Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.

Complex-order dynamics in hexapod locomotion

This paper studies the dynamics of foot–ground interaction in hexapod locomotion systems. For that objective the robot motion is characterized in terms of several locomotion variables and the ground is modelled through a non-linear spring-dashpot system, with parameters based on the studies of soil mechanics. Moreover, it is adopted an algorithm with foot-force feedback to control the robot locomotion. A set of model-based experiments reveals the influence of the locomotion velocity on the foot–ground transfer function, which presents complex-order dynamics.

Fractional order electromagnetics

The Maxwell equations constitute a formalism for the development of models describing electromagnetic phenomena. The four Maxwell laws have been adopted successfully in many applications and involve only the integer order differential calculus. Recently, a closer look for the cases of transmission lines, electrical motors and transformers, that reveal the so-called skin effect, motivated a new perspective towards the replacement of classical models by fractional-order mathematical descriptions. Bearing these facts in mind this paper addresses the concept of static fractional electric potential. The fractional potential was suggested some years ago. However, the idea was not fully explored and practical methods of implementation were not proposed. In this line of thought, this paper develops a new approximation algorithm for establishing the fractional order electrical potential and analyzes its characteristics.

Aplicação de métodos estocásticos na gestão de custos de construção de condutas de saneamento

O trabalho apresentado centra-se na determinação dos custos de construção de condutas de pequenos e médios diâmetros em Polietileno de Alta Densidade (PEAD) para saneamento básico, tendo como base a metodologia descrita no livro Custos de Construção e Exploração – Volume 9 da série Gestão de Sistemas de Saneamento Básico, de Lencastre et al. (1994). Esta metodologia descrita no livro já referenciado, nos procedimentos de gestão de obra, e para tal foram estimados custos unitários de diversos conjuntos de trabalhos. Conforme Lencastre et al (1994), “esses conjuntos são referentes a movimentos de terras, tubagens, acessórios e respetivos órgãos de manobra, pavimentações e estaleiro, estando englobado na parte do estaleiro trabalhos acessórios correspondentes à obra.” Os custos foram obtidos analisando vários orçamentos de obras de saneamento, resultantes de concursos públicos de empreitadas recentemente realizados. Com vista a tornar a utilização desta metodologia numa ferramenta eficaz, foram organizadas folhas de cálculo que possibilitam obter estimativas realistas dos custos de execução de determinada obra em fases anteriores ao desenvolvimento do projeto, designadamente numa fase de preparação do plano diretor de um sistema ou numa fase de elaboração de estudos de viabilidade económico-financeiros, isto é, mesmo antes de existir qualquer pré-dimensionamento dos elementos do sistema. Outra técnica implementada para avaliar os dados de entrada foi a “Análise Robusta de Dados”, Pestana (1992). Esta metodologia permitiu analisar os dados mais detalhadamente antes de se formularem hipóteses para desenvolverem a análise de risco. A ideia principal é o exame bastante flexível dos dados, frequentemente antes mesmo de os comparar a um modelo probabilístico. Assim, e para um largo conjunto de dados, esta técnica possibilitou analisar a disparidade dos valores encontrados para os diversos trabalhos referenciados anteriormente. Com os dados recolhidos, e após o seu tratamento, passou-se à aplicação de uma metodologia de Análise de Risco, através da Simulação de Monte Carlo. Esta análise de risco é feita com recurso a uma ferramenta informática da Palisade, o @Risk, disponível no Departamento de Engenharia Civil. Esta técnica de análise quantitativa de risco permite traduzir a incerteza dos dados de entrada, representada através de distribuições probabilísticas que o software disponibiliza. Assim, para por em prática esta metodologia, recorreu-se às folhas de cálculo que foram realizadas seguindo a abordagem proposta em Lencastre et al (1994). A elaboração e a análise dessas estimativas poderão conduzir à tomada de decisões sobre a viabilidade da ou das obras a realizar, nomeadamente no que diz respeito aos aspetos económicos, permitindo uma análise de decisão fundamentada quanto à realização dos investimentos.

Numerical analysis of the initial conditions in fractional systems

Fractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of adequate values for the initial conditions in numerical simulations.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Fractional order modelling of dynamic backlash

This paper studies the dynamical properties of systems with backlash and impact phenomena. This type of non-linearity can be tackled in the perspective of the fractional calculus theory. Fractional and integer order models are compared and their influence upon the emerging dynamics is analysed. It is demonstrated that fractional models can memorize dynamical effects due to multiple micro-collisions.

Numerical analysis of the initial conditions in fractional systems

Fractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of adequate values for the initial conditions in numerical simulations.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

A Fractional Perspective to the Bond Graph Modelling of World Economies

Inspired in dynamic systems theory and Brewer’s contributions to apply it to economics, this paper establishes a bond graph model. Two main variables, a set of inter-connectivities based on nodes and links (bonds) and a fractional order dynamical perspective, prove to be a good macro-economic representation of countries’ potential performance in nowadays globalization. The estimations based on time series for 50 countries throughout the last 50 decades confirm the accuracy of the model and the importance of scale for economic performance.

Modeling Vegetable Fractals by means of Fractional-order Equations

This paper characterizes four ‘fractal vegetables’: (i) cauliflower (brassica oleracea var. Botrytis); (ii) broccoli (brassica oleracea var. italica); (iii) round cabbage (brassica oleracea var. capitata) and (iv) Brussels sprout (brassica oleracea var. gemmifera), by means of electrical impedance spectroscopy and fractional calculus tools. Experimental data is approximated using fractional-order models and the corresponding parameters are determined with a genetic algorithm. The Havriliak-Negami five-parameter model fits well into the data, demonstrating that classical formulae can constitute simple and reliable models to characterize biological structures.

Fractional Dynamics in the Rayleigh's Piston

This paper studies the dynamics of the Rayleigh piston using the modeling tools of Fractional Calculus. Several numerical experiments examine the effect of distinct values of the parameters. The time responses are transformed into the Fourier domain and approximated by means of power law approximations. The description reveals characteristics usual in Fractional Brownian phenomena.

Fractional State Space Analysis of Economy Systems

This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners’ modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis.

Challenges in the creation and development of a mathematics MOOC

The fast development of distance learning tools such as Open Educational Resources (OER) and Massive Open Online Courses (MOOC or MOOCs) are indicators of a shift in the way in which digital teaching and learning are understood. MOOC are a new style of online classes that allow any person with web access, anywhere, usually free of charge, to participate through video lectures, computer graded tests and discussion forums. They have been capturing the attention of many higher education institutions around the world. This paper will give us an overview of the “Introduction to Differential Calculus” a MOOC Project, created by an engaged volunteer team of Mathematics lecturers from four schools of the Polytechnic Institute of Oporto (IPP). The MOOC theories and their popularity are presented and complemented by a discussion of some MOOC definitions and their inherent advantages and disadvantages. It will also explore what MOOC mean for Mathematics education. The Project development is revealed by focusing on used MOOC structure, as well as the quite a lot of types of course materials produced. It ends with a presentation of a short discussion about problems and challenges met throughout the development of the project. It is also our goal to contribute for a change in the way teaching and learning Mathematics is seen and practiced nowadays, trying to make education more accessible to as many people as possible and increase our institution (IPP) recognition.