Notes on Newton-Euler formulation of robotic manipulators

The equations corresponding to Newton-Euler iterative method for the determination of forces and moments acting on the rigid links of a robotic manipulator are given a new treatment using composed vectors for the representation of both kinematical and dynamical quantities. It is shown that Lagrange equations for the motion of a holonomic system are easily found from the composed vectors defined in this note. Application to a simple model of an industrial robot shows that the method developed in these notes is efficient in solving the dynamics of a robotic manipulator. An example is developed, where it is seen that with the application of appropriate control moments applied to each arm of the robot, starting from a given initial position, it is possible to reach equilibrium in a final pre-assigned position.

Deconvolução de Euler: passado, presente e futuro – um tutorial

Neste tutorial apresentamos uma revisão da deconvolução de Euler que consiste de três partes. Na primeira parte, recordamos o papel da clássica formulação da deconvolução de Euler 2D e 3D como um método para localizar automaticamente fontes de campos potenciais anômalas e apontamos as dificuldades desta formulação: a presença de uma indesejável nuvem de soluções, o critério empírico usado para determinar o índice estrutural (um parâmetro relacionado com a natureza da fonte anômala), a exeqüibilidade da aplicação da deconvolução de Euler a levantamentos magnéticos terrestres, e a determinação do mergulho e do contraste de susceptibilidade magnética de contatos geológicos (ou o produto do contraste de susceptibilidade e a espessura quando aplicado a dique fino). Na segunda parte, apresentamos as recentes melhorias objetivando minimizar algumas dificuldades apresentadas na primeira parte deste tutorial. Entre estas melhorias incluem-se: i) a seleção das soluções essencialmente associadas com observações apresentando alta razão sinal-ruído; ii) o uso da correlação entre a estimativa do nível de base da anomalia e a própria anomalia observada ou a combinação da deconvolução de Euler com o sinal analítico para determinação do índice estrutural; iii) a combinação dos resultados de (i) e (ii), permitindo estimar o índice estrutural independentemente do número de soluções; desta forma, um menor número de observações (tal como em levantamentos terrestres) pode ser usado; iv) a introdução de equações adicionais independentes da equação de Euler que permitem estimar o mergulho e o contraste de susceptibilidade das fontes magnéticas 2D. Na terceira parte apresentaremos um prognóstico sobre futuros desenvolvimentos a curto e médio prazo envolvendo a deconvolução de Euler. As principais perspectivas são: i) novos ataques aos problemas selecionados na segunda parte deste tutorial; ii) desenvolvimento de métodos que permitam considerar interferências de fontes localizadas ao lado ou acima da fonte principal, e iii) uso das estimativas de localização da fonte anômala produzidas pela deconvolução de Euler como vínculos em métodos de inversão para obter a delineação das fontes em um ambiente computacional amigável.

Implicit Taylor methods for stiff stochastic differential equations

In this paper we discuss implicit Taylor methods for stiff Ito stochastic differential equations. Based on the relationship between Ito stochastic integrals and backward stochastic integrals, we introduce three implicit Taylor methods: the implicit Euler-Taylor method with strong order 0.5, the implicit Milstein-Taylor method with strong order 1.0 and the implicit Taylor method with strong order 1.5. The mean-square stability properties of the implicit Euler-Taylor and Milstein-Taylor methods are much better than those of the corresponding semi-implicit Euler and Milstein methods and these two implicit methods can be used to solve stochastic differential equations which are stiff in both the deterministic and the stochastic components. Numerical results are reported to show the convergence properties and the stability properties of these three implicit Taylor methods. The stability analysis and numerical results show that the implicit Euler-Taylor and Milstein-Taylor methods are very promising methods for stiff stochastic differential equations.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

Métodos numéricos para o retoque digital

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Modelo distribuído aplicado à análise de evaporadores do tipo tubo aletado

Pós-graduação em Engenharia Mecânica - FEIS

Explicit and Implicit Methods In Solving Differential Equations

Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.

Modelagem estocástica da dispersão axial: aplicação em um reator tubular de polimerização.

Reatores tubulares de polimerização podem apresentar um perfil de velocidade bastante distorcido. Partindo desta observação, um modelo estocástico baseado no modelo de dispersão axial foi proposto para a representação matemática da fluidodinâmica de um reator tubular para produção de poliestireno. A equação diferencial foi obtida inserindo a aleatoriedade no parâmetro de dispersão, resultando na adição de um termo estocástico ao modelo capaz de simular as oscilações observadas experimentalmente. A equação diferencial estocástica foi discretizada e resolvida pelo método Euler-Maruyama de forma satisfatória. Uma função estimadora foi desenvolvida para a obtenção do parâmetro do termo estocástico e o parâmetro do termo determinístico foi calculado pelo método dos mínimos quadrados. Uma análise de convergência foi conduzida para determinar o número de elementos da discretização e o modelo foi validado através da comparação de trajetórias e de intervalos de confiança computacionais com dados experimentais. O resultado obtido foi satisfatório, o que auxilia na compreensão do comportamento fluidodinâmico complexo do reator estudado.

The composite Euler method for stiff stochastic differential equations

In this paper we present the composite Euler method for the strong solution of stochastic differential equations driven by d-dimensional Wiener processes. This method is a combination of the semi-implicit Euler method and the implicit Euler method. At each step either the semi-implicit Euler method or the implicit Euler method is used in order to obtain better stability properties. We give criteria for selecting the semi-implicit Euler method or the implicit Euler method. For the linear test equation, the convergence properties of the composite Euler method depend on the criteria for selecting the methods. Numerical results suggest that the convergence properties of the composite Euler method applied to nonlinear SDEs is the same as those applied to linear equations. The stability properties of the composite Euler method are shown to be far superior to those of the Euler methods, and numerical results show that the composite Euler method is a very promising method. (C) 2001 Elsevier Science B.V. All rights reserved.

A contractive method for computing the stationary solution of the euler equation

A contractive method for computing stationary solutions of intertemporal equilibrium models is provide. The method is is implemented using a contraction mapping derived from the first-order conditions. The deterministic dynamic programming problem is used to illustrate the method. Some numerical examples are performed.

Wavelets And Adaptive Grids For The Discontinuous Galerkin Method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method. © Springer 2005.