Isomorphism Identification of Planar Multiple Joint Kinematic Chains by Circuit Analysis Method

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 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 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.

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

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.

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.

In order to solve a baffling problem of kinematic chain isomorphism identification, proceeded from the isomorphism’s principles of graph theory, a new method for detecting isomorphism among planar kinematic chains using the standardization adjacent matrix is presented in this paper. The general course of adjacent matrix standardization processing and numbering principle of node are introduced, the implementation of this new method is illustrated with an example, it is showed that this new method can be accurately and effectively performed.

A Tonpilz type piezoelectric ceramic ultrasonic transducer is investigated through the methods of equivalent circuit and finite element analysis(FEA). The emphasis of this paper lies in the difference between the two methods. Essentially, the piezoelectric constitutive equation is the common law of them. Derived from this equation, the Mason’s equivalent circuit and the frequency equation of the piezoelectric transducer is obtained, and then the finite element control equation is obtained. A mason’s equivalent circuit model and a 1/4 symmetry finite element model of the Tonpilz type piezoelectric ceramic ultrasonic transducer are both constructed. A comparison about the frequency performance and the admittance performance which are respectively obtained from the two methods and the corresponding test shows that, the method of FEA is more quickly and accurately, and its analysis error is controlled within 5%. What is worth to mention is that some performance and the vibration of the transducer have been predicted distinctly through FEA. So FEA is suitable to the piezoelectric transducer. A transducer prototype which is made according to the analysis result has a good performance and satisfying applied requirements.

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

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.

A probabilistic model and methods to determine the means and variances of the velocity and acceleration of stochastically-defined planar pin jointed kinematic chains are presented. The presented model considers the effect of tolerances on link length and radial clearance and uncertainty of pin location as a net effect on the link’s effective length. The determination of the mean values and variances of the output variables requires the calculation of sensitivities of secondary variables with respect to the random variables. It is shown that this computation is straightforward and can be accomplished by a conventional kinematic analysis package. Thus, the concepts of tolerance and clearance have been captured by the model and analysis. The only input data is the nominal linkage model and statistical information. The “effective link length” model is shown to be applicable to both analytical solution and Monte Carlo simulation. The results from both methods are compared. This paper solves the higher-order kinematics problem for the probabilistic design analysis of stochastically defined mechanisms.

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.

In this chapter, the problems of the first chapter are fully solved, in detail, step-by-step, and with multiple methods, by using mesh analysis, nodal analysis, and heuristic techniques. In all the problems’ solutions, Kirchhoff current law (KCL) and Kirchhoff voltage law (KVL) as well as the concepts of mesh current, node voltage, supermesh, and supernode are clearly described. Moreover, in the application of circuit theorems, the concepts of open circuit voltage, short-circuit current, Thevenin voltage, Thevenin resistance, Norton current, and Norton resistance are explained. In addition, this chapter shows how to deal with circuit components like resistors, dependent current and voltage sources, and independent current and voltage sources.