Kinematic and dynamic analysis of open‐loop mechanical systems using non‐linear recursive formulation

Abstract

In this paper, a non-linear recursive formulation is developed for kinematic and dynamic analysis of open-loop mechanical systems. The non-linear equations of motion are developed for deformable links that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars and matrices that depend on the spatial co-ordinates as well as the assumed displacement field, and these time invariant quantities represent the dynamic coupling between the rigid-body modes and elastic deformations. A new recursive formulation is presented for solving equations of motion for open-loop chains consisting of interconnected rigid and deformable open-loop mechanical systems. This formulation is expressed by the recursive relationships and the generalized non-linear equations for deformable mechanical systems to obtain a large system of loosely coupled equations of motion. The main processor program consists of three main modules: constraint module, mass module and force module. The constraint module is used to numerically evaluate the relationship between the absolute and joint accelerations. The mass module is used to numerically evaluate the system mass matrix as well as the non-linear Coriolis and centrifugal forces associated with the absolute, joint and elastic co-ordinates. Simultaneously, the force module is used to numerically evaluate the generalized external and elastic forces associated with the absolute, joint and elastic co-ordinates. Computational efficiency is achieved by taking advantage of the structure of the resulting system of loosely coupled equations. The solution techniques used in this investigation yield a much smaller operations count and can more efficiently implement in any computer. The algorithms and solutions presented in this paper are illustrated by using an industrial robotic manipulator system. The numerical results using this formulation are also presented and discussed in this paper. Copyright © 2006 John Wiley & Sons, Ltd.