Fernando B. Duarte

Chaotic Phenomena and Fractional-Order Dynamics in the Trajectory Control of Redundant Manipulators

Redundant manipulators have some advantages when compared with classicalarms because they allow the trajectory optimization, both on the freespace and on the presence of obstacles, and the resolution ofsingularities. For this type of arms the proposed kinematic controlalgorithms adopt generalized inverse matrices but, in general, thecorresponding trajectory planning schemes show important limitations.Motivated by these problems this paper studies the chaos revealed by thepseudoinverse-based trajectory planning algorithms, using the theory offractional calculus.

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 DYNAMICS IN FINANCIAL INDICES

The goal of this study is the analysis of the dynamical properties of financial data series from 32 worldwide stock market indices during the period 2000–2009 at a daily time horizon. Stock market indices are examples of complex interacting systems for which a huge amount of data exists. The methods and algorithms that have been explored for the description of physical phenomena become an effective background in the analysis of economical data. In this perspective are applied the classical concepts of signal analysis, Fourier transform and methods of fractional calculus. The results reveal classification patterns typical of fractional dynamical systems.

Pseudoinverse trajectory control of redundant manipulators: a fractional calculus perspective

Redundant manipulators have some advantages when compared with classical arms because they allow the trajectory optimization, both on the free space and on the presence of obstacles, and the resolution of singularities. For this type of arms the proposed kinematic control algorithms adopt generalized inverse matrices but, in general, the corresponding trajectory planning schemes show important limitations. Motivated by these problems this paper studies the chaos revealed by the pseudoinverse-based trajectory planning algorithms, using the theory of fractional calculus.

Multidimensional Scaling Analysis of Stock Market Indexes

This paper applies multidimensional scaling techniques and Fourier transform for visualizing possible time-varying correlations between 25 stock market values. The method is useful for observing clusters of stock markets with similar behavior.

Chaos dynamics in the trajectory control of redundant manipulators

Several kinematic control algorithms for redundant manipulators adopt generalized inverse matrices. In this line of thought, the generalized inverse control scheme is tested through experiments that reveal the difficulties that often arise. Motivated by these problems this paper studies kinematics and dynamics of the trajectory planning scheme. The experiment confirm the nonlinear and chaotic performance of the algorithm and gives a deeper insight towards the future development of superior trajectory control systems.

FRACTIONAL DYNAMICS IN THE DESCRIBING FUNCTION ANALYSIS OF NONLINEAR FRICTION

Abstract This paper studies the describing function (DF) of systems constituted by a mass subjected to nonlinear friction. The friction force is decomposed in three components namely, the viscous, the Coulomb and the static forces. The system dynamics is analyzed in the DF perspective revealing a fractional-order behaviour. The reliability of the DF method is evaluated through the signal harmonic content and the limit cycle prediction.

A trajectory planning algorithm for redundant manipulators

Redundant manipulators have some advantages when compared with classical arms because they allow trajectory optimization, both in free space and in the presence of obstacles, and the resolution of singularities. For this type of manipulators, several kinematic control algorithms adopt generalized inverse matrices. In this line of thought, the generalized inverse control scheme is tested through several experiments that reveal the difficulties that often arise. Motivated by these problems the paper presents a method that optimizes the manipulability through a least square rational function approximation, to determine the joints positions and studies the chaos revealed by the kinematics trajectory planning scheme, as well as its influence on the dynamics, when controlling redundant and hyper-redundant manipulators. The experiments confirm the superior performance of the proposed algorithm, for redundant and hyper-redundant manipulators, revealing several properties and give a deeper insight towards the future development of superior trajectory control algorithms.

Analysis of Stock Market Indices with Multidimensional Scaling and Wavelets

Stock market indices (SMIs) are important measures of financial and economical performance. Considerable research efforts during the last years demonstrated that these signals have a chaotic nature and require sophisticated mathematical tools for analyzing their characteristics. Classical methods, such as the Fourier transform, reveal considerable limitations in discriminating different periods of time. This paper studies the dynamics of SMI by combining the wavelet transform and the multidimensional scaling (MDS). Six continuous wavelets are tested for analyzing the information content of the stock signals. In a first phase, the real Shannon wavelet is adopted for performing the evaluation of the SMI dynamics, while their comparison is visualized by means of the MDS. In a second phase, the other wavelets are also tested, and the corresponding MDS plots are analyzed.

Power law analysis of financial index dynamics

Power law (PL) and fractional calculus are two faces of phenomena with long memory behavior. This paper applies PL description to analyze different periods of the business cycle. With such purpose the evolution of ten important stock market indices (DAX, Dow Jones, NASDAQ, Nikkei, NYSE, S&P500, SSEC, HSI, TWII, and BSE) over time is studied. An evolutionary algorithm is used for the fitting of the PL parameters. It is observed that the PL curve fitting constitutes a good tool for revealing the signal main characteristics leading to the emergence of the global financial dynamic evolution.