Laplace and Extended Adjacency Matrices for Isomorphism Detection of Kinematic Chains Using the Characteristic Polynomial Approach

2005 ◽  
Author(s):  
Rajesh Pavan Sunkari ◽  
Linda C. Schmidt
Keyword(s):  
Characteristic Polynomial ◽  
Spectral Properties ◽  
Kinematic Chain ◽  
Detection Algorithm ◽  
Isomorphism Problem ◽  
Characteristic Polynomials ◽  
Kinematic Chains ◽  
Laplace Matrix ◽  
Adjacency Matrices ◽  
Single Matrix

The kinematic chain isomorphism problem is one of the most challenging problems facing mechanism researchers. Methods using the spectral properties, characteristic polynomial and eigenvectors, of the graph related matrices were developed in literature for isomorphism detection. Detection of isomorphism using only the spectral properties corresponds to a polynomial time isomorphism detection algorithm. However, most of the methods used are either computationally inefficient or unreliable (i.e., failing to identify non-isomorphic chains). This work establishes the reliability of using the characteristic polynomial of the Laplace matrix for isomorphism detection of a kinematic chain. The Laplace matrix of a graph is used extensively in the field of algebraic graph theory for characterizing a graph using its spectral properties. The reliability in isomorphism detection of the characteristic polynomial of the Laplace matrix was comparable with that of the adjacency matrix. However, using the characteristic polynomials of both the matrices is superior to using either method alone. In search for a single matrix whose characteristic polynomial unfailingly detects isomorphism, novel matrices called the extended adjacency matrices are developed. The reliability of the characteristic polynomials of these matrices is established. One of the proposed extended adjacency matrices is shown to be the best graph matrix for isomorphism detection using the characteristic polynomial approach.

An Efficient Algorithm for Finding Optimum Code Under the Condition of Incident Degree

1992 ◽  
Author(s):  
T. J. Jongsma ◽  
W. Zhang
Keyword(s):  
Computer Program ◽  
Efficient Algorithm ◽  
Kinematic Chain ◽  
Isomorphism Problem ◽  
Kinematic Chains ◽  
Adjacency Matrices ◽  
Set Up

Abstract This paper deals with the identification of kinematic chains. A kinematic chain can be represented by a weighed graph. The identification of kinematic chains is thereby transformed into the isomorphism problem of graph. When a computer program has to detect isomorphism between two graphs, the first step is to set up the corresponding connectivity matrices for each graph, which are adjacency matrices when considering adjacent vertices and the weighed edges between them. Because these adjacency matrices are dependent of the initial labelling, one can not conclude that the graphs differ when these matrices differ. The isomorphism problem needs an algorithm which is independent of the initial labelling. This paper provides such an algorithm.

Linkage Characteristic Polynomials: Definitions, Coefficients by Inspection

1981 ◽  
Vol 103 (3) ◽  
pp. 578-584 ◽  
Author(s):  
H. S. Yan ◽  
A. S. Hall
Keyword(s):  
Structural Analysis ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Kinematic Chain ◽  
Companion Paper ◽  
Characteristic Polynomials ◽  
Kinematic Chains ◽  
Analysis And Synthesis

A linkage characteristic polynomial is defined as the characteristic polynomial of the adjacency matrix of the kinematic graph of the kinematic chain. Some terminology and definitions, needed for discussions to follow in a companion paper, are stated. A rule from which all coefficients of the characteristic polynomial of a kinematic chain can be identified by inspection, based on the interpretation of a graph determinant, is derived and presented. This inspection rule interprets the topological meanings behind each characteristic coefficient, and might have some interesting possible uses in studies of the structural analysis and synthesis of kinematic chains.

A New Multivalued Neural Network for Isomorphism Identification of Kinematic Chains

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Gloria Galán-Marín ◽  
Domingo López-Rodríguez ◽  
Enrique Mérida-Casermeiro
Keyword(s):  
Neural Network ◽  
Intelligent Systems ◽  
Initial Conditions ◽  
Graph Isomorphism ◽  
Kinematic Chain ◽  
Parameter Tuning ◽  
Isomorphism Problem ◽  
Kinematic Chains ◽  
Graph Isomorphism Problem ◽  
Better Than

A lot of methods have been proposed for the kinematic chain isomorphism problem. However, the tool is still needed in building intelligent systems for product design and manufacturing. In this paper, we design a novel multivalued neural network that enables a simplified formulation of the graph isomorphism problem. In order to improve the performance of the model, an additional constraint on the degree of paired vertices is imposed. The resulting discrete neural algorithm converges rapidly under any set of initial conditions and does not need parameter tuning. Simulation results show that the proposed multivalued neural network performs better than other recently presented approaches.

scholarly journals Alpha Adjacency: A generalization of adjacency matrices

2019 ◽  
Vol 35 ◽  
pp. 365-375
Author(s):  
Matt Hudelson ◽  
Judi McDonald ◽  
Enzo Wendler
Keyword(s):  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Average Characteristic ◽  
Adjacency Matrices ◽  
Skew Adjacency Matrix

B. Shader and W. So introduced the idea of the skew adjacency matrix. Their idea was to give an orientation to a simple undirected graph G from which a skew adjacency matrix S(G) is created. The -adjacency matrix extends this idea to an arbitrary field F. To study the underlying undirected graph, the average -characteristic polynomial can be created by averaging the characteristic polynomials over all the possible orientations. In particular, a Harary-Sachs theorem for the average-characteristic polynomial is derived and used to determine a few features of the graph from the average-characteristic polynomial.

Linkage Characteristic Polynomials: Assembly Theorems, Uniqueness

1982 ◽  
Vol 104 (1) ◽  
pp. 11-20 ◽  
Author(s):  
H. S. Yan ◽  
A. S. Hall
Keyword(s):  
Characteristic Polynomial ◽  
Computer Aided Design ◽  
Characteristic Polynomials ◽  
Topological Information ◽  
Automated Recognition ◽  
Practical Applications ◽  
Kinematic Chains ◽  
Computer Aided ◽  
Aided Design ◽  
Insight Into

Several assembly theorems, for obtaining the linkage characteristic polynomial for a complex chain through a series of steps involving the known polynomials for subunits of the chain, are derived and presented. These theorems give insight into how the topological information concerning the linkage is stored in the polynomial and might contribute to the automated recognition of linkage structure in generalized computer-aided design programs. Based on graph theory, the characteristic polynomial cannot characterize the graph up to isomorphism. However, for practical applications in the field of linkage mechanisms, it is extremely likely that the characteristic polynomials are unique for closed connected kinematic chains without any overconstrained subchains.

scholarly journals A new graph product and its spectrum

1978 ◽  
Vol 18 (1) ◽  
pp. 21-28 ◽  
Author(s):  
C.D. Godsil ◽  
B.D. McKay
Keyword(s):  
Characteristic Polynomial ◽  
Spectral Properties ◽  
Characteristic Polynomials ◽  
Factor Graphs ◽  
Graph Product

A new graph product is introduced, and the characteristic polynomial of a graph so–formed is given as a function of the characteristic polynomials of the factor graphs. A class of trees produced using this product is shown to be characterized by spectral properties.

Loop Based Detection of Isomorphism Among Chains, Inversions and Type of Freedom in Multi Degree of Freedom Chain

1999 ◽  
Vol 122 (1) ◽  
pp. 31-42 ◽  
Author(s):  
A. C. Rao ◽  
V. V. N. R. Prasad Raju Pathapati
Keyword(s):  
Unsolved Problem ◽  
Kinematic Chain ◽  
Computational Effort ◽  
Degree Of Freedom ◽  
Complete List ◽  
Structural Synthesis ◽  
Kinematic Chains ◽  
Adjacency Matrices ◽  
The Creation ◽  
A Chain

Structural synthesis of kinematic chains usually involves the creation of a complete list of kinematic chains, followed by a isomorphism test to discard duplicate chains. A significant unsolved problem in structural synthesis is the guaranteed precise elimination of all isomorphs. Many methods are available to the kinematician to detect isomorphism among chains and inversions but each has its own shortcomings. Most of the study to detect isomorphism is based on link-adjacency matrices or their modification but the study based on loops is very scanty although it is very important part of a kinematic chain.  Using the loop concept a method is reported in this paper to reveal simultaneously chain is isomorphic, link is isomorphic, and type of freedom with no extra computational effort. A new invariant for a chain, called the chain loop string is developed for a planar kinematic chain with simple joints to detect isomorphism among chains. Another invariant called the link adjacency string is developed, which is a by-product of the same method to detect inversions of a given chain. The proposed method is also applicable to know the type of freedom of a chain in case of multi degree of freedom chains. [S1050-0472(00)70801-4]

ISOMORPHISM AND INVERSIONS OF KINEMATIC CHAINS UP TO TEN LINKS USING DEGREES OF FREEDOM OF KINEMATIC PAIRS

2008 ◽  
Vol 05 (02) ◽  
pp. 329-339 ◽  
Author(s):  
ALI HASAN ◽  
R. A. KHAN
Keyword(s):  
Characteristic Polynomial ◽  
Degrees Of Freedom ◽  
Kinematic Chain ◽  
Connectivity Matrix ◽  
Single Degree Of Freedom ◽  
Absolute Value ◽  
Kinematic Chains ◽  
Single Degree ◽  
Polynomial Coefficients ◽  
The Family

This paper presents a new method for identifying the distinct mechanisms (DMs) from a given kinematic chain (KC). The KCs are represented in the form of the weighted physical connectivity matrix (WPCM). Two structural invariants derived from the characteristic polynomials of the WPCM of the KC are the sum of absolute characteristic polynomial coefficients (∑WPCM) and the maximum absolute value of the characteristic polynomial coefficient (MWPCM). They have been used as the composite identification number of a KC and mechanism. This is capable of detecting DMs in all types of simple jointed planar KCs up to ten links having the same or different kinematic pairs (KPs). The proposed method has been tested successfully in identifying all the DMs derived from the family of single degree of freedom (dof) KCs up to ten links. Also, it does not require any test for isomorphism separately. This study will help the designer to select the best KC and mechanisms to perform the specified task at the conceptual stage of design.

Structure Based Grading of Kinematic Chains

2014 ◽  
Vol 575 ◽  
pp. 501-506 ◽  
Author(s):  
Shubhashis Sanyal ◽  
G.S. Bedi
Keyword(s):  
Structural Properties ◽  
Structural Property ◽  
Kinematic Chain ◽  
Degree Of Freedom ◽  
Kinematic Chains ◽  
Structural Differences ◽  
The Individual ◽  
Independent Loops

Kinematic chains differ due to the structural differences between them. The location of links, joints and loops differ in each kinematic chain to make it unique. Two similar kinematic chains will produce similar motion properties and hence are avoided. The performance of these kinematic chains also depends on the individual topology, i.e. the placement of its entities. In the present work an attempt has been made to compare a family of kinematic chains based on its structural properties. The method is based on identifying the chains structural property by using its JOINT LOOP connectivity table. Nomenclature J - Number of joints, F - Degree of freedom of the chain, N - Number of links, L - Number of basic loops (independent loops plus one peripheral loop).

Combined Graph Layout Algorithms for Automated Sketching of Kinematic Chains

2012 ◽  
Author(s):  
Martín A. Pucheta ◽  
Nicolás E. Ulrich ◽  
Alberto Cardona
Keyword(s):  
Computational Cost ◽  
Kinematic Chain ◽  
Initial Position ◽  
Graph Representation ◽  
Combinatorial Algorithms ◽  
Graph Layout ◽  
Kinematic Chains ◽  
Aesthetic Characteristics ◽  
Edge Crossings ◽  
Independent Loops

The graph layout problem arises frequently in the conceptual stage of mechanism design, specially in the enumeration process where a large number of topological solutions must be analyzed. Two main objectives of graph layout are the avoidance or minimization of edge crossings and the aesthetics. Edge crossings cannot be always avoided by force-directed algorithms since they reach a minimum of the energy in dependence with the initial position of the vertices, often randomly generated. Combinatorial algorithms based on the properties of the graph representation of the kinematic chain can be used to find an adequate initial position of the vertices with minimal edge crossings. To select an initial layout, the minimal independent loops of the graph can be drawn as circles followed by arcs, in all forms. The computational cost of this algorithm grows as factorial with the number of independent loops. This paper presents a combination of two algorithms: a combinatorial algorithm followed by a force-directed algorithm based on spring repulsion and electrical attraction, including a new concept of vertex-to-edge repulsion to improve aesthetics and minimize crossings. Atlases of graphs of complex kinematic chains are used to validate the results. The layouts obtained have good quality in terms of minimization of edge crossings and maximization of aesthetic characteristics.

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