Theory Of Linear Transformations Research Articles

Automatic synthesis of arthrokinematically compatible exoskeletons. A case study on its application on a shoulder occupational exoskeleton

The synthesis of kinematic structures compatible with the human anatomy and complexity is not a straightforward problem. The resulting solution space has a large amount of possible combinations and finding one that is suitable, usually comes down to the designer’s expertise, often leading to sub-optimal solutions. Automatic synthesis of wearable devices is a field that can help on providing the designer with an atlas of solutions to greatly simplify and improve this process. However, existing methods up to now, often require unrealistic simplifications to reduce the number of solutions produced by the machine, to a reasonable amount that can be useful for the designer. The objective of this work is to present a new method for the type synthesis of arthrokinematically compatible exoskeletons, that uses graph and linear transformation theory to overcome the limitations of the existing methods, expanding their applicability to the 3D space. Spatial kinematically compatible shoulder exoskeletons have been generated with an algorithm that uses 99% less symbolic calculations in comparison with the state of the art, reducing the complexity of the process and enabling its use in real case scenarios.

Maximal nonnegative and $\theta$-accretive extensions of a positive definite linear relation

Let $L_{0}$ be a closed linear positive definite relation ("multivalued operator") in a complex Hilbert space. Using the methods of the extension theory of linear transformations in a Hilbert space, in the terms of so called boundary value spaces (boundary triplets), i.e. in the form that in the case of differential operators leads immediately to boundary conditions, the general forms of a maximal nonnegative, and of a proper maximal $\theta$-accretive extension of the initial relation $L_{0}$ are established.

Application of Linear Algebra in Real Life

Linear algebra is an important basic course in university mathematics, finite dimensional vector space and linear transformation theory mainly on matrix theory, and the combination of matrix. Concept in linear algebra is defined directly by the mathematical symbol, rarely by example to import. How to make students master the abstract and difficult course, first of all is to let the students find the course in our daily life, to stimulate interest in learning.

An Explicit Analysis Method for Walking-Beam Oil Pumping Unit

Based on the theory of linear transformation and vector operations, an explicit analysis method is proposed for walking-beam oil pumping unit, which overcomes the difficulties of traditional method: roots discriminating of quadratic equations method and angle quadrant discriminating trigonometric functions method. The kinematics and torque factor of suspension point, and the crank torque of the three kinds of walking-beam pumping units are calculated in a new way according to the rule of power conservation, which avoids the complex calculation of the force bore by each rod. The validity of this proposed method is illustrated with a numerical example.

Parallel mechanisms with decoupled rotation of the moving platform in planar motion

The current article presents a new family of T2R1-type spatial parallel mechanisms (PMs) with decoupled and unlimited rotation of the moving platform in planar motion. The moving platform performs two independent translations (T2) and one independent unlimited rotation (R1) whose axis is perpendicular to the plane of translations. A method is proposed for structural synthesis of T2R1-type PMs based on the theory of linear transformations. The moving platform has unlimited rotational capabilities and is decoupled with respect to translational motion. To the best of the author's knowledge, this article presents for the first time solutions of T2R1-type spatial PMs with decoupled and unlimited rotation of the moving platform in planar motion.

Structural synthesis of 5 DoFs 3T2R parallel manipulators with prismatic actuators on the base

A method is presented to synthesize 5 degrees of freedom (DoFs) of 3 translational and 2 rotational (3T2R) parallel kinematic structures. This method is based on the theory of linear transformation and geometrical analysis. Central to this method is a set of novel 5 DoFs 3T2R parallel mechanisms (PMs). Based on the legs configuration, the generated mechanisms are classified. Moreover, the promising mechanisms of each class are introduced with respect to some criteria, i.e.: (a) degree of coupling between the actuators and degrees of freedom; (b) easy kinematics and control command; (c) easy construction (or low cost construction); and, (d) manufacturability. With reference to these criteria, some discussions are given on the promising mechanisms. Finally, to demonstrate the role of legs configuration in relation to the characteristics of a manipulator, the kinematic analysis of two of the introduced mechanisms is compared.

Structural synthesis of maximally regular T3R2-type parallel robots via theory of linear transformations and evolutionary morphology

SUMMARYThe paper presents structural synthesis of maximally regular T3R2-type parallel robotic manipulators (PMs) with five degrees of freedom. The moving platform has three independent translations (T3) and two rotations (R2). A method is proposed for structural synthesis of maximally regular T3R2-type PMs based on the theory of linear transformations and evolutionary morphology. A one-to-one correspondence exists between the actuated joint velocity space and the external velocity space of the moving platform. The Jacobian matrix mapping the two vector spaces of maximally regular T3R2-type PMs presented in this paper is the 5×5 identity matrix throughout the entire workspace. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission capabilities. Kinematic analysis of maximally regular parallel robots is trivial and no computation is required for real-time control. This paper presents in a unified approach the structural synthesis of PMs with five degrees of freedom with decoupled and uncoupled motions, along with the maximally regular solutions.

The Reproducing Kernel Structure Arising from a Combination of Continuous and Discrete Orthogonal Polynomials into Fourier Systems

We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier-type systems.We prove Ismail’s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh’s theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical, their continuous q-extensions and generalizations. As a byproduct of this approach, a new class of sampling theorems is obtained, as well as Neumann-type expansions in Bessel and q-Bessel functions.

On completely irreducible operators

Looking back to Linear Algebra, to transform a matrix into a Jordan standard form is really situated at the centre in the theory of linear transformation. It is also a prototype in the spectral theory of bounded linear operators on infinite dimensional space. One naturally asks: What is a suitable analogue of Jordan blocks in the set of all bounded linear operators on an infinite dimensional Hilbert space? We consider that a suitable analogue is the completely irreducible operator, or BIR operator in brief. In 1970’s Z.J. Jiang [JZJ, 1, 2] indicated the importance of investigating BIR operators and proposed some questions concerning BIR operators and the operator structure at the functional analysis seminar of Jilin University. In the past fifteen years some results were obtained by the seminar’s collective efforts. In this paper we briefly sum up the works on BIR operators that have been done by our seminar.

Structural synthesis of fully-isotropic parallel robots with Schönflies motions via theory of linear transformations and evolutionary morphology

Abstract The paper presents structural synthesis of fully-isotropic parallel robotic manipulators (PMs) with Schonflies motions. The moving platform of a parallel manipulator with Schonflies motions (PMSM) has four degrees of freedom, which are three independent translations and one rotation about an axis of fixed direction. A method is proposed for structural synthesis of fully-isotropic PMSMs based on the theory of linear transformations and the evolutionary morphology. A one-to-one correspondence exists between the actuated joint velocity space and the external velocity space of the moving platform. The Jacobian matrix mapping the two vector spaces of fully-isotropic PMSMs presented in this paper is the identity 4 × 4 matrix throughout the entire workspace. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission capabilities. The synthesis method proposed in this paper allows us to obtain structural solutions of PMSMs with decoupled and uncoupled motions, along with the fully-isotropic solutions in a systematic manner. Overconstrained/isostatic solutions with elementary/complex and identical/different legs are obtained. Uncoupled and fully-isotropic PMSMs have the advantage of simple command and important energy-saving due to the fact that, for a unidirectional motion, only one motor works as in a serial translational manipulator.

General Algebra and Linear Transformations Preserving Matrix Invariants

The interrelations between the theory of linear transformations preserving matrix invariants and different branches of mathematics are surveyed here. The preferences are given for those methods and motivations to study these transformations that arise from general algebra.

Mobility and spatiality of parallel robots revisited via theory of linear transformations

Abstract Parallel robots are extensively used for various applications including manipulation, machining, guiding, testing and control. The mechanical architecture of parallel robots is based on parallel mechanisms in which a mobile platform is connected to a reference element by at least two legs. Mobility and spatiality are the main structural and kinematic parameters of a parallel robot. These two parameters are defined via the theory of linear transformation and can be easily determined by inspection using the definitions, properties and theorems introduced in this paper. An analytical method to compute these parameters is also presented just for verification and for a better understanding of their meanings. The new formalism presented in this paper is based on spatiality of an elementary open kinematic chain and relative spatiality between two elements of a closed kinematic chain. As far as we are aware, this paper demonstrates for the first time a new formula for calculation of general (full-cycle) mobility of parallel robots that overcomes the drawbacks of Chebychev–Grubler–Kutzbach's mobility criterion largely used for mobility calculation of multi-loop mechanisms. This new formula is easily applicable to parallel robotic manipulators with elementary or complex legs and mobility calculation does not involve the setting up of instantaneous constraint systems associated to the parallel mechanism.

Chebychev–Grübler–Kutzbach's criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations

The paper presents a critical review of the well known Chebychev–Grübler–Kutzbach's criterion for global mobility calculation of multi-loop mechanisms. We emphasize on the limits of this criterion by applying it to three parallel robots and ascertaining that the results are erroneous. In fact, this criterion does not fit for many classical mechanisms or modern parallel robots. We explain why this criterion does not work for certain mechanisms and we accurately limit its applicability. We found our proof on the theory of linear transformations. As far as we are aware this paper presents for the first time an accurate applicability limitation of this well known criterion with huge and long-time utilisation in analysis and synthesis of mechanisms.

Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations

Abstract The paper presents a new structural synthesis approach of fully-isotropic translational parallel robotic manipulators (TPMs) based on the theory of linear transformations. A TPM is a 3-DOF (degree of freedom) parallel mechanism whose output link, called platform, can achieve three independent orthogonal translational motions with respect to the fixed base. The manipulators presented in this paper have three legs connecting the moving platform and the base (fixed platform). Only one kinematic pair per leg is actuated by a linear motor situated on the fixed base. A one-to-one correspondence exists between the actuated joint space and the operational space of the moving platform. The Jacobian matrix of fully-isotropic TPMs presented in this paper is the identity 3 × 3 diagonal matrix throughout the entire workspace. The synthesis method proposed in this paper allows us to obtain all structural solutions of fully-isotropic TPMs in a systematic manner. Overconstrained/isostatic solutions with elementary/complex and identical/different legs are obtained. Fully-isotropic TPMs have the advantage of simple command and important energy-saving due to the fact that, for a unidirectional translation, only one motor works as in a serial translational manipulator.

Linear Transformation Theory of Quantum Field Operators and Its Applications**The project supported by the Youth Sci-Tech “Qimingxing” Program of Shanghai, Foundation for University Key Teacher of Ministry of Education and Youth Science Foundation of Shanghai Higher Education

We extend the linear quantum transformation theory to the case of quantum field operators. The corresponding general transformation expressions of CPT transformations and gauge field transformations are considered as its applications.

Dualizing the Poisson summation formula.

If f(x) and g(x) are a Fourier cosine transform pair, then the Poisson summation formula can be written as 2sumfrominfinityn = 1g(n) + g(0) = 2sumfrominfinityn = 1f(n) + f(0). The concepts of linear transformation theory lead to the following dual of this classical relation. Let phi(x) and gamma(x) = phi(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = sumfrominfinityn = 1phi(n/x)/x - integralinfinity0phi(t)dt and G(x) = sumfrominfinityn = 1gamma (n/x)/x - integralinfinity0 gamma(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the "discrepancy" of phi because it is the error in estimating the integral phi of by its Riemann sum with the constant mesh spacing 1/x.

Dynamic workspace analysis of multiple cooperating robot arms

The authors focus on the investigation of maximum applicable force/torque by multiple robot arms in coordinated motion, i.e., the dynamic workspace analysis of multiple cooperating robot arms. Simple procedures to solve the dynamic workspace analysis problem for multiple cooperating robot arms with or without the internal force/torque constraint are presented. These procedures model the cooperating robot arms as a closed mechanical chain and are based on the theory of linear transformations and the properties of mechanics. For low degree-of-freedom robot arms, the maximum force/torque problem can be solved by geometrically using the proposed procedures, as illustrated by examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The measurement principle for evaluating the performance of drugs and cosmetics by skin impedance

Numerous studies have been carried out on the application of skin impedance to biological measurement. Measurement is possible because a difference will result due to the objective factor of the impedance, and the reference impedance (which does contain the objective factor). Emphasis has been placed, in the past, on the relationship between the change in skin impedance and objective factors such as disease, and there are only a few studies on the principle of measurement as the establishment of a reference impedance and a data-analysis method. (a) A method for setting a reference level based on identity transformation of skin impedance, (b) linear-transformation theory for data analysis, and (c) rate of change triangle expression of the results obtained, are proposed, and a basic principle of measurement has been established. Performance evaluation of cosmetics was carried out as an example of the application of this principle.

Stochastic properties of a linear econometric model of the Federal Republic of Germany

This paper investigates the cyclical behavior of four alternative estimates of the van derWerf-Beckmann-Uebe model under the usual assumptions that the exogenous variables are fixed and that the disturbances are serially uncorrelated. As this model is linear we use a spectral analytical approach based on the theory of linear transformations of stochastic processes. We find that time paths of the endogenous variables implied by the disturbances of the four estimates of the model have different shapes in regard to periodicity and variability. But in all cases these cycles are longer than the generally observed five-year business cycle of the Federal Republic of Germany.

Algebraic Model of Electrical Network

The set of all possible electrical states of a network is modeled using linear algebra and group theory. The model which is the multidimensional linear space is established, described, and verified. The formulas are derived making possible numerical calculations and network analysis. All possible methods of network solution are modeled and discussed using linear transformation theory. The degree of freedom of a network is defined and described. Finally, the classification of networks is proposed as far as topological structures and methods of solution are concerned. The theory developed makes possible a more general approach and solution to power system problems.