Construction of Baranov Trusses Using a Single Universal Construction Rule

2016 ◽  
Author(s):  
Elad Hahn ◽  
Offer Shai
Keyword(s):  
Graph Theory ◽  
Construction Rule ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Mathematical Basis ◽  
Synthesis Of Mechanisms ◽  
Complete Set ◽  
Universal Construction ◽  
Planar Linkages ◽  
Mechanism Theory

The kinematic structure of Baranov trusses has been widely studied in the field of mechanism theory. Baranov trusses are seen as the fundamental planar linkages which are a basis for all other planar linkages. As such, they have been used for synthesis of mechanisms as well as their analysis. However, up until now only a limited number Baranov trusses are known and cataloged. In this paper, a method is proposed for generation of Baranov trusses using a new graph representation suitable for linkages of the sort. This method, named the Universal construction rule, is capable of generating a complete set of all feasible Baranov trusses with any number of links. The method has been proven using a mathematical basis from rigidity theory. It is based on the correspondence between Baranov trusses and Assur groups, which are reformulated in terms of graph theory to be known as Assur graphs.

scholarly journals Kinematic Analysis of Tendon-Driven Robotic Mechanisms Using Graph Theory

1989 ◽  
Vol 111 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Lung-Wen Tsai ◽  
Jyh-Jone Lee
Keyword(s):  
Graph Theory ◽  
Kinematic Analysis ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Kinematic Chains ◽  
Epicyclic Gear ◽  
Gear Trains ◽  
Robotic Devices

The kinematic structure of tendon-driven robotic mechanisms has been investigated with the aid of graph theory. The correspondence between the graph representation of the kinematic structure and the mechanism has been established. We have shown that the kinematic structure of tendon-driven kinematic chains is similar to that of epicyclic gear trains. We also have shown that, using the concept of fundamental circuits, the displacement equations of tendon-driven robotic mechanisms can be systematically derived from the kinematic structure. The theory has been demonstrated by the kinematic analysis of three articulated robotic devices.

A Single Universal Construction Rule for the Structural Synthesis of Mechanisms

2016 ◽  
Author(s):  
Elad Hahn ◽  
Offer Shai
Keyword(s):  
Synthesis Method ◽  
Building Blocks ◽  
Construction Rule ◽  
Synthesis Methods ◽  
Rigidity Theory ◽  
Structural Synthesis ◽  
Graph Representations ◽  
Synthesis Of Mechanisms ◽  
Different Types ◽  
Universal Construction

In the field of structural synthesis of mechanisms several synthesis methods have been developed using different approaches. One of the more interesting approaches was that of bottom-up construction via the combination of modular structural groups, known as Assur groups. This approach is combined with new graph representations of mechanisms taken from rigidity theory, capable of representing all the different types of planar and spatial mechanisms. With the strong mathematical base of rigidity theory, a new synthesis method is proposed based on Assur groups, which are reformulated in terms of graph theory and renamed Assur Graphs. Using a single universal construction rule, Assur Graphs of different types and of any number of links are constructed, creating a complete set of building blocks for the synthesis of feasible mechanisms. As its name implies, the single universal construction is applicable for mechanisms of all types of joints and links, for planar or spatial motion.

An Algorithm for Automatic Sketching of Planar Kinematic Chains

1985 ◽  
Vol 107 (1) ◽  
pp. 106-111 ◽  
Author(s):  
D. G. Olson ◽  
T. R. Thompson ◽  
D. R. Riley ◽  
A. G. Erdman
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Kinematic Chain ◽  
Graph Representation ◽  
Synthesis Process ◽  
Line Graphs ◽  
Type Synthesis ◽  
Kinematic Chains ◽  
Synthesis Of Mechanisms ◽  
Computer Aided

One of the problems encountered in attempting to computerize type synthesis of mechanisms is that of automatically generating a computer graphics display of candidate kinematic chains or mechanisms. This paper presents the development of a computer algorithm for automatic sketching of kinematic chains as part of the computer-aided type synthesis process. Utilizing concepts from graph theory, it can be shown that a sketch of a kinematic chain can be obtained from its graph representation by simply transforming the graph into its line graph, and then sketching the line graph. The fundamentals of graph theory as they relate to the study of mechanisms are reviewed. Some new observations are made relating to graphs and their corresponding line graphs, and a novel procedure for transforming the graph into its line graph is presented. This is the basis of a sketching algorithm which is illustrated by computer-generated examples.

A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains

2002 ◽  
Vol 124 (4) ◽  
pp. 662-675 ◽  
Author(s):  
V. V. N. R. Prasad Raju Pathapati ◽  
A. C. Rao
Keyword(s):  
Graph Isomorphism ◽  
Planetary Gear ◽  
Degree Of Freedom ◽  
Gear Train ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Graph Representations ◽  
Planetary Gear Trains ◽  
Gear Trains ◽  
A New Technique

The most important step in the structural synthesis of planetary gear trains (PGTs) requires the identification of isomorphism (rotational as well as displacement) between the graphs which represent the kinematic structure of planetary gear train. Previously used methods for identifying graph isomorphism yielded incorrect results. Literature review in this area shows there is inconsistency in results from six link, one degree-of-freedom onwards. The purpose of this paper is to present an efficient methodology through the use of Loop concept and Hamming number concept to detect displacement and rotational isomorphism in PGTs in an unambiguous way. New invariants for rotational graphs and displacement graphs called geared chain hamming strings and geared chain loop hamming strings are developed respectively to identify rotational and displacement isomorphism. This paper also presents a procedure to redraw conventional graph representation that not only clarifies the kinematic structure of a PGT but also averts the problem of pseudo isomorphism. Finally a thorough analysis of existing methods is carried out using the proposed technique and the results in the category of six links one degree-of-freedom are established and an Atlas comprises of graph representations in conventional form as well as in new form is presented.

A Model for Mechanism Data Storage

1992 ◽  
Author(s):  
Wan Wang
Keyword(s):  
Data Storage ◽  
Data Model ◽  
Computer Time ◽  
Kinematic Structure ◽  
Topological Graph ◽  
Complete Set ◽  
Isomorphism Identification

Abstract A data model for kinematic structure of mechanisms and its coding principle are proposed, based on the topological graph and contract graph. In the model every basic chain is mapped by a code of 5 decimal digits and a mechanism is mapped by a set of code of basic chains. The model occupies minimal memory, and contains a complete set of useful primary parameters of structure, and significantly reduce computer time for isomorphism identification.

scholarly journals Denavit-Hartenberg Coordinate System for Robots with Tree-like Kinematic Structure

2016 ◽  
Vol 5 (4) ◽  
pp. 244
Author(s):  
Alexander Kovalchuk ◽  
F. Akhmetova
Keyword(s):  
Graph Theory ◽  
Coordinate System ◽  
Kinematic Chain ◽  
Kinematics And Dynamics ◽  
Kinematic Structure ◽  
Coordinate Systems ◽  
Reachability Graph ◽  
Joint Application ◽  
Linear Chains ◽  
Use Efficiency

<p class="MDPI17abstract"><span lang="EN-US">The paper presents a modified Denavit-Hartenberg coordinate system resulted from joint application of graph theory and the Denavit-Hartenberg coordinate system, which was developed to describe the kinematics of robot actuators with a linear open kinematic chain. It allows forming mathematical models of actuating mechanisms for the robots with tree-like kinematic structures. The work introduces the concept of primary and auxiliary coordinate systems. It considers an example of making the links’ reachability matrix and reachability graph for the tree-like actuating mechanism of a robotic mannequin. The use efficiency of the proposed modified Denavit-Hartenberg coordinate system is illustrated by the examples giving the mathematical description of the kinematics and dynamics of specific robots’ tree-like actuating mechanisms discussed in the previously published papers. It is shown that the proposed coordinate system can also be successfully applied to describe the actuating mechanisms of robots with a linear open kinematic chain, which is a particular case of the tree-like kinematic structure. The absence of branching joints in it does not require introducing auxiliary coordinate systems and the parameters f(i) and ns(i) are necessary only for the formal notation of equations, which have similar forms for the tree-like and linear chains. In this case, the modified and traditional coordinate systems coincide.</span></p>

scholarly journals COLLEGE STUDENT’S ERROR ANALYSIS BASED ON THEIR MATHEMATICAL CONNECTIONS ON GRAPH REPRESENTATION

2019 ◽  
Vol 4 (1) ◽  
pp. 18
Author(s):  
I ketut Suastika ◽  
Vivi Suwanti
Keyword(s):  
Graph Theory ◽  
College Student ◽  
Real Life ◽  
Graph Representation ◽  
High Ability ◽  
Pilot Studies ◽  
Graph Representations ◽  
Mathematical Connections ◽  
Life Problems ◽  
Basic Concepts

This study is investigates the college student’s errors on their graph representations making based on the mathematical connections indicators. Pilot studies were conducted with 4 college students of middle to high ability in Graph Theory class. Data analyze revealed that top 3 subject’s errors are 1) Finding the relations of a representations to it’s concepts and procedures, 2) Applying mathematics in other sciences or real life problems, and 3) Finding relations among procedures of the equivalent representations. Their lack of graph concepts understanding and it’s connections plays the major role in their errors. They failed at recognizing and choosing the suitable properties of graph which able to detect the error of their graph representation. So, in order to decrease college student errors in graph representations, we need to strengthen their basic concepts and its connections.

A Dyad Search Algorithm for Solving Planar Linkages Using the Dyad Method

1992 ◽  
Author(s):  
Lofti Romdhane
Keyword(s):  
Search Algorithm ◽  
Class Ii ◽  
Graph Representation ◽  
Step Size ◽  
Lower Pair ◽  
Fast Solution ◽  
Complete Automation ◽  
Newton Raphson ◽  
Dyad Analysis ◽  
Planar Linkages

Abstract Based on graph representation of planar linkages, a new algorithm was developed to identify the different dyads of a mechanism. A dyad or class II group, is composed of two binary links connected by either a revolute (1) or a slider (0) pair with provision for attachment to other links by lower pair connectors located at the end of each link. There are five types of dyads: the D111, D101, D011, D001, and D010. The dyad analysis of a mechanism is predicated on the ability to construct the system from one or more of the five binary structure groups or class II groups. If the mechanism is complicated and several dyads are involved, the task of identifying these dyads by inspection could be difficult and time consuming for the user. This algorithm allows a complete automation of this task. This algorithm is based on the Dijkstra’s algorithm, for finding the shortest path in a graph, and it is used to develop a computer program, called KAMEL: Kinematic Analysis of MEchanical Linkages, and implemented on an IBM-PC PS/2 model 80. When compared to algorithmic methods, like the Newton-Raphson, the dyad method proved to be a very efficient one and requires as little as one tenth of the time needed by the method using Newton-Raphson algorithm. Moreover, the dyad method yields the exact solution of the position analysis and no initial estimates are needed to start the analysis. This method is also insensitive to the value of the step-size crank rotation, therefore, allowing a very accurate and fast solution of the mechanism at any position of the input link.

Dynamic analysis of planar elastic mechanisms using the dyad method

2003 ◽  
Vol 217 (1) ◽  
pp. 1-14 ◽  
Author(s):  
L Romdhane ◽  
H Dhuibi ◽  
H Bel Hadj Salah
Keyword(s):  
Class Ii ◽  
Graph Representation ◽  
Elastic Mechanism ◽  
Complete Automation ◽  
Newton Raphson ◽  
Set Up ◽  
Dyad Analysis ◽  
Planar Linkages ◽  
Raphson Method ◽  
Good Agreement

Based on graph representation of planar linkages, a new algorithm has been developed to identify the different dyads of a mechanism. A dyad, or class II group, is composed of two binary links connected by either a revolute (1) or a slider (0) pair, with provision for attachment of other links by lower pair connectors located at the end of each link. There are five types of dyad: D111, D101, D011, D001 and D010. The dyad analysis of a mechanism is predicated on the ability to construct the system from one or more of the five binary structure groups or class II groups. If the mechanism is complicated and several dyads are involved, the task of identifying these dyads, by inspection, can be difficult and time consuming for the user. This algorithm allows complete automation of this task. It is based on Dijkstra's algorithm for finding the shortest path in a graph. When compared with algorithmic methods, such as the Newton-Raphson method, the dyad method proved to be a very efficient one and requires as little as one-tenth of the time needed by the method using the Newton-Raphson algorithm. The second part of this work presents an extension of the dyad method to non-rigid or elastic mechanisms. Here also, this method is predicated on the ability to subdivide the elastic mechanism into elastic dyads. The solution for each type of elastic dyad is derived and can be applied to each dyad in the mechanism. Therefore, a solution of the complete elastic mechanism is possible when the mechanism is made of dyads only. This method makes a powerful and simple tool for analysing complex elastic mechanisms. Moreover, the complexity of the model does not increase as the mechanism becomes more complex. The D111 dyad is taken as an example to demonstrate this method. A finite element (FE) analysis was made for this type of dyad, and an experimental set-up was built to validate the analysis. The dyad-FE results were in good agreement with the experimental ones.

An Application of Boolean Algebra to the Motion of Epicyclic Drives

1971 ◽  
Vol 93 (1) ◽  
pp. 176-182 ◽  
Author(s):  
F. Freudenstein
Keyword(s):  
Boolean Algebra ◽  
Dynamic Analysis ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Computer Aided ◽  
Canonical Graph

The kinematic structure of epicyclic drives has been investigated with the aid of Boolean algebra. The correspondence between the graph representation of the structure, the mechanism, and the form of the displacement equations has been derived. A canonical graph representation has been given. A method, believed to be novel, is described for the determination of the algebraic displacement equations by inspection, directly from the kinematic structure. The theory can be applied similarly to dynamic analysis and computer-aided sketching and animation.

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