INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1211–1224 Published online 9 November 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1353 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Nonstandard numerical schemes for modeling a 2-DOF serial robot with rotational spring–damper–actuators Miguel D´ ıaz-Rodr´ ıguez 1, ∗, † Gilberto Gonz´ alez-Parra 2 and Abraham J. Arenas 3 1 N´ ucleo, La Hechicera, Departamento de Tecnolog´ ıa y Dise˜ no, Facultad de Ingenier´ ıa, Universidad de los Andes, M´ erida 5101a, Venezuela 2 Departamento de C´ alculo, Universidad de los Andes, M´ erida, Venezuela 3 Departamento de Matem´ aticas y Estad´ ıstica, Universidad de C´ ordoba, Monter´ ıa, Colombia SUMMARY 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. Received 19 May 2009; Revised 7 September 2009; Accepted 17 September 2009 KEY WORDS: nonstandard difference schemes; serial robots; equations of motion; numerical solution 1. INTRODUCTION In engineering and other sciences, many problems are modeled using systems of nonlinear differ- ential equations of initial value. However, due to the strong nonlinearity and the coupling of the differential equations, exact solutions are usually complicated or impossible to determinate. For strong nonlinearity systems, numerical methods are commonly used. The most traditional approach to solve models with strong nonlinearity is to adopt Runge– Kutta-type numerical schemes. However, using such schemes to solve nonlinear dynamic systems may result in a sudden growth of the iterative values, which leads to a floating-point overflow within a few time steps, or a wrong solution. Thus, the usage of these traditional schemes of finite differences to solve numerical systems of nonlinear differential equations of initial value raises questions such as what is the truncation error or the region of stability. For instance, forward Euler, ∗ Correspondence to: Miguel D´ ıaz-Rodr´ ıguez, N´ ucleo La Hechicera, Departamento de Tecnolog´ ıa y Dise˜ no, Facultad de Ingenier´ ıa, Universidad de los Andes, M´ erida 5101a, Venezuela. † E-mail: dmiguel@ula.ve Copyright 2009 John Wiley & Sons, Ltd.