Automatic Structural Synthesis of Planar Multiple Joint Kinematic Chains

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.

The family of planar kinematic chains with two multiple joints

In this paper, isomorphism identification algorithm of planar multiple joint kinematic chains by circuit analysis method has been presented. Firstly, topological model is introduced. Then, corresponding equivalent circuit model is established to obtain node voltage sequence; By the solved node voltage sequence, an algorithm to identify isomorphic kinematic is obtained. Lastly, examples are carried out to show this useful method.

There is no dearth of methods to test isomorphism amongst kinematic chains. Search for a computationally easier, logically simple and unique method is still on. Present work is in quest of a reliable test to detect isomorphism among kinematic chains. Work presented here is more versatile as it incorporates more features of the kinematic chain which were not included earlier such as number and type of links, their relative dispositions in the kinematic chain, nature of adjacent links etc. The method proposed is based on the concept of pseudo-probability (pseudo means it appears to be, but not exactly. The approach does not follow in-toto the principles of probability and considerable liberty has been taken in interpreting the word probability hence the word pseudo is used along with the probability schemes). Using the resemblance of different coloured balls in an urn for the number and type of links in a kinematic chain, a matrix (named P-Matrix) representing the kinematic chain in totality is generated. For the sake of comparison a numerical scheme named, pseudo probability scheme, P-Scheme, is developed from the above P-Matrix and is used for testing isomorphism. In fact the method is more powerful in the sense that each row of the proposed P-Matrix is capable of representing the respective kinematic chain distinctly and can be used to compare the kinematic chains with same link assortments, uniquely. The proposed method, besides possessing the potential of testing the isomorphism among simple-joint, single degree of freedom kinematic chains is also capable of multi degrees of freedom and multiple-joint kinematic chains.

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.

This chapter introduces an original theory of loop algebra and its application in isomorphism identification, rigid sub-chain detection, and atlas database of kinematic chains. Introduced first is the unified topology modeling of planar kinematic chains with simple joints, multiple joints, and geared (cam) joints. Based on the array representation of loops in the topological graphs of kinematic chains, basic loop operations are introduced, and the loop algebra is established. The most important problem of isomorphism identification in the automatic structural synthesis of kinematic chains is presented by finding a unique representation of topological graphs. Finally, the digital atlas database for the topological graphs of kinematic chains is also provided.

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

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.

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.

Constrained motion interpolation for planar open kinematic chains

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.

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

A computerized methodology for structural synthesis of kinematic chains: Part 1— Formulation

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.