Recent applications of fractional calculus on advanced modeling of (bio)mechanical systems

Abstract

In recent years, there have been extensive research activities related to applications of fractional calculus (FC), in nonlinear dynamics, mechatronics, bioengineering as well as control theory,[1,2].The theory of FC is a well-adapted tool to the modelling of many physical phenomena, allowing the description to take into account some peculiarities that classical integer-order model simply neglect. Fractional order calculus is the general expansion of integer-order calculus and is considered as one of the novel topics for modelling dynamical systems in different applications. Fractional derivatives are often used to describe viscoelastic properties of advanced materials, dissipative forces in structural dynamics, or rheological properties of various materials. Spring, damper, or actuator elements are often used in multibody systems to improve their characteristics and such systems can be found in (bio)robotics/ mechatronics/adaptronics. In particular, dynamics of multibody systems involving generalized forces of a spring/spring-pot/ actuator (SSPA) elements can be modeled with fractional order derivatives of a composite function where elements can be used in order to represent smart materials and components in multibody adaptronic systems by introducing new tuning parameter, [3]. Dynamic equations are given as Lagrange equations of the second kind using Rodrigues approach with external generalized forces of the gravity, motor-torque, and a fractional order SSP or SSPA element. Generalized forces of an element are derived by using the principle of virtual work and force-displacement relation of the fractional order Kelvin-Voigt model. Also,the numerical scheme for solving fractional order differential equations is applied Atanackovic-Stankovic formula [4] in order to approximate fractional order derivative of a composite function. Finally, numerical example for the multibody system with three degrees of freedom is presented where the results obtained for generalized forces are compared for different values of parameters in the fractional order derivative model. In addition, such fractional derivative models can be utilized for characterization of magnetorheological (MR) fluids,[5]. Hence, using the appropriate constitutive equations, fractional calculus approach can be useful in modeling of MR elements connected into multibody systems as well. Also, one of the main challenges in biomechanics is the modelling of soft tissues. Beside, memristive systems are also used for modeling of biomechanical systems. Namely, memristor as nonlinear element was postulated by Chua in 1971 by analyzing mathematical relations between pairs of fundamental circuit variables and realized by HP laboratory in 2008, [6]. Nowadays, the memristive systems have a wide range of applications in modeling of the bioelectromechanical properties of soft tissues and so on. On the other hand, we can enlarge the family of elementary circuit elements, to fractional circuit elements in order to model many irregular and exotic nondifferentiable phenomena which are common and dominant to the nonlinear dynamics of many biological, molecular systems and nanodevices. In this work, it is used the concept of nonintegerorder memristive elements for biomehanical modelling of soft tissues, here bioelectrical properties of human skin. In literature, some models of human skin based on classical memristive approach are obtained but further improvements to the memristive models are possible. It is known that a constant-phase element (CPE) cannot be replaced by a finite number of conventional passive circuit elements (RLC). Here,we investigate which fractional memristive system is the most suitable for modeling bioimpedance properties of human skin. Particularly, we can compare obtained models for human skin based on fractional and integer memristive elements as well as fractional distributed-order modified Cole model of human skin [2]. In addition, computational models are developed and presented.