Systems of Navier-Stokes equations on cantor sets

We present systems of Navier-Stokes equations on Cantor sets, which are described by the local fractional vector calculus. It is shown that the results for Navier-Stokes equations in a fractal bounded domain are efficient and accurate for describing fluid flow in fractal media.

Um novo gerador central de padrões de locomoção para geração de trajectórias em tempo real de robôs quadrúpedes

Neste trabalho estuda-se a geração de trajectórias em tempo real de um robô quadrúpede. As trajectórias podem dividir-se em duas componentes: rítmica e discreta. A componente rítmica das trajectórias é modelada por uma rede de oito osciladores acoplados, com simetria 4 2 Z  Z . Cada oscilador é modelado matematicamente por um sistema de Equações Diferenciais Ordinárias. A referida rede foi proposta por Golubitsky, Stewart, Buono e Collins (1999, 2000), para gerar os passos locomotores de animais quadrúpedes. O trabalho constitui a primeira aplicação desta rede à geração de trajectórias de robôs quadrúpedes. A derivação deste modelo baseia-se na biologia, onde se crê que Geradores Centrais de Padrões de locomoção (CPGs), constituídos por redes neuronais, geram os ritmos associados aos passos locomotores dos animais. O modelo proposto gera soluções periódicas identificadas com os padrões locomotores quadrúpedes, como o andar, o saltar, o galopar, entre outros. A componente discreta das trajectórias dos robôs usa-se para ajustar a parte rítmica das trajectórias. Este tipo de abordagem é útil no controlo da locomoção em terrenos irregulares, em locomoção guiada (por exemplo, mover as pernas enquanto desempenha tarefas discretas para colocar as pernas em localizações específicas) e em percussão. Simulou-se numericamente o modelo de CPG usando o oscilador de Hopf para modelar a parte rítmica do movimento e um modelo inspirado no modelo VITE para modelar a parte discreta do movimento. Variou-se o parâmetro g e mediram-se a amplitude e a frequência das soluções periódicas identificadas com o passo locomotor quadrúpede Trot, para variação deste parâmetro. A parte discreta foi inserida na parte rítmica de duas formas distintas: (a) como um offset, (b) somada às equações que geram a parte rítmica. Os resultados obtidos para o caso (a), revelam que a amplitude e a frequência se mantêm constantes em função de g. Os resultados obtidos para o caso (b) revelam que a amplitude e a frequência aumentam até um determinado valor de g e depois diminuem à medida que o g aumenta, numa curva quase sinusoidal. A variação da amplitude das soluções periódicas traduz-se numa variação directamente proporcional na extensão do movimento do robô. A velocidade da locomoção do robô varia com a frequência das soluções periódicas, que são identificadas com passos locomotores quadrúpedes.

Robôs quadrúpedes: geração de trajectórias em tempo real usando sistemas dinâmicos não-lineares

A geração de trajectórias de robôs em tempo real é uma tarefa muito complexa, não existindo ainda um algoritmo que a permita resolver de forma eficaz. De facto, há controladores eficientes para trajectórias previamente definidas, todavia, a adaptação a variações imprevisíveis, como sendo terrenos irregulares ou obstáculos, constitui ainda um problema em aberto na geração de trajectórias em tempo real de robôs. Neste trabalho apresentam-se modelos de geradores centrais de padrões de locomoção (CPGs), inspirados na biologia, que geram os ritmos locomotores num robô quadrúpede. Os CPGs são modelados matematicamente por sistemas acoplados de células (ou neurónios), sendo a dinâmica de cada célula dada por um sistema de equações diferenciais ordinárias não lineares. Assume-se que as trajectórias dos robôs são constituídas por esta parte rítmica e por uma parte discreta. A parte discreta pode ser embebida na parte rítmica, (a.1) como um offset ou (a.2) adicionada às expressões rítmicas, ou (b) pode ser calculada independentemente e adicionada exactamente antes do envio dos sinais para as articulações do robô. A parte discreta permite inserir no passo locomotor uma perturbação, que poderá estar associada à locomoção em terrenos irregulares ou à existência de obstáculos na trajectória do robô. Para se proceder á análise do sistema com parte discreta, será variado o parâmetro g. O parâmetro g, presente nas equações da parte discreta, representa o offset do sinal após a inclusão da parte discreta. Revê-se a teoria de bifurcação e simetria que permite a classificação das soluções periódicas produzidas pelos modelos de CPGs com passos locomotores quadrúpedes. Nas simulações numéricas, usam-se as equações de Morris-Lecar e o oscilador de Hopf como modelos da dinâmica interna de cada célula para a parte rítmica. A parte discreta é modelada por um sistema inspirado no modelo VITE. Medem-se a amplitude e a frequência de dois passos locomotores para variação do parâmetro g, no intervalo [-5;5]. Consideram-se duas formas distintas de incluir a parte discreta na parte rítmica: (a) como um (a.1) offset ou (a.2) somada nas expressões que modelam a parte rítmica, e (b) somada ao sinal da parte rítmica antes de ser enviado às articulações do robô. No caso (a.1), considerando o oscilador de Hopf como dinâmica interna das células, verifica-se que a amplitude e frequência se mantêm constantes para -50.2. A extensão do movimento varia de forma directamente proporcional à amplitude. No caso das equações de Morris-Lecar, quando a componente discreta é embebida (a.2), a amplitude e a frequência aumentam e depois diminuem para - 0.170.5 Pode concluir-se que: (1) a melhor forma de inserção da parte discreta que menos perturbação insere no robô é a inserção como offset; (2) a inserção da parte discreta parece ser independente do sistema de equações diferenciais ordinárias que modelam a dinâmica interna de cada célula. Como trabalho futuro, é importante prosseguir o estudo das diferentes formas de inserção da parte discreta na parte rítmica do movimento, para que se possa gerar uma locomoção quadrúpede, robusta, flexível, com objectivos, em terrenos irregulares, modelada por correcções discretas aos padrões rítmicos.

Bipedal Locomotion: A Fractional CPG for Generating Patterns

Proceedings of the 10th Conference on Dynamical Systems Theory and Applications

Combined mutation differential evolution to solve systems of nonlinear equations

This paper presents a differential evolution heuristic to compute a solution of a system of nonlinear equations through the global optimization of an appropriate merit function. Three different mutation strategies are combined to generate mutant points. Preliminary numerical results show the effectiveness of the presented heuristic.

Control and dynamics of fractional order systems

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades due to the progress in the area of nonlinear dynamics. This article discusses several applications of fractional calculus in science and engineering, namely: the control of heat systems, the tuning of PID controllers based on fractional calculus concepts and the dynamics in hexapod locomotion.

Control and Dynamics of Fractional Order Systems

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades due to the progress in the area of nonlinear dynamics. This article discusses several applications of fractional calculus in science and engineering, namely: the control of heat systems, the tuning of PID controllers based on fractional calculus concepts and the dynamics in hexapod locomotion.

Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

On the fractional order control of heat systems

The differentiation of non-integer order has its origin in the seventeenth century, but only in the last two decades appeared the first applications in the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated namely the fractional PID and the Smith predictor. Extensive simulations are presented assessing the performance of the proposed fractional-order algorithms.

Project, modelling and simulation of air stripping columns for the treatment of groundwater contaminated with volatile organic compounds

Volatile organic compounds are a common source of groundwater contamination that can be easily removed by air stripping in columns with random packing and using a counter-current flow between the phases. This work proposes a new methodology for the column design for any particular type of packing and contaminant avoiding the necessity of a pre-defined diameter used in the classical approach. It also renders unnecessary the employment of the graphical Eckert generalized correlation for pressure drop estimates. The hydraulic features are previously chosen as a project criterion and only afterwards the mass transfer phenomena are incorporated, in opposition to conventional approach. The design procedure was translated into a convenient algorithm using C++ as programming language. A column was built in order to test the models used either in the design or in the simulation of the column performance. The experiments were fulfilled using a solution of chloroform in distilled water. Another model was built to simulate the operational performance of the column, both in steady state and in transient conditions. It consists in a system of two partial non linear differential equations (distributed parameters). Nevertheless, when flows are steady, the system became linear, although there is not an evident solution in analytical terms. In steady state the resulting system of ODE can be solved, allowing for the calculation of the concentration profile in both phases inside the column. In transient state the system of PDE was numerically solved by finite differences, after a previous linearization.

A New Approach in the Design of Air Stripping Columns for the Treatment of Groundwater Contaminated With Volatile Organic Compounds

Volatile organic compounds are a common source of groundwater contamination that can be easily removed by air stripping in columns with random packing and using a counter-current flow between the phases. This work proposes a new methodology for column design for any type of packing and contaminant which avoids the necessity of an arbitrary chosen diameter. It also avoids the employment of the usual graphical Eckert correlations for pressure drop. The hydraulic features are previously chosen as a project criterion. The design procedure was translated into a convenient algorithm in C++ language. A column was built in order to test the design, the theoretical steady-state and dynamic behaviour. The experiments were conducted using a solution of chloroform in distilled water. The results allowed for a correction in the theoretical global mass transfer coefficient previously estimated by the Onda correlations, which depend on several parameters that are not easy to control in experiments. For best describe the column behaviour in stationary and dynamic conditions, an original mathematical model was developed. It consists in a system of two partial non linear differential equations (distributed parameters). Nevertheless, when flows are steady, the system became linear, although there is not an evident solution in analytical terms. In steady state the resulting ODE can be solved by analytical methods, and in dynamic state the discretization of the PDE by finite differences allows for the overcoming of this difficulty. To estimate the contaminant concentrations in both phases in the column, a numerical algorithm was used. The high number of resulting algebraic equations and the impossibility of generating a recursive procedure did not allow the construction of a generalized programme. But an iterative procedure developed in an electronic worksheet allowed for the simulation. The solution is stable only for similar discretizations values. If different values for time/space discretization parameters are used, the solution easily becomes unstable. The system dynamic behaviour was simulated for the common liquid phase perturbations: step, impulse, rectangular pulse and sinusoidal. The final results do not configure strange or non-predictable behaviours.

Multiple Roots of Systems of Equations by Repulsion Merit Functions

In this paper we address the problem of computing multiple roots of a system of nonlinear equations through the global optimization of an appropriate merit function. The search procedure for a global minimizer of the merit function is carried out by a metaheuristic, known as harmony search, which does not require any derivative information. The multiple roots of the system are sequentially determined along several iterations of a single run, where the merit function is accordingly modified by penalty terms that aim to create repulsion areas around previously computed minimizers. A repulsion algorithm based on a multiplicative kind penalty function is proposed. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.

Symmetry and order parameter dynamics of the human odometer

Bipedal gaits have been classified on the basis of the group symmetry of the minimal network of identical differential equations (alias cells) required to model them. Primary bipedal gaits (e.g., walk, run) are characterized by dihedral symmetry, whereas secondary bipedal gaits (e.g., gallop-walk, gallop- run) are characterized by a lower, cyclic symmetry. This fact has been used in tests of human odometry (e.g., Turvey et al. in P Roy Soc Lond B Biol 276:4309–4314, 2009, J Exp Psychol Hum Percept Perform 38:1014–1025, 2012). Results suggest that when distance is measured and reported by gaits from the same symmetry class, primary and secondary gaits are comparable. Switching symmetry classes at report compresses (primary to secondary) or inflates (secondary to primary) measured distance, with the compression and inflation equal in magnitude. The present research (a) extends these findings from overground locomotion to treadmill locomotion and (b) assesses a dynamics of sequentially coupled measure and report phases, with relative velocity as an order parameter, or equilibrium state, and difference in symmetry class as an imperfection parameter, or detuning, of those dynamics. The results suggest that the symmetries and dynamics of distance measurement by the human odometer are the same whether the odometer is in motion relative to a stationary ground or stationary relative to a moving ground.

Fractional order modelling of zero length column desorption response for adsorbents with variable particle sizes

This manuscript analyses the data generated by a Zero Length Column (ZLC) diffusion experimental set-up, for 1,3 Di-isopropyl benzene in a 100% alumina matrix with variable particle size. The time evolution of the phenomena resembles those of fractional order systems, namely those with a fast initial transient followed by long and slow tails. The experimental measurements are best fitted with the Harris model revealing a power law behavior.

Fractional order modelling of fractional-order holds

Discrete time control systems require sample- and-hold circuits to perform the conversion from digital to analog. Fractional-Order Holds (FROHs) are an interpolation between the classical zero and first order holds and can be tuned to produce better system performance. However, the model of the FROH is somewhat hermetic and the design of the system becomes unnecessarily complicated. This paper addresses the modelling of the FROHs using the concepts of Fractional Calculus (FC). For this purpose, two simple fractional-order approximations are proposed whose parameters are estimated by a genetic algorithm. The results are simple to interpret, demonstrating that FC is a useful tool for the analysis of these devices.