Nonstandard numerical schemes for modeling a 2-DOF serial robot with rotational spring-damper-actuators

Abstract

In this paper we construct and develop different nonstandard finite difference schemes for a 2 degree of freedom serial robot with rotational spring–damper–actuators (RSDA). The mathematical model of the system is developed using Gibbs–Appell (G–A) equation of motion and the resulting symbolic expressions are written in configuration-space form allowing the exploitation of some characteristics related to centrifugal and coriolis forces such that nonlocal approximations can be applied to the quadratic and product joint velocity terms. To obtain numerical solutions of these highly nonlinear and coupled differential equations is not a straightforward task. Here, we show that nonstandard finite difference schemes increase the numerical stability region for the time step size. Our schemes can be divided into two approaches: the first is when denominators of the discrete derivatives are defined using nontraditional functions of the time step size in order to ensure that the fixed (equilibrium) points of the resulting discrete system has the same stability properties as those of the original system. In the second approach, the nonlinear terms are replaced by nonlocal discrete representations. Numerical comparisons of these numerical schemes are performed with Runge–Kutta-type methods in order to observe the advantages of the nonstandard difference schemes. Numerical simulations of the dynamics of robotic systems with these nonstandard numerical schemes give more reliable and robust results. Copyright © 2009 John Wiley & Sons, Ltd.