Variable-, fractional-order simple model of the robot arm

Abstract

It 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.[...]