Seri ve paralel maniplatörlerin lineer graf teorisi yaklaşımı ile modellenmesi
A Systems approach to serial and parallel manipulators using graph-theoretic models
- Tez No: 19261
- Danışmanlar: PROF.DR. ALİ NUR GÖNÜLEREN
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1991
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 96
Özet
ÖZET Bu tezde, yaygın olarak elektrik devrelerinin ana lizinde kullanılan, son yıllarda da bir boyutlu ve kıs men üç boyutlu mekanik sistemlerin modellenmesinde kullanılmaya başlanan“Lineer Graf Teorisi”yöntemlerinin, üç boyutlu hareket yapan robot maniplatörlerin model lenmesinde kullanılabilirliği araştırılmıştır. Bu an lamda, seri açık kinematik zincir meydana getiren sis temlerin analizi için lineer graf teorisi yöntemlerin den yararlanılarak elde edilen bir model tanıtılmıştır. Aynı sistematik analiz yöntemleri kullanılarak, kapalı kinematik zincir oluşturan iki planar robot kolundan meydana gelen paralel bir sistem için, aynı model elde edilmiş ve incelenen bu matematik modelin genel amaçlı olarak kullanılabilirliği tartışılmıştır. Ayrıca analitik hesaplamaların zorluğu gözönüne alınarak, bilgisayarda programlanabilecek rekürsif bir hesaplama yöntemi, seri ve paralel robot sistemleri için tanıtılmıştır. Bu yöntem kullanılarak, iki eklem li, planar, SCARA tipi bir seri maniplatörün analizi bilgisayarda gerçekleştirilmiştir. Tezde yararlanılan sistematik analiz yöntemleri, robot sistemlerin modellenmesi konusunda bilinen ve literatürde karşılaşılan bir yöntem değildir. Bunun nedeni, bu konu ile uğraşanların kesitleme denklemleri ve elemanların enerji denklemleri gibi alışılagelmiş klasik formülasyon metodlarını kullanmalarıdır. Burada yapılan çalışma, robot sistemlerin modellenmesi konusunda çalışanların, elektrik ve elektronik mühendisliğinde çok iyi bilinen ve kullanılan analiz yöntemlerin den de kolaylıkla yararlanabileceklerini ortaya koymak tadır.
Özet (Çeviri)
SUMMARY A SYSTEMS APPROACH TO SERIAL AND PARALLEL MANIPULATORS USING GRAPH-THEORETIC MODELS Robot manipulators are being used in various fields such as industrial processing, space and under water research, medicine, and nuclear power generation in ever increasing numbers. The necessity of using this devices in hazardous environments, the need to perform precision tracking tasks, and the difference between the employee wages and the cost of operating a robot are the main factors influencing this growth rate. An industrial robot is a general purpose compu ter-controlled manipulator consisting of several rigid links connected in series by revolute or prismatic joints. In the analysis of these robot systems consis ting of three dimensional mechanical components, the problem is to determine the mathematical models of multi-terminal mechanical components. On the other hand, the formulation of system equation or the equa tion of motion of three dimensional mechanical sys tems containing rigid bodies, establishement of a complete and compact mathematical model of the rigid bodies plays an important role. In the past three decade, the formulation techniques developed and used in the study of electrical networks, based on linear graph theory, have conveniently been applied to one dimensional and also some restricted classes of higher dimensional mechanical systems. Recently, based on the same approach these techniques have been extended to three dimensional systems. The success of this app roach however relies on the availability of a complete and adequate mathematical model of the rigid body valid in the three dimensional motion. VIDue to the present research, linear graph theory is applicable to all types of lumped-parameter systems, and in particular, to electrical, mechanical, hydraulic and heat-transfer systems. The key step in this evolu- ation was presented by Koenig and Blackwell, which established unambigously the link between linear graph theory and physical systems [11. Thus linear graph theory was assigned the status of a fundemental en gineering discipline for formulating the mathematical characteristics of all types of discrete physical systems without recourse to the possible use of physi cal analogies between them. Later, the much broader concept of system theory based on graph-theoretic concepts was developed, but most of application were confined to electrical systems [23-C5]. At this stage of development, the analy sis of mechanical systems using graph-theoretic con- concepts was limited to the one-dimensional case and specifically did not involve the general motion of rijid bodies. However in 1971, Andrews [1] presented his vector-network techniques to extend the range of application of linear graphs to include three-dimen sional physical quantities { spatial vectors ) [6], This study of three-dimensional motion was nevertheless confined mostly to point massesand did not place sufficient emphasis on the general framework of system theory developed in C2]-[5]. Following this, the si mulation of planar systems with rotational inertia was accomplished by Rogers and Andrews [7], [8]. The study of kinematics based on graph-theoretic model ( GTM }, which was hitherto untouched, was presented by Singhal and Ke savan [93-C10]; a recent study of a planar four-bar linkage in which the formulation was based on the use of nodal variables and the solution technique, based on the Newton-Raphson method has greatly enhanced the conceptual framework of the graph graph-theoretic method. The advantage of using GTM' s is that the curcuit postulate in the accross variables as well as the ver tex postulate in the through variables are explicitly VIIavailable for formulation. In contrast to the classi cal methods of formulation, which start with the cutset postulate and the energy equations of components, it was proceed here instead from the cutset and circuit postulate and the terminal equation of the components. Since the cutset and curcuit postulates imply the prin ciple of conservation of energy, the latter principle is not utilized explicitly in the present formulation. In fact, this central fact captures the very essence of GTM's, which lay emphasis on the structural features of the system, rather than on an invar iance principle such as the conservation of energy. In the systems approach, the model for a system component is formed by the use of two distinct piece of informations {i} The Terminal Graph, indicating the terminal pairs ( ports ) of the components and the manner of connections of the associated instruments, real or conceptual, to measure a pair of complementary ( an accross and a through ) variables at each port to describe the physical behavior of the component. (ii) The Terminal Equations, giving the relation ships between all the measured port accross and through variables. These relations are also known as Constitu tive Equations of the component. The terminal graph may or may not be connected, however, it does not contain any curcuit. For a mechanical component the terminal variables are not scalar as in the case of electrical component, rather they are vectorial quantities. Therefore, the corres ponding terminal graph can properly be named as Vector Terminal Graph which is equivalent, in the most general case, to six usual ( scalar } terminal graphs all of which are of identical topological form and each of which corresponding to the x,y,z components of the translational and rotational terminal variables. In VIIIa translational motion, the terminal variables are the linear velocity and the force while in a rotational motion these variables are the angular velocity and torque ( torsional moment }. Prom this point of view, even a simple 2-terminal ( 1-port ) mechanical element, such as an elastic beam with one terminal rigidly attached to a reference, can be represented as a scalar 6-port element, if the terminal variables are resolved into their components. The above choice for the termi nal variables as velocities and forces { torques ) for mechanical components is useful in establishing the state equatons of a given mechanical system. This choice also implies that the product of two comple- lementary variables is necessarily represent mecha nical power. This property makes it possible to define such components as perfect couplers or instantaneously powerless ( non energic ) multy-terminal components. On the other hand, in the formulation of equations of electrical network containing non-linear elements, in general, it becomes necessary to consider the integral of either or both of the accross and through variables. Since in three-dimensional motion, the kinematical relations are inherently non-linear, in the final set of state equations of motion at least one or both of the integral variables i.e., displacement and momentum ( linear and angular }, will appear. In the classical formulation of equations, however momentum variables are eliminated through the formulation procedure yielding the equations of motion in the form of a set of second order differantial equations in the displacement variables. In this thesis, special systems of rijid bodies used in robotic applications are considered. In this applications, each rigid body is regarded as a 2-port component, and the main structure presented by inter connection of this components in the forms of open and closed kinematic chain. To be able to use the system of rigid bodies under various operating conditions, it is modelled as a multi-port component and a closed form of IXavailable for formulation. In contrast to the classi cal methods of formulation, which start with the cutset postulate and the energy equations of components, it was proceed here instead from the cutset and circuit postulate and the terminal equation of the components. Since the cutset and curcuit postulates imply the prin ciple of conservation of energy, the latter principle is not utilized explicitly in the present formulation. In fact, this central fact captures the very essence of GTM's, which lay emphasis on the structural features of the system, rather than on an invar iance principle such as the conservation of energy. In the systems approach, the model for a system component is formed by the use of two distinct piece of informations {i} The Terminal Graph, indicating the terminal pairs ( ports ) of the components and the manner of connections of the associated instruments, real or conceptual, to measure a pair of complementary ( an accross and a through ) variables at each port to describe the physical behavior of the component. (ii) The Terminal Equations, giving the relation ships between all the measured port accross and through variables. These relations are also known as Constitu tive Equations of the component. The terminal graph may or may not be connected, however, it does not contain any curcuit. For a mechanical component the terminal variables are not scalar as in the case of electrical component, rather they are vectorial quantities. Therefore, the corres ponding terminal graph can properly be named as Vector Terminal Graph which is equivalent, in the most general case, to six usual ( scalar } terminal graphs all of which are of identical topological form and each of which corresponding to the x,y,z components of the translational and rotational terminal variables. In VIII
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