descriptive method is that in the canonical adjacency matrix set the element number is reduced dramatically, usually to only one. After that, an effective method to identify isomorphism of kinematic chains is given. Finally, some typical examples of isomorphism identification including two 28-vertex topological graphs are presented in this paper.

A mixed isomorphism approach for kinematic structure enumeration graphs based on intelligent design and manufacturing

When using graph theory for kinematic structure enumeration, the isomorphism identification of graphs is an important and complicated problem. The problem is known to be an NP-complete problem. This paper presents a mixed algorithm based on a mapping property, a genetic algorithm, and a simulated annealing algorithm for the isomorphism identification problem. A validity encoding scheme was developed by considering the mapping relationship between two graphs, and some reset measures for the crossover and mutation operators were developed based on the characteristics in which the encoding cell could not be reiterated. A simulated annealing algorithm was introduced into the mixed algorithm to prevent premature convergence in resolution, and some other measures were developed for improving the efficiency, based on the parameter selection. An example shows that the mixed algorithm is a valid algorithm for the isomorphism identification of kinematic structure graphs in mechanism design. It will be a reliable isomorphism identification algorithm for intelligent computer-aided design (CAD) and manufacturing (CAM).

Isomorphism Identification of Kinematic Chains Using Novel Evolutionary Approaches

This paper presents a new method to isomorphism identification based on two novel evolutionary approaches—ant algorithm (AA) and artificial immune system (AIS). Salient features of the two evolutionary approaches are their efficient, robust and general-purpose algorithms for isomorphism identification despite its nondeterministic polynomial (NP) hard nature. First, based on the rearrangement of the vertexes in kinematic chains, the isomorphism identification of kinematic chains is transformed into a degree-reducible traveling salesman problem (TSP), so that the dimension and complexity can be largely decreased. Then AA and AIS algorithms are adopted to solve the transformed TSP. At last, characteristics of the two evolutionary approaches are discussed based on case studies.

Comments on mechanism kinematic chain isomorphism identification using adjacent matrices

A number of theories on the relationship between adjacent matrices of isomorphic mechanism kinematic chains have been investigated. A particular published theory about mechanism kinematic chain isomorphism using adjacent matrices has been revised, after errors in the original theory were discovered. Subsequently, the necessary and sufficient conditions of the eigenvalues and eigenvectors of adjacent matrices for isomorphic kinematic chains have been proven rigorously. A new procedure to identify isomorphic chains has been developed and presented. With this new procedure it is only necessary to compare eigenvalues and several eigenvectors of adjacent matrices of isomorphic kinematic chains to identify the isomorphic chains. Some examples have been provided to demonstrate how to use the theory.

A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains

The most important step in the structural synthesis of planetary gear trains (PGTs) requires the identification of isomorphism (rotational as well as displacement) between the graphs which represent the kinematic structure of planetary gear train. Previously used methods for identifying graph isomorphism yielded incorrect results. Literature review in this area shows there is inconsistency in results from six link, one degree-of-freedom onwards. The purpose of this paper is to present an efficient methodology through the use of Loop concept and Hamming number concept to detect displacement and rotational isomorphism in PGTs in an unambiguous way. New invariants for rotational graphs and displacement graphs called geared chain hamming strings and geared chain loop hamming strings are developed respectively to identify rotational and displacement isomorphism. This paper also presents a procedure to redraw conventional graph representation that not only clarifies the kinematic structure of a PGT but also averts the problem of pseudo isomorphism. Finally a thorough analysis of existing methods is carried out using the proposed technique and the results in the category of six links one degree-of-freedom are established and an Atlas comprises of graph representations in conventional form as well as in new form is presented.

A new method to mechanism kinematic chain isomorphism identification

A new method based on eigenvectors and eigenvalues is developed in this paper to identify isomorphism of mechanics kinematic chain. Kinematic chains are firstly represented by adjacent matrices. By comparing the eigenvalues and corresponding eigenvectors of adjacent matrices, the isomorphism of mechanism kinematics chain can easily be identified. The relationships in the mechanisms can also be obtained. A proof on this method is given in this paper. Some examples are provided to demonstrate the effectiveness of this method.

An artificial neural network approach to mechanism kinematic chain isomorphism identification

A new method based on an artificial neural network (ANN) technique is presented in this paper to identify the isomorphism of the mechanism kinematic chain. The mechanism kinematic chain is represented by a graph. The Hopfield–Tank ANN model for the graph isomorphism identification is developed, which includes the construction of an energy function and a dynamic equation. A case study is provided to demonstrate the preliminary success of the approach. An important motivation of this study is based on an observation that existing methods have not provided robust solutions, because of the NP hard nature of this problem.

Displacement isomorphism of planetary gear trains

This paper presents an efficient method for the identification of the displacement isomorphism of planetary gear trains with any number of gear pairs. For every plantary gear train, the corresponding displacement graph and rotation graph are used to characterize its kinematic structure. Next, a mathematical representation, called the Structural Code, is proposed to represent the topological structure of the displacement graph and rotation graph of a planetary gear train. By comparing the corresponding Structural codes, the linear and rotational displacement isomorphism of planetary gear trains can be identified in an unambiguous way.

On Identification and Canonical Numbering of Pin-Jointed Kinematic Chains

This paper deals with the development of a standard code for the unique representation of pin-jointed kinematic chains based on graph theory. Salient features of this method include the development of an efficient and robust algorithm for the identification of isomorphism in kinematic chains; the formulation of a unified procedure for the analysis of symmetry in kinematic chains; and the utilization of symmetry in the coding process resulting in the unique well-arranged numbering of the links. This method is not restricted to simple jointed kinematic chains only, and it can be applied to any kinematic chain which can be represented as simple graphs including open jointed and multiple jointed chains. In addition, the method is decodable as the original chain can be reconstructed unambiguously from the code values associated with the chains.

Identification of isomorphism among kinematic chains and inversions using link's adjacent-chain-table

A completely new concept about the kinematic chain, called the link's adjacent-chain table (shortened as ACT), is originated in this paper. It is an invariant which can be used to describe the topological relationships between links in the kinematic chains (shortened as KC). In comparison with the traditional representation of the KC, i.e. with the adjacent matrix, link's ACT is much more audio-visual, simpler and more directly in describing the topological relationship among links. Using the link's ACT, the isomorphism of the KC and inversions can be easily determined. Among the existing methods for the identification of the isomorphic graphs, link's ACT take the least time in calculation. So its introduction of the link's ACT leads to an effectual solution to identify the isomorphism of the KC. It is also felt that the link's ACT reveals at glance how many inversions are possible out of a given chain.

The Degree Code—A New Mechanism Identifier

An important step in the structural synthesis of mechanisms requires the identification of isomorphism between the graphs which represents the mechanism topology. Previously used methods for identifying graph isomorphism either yield incorrect results for some cases or their algorithms are computationally inefficient for this application. This paper describes a new isomorphism identification method which is well suited for the automated structural synthesis of mechanisms. This method uses a new and compact mathematical representation for a graph, called the Degree Code, to identify graph isomorphism. Isomorphic graphs have identical Degree Codes; nonisomorphic graphs have distinct Degree Codes. Therefore, by examining the Degree Codes of the graphs, graph isomorphism is easily and correctly identified. This Degree Code algorithm is simpler and more efficient than other methods for identifying isomorphism correctly. In addition, the Degree Code can serve as an effective nomenclature and storage system for graphs or mechanisms. Although this identification scheme was developed specifically for the structural synthesis of mechanisms, it can be applied to any area where graph isomorphism is a critical issue.

An optimally data efficient isomorphism inference algorithm

The time, space, and data complexity of an optimally data efficient isomorphism identification algorithm are presented. The data complexity, the amount of data required for an inference algorithm to terminate, is analyzed and shown to be the minimum possible for all possible isomorphism inference algorithms. The minimum data requirement is shown to be ⌈log 2 ( n )⌉, and a method for constructing this minimal sequence of data is presented. The average data requirement is shown to be approximately 2 log 2 ( n ). The time complexity is O ( n 2 log 2 ( n )) and the space requirement is O ( n 2 )