Modelling of Advanced Robotic Systems with Fractional order hereditary/actuator elements

Abstract

The investigation into the dynamics of robotic and complex mechanical systems has been an active topic of research for many years. The modelling complex rigid multibody systems (RBS) using symbolic equations can provide many advantages over the more widely-used numerical methods of modeling these systems. In this contribution, we propose using procedure for recursive symbolic form computation of the complete dynamics of robotic systems with the open kinematic chain structures using Rodriquez approach for matrices of coordinate transformations. Dynamic equations are given as Lagrange equations of the second kind in the covariant form with external generalized forces of the gravity, motor-torque, viscous and spring. On the other side, the use of adaptive (hereditary/actuator: viscoelastic element with an actuator, piezo-viscoelastic,thermo-viscoelastic and magneto-rheologic) elements in complex RBS can be significant for the additional control of these systems and for reducing undesirable vibrations. Recently, fractional calculus (FC) has attracted an increased attention of scientific society. The fractional integro-differential operators are a generalization of integration and derivation to fractional operators where fractional derivatives (FD) are often used to describe viscoelastic, rheological properties of advanced materials and dissipative forces in structural dynamics. Here, modeling of dynamics of multibody systems involving generalized forces of a spring/spring-pot/actuator (SSPA) and MR elements modeled with fractional order derivatives, including recently new obtained definitions of FD, is studied. The system is defined as a discrete material system where in the force–displacement relation, mass of the element is neglected. Generalized forces of an element are derived by using the principle of virtual work and force–displacement relation of the fractional order Kelvin–Voigt/Zener model. Finally, the results obtained for generalized forces are compared for different values of parameters in the fractional order derivative model.