Planar Kinematic Chains Research Articles

Hierarchical clustering approach for determination of isomorphism among planar kinematic chains and their derived mechanisms

The problem of isomorphism among kinematic chains and their derived mechanism has been a hot area of research from last several years. The researchers so far have proposed many methods which are mainly based on characteristic polynomial and some code based methods to test the isomorphism among kinematic chains. In this present communication a hierarchical clustering based computerized method is proposed for the above said problem and it is tested for planar kinematic chains upto twelve links without any counter examples. In this method a hierarchical clustering algorithm is also developed for the identification of distinct mechanism derived from kinematic chains. In this method kinematic chains are represented in the form of weighted squared shortest path distance matrix and this matrix is further transformed in the form of tree or dendrogram with the help of hierarchical clustering algorithm. This algorithm directly gives the number of distinct mechanism derived from a given kinematic chain. The cophenetic correlation coefficient of dendrogram is used as an index for isomorphism identification among kinematic chains. The proposed method is efficient and accurate and only one matrix for a given kinematic chain is developed for the determination of distinct mechanisms. This method is successfully examined for one degree of freedom, 6, 8, 10, 12 links planar kinematic chains, 9 links two degree of freedom and 10 links three degree of freedom planar kinematic chains. The computer algorithm for the proposed method has been proposed which can easily be converted into a computer program. These results are useful for designers to detect isomorphism in mechanisms derived from kinematic chains and duplication among kinematic chains.

Isomorphism Identification of Planar Kinematic Chains Based on Characteristic Arrays and Computation

At present, the research of isomorphism identification of planar kinematic chains has reached a great improvement, however, most of the methods are proved so complex, inefficient as well as unreliable that the research is hindered and restricted a lot. At the same time, especially with the appearance of the planar kinematic chains with multiple joints, there is less research about the methods. Hence, in the paper, based on the method of kinematic chains with characteristic arrays, one method of isomorphism identification of kinematic chains with single joint as well as multiple joints is proposed presently. What is more, a software based on this new method proposed is developed with Visual Basic. In the end of the paper, an example is analysed in detail with hand computation and this software, through which the method is proved simple, efficient as well as reliable.

Representations & Isomorphism Identification of Planar Kinematic Chains with multiple Joints Based on the Converted Adjacent Matrix

Handling multiple-joint and isomorphism identification of planar kinematic chains with multiple-joints is a hard work in the analysis and synthesis of mechanisms.A new approach is suggested to convert the multiple-joints to some parallel simple-joints by using a Pin-link to adjoin all the links on a multiple-joint.The converted adjacent matrix is defined to represent the kinematic chains with multiple-joints.According to some characteristic parameters of the converted adjacent matrix,a new method for isomorphism identification of planar kinematic chains is proposed.Some application examples show that the converted adjacent matrix can conveniently represent planar kinematic chains with multiple-joints,and the Eigen values and eigenvectors of the converted adjacent matrix can be used to identify the isomorphism of planar kinematic chains effectively and efficiently.A new way is provided to the analysis and synthesis of mechanisms with multiple-joints.

Synthesis of the whole family of planar 1-DOF kinematic chains and creation of their atlas database

Obtaining all the independent kinematic structures of mechanisms is very helpful for the selection of the most suitable candidate at the conceptual design stage of mechanisms. This paper proposes a fully-automatic method to synthesize the whole family of planar 1-DOF (degrees of freedom) kinematic chains with different links, and the corresponding atlas databases containing all the classified topological graphs are also established. The topological graph and the contracted graph are used to represent the topological structures of kinematic chains. The contracted graphs and valid contracted graphs of 1-DOF kinematic chains with different numbers of links are synthesized first. All the topological graphs are then synthesized from the corresponding valid contracted graphs. The atlas database of 1-DOF kinematic chains with 6, 8, 10, 12 and 14 links, which contains all the valid topological graphs classified by their structure characteristics, is also established and illustrated at the end of the paper, proving the effectiveness of the method.

Modified joint connectivity approach for identification of topological characteristics of planar 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.

Synthesis of a complete set of contracted graphs for planar non-fractionated simple-jointed kinematic chains with all possible DOFs

Abstract The contracted graph shows the primary topological structure of kinematic chains and is the foundation for constructing kinematic structures of mechanisms. In this paper, a fully-automatic method is proposed to synthesize a complete set of contracted graphs for planar non-fractionated simple-jointed kinematic chains with all possible degrees of freedom. First, based on the 4-parameter index and link assortment array, the relationships of link assortment arrays and non-fractionated contracted graphs for different types of kinematic chains are revealed. Then, the general steps to generate the adjacency matrices of contracted graphs from the synthesis equation set are given. Efficient methods to detect fractionated structures and isomorphism are also addressed. Finally, a human–machine interactive synthesis program is developed, and the complete list of both contracted graphs and valid contracted graphs for planar non-fractionated simple-jointed kinematic chains with up to 19 links and all possible degrees of freedom is presented for the first time.

Impact of a Planar Kinematic Chain with Granular Matter

The theoretical model of a kinematic chain impacting granular matter is studied. The force of the granular medium acting on the chain is a linear superposition of a static (depth-dependent) resistance force and a dynamic (velocity-dependent) frictional force. This resistance force is opposed to the direction of the velocity of the immersed chain. We present two methods (one using EventLocator and the other using FixedStep) for the problem. As examples, a single and a double pendulum are simulated using different initial impact velocity conditions. We analyze how rapidly the kinematic chain impacting the granular medium slows upon collision. For the analyzed cases the kinematic chain under high impact force (higher initial velocity) comes to rest faster in the granular matter than the same body under low impact force (lower initial velocity).

A method for identification of isomorphism and structural properties of kinematic chains

The present work deals with the structural analysis and synthesis of kinematic chains of mechanisms and robot manipulators. Four key problems, detection of isomorphism among kinematic chain, identification of isomorphism among inversions of given kinematic chain, identification of rigid sub-chain and type of degree of freedom of kinematic chains are considered. A method based on the flow path matrices of kinematic chains is proposed and two new structural invariants, sum of the absolute values of the characteristic polynomial coefficients (SCPC) and maximum absolute value of the characteristic polynomial coefficient (MCPC) are determined using software MATLAB. These invariants are used as the composite identification number of a kinematic chain and mechanisms. The advantage of this method is that it covers all planar kinematic chains of one, two and three degree of freedom at once for identification of isomorphism and for structural properties.

Research on Kinematic Chain Isomorphism Identification Method Based on the Standardization Adjacent Matrix

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.

Series Solution for Finite Displacement of Planar Four-Bar Linkages

A time-varying instantaneous screw characterizes the motion of a rigid body. The kinematic differential equation expresses the path taken by any point on that rigid body in terms of this screw. Therefore, when a revolute joint is attached to a moving link in a planar kinematic chain, the path taken by the center of that revolute joint is the solution to such an equation. The instantaneous screw of a link in that chain is in turn determined by the action of the joints connecting that link to ground, where the contribution of each joint to that instantaneous screw is determined by its actuation rate and center point. Substituting power series expansions for joint rates into the kinematic differential equations for joint centers, and expressing loop closure as a linear constraint on the instantaneous screws of the links, a recurrence relation is established that solves for the coefficients in those power series. The resulting solution is applied to determine the equilibrium pendulum tilt of the United Aircraft TurboTrain. Comparing that power series approximation with an exact kinematic analysis shows convergence properties of the series.

Constrained motion interpolation for planar open kinematic chains

This paper deals with the problem of motion interpolation of an object as part of planar open kinematic chains . Instead of synthesizing the interpolating motions in the joint space of a planar chain, this paper presents a method for synthesizing smooth motions in the workspace of the chain. The resulting motions are piecewise rational and therefore more compatible with the existing NURBS based CAD systems. The paper brings together the well-known kinematics of planar two- and three-DOF kinematic chains with revolute and/or prismatic joints and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions. Both C 1 and C 2 continuous interpolations are considered. The results have applications in Cartesian motion planning in robotics and task specification for the task driven design of robots and mechanisms.

Unified Topological Representation Models of Planar Kinematic Chains

This paper attempts to establish unified topological models and corresponding mathematical representations for planar simple joint, multiple joint, and geared (cam) kinematic chains. First, the conventional topological representation models of kinematic chains are introduced. Then, new topological models of multiple joint and geared (cam) kinematic chains, which are derived from the topological model of simple joint kinematic chains, are presented. The characteristics of the new topological graphs and their associations with the topological graph of simple joint kinematic chains are also addressed. The most important merit of the new topological graphs is that it makes it much easier to undertake unified structure synthesis and further to establish conceptual design platform for various planar mechanisms. Synthesis examples of both multiple joint and geared chains are given, which show the effectiveness of the unified topological models.

Multiple impacts of a planar kinematic chain with a granular matter

The theoretical model of a planar kinematic chain with multiple points impacting with a granular matter is studied. The force acting on the links penetrating the granular media is a linear superposition of a static (depth-dependent) resistance force and a dynamic (velocity-dependent) frictional force. A general algorithm for the m impacts for an n planar kinematic chain is presented. The complete solution of a two link chain with two impact points is simulated using different initial impact velocities and different impact geometries. We analyze how rapidly the ends of both links impacting a granular media slow upon collision. For most of the analyzed cases the tips of the kinematic chain under high impact force (higher initial velocity) come to rest faster in a granular matter than the same body under low impact force (lower initial velocity). There are also some exceptions to this rule.

Enumeration of kinematic chains and mechanisms

A problem still unsolved in kinematics is the enumeration of a complete list of kinematic chains and mechanisms without isomorphisms and without degenerate chains that operate in any screw system. In this paper, a method for the enumeration of kinematic chains without isomorphisms and degenerate chains for all screw systems and a new method of enumeration of kinematic chain inversions (i.e. mechanisms) based on group theory techniques are presented. New concepts of the group theory are introduced and a new method of enumeration of inversions is presented. Kinematic inversions are related to the symmetries of the chain which can be identified analysing the corresponding graph. The symmetry of a graph can be identified for the group of automorphisms of the graph and its orbits provides sets of vertices (links) that are in the same equivalence classes, i.e. they have the same properties of symmetry. The main definitions of group theory and examples of application of the new method of enumeration of inversions are presented. New results are obtained and divided in two classes: original results in non-planar screw systems (λ≠3) and results in agreement with a previously published list for planar kinematic chains and planar mechanisms (inversions). Two tables (1 and 3) are provided, which are up-to-date lists of kinematic chains and mechanisms for several screw systems.

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

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.

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.

Basic Links Structure-tree Method of Type Synthesis for Planar Kinematic Chains

Based on a study of the structural composition of planar closed kinematic chains,the basic links structure-tree method is presented to complete the type synthesis of kinematic chains.For kinamatic chains with given number of links and degrees of freedom, the constant sequences,such as link sequence,contracted link sequence,link degree sequence and basic links connective sequence,are first of all achieved by synthesis on the basis of the contracted link adjacency matrix.And then,the approaches and steps are induced to complete type synthesis for kinematic chains based on the contracted link adjacency matrix,and also the basic links structure-tree is defined.The basic links structrue matrix of the kinamatic chains is achieved by synthesizing.And a further study of approaches to matching contracted links is also made for synthsizing the connetive matrix of the kinematic chains.As a result of using the basic links structure-tree method in the process of synthesizing,the kinematic chains are classified in accordance with the structural invariants such as the degrees of links,the numbers of kinematic pairs and connective relation between links.So the type synthesis is charaterized by strong regular pattern and high efficiency,and its results are advantegeous to reducing the operation during the follow-up rigid sub-chains eliminating and isomorphinsim identifying.And the method is especially suitable for the computer to achieve automatic type synthesis for the planar closed kinematic chains with multiple links and multiple degrees of freedom.And also it is successfully used in the type synthesis for the planar closed kinematic chains with 12 links and 12-below-links.As an example,the type synthesis method is demonstrated with the aid of kinematic chains with 10 links and single degree of freedom.

New Method to Detect Isomorphism in Kinematic Chains

The present work deals with the problem of detection of isomorphism, which is frequently encountered in structural synthesis of kinematic chains (KC). A new method based on theoretic approach, easy to compute and reliable is suggested in this paper. The KC are represented in the form of the Joint-Joint [JJ] matrices. Two structural invariants derived from the characteristic polynomials of the [JJ] matrix of the KC are the sum of absolute characteristic polynomial coefficients (shortened as ΣJJ) and maximum absolute value of the characteristic polynomial coefficient (shortened as MJJ). These invariants are used as the composite identification number of a KC and mechanisms. It is capable of detecting isomorphism in all types of planar kinematic chains. This study will help the designer to select the best KC and mechanisms to perform the specified task and avoid duplication at the conceptual stage of design. The application this study is in research and development industries. The proposed method has been explained with the help of examples. The KC of 1-F, 8-links, and 10-links as well as 2-F, 9-link has been tested and the results are in complete agreement with the available literature of structural synthesis.

A Parallelogram-Based Parallel Manipulator for Schönflies Motion

A parallelogram-based 4 degrees-of-freedom parallel manipulator is presented in this paper. The manipulator can generate the so-called Schönflies motion that allows the end effector to translate in all directions and rotate around an axis parallel to a fixed direction. The theory of group of displacements is applied in the synthesis of this manipulator, which employs parallelograms in every limb. The planar parallelogram kinematic chain provides a high rotational capability and an improved stiffness to the manipulator. This paper shows the kinematic analysis of the manipulator, including the closed-form resolution of the forward and inverse position problems, the velocity, and the singularity analysis. Finally, a prototype of the manipulator, adding some considerations about its singularity-free design, and some technical applications in which the manipulator can be used are presented.

Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm

Synthesis of kinematic chains can be viewed as the enumeration of a certain class of graphs. Very efficient algorithms, using group theoretic techniques, exist for exhaustive isomorph-free generation of certain classes of combinatorial objects, which either eliminate or restrict the explicit isomorphism detection. An algorithm belonging to one particular class called McKay-type, in combination with an efficient degeneracy testing algorithm, is used for the synthesis of planar mechanisms. The existing results are reexamined and the discrepancies are reconciled.