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- Kinematic evaluation of robotic biped locomotion systemsPublication . Silva, Filipe M.; Tenreiro Machado, J. A.This paper presents the kinematic study of robotic biped locomotion systems. The main purpose is to determine the kinematic characteristics and the system performance during walking. For that objective, the prescribed motion of the biped is completely characterised in terms of five locomotino variables: step length, hip height, maximum hip ripple, maximum foot clearance and link lengths. In this work, we propose three methods to quantitatively measure the performance of the walking robot: perturbation analysis, lowpass frequency response and locomobility measure. These performance measures are discussed and compared in determining the robustness and effectiveness of the resulting locomotion.
- Integer and Fractional Order Entropy Analysis of Earthquake Data-seriesPublication . Lopes, António M.; Machado, J.A.TenreiroThis paper studies the statistical distributions of worldwide earthquakes from year 1963 up to year 2012. A Cartesian grid, dividing Earth into geographic regions, is considered. Entropy and the Jensen–Shannon divergence are used to analyze and compare real-world data. Hierarchical clustering and multi-dimensional scaling techniques are adopted for data visualization. Entropy-based indices have the advantage of leading to a single parameter expressing the relationships between the seismic data. Classical and generalized (fractional) entropy and Jensen–Shannon divergence are tested. The generalized measures lead to a clear identification of patterns embedded in the data and contribute to better understand earthquake distributions.
- Performance analysis of multi-legged systemsPublication . Silva, Manuel; Tenreiro Machado, J. A.; Lopes, António M.This paper studies periodic gaits of multi-legged robot locomotion systems based on dynamic models. The purpose is to determine the system performance during walking and the best set of locomotion variables. For that objective the prescribed motion of the robot is completely characterized in terms of several locomotion variables such as gait, duty factor, body height, step lenght, stroke pitch, foot clearance, legs link lengths, foot-hip offset, body and legs mass and cycle time. In this perspective, we formulate four performance measures of the walking robot namely, the locomobility of the foot, the mean absolute power, the mean power dispersion and the mean power lost in the joint actuators per walking distance. A set of model-based experiments reveals the influence of the locomotion variables in the proposed indices.
- Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional CalculusPublication . Machado, J. A. Tenreiro; Lopes, António M.In this paper we study several natural and man-made complex phenomena in the perspective of dynamical systems. For each class of phenomena, the system outputs are time-series records obtained in identical conditions. The time-series are viewed as manifestations of the system behavior and are processed for analyzing the system dynamics. First, we use the Fourier transform to process the data and we approximate the amplitude spectra by means of power law functions. We interpret the power law parameters as a phenomenological signature of the system dynamics. Second, we adopt the techniques of non-hierarchical clustering and multidimensional scaling to visualize hidden relationships between the complex phenomena. Third, we propose a vector field based analogy to interpret the patterns unveiled by the PL parameters.
- Intrinsic fractal dynamics in the respiratory system by means of pressure-volume loopsPublication . Ionescu, Clara M.; Machado, J. A. TenreiroThis contribution presents novel concepts for analysis of pressure–volume curves, which offer information about the time domain dynamics of the respiratory system. The aim is to verify whether a mapping of the respiratory diseases can be obtained, allowing analysis of (dis)similarities between the dynamical pattern in the breathing in children. The groups investigated here are children, diagnosed as healthy, asthmatic, and cystic fibrosis. The pressure–volume curves have been measured by means of the noninvasive forced oscillation technique during breathing at rest. The geometrical fractal dimension is extracted from the pressure–volume curves and a power-law behavior is observed in the data. The power-law model coefficients are identified from the three sets and the results show that significant differences are present between the groups. This conclusion supports the idea that the respiratory system changes with disease in terms of airway geometry, tissue parameters, leading in turn to variations in the fractal dimension of the respiratory tree and its dynamics.
- Theory of fractional integrals and derivatives: application to motion controlPublication . Tenreiro Machado, J. A.The theory of fractional derivatives and integrals (FDI's) is still in a research stage but recent progresses in the area of chaos reveal promissing aspects for future developments. In the field of automatic control systems some preliminary results are restricted to the frequency domain. In this paper a novel method for the FDI approximation is presented. The proposed algorithms adopt the time domain which makes them well suited for z-transform analysis and digital implementation. Based on the new concepts the paper shows that classical P, I and D actions are special cases of a broader paradigm.
- Self-similarity principle: the reduced description of randomnessPublication . Nigmatullin, Raoul R.; Machado, J. A. Tenreiro; Menezes, RuiA new general fitting method based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth’s mean temperature surface data versus time) and for these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.
- Fractional-order derivative approximations in discrete-time control systemsPublication . Machado, J. A. TenreiroThe theory of fractional calculus goes back to he beginning of the theory of differential calculus but its inherent complexity postponed the application of the associated concepts. in the last decade the progress in the areas of chaos and fractals revealed subtle relationships with the fractional calculus leading to an increasing interest in the development of the new paradigm. in the area of automatic control preliminay work has already been carried out but the proposed algorithms are restricted to the frequency domain. The paper discusses the design of fractional-order discrete-time controllers. the algorithms and discrete-time implementation.
- Root-locus practical sketching rules for fractional-order systemPublication . Lopes, António M.; Machado, J. A. TenreiroFor integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.
- Fractional calculus: application in modelling and controlPublication . Machado, J. A. TenreiroThis contribution introduces the fractional calculus (FC) fundamental mathematical aspects and discuses some of their consequences. Based on the FC concepts, the chapter reviews the main approaches for implementing fractional operators and discusses the adoption of FC in control systems. Finally are presented some applications in the areas of modeling and control, namely fractional PID, heat diffusion systems, electromagnetism, fractional electrical impedances, evolutionary algorithms, robotics, and nonlinear system control.