Some Problems in the Theory of Engineering Systems (Geometric Approaches)

Alexander Milnikov and Archil Prangishvil
Georgian Technical University, Tbilisi, Georgia

Series: Systems Engineering Methods, Developments and Technology
BISAC: TEC000000

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Edited by I Leslie Rubin, Robert J Geller, Abby Mutic, Benjamin A Gitterman, Nathan Mutic, Wayne Garfinkel, Claire D Coles, Kurt Martinuzzi, and Joav Merrick

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It has been proven that to any electric circuit there exists a corresponding two pairs of conjugate linear vector spaces. One of these pairs is generated by a homological group, while the other is cohomological. A new method of analysis of mechanical and electric circuits is proposed, which consists of representing the main variables and matrices of oscillatory circuits in terms of many-dimensional tensor objects. A solution is obtained for the problem of defining eigenvalues of pure-loop and pure-node circuits.

It is shown that the latter circuits are pairwise equal to impedances’ values of individual branches (to eigenvalues of a primitive circuit). A simple technique of separating roots of characteristic polynomials of many-dimensional systems with concentrated parameters is found. New matrix relations are obtained for loop and node matrices which make it possible to reduce the calculation of determinants of these matrices to the calculation of determinants of matrices of lower orders. A new method is developed for defining a full range of eigenvalues of linear oscillatory systems with a great number of degrees of freedom.

The notion of three-dimensional generalized rotations has been introduced. Relations between the parameters of the spinor representation of a group of three-dimensional generalized rotations and the coordinates of the initial and terminal points of rotation have been obtained. The simple relations between the elements of a three-dimensional orthogonal matrix of the basic representation and the Euler angles, on the one hand, and the coordinates of the initial and terminal points of rotation, on the other hand were derived. The spinor method of solution of inverse kinematic problem for spatial mechanisms with spherical pairs has been developed and the corresponding algorithm has been proposed. The obtained results permitted to reduce the actual three-dimensional problem of spatial motion control to the one-dimensional problem; simple adaptive algorithms are suggested, by means of which various partial problems on the terminal control are solved under various terminal conditions. New algorithms of control of spatial rotations of manipulating robots are studied. (Imprint: Nova)

Preface

List of Figures

List of Tables

Part I. Tensor-Geometric Methods for Problems of Circuit Theory

Chapter 1. Circuit Tensor Theory

Chapter 2. Topological Fundamentals of the Tensor Circuit Theory

Chapter 3. Topological Invariants of Circuits (Power Invariants)

Chapter 4. Operations of Node Division and Connection

Chapter 5. Theoretical Foundations of the Method of Solving the Complete Problem on Eigenvalues for Multidimensional Oscillatory Systems

Chapter 6. Some New Results on the Estimation of Roots in Characteristic Polynomials

Chapter 7. Algorithm of the New Method of Solution of the Generalized Problem on Eigenvalues of Multidimensional Electrical Circuits

Chapter 8. Examples

PART II. New Spinorial Method of Spatial Rotations Terminal Control

Chapter 9. Solution of Inverse and Theory of Spatial Rotations Group

Chapter 10. Solution of the Inverse Problem of Kinematics of Multi-Joint Spatial Mechanisms

Chapter 11. A New Algorithm of Spatial Rotations Terminal Control

Chapter 12. Control of Terminal States of Spatial Rotations of Robot-Manipulators

References

Appendix

Index

"Unconventional approach to the solution of the Inverse Problem of Kinematics – a classical problem of Mechanics which leads to the development of new approach to the Terminal Control of mechanical objects movement. The present method has both theoretical and practical value. The novelty of the research consists in the use of spinor geometry's mathematical apparatus. The latter gave an efficient and simple solution of Inverse Problem of Kinematics, which should be useful in practical problems of robotics." - Tamaz Natriashvili, Professor, Doctor of Sciences, Principal of R. Dvali Research Institute of Machine Mechanics

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