Timoshenko Beam Theory based Dynamic Modeling of Lightweight Flexible Link Robotic Manipulators

Abstract

An investigation into the development of flexible link robot manipulators mathematical models, with a high modeling accuracy, using Timoshenko beam theory concepts has been presented. The emphasis has been, essentially, set on obtaining accurate and complete equations of motion that display the most relevant aspects of structural properties inherent to the modeled lightweight flexible robotic structure. In particular, two important damping mechanisms: internal structural viscoelasticity effect (Kelvin-Voigt damping) and external viscous air damping have been included in addition to the classical effects of shearing and rotational inertia of the elastic link cross-section. To derive a closed-form finite-dimensional dynamic model for the planar lightweight robot arm, the main steps of an energetic deriving procedure based on the Lagrangian approach combined with the assumed modes method has been proposed. An illustrative application case of the general presentation has been rigorously highlighted. As a contribution, a new comprehensive mathematical model of a planar single link flexible manipulator considered as a shear deformable Timoshenko beam with internal structural viscoelasticity is proposed. On the basis of the combined Lagrangian-Assumed Modes Method with specific accurate boundary conditions, the full development details leading to the establishment of a closed form dynamic model have been explicitly given. In a coming work, a digital simulation will be performed in order to reveal the vibrational behavior of the modeled system and the relation between its dynamics and its parameters. It is also planned to do some comparative studies with other dynamic models. The mathematical model resulting from this work could, certainly, be quite suitable for control purposes. Moreover, an extension to the multi-link case, requiring very high modeling accuracy to avoid the cumulative errors, should be a very good topic for further investigation.