An innovative method for robots modeling and simulation

INTRODUCTION. The robot dynamics modeling and simulation problem has been studied for the last three decades intensively. In particular, the forward dynamics problem of a robot is a very relevant issue, which there is still to say about in terms of efficient computation algorithms, that can be also simple to understand, to develop and to implement, above all for practical robots, robots with many links and/or with flexible links (Featherstone, 1987), (Featherstone and Orin, 2000), (Sciavicco and Siciliano, 2000), (Khalil and Dombre, 2002). Indeed, in these cases the methods based on the classical Lagrange formulation give rise to an analytical model with numerous terms that may be difficult to use. The methods based on the Newton-Euler formulation are not very easy to apply and do not provide easily manageable analytical formulae, even if they are efficient from a computational point of view (Featherstone, 1987). An important contribution to solve the previous problems is given (Celentano, 2006), (Celentano and Iervolino, 2006), (Celentano and Iervolino, 2007) by a new, simple and efficient methodology of analysis, valid for all of robots, that makes use of a mathematical model containing a lower number of algebraic terms and that allows computing, with a prescribed maximum error, the gradient of the kinetic energy starting from the numerical knowledge of the only inertia matrix rather than using, as usually found in the literature, complex analytical calculations of the closed-form expression of this matrix. This result is very strong because it allows solving the forward dynamics problem of a robot in a simple and efficient manner, by analytically or numerically computing the inertia matrix and the potential energy gradient only. Moreover, this method allows students, researchers and professionals, with no particular knowledge of mechanics, to easily model planar and spatial robots with practical links. From this methodology follows also a simple and efficient algorithm for modeling flexible robots dividing the links into rigid sublinks interconnected by equivalent elastic joints and approximating and/or neglecting some terms related to the deformation variables. In this chapter some of the main results stated in (Celentano, 2006), (Celentano and Iervolino, 2006), (Celentano and Iervolino, 2007) are reported. In details, in Section II the new integration scheme for robots modeling, based on the knowledge of the inertia matrix and of the potential energy only, is reported (Celentano and Iervolino, 2006). In Section III, for planar robots with revolute joints, theorems can be introduced and demonstrated to provide a sufficiently simple and efficient method of expressing both the inertia matrix and the gradient of the kinetic energy in a closed and elegant analytical form. Moreover, the efficiency of the proposed method is compared to the efficiency of the Articulated-Body method, considered one of the most efficient Newtonian methods in the literature (Celentano and Iervolino, 2006). In Section IV, for spatial robots with generic shape links and connected, for the sake of brevity, with spherical joints, several theorems are formulated and demonstrated in a simple manner and some algorithms that allow efficiently computing, analytically the inertia matrix, analytically or numerically the gradient of the kinetic and of the gravitational energy are provided. Furthermore, also in this case a comparison of the proposed method in terms of efficiency with the Articulated-Body one is reported (Celentano and Iervolino, 2007). In Section V some elements of flexible robots modeling, that allow obtaining, quite simply, accurate and efficient, from a computational point of view, finite-dimensional models, are provided. Moreover, a significant example of implementation of the proposed results is presented (Celentano, 2007). Finally, in Section VI some conclusions and future developments are reported.