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

Abstract. The similarity recognition of kinematic chains (KCs) is helpful for improving the efficiency of configuration synthesis, which has been paid more and more attention in recent years. The existing recognition methods are divided into the definition method and feature constant method. Among them, the definition method is difficult to adopt in practice because of its long operation time, especially when the number of similar vertices in KCs is large. In this paper, the new concepts of a loop tree (LT) and a loop tree matrix (LTM) have been proposed, which improve the efficiency of similarity recognition. This method is applied on the complete structure of the following: 8-link with 1 DOF (degree of freedom), 9-link with 2 DOF, 10-link with 1 DOF, 12-link with 1 DOF, 13-link with 2 DOF, 14-link with 3 DOF, 15-link with 4 DOF planar single-joint KCs, and contracted graphs with up to six independent loops. All results are verified by the definition method to prove the good applicability, reliability, and efficiency of the proposed method. Simultaneously, the application case of the similarity recognition in a mechanism creation is given to provide a reference for an innovative design.

The complete set of 12-bar planar simple-jointed kinematic chains is enumerated using the stratified or hierarchical representation, of Fang and Freudenstien (1990) in which post-priori isomorphism testing is not needed. This is achieved by directly enumerating the simplified and contracted graphs from which the kinematic chains are built up. The number obtained is compared with those from previous studies, and the list is catagorized by the link sets and by the numbers of major vertices and edges in the simplified graphs. A major feature of the technique is the use of an efficient rote-based technique for eliminating degenerate chains.

The family of planar kinematic chains with two multiple joints

It is of great importance in the conceptual creative design of mechanical systems to synthesize as many feasible kinematic structures of mechanisms as possible. However, the methods for the structural synthesis of multiple joint kinematic chains are seldom addressed in literature even though they are widely used in various mechanical products. This paper proposes an automatic method to synthesize planar multiple joint kinematic chains. First, the bicolor topological graph and the bicolor contracted graph are introduced to represent the topological structures of multiple joint kinematic chains. Then, the characteristic number string of bicolor topological graphs is proposed and used to efficiently detect isomorphism in the synthesis progress. Finally, a systematic method for the synthesis of kinematic chains with one multiple joint is proposed, and the whole families of multiple joint kinematic chains with up to 16 links and all possible degrees of freedom are synthesized for the first time.

In a kinematic chain, the links are connected to each other through joints. The connectivity of a joint indicates the number of joints to which it is connected. The connectivity level of a joint indicates the distance by which it is separated from the adjacent joints. The concept of joint connectivity and its application to detect isomorphism among kinematic chains and their inversions has been already reported by authors. The method utilizes the connectivity of joints at different levels to detect isomorphism and inversions among planar kinematic chains. The method is applied to eight-, nine-, and ten-link planar kinematic chains. The results so obtained are in agreement with those available in the literature. In this study, the method is further improved by incorporating the type of joint to make it more effective for the detection of isomorphism and distinct inversions. A joint connectivity table completely representing the kinematic chain is proposed. The application of the method is extended for the determination of additional topological characteristics of chains such as categorization of kinematic chains and selection of preferred frame, input and output links for function and path generation. The concept of ‘Motion Transfer Ability’ is introduced and utilized to develop numerical measures for comparing and categorizing the chains at the synthesis stage of mechanism design for a specific application. The method was successfully tested on planar kinematic chains with single and multiple degrees of freedom and the results for eight- and nine-link kinematic chains are appended.

A number of distinct or non-isomorphic kinematic chains exist for a specified number of links and joints. For example, sixteen distinct chains can be obtained with eight links and two hundred and thirty chains with ten links having a single degree of freedom. Similarly, many space mechanisms can be formed with four links and joints having different degrees of freedom. So far no measure is available to know which of these possesses greater mobility or flexibility. Flexibility is not to be confused with the degree of freedom. Intuitively one feels that a six-link chain has greater flexibility than a four-bar chain both having the same degrees of freedom. Though the mobility of a chain increases with the number of links one is not sure how the structural arrangement, type of links and joints, their numbers and sequence etc. influence the same. Combining graph theory with the concepts of probability, simple formulae are developed to investigate the relative merits of spatial and planar kinematic chains. The greater the flexibility or mobility of the chain, the higher is the ability to meet the motion requirements, i.e., a chain having greater entropy can be expected, say, to reproduce a given function more accurately.

The structural synthesis of Baranov trusses is an open problem even though it finds its wide usage in design and analysis of mechanisms and robotics. A Baranov truss is a planar zero DOF kinematic chain, which contains no rigid subchains. The present work aims at synthesising the complete collection of N - link Baranov trusses by adding a binary link to (N – 1) - link, 1 degree of freedom (DoF) simple jointed kinematic chains. From this collection, chains which contain degenerate sub-chains are identified and removed as they are not Baranov truss by definition. Next, an isomorphism test is carried out to identify distinct chains and resulting complete collection of N-link Baranov trusses is reported.

This paper presents a new viewpoint about structural composition of planar kinematic chains: single-opened-chains, which is composed of binary links, are regarded as basic structural units of mechanisms. The constraint characteristics (the constraint factors, Δj) of single opened chains and the constraint characteristics (the coupled degree κ and the κ-algorithm) of mechanical networks are presented. Thus a kinematic chain with v independent loops is regarded to be composed of one basic link and v single-opened-chains in regular sequence. Based on the ordered single-opened-chains and the topological constraints characteristics of mechanisms, a powerful new method for structural synthesis of planar kinematic chains with plane and nonplane linear graphs has been developed. This permits the development of a highly efficient and completely automatic program for the computer-generated enumeration of structural types of mechanisms. The method is illustrated by applying to the case of kinematic chains with up to six independent loops on a personal computer. The ordered single-opened-chains and the topological characteristics are used for setting up a new unified model for structics, kinematics and dynamics of planar mechanisms.

A new atlas for 8-bar kinematic chains with up to 3 prismatic pairs using Joint Sorting Code

Constrained motion interpolation for planar open kinematic chains

The use of finite symmetry groups to enumerate basic kinematic chains has been extended by making use of the subgroup structure of the symmetry groups. This refinement not only makes the enumeration of kinematic chains considerably more efficient, but also permits the enumeration of all distinct inversions of each of the chains found. As examples, all chains with 2, 3, 4, and 5 independent loops and 1, 2, or 3 degrees of freedom and their inversions have been found on a personal computer.

A new matrix, called the joint-joint relation matrix, is defined for representing the structure of a kinematic chain. An algorithm has been developed for deriving the joint-joint distance matrix of a kinematic chain from its joint-joint relation matrix with the help of the computer. A new structural invariant, called the Arranged Sequence of Total Distance Ranks of all the Joints, and based on the concept of distance, has been developed for a kinematic chain, leading to the development of a simple methodology for detecting isomorphism among planar kinematic chains with simple joints.

Some planar redundantly closed mechanisms (RCMs) have better dexterity, less singular configuration, and higher stiffness. In this paper, the derivation of valid topology graphs (TGs) of some planar four degrees of freedom (4DOF) RCMs is studied based on the contracted graph (CG), arrays, and topology graph with digits (DTG). First, some CGs without any binary links are constructed for the planar 4DOF RCMs, some curves with only binary links are distributed over CGs, and some valid TGs of the planar 4DOF RCMs are derived. Second, a complicated derivation of TG is transformed into an easy derivation of array and DTG, and some programs are compiled in VISUAL BASIC; all valid arrays corresponding to nonisomorphic TGs are derived, and some invalid arrays corresponding to the isomorphic TGs and invalid TGs are determined and removed by the compiled programs. Third, many valid TGs of the planar 4DOF RCMs with various basic links are derived from valid arrays and DTGs. Finally, some application examples are illustrated.

In this paper, a new method is proposed for structural synthesis of planar closed kinematic chains with all the possible kinds of multiple joints according to given number K-independent loops up to maximum multiple joint factor Vmax(K). At first, the new procedure for structural synthesis of non-fractionated and fractionated multiple joint kinematic chains based on the combination of corresponding simple and multiple joints is presented. Then the atlas database with all the possible kinds of multiple joints containing various 24 (case K = 4) multiple joint non-fractionated and fractionated structures is established and illustrated. Next, all the possible multiple joint assortments ([M.J.A.]) for non-fractionated as well as fractionated 10-link 1-DOF multiple joint kinematic chains with up to pentagonal links with K = 4 and up to Vmax = 6 are synthesized and separated for the first time. Finally, the complete structural analysis of multiple joint kinematic chains for industrial application in 10-link manipulators designed from the atlas of synthesized multiple joint kinematic chains is conducted.

Our goal is to present a general approach for the planar chains with rotational links with clearances.This approach is realized in a multibody style, the main problem being the determination of the matrix of constraints.IntroductionThe application of the multibody type methods [1-13], made possible the elaboration of certain general algorithms for the numerical, dynamical and elasto-dynamical calculation of the kinematical and dynamical parameters for the mechanical systems' motions.These algorithms can be applied both to the determined kinematical mechanical systems with and without friction [4,9,14], respectively with one degree of mobility, and to the systems with one degree of mobility and a single motor element as, for instance, [2,7,12], the mechanical convertor of torque created by G. Constantinescu.The applications of the multibody type methods for the studying of the planar mechanical systems having articulations with clearances (rotational kinematical joints) [3,4,6,10,14], assumes the inserting of certain virtual without mass elements, which leads to singular matrix of inertia and, consequently, it does not permit the separation of the general system of equations in two systems from which result, in order, the time history of the reactions and then the time history of the kinematical parameters.In this paper we elaborate a multibody type method based on a new form of the matrix of constraints, method that permits the numerical dynamical study of the planar systems with rotational joints with or without clearance, with one or several degrees of freedom.