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- Fractional Dynamics: A Statistical PerspectivePublication . Tenreiro Machado, J. A.; Galhano, AlexandraFractional calculus is a mathematical paradigm that has been increasingly adopted to describe the dynamics of systems with hereditary characteristics, or that reflect an average of a large population of microelements. In this line of thought, this article analyzes the statistical dynamics of a system composed of a large number of micromechanical masses with backlash and impacts. We conclude that, while individual dynamics of each element has an integer-order nature, the global dynamics reveal the existence of both integer and fractional dynamics.
- Kinematic Robutness of Manipulating SystemsPublication . Galhano, Alexandra; Tenreiro Machado, J. A.Present day mechanical manipulators have poor performances when compared with the human arm. In fact, joint-driven manipulators are not efficient due to the high actuator requirements imposed by the transients of the operational tasks. muscle-actuated arms are superior because the anatomic levers adapt the manipulating exigences to the driving linear actuators. The kinematic analysis of these systems highlights its main properties and constitutes a step towards the design of new mechanical biological-like robotic structures adopting linear actuators replacing the standard rotational joint-driving motor systems.
- Fractional dynamics: a statistical perspectivePublication . Tenreiro Machado, J. A.; Galhano, AlexandraFractional calculus is a mathematical paradigm that has been increasingly adopted to describe the dynamics of systems with hereditary characteristics, or that reflect an average of a large population of micro elements. In this line of thought, this article analyzed the statistical dynamics of a system composed of a large number of micro-mechanical masses with backlash and impacts. We conclude that, while individual dynamics of each element has an integer order nature, the global dynamics reveal the existence of both integer and fractional dynamics.
- Approximating fractional derivatives through the generalized meanPublication . Machado, J. A. Tenreiro; Galhano, Alexandra; Oliveira, Anabela; Tar, József K.This paper addresses the calculation of fractional order expressions through rational fractions. The article starts by analyzing the techniques adopted in the continuous to discrete time conversion. The problem is re-evaluated in an optimization perspective by tacking advantage of the degree of freedom provided by the generalized mean formula. The results demonstrate the superior performance of the new algorithm.
- Electric Fractional Order PotencialPublication . Tenreiro Machado, J. A.; Galhano, AlexandraIn this study we apply the concept of fractional calculus to electromagnetism and we develop a new fractional order approximation method to the electrical potential
- Fractional order dynamical systems and its applicationsPublication . Tenreiro Machado, J. A.; Barbosa, Ramiro; Jesus, Isabel S.; Silva, Manuel; Figueiredo, Lino; Reis, Cecília; Marcos, Maria da Graça; M. Afonso, Luís; Galhano, Alexandra; Duarte, Fernando B.; Lima, Miguel L.; Pires, Eduardo S.; Ferreira, Nuno M. FonsecaThis article illustrates several applications of fractional calculus (FC) in science and engineering. It has been recognized the advantageous use of this mathematical tool in the modeling and control of many dynamical systems. In this perspective, this paper investigates the use of FC in the following fields: Controller tuning; Electrical systems; Traffic systems; Digital circuit synthesis; Evolutionary computing; Redundant robots; Legged robots; Robotic manipulators; Nonlinear friction; Financial modeling.
- A fractional calculus perspective in electromagneticsPublication . Tenreiro Machado, J. A.; Jesus, Isabel S.; Galhano, AlexandraSome experimentation with magnets was beginning in the late 19th century. By then reliable batteries had been developed and the electric current was recognized as a stream of charge particles. Maxwell developed a set of equations expressing the basic laws of electricity and magnetism, and demonstrated that these two phenomena are complementary aspects of electromagnetism. He showed that electric and magnetic fields travel through space, in the form of waves, at a constant velocity. Maxwell is generally regarded as the nineteenth century scientist who had the greatest influence on twentieth century physics, making contributions to the fundamental models of nature. Bearing these ideas in mind, in this study we apply the concept of fractional calculus and some aspects of electromagnetism, to the static electric potential, and we develop a new fractional order approximation method to the electrical potential.
- On the statistical and Fourier modelling of robot motionPublication . Tenreiro Machado, J. A.; Galhano, AlexandraA new method for the study and optimization of manipulator trajectories is developed. The novel feature resides on the modelling formulation. Standard system descriptions are based on a set of differential equations which, in general, require laborious computations and may be difficul to analyse. Moreover, the derived algorithms are suited to "deterministic" tasks, such as those appearing in a repetitive work, and are not well adapted to a "random" operation that occurs in intelligent systems interacting with a non-structured and changing environment. These facts motivate the development of alternative models based on distinct concepts. The proposed embedding of statistics and Fourier transform gives a new perspective towards the calculation and optimization of the robot trajectories in manipulating tasks.
- A new method for approximating fractional derivatives: application in non-linear controlPublication . Tenreiro Machado, J. A.; Galhano, AlexandraThe theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received attention only recently. In the area of automatic control some work was developed but the proposed algorithms are still in a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear system control.
- Application of fractional calculus in the system modelling and controlPublication . Tenreiro Machado, J. A.; Barbosa, Ramiro; Silva, Manuel; Figueiredo, Lino; Jesus, Isabel S.; Galhano, AlexandraFractional 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 chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses a FC perspective in the study of the dynamics and control of several systems.
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