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Pozycja Mobile device path planning basing on the fractional potential(Wydawnictwo Politechniki Łódzkiej, 2015) Ostalczyk, Piotr; Bąkała, Marcin; Duch, Piotr; Sankowski, DominikThe aim of the paper is to develop new quick and safe algorithms used for mobile device path planning [1,2,6,7,8,10,13,19]. The path should lead the device to some predefined target without striking the obstacles. In the proposed algorithms, applying fractional-calculus [12,14-18,22] where the frac- . tional-order differences [17,18] are defined one evaluates the generalised fractional potential [3,5,11,20,21] defining the level of an obstacle danger. Autonomous mobile devices equipped with safe paths creation abilities are valuable in different real-life situations. One can mention catastrophe areas with different types of obstacles of different obstacle danger. Applying the fractional calculus with mentioned earlier integrals of any, fractional orders one can create 3D space with better indication of the danger zones. Next one defines an optimal, safe path.[...]Pozycja Variable-, fractional-order simple model of the robot arm(Wydawnictwo Politechniki Łódzkiej, 2015) Bąkała, Marcin; Duch, Piotr; Sankowski, Dominik; Ostalczyk, PiotrIt is well known that to synthesise a satisfactory control algorithm of a dynamical system a ,,good" mathematical model should be evaluated [5]. Also there is a trade off between the model accuracy and its complexity. In robotics the dynamical models of robot arms derived by Euler-Lagrange equations or Euler - Lagrange formalism are known [9]. Their complexity expressed by a large number of relevant parameters, often difficult to estimate forces a search for new solutions. The methodology proposed in the present paper allows one to find a simple model described by a variable-, fractional-order difference equation. The fractional calculus as a new mathematical tool generalising the commonly known calculus is becoming more and more popular in different research areas, ranging from electrical engineering, electromagnetism, electrochemistry, electronics, mechanics, rheology to biophysics and economy [6, 7, 8]. It seems to be very useful in complex unmodelable processes, for instance, in description the transient behaviour of a robot arm. The paper is organized as follows. In Section 2 the definition of the variable-, fractional-order backward difference is introduced. Next some discussion concerning the mathematical models of robot arms are given. Some rules for a choice of the differentiation order functions are given. Finally measured and simulated step responses of the robot arm are presented.[...]Pozycja Variable-, fractional-order digital filter - application(Wydawnictwo Politechniki Łódzkiej, 2015) Duch, Piotr; Bąkała, Marcin; Sankowski, Dominik; Ostalczyk, Piotr[...] This chapter is organised as follows. First, in Section 2 fundamental definition of the VFOBD is given. Next the VFO-DF fundamental structure is proposed. In Section 3 two equivalent descriptions of the VFO-DF a filter response is presented. finally in Section 4 the VFO-DF responses is evaluated and compared to the filtering effects of the classical first-order linear filter.[...]Pozycja Variable-, fractional-order digital filter theoretical background(Wydawnictwo Politechniki Łódzkiej, 2015) Ostalczyk, PiotrDigital filtering is a subject of scientific investigations [1-3], [7,9], [11-13] from over than 50 years. Many proposed algorithms for digital filtering has already been recognized as a classic solutions. Here one can mention IIR and FIR filter types [l]. All filters algorithms base on the backward difference notion [5] used to its description. Since the early seventies of the last century a growing focus on the generalisation of the backward difference notion can be seen. The fractional-order backward difference [4] (FOBD) is a discrete counterpart of the fractional-order derivative [6,7]. In this work, a further generalization of the FOBD named variable-, fractional-order backward difference [10] (VFOBD) is applied to the digital filtering performed by the variable-, fractional-order digital filter (VFO-DF). A paper is organised as follows. First, in Section 2 fundamental definitions of the VFOBD is given. Next two equivalent descriptions of the VFODF are introduced. Finally some remarks concerning the order functions are given.[...]Pozycja Variable-, fractional-order digital filter - application(Wydawnictwo Politechniki Łódzkiej, 2014) Duch, Piotr; Bąkała, Marcin; Sankowski, Dominik; Ostalczyk, Piotr[...] This chapter is organised as follows. First, in Section 2 fundamental definition of the VFOBD is given. Next the VFO-DF fundamental structure is proposed. In Section 3 two equivalent descriptions of the VFO-DF a filter response is presented. Finally in Section 4 the VFO-DF responses is evaluated and compared to the filtering effects of the classical first-order linear filter.[...]Pozycja Variable-, fractional-order digital filter theoretical background(Wydawnictwo Politechniki Łódzkiej, 2014) Ostalczyk, PiotrIn this chapter variable-, fractional-order digital filters are considered. Two equivalent filter description methods are presented. Also the filter response to any discrete-time signal is given due to both introduced description methods. Finally some remarks concerning the order functions selection are given basing the frequency characteristics.