Offer Shai

Extension of Graph Theory to the Duality Between Static Systems and Mechanisms

This paper is a study of the duality between the statics of a variety of structures and the kinematics of mechanisms. To provide insight into this duality, two new graph representations are introduced; namely, the flow line graph representation and the potential line graph representation. The paper also discusses the duality that exists between these two representations. Then the duality behveen a static pillar system and a planar linkage is investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. A compound planetary gear train is shown to be dual to the special case of a statically determinate beam and the duality between a serial robot and a platform-type robot, such as the Stewart platform, is explained. To show that the approach presented here can also be applied to more general robotic manipulators, the paper includes a two-platform robot and the dual spatial linkage. The dual transformation is then used to check the stability of a static system and the stationary, or locked, positions of a linkage. The paper shows that two novel platform systems, comprised of concentric spherical platforms inter-connected by rigid rods, are dual to a spherical six-bar linkage. The dual transformation, as presented in this paper, does not require the formulation and solution of the governing equations of the system under investigation. This is an original contribution to the literature and provides an alternative technique to the synthesis of structures and mechanisms. To simplify the design process, the synthesis problem can be transformed from the given system to the dual system in a straightforward manner.

The duality relation between mechanisms and trusses

Abstract The work reported here was derived as a part of a research program aimed at investigating various discrete mathematical representations for a wide variety of engineering problems. For the results in this paper, the starting point was to represent trusses and mechanisms using graph theory and matroid theory, then using knowledge and algorithms embedded in these theories, to develop analysis methods and understanding of those two kinds of engineering systems. The research program was approached by searching for mathematical representations which may be useful for representing engineering problems. For each chosen representation, the attributes of the representation were mapped, or matched, to attributes of the engineering system considered. Properties and algorithms embedded within the representation were then applied to the corresponding attributes of the engineering system. In this way, new properties of the engineering system were found, together with new techniques to derive known properties. This paper deals exclusively with the representations applied to trusses and mechanisms. The paper first shows how plane axial force trusses and mechanisms can be represented as graphs and then analyzed using the language and algorithms of graph theory. These sections are based upon material, which has already been reported in the literature, but also include many new details in the solution methods. Having defined the dual of a planar graph, the paper shows that, in general, the graph representations of mechanisms and trusses are mathematically dual. From the perspective of this duality a number of classes of trusses and mechanisms can be defined. The duality is demonstrated for plane statically determinate trusses, and for the corresponding class of mechanisms. This direction of investigation enhances understanding of the connection between static and kinematic systems, and enables utilization of knowledge from one field to solve problems in the other.

Graph theory representations of engineering systems and their embedded knowledge

Abstract The discrete mathematical representations of graph theory, augmented by theorems of matroid theory, were found to have elements and structures isomorphic with those of many different engineering systems. The properties of the mathematical elements of those graphs and the relations between them are then equivalent to knowledge about the engineering system, and are hence termed “embedded knowledge”. The use of this embedded knowledge is illustrated by several examples: a structural truss, a gear wheel system, a mass-spring-dashpot system and a mechanism. Using various graph representations and the theorems and algorithms embedded within them, provides a fruitful source of representations which can form a basis upon which to extend formal theories of reformulation.

Combinatorial characterization of the Assur graphs from engineering

Abstract We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. This paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting the recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from ‘minimal’ linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work.

Utilization of the dualism between determinate trusses and mechanisms

Abstract Current paper presents a continuation of a previously published one, in which a mutual duality connection between determinate trusses and mechanisms has been established and proved. The dualism argued in these papers states that for every determinate truss there exists a corresponding dual mechanism and vice versa. This results in coincidence of the statical analysis procedure of the former with the kinematical analysis procedure of the latter. The new relation has opened up new ways of research and practical application, to which the current paper is dedicated. Among the applications presented in the paper there are: establishing connections between known methods in statics and kinematics; deriving new methods in structural mechanics from machine theory: a method for truss decomposition to components, dual vector resolution method, methods for checking the stability of structures; deriving methods in machine theory from structural analysis: dual Hennebergs method, a method for checking the mobility of mechanisms and new systematic design techniques based on the dualism connection.

A Study of the Duality Between Planar Kinematics and Statics

This paper provides geometric insight into the correlation between basic concepts underlying the kinematics of planar mechanisms and the statics of simple trusses. The implication of this correlation, referred to here as duality, is that the science of kinematics can be utilised in a systematic manner to yield insight into statics, and vice versa. The paper begins by introducing a unique line, referred to as the equimomental line, which exists for two arbitrary coplanar forces. This line. where the moments caused by the two forces at each point on the line are equal, is used to define the direction of a face force which is a force variable acting in a face of a truss. The dual concept of an equimomental line in kinematics is the instantaneous center of zero velocity (or instant center) and the paper presents two theorems based on the duality between equimomental lines and instant centers. The first theorem, referred to as the equimomental line theorem, states that the three equimomental lines defined by three coplanar forces must intersect at a unique point. The second theorem states that the equimomental line for two coplanar forces acting on a truss, with two degrees of indeterminacy, must pass through a unique point. The paper then presents the dual Kennedy theorem for statics which is analogous to the well-known Aronhold-Kennedy theorem in kinematics. This theorem is believed to be an original contribution and provides a general perspective of the importance of the duality between the kinematics of mechanisms and the statics of trusses. Finally, the paper presents examples to demonstrate how this duality provides geometric insight into a simple truss and a planar linkage. The concepts are used to identify special configurations where the truss is not stable and where the linkage loses mobility (i.e., dead-center positions).

Geometric properties of Assur graphs

In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks-Assur graphs-which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam.

Transforming engineering problems through graph representations

Abstract The paper introduces a general approach for solving engineering problems by transforming them to problems in other engineering fields, through general discrete mathematical models called graph representations. The idea of the method is first to raise the problem to an abstract mathematical level of graph representations. At that abstract level, either the solution is found through the tools of graph theory, such as the graph duality principle, or the problem is transformed further to another engineering domain, where the problem is associated with a known solution. The paper demonstrates a number of applications of the approach, among them: deriving new concepts in engineering; establishing new engineering designs; analyzing complicated engineering systems, and a generic treatment of both analysis and design. The paper draws the correlation of the suggested approach to known AI topics: representation change and classification problem solving method.

Modeling of caterpillar crawl using novel tensegrity structures

Caterpillars are soft-bodied animals. They have a relatively simple nervous system, and yet are capable of exhibiting complex movement. This paper presents a 2D caterpillar simulation which mimics caterpillar locomotion using Assur tensegrity structures. Tensegrity structures are structures composed of a set of elements always under compression and a set of elements always under tension. Assur tensegrities are a novel sub-group of tensegrity structures. In the model, each caterpillar segment is represented by a 2D Assur tensegrity structure called a triad. The mechanical structure and the control scheme of the model are inspired by the biological caterpillar. The unique engineering properties of Assur tensegrity structures, together with the suggested control scheme, provide the model with a controllable degree of softness-each segment can be either soft or rigid. The model exhibits several characteristics which are analogous to those of the biological caterpillar. One such characteristic is that the internal pressure of the caterpillar is not a function of its size. During growth, body mass is increased 10 000-fold, while internal pressure remains constant. In the same way, the model is able to maintain near constant internal forces regardless of size. The research also suggests that caterpillars do not invest considerably more energy while crawling than while resting.

CHECKING MOBILITY AND DECOMPOSITION OF LINKAGES VIA PEBBLE GAME ALGORITHM

The decomposition of linkages into Assur graphs (Assur groups) was developed by Leonid Assur in 1914 - to decompose a linkage into fundamental minimal components for the analysis and synthesis of the linkages. In the paper, some new results and new methods are introduced for solving problems in mechanisms, bringing in methods from the rigidity theory community. Using these techniques, an investigation of Assur graphs and the decomposition of linkages has reworked and extended the decomposition using the well developed mathematical concepts from theory of rigidity and directed graphs. We recall some vocabulary and provide an efficient algorithm for decomposing 2dimensional linkages into Assur components using strongly connected decompositions of graphs and a fast combinatorial Pebble Game Algorithm, which has been recently used in the study of rigidity and flexibility of structures and in fast analysis of large biomolecular structures such as proteins. Working on a one degree of freedom mechanism, we apply our algorithm to give the Assur decomposition. The Pebble Game Algorithm on such a mechanism is presented, along with an overview of the key properties and advantages of this elegant algorithm. We show how the pebble game algorithm can be used in the analysis and synthesis of linkages to mechanical engineering community. Core techniques and algorithms easily generalize to 3-dimensional structures, and can be further adapted to entire suite of other (bodybar) types of kinematic structures.