On the dynamic modelling of flexible manipulators

Abstract

The manipulators of interest are those that can be idealized as beams and attached to a rotating or nonrotating hub. These manipulators are mainly observed in robotic applications where they are used to transfer parts or objects from one point to another. They are also widely used in automotive and aerospace industries for activities such as spray painting and welding. Traditionally, these manipulators are usually rigid and heavy. However, the need for improved power consumption and efficiency has motivated the use of modern materials and manufacturing methods to construct flexible and lightweight manipulators. The attendant problems have been the increased complexity in the system dynamics and control. A recent survey on the subject of dynamics and control of flexible manipulators has been presented by Dwivedy and Eberhard [7]. Flexible manipulators are either singleor multilink. A more detailed model, especially in the multi-link scenario, includes the modelling of the motors and joints as demonstrated in Refs. [3] and [11]. Each link is modelled using either Euler-Bernoulli beam theory or Timoshenko beam theory. The single-link model is presented in this paper. The development of the system governing equations is usually based on Newton-Euler method or the energy methods of Lagrange or Hamilton’s principle. Given that each link is a continuum, the problem is simplified by a finite dimensionalization process which is often approached via the assumed mode method or finite element method. In the former, the field variable is expanded as the sum of the products of eigenfunctions and undetermined parameters. The most commonly used eigenfunctions are those that relate to the nonrotating system. There are instances where the lumped parameter models are implemented. We examine three issues and organize the paper accordingly. The first is the question of reference frame(s) selection to describe the system dynamics. Here we discuss the classical clamped (also called pseudo-clamped) and the nonclassical pseudo-pinned and pseudopinned-pinned reference frames. The second problem of interest is the determination of the characteristics function of a given system, its eigenfunctions and orthogonality conditions. There are many reasons to seek closed-form expressions of the characteristics function and eigenfunctions. Apart from the potential to provide insights into the influence of some design parameters, closed-form expressions lead to smaller vector space when compared to finite element dimensionalization, a highly desirous characteristic in control implementation. In presenting the closed-form expressions, we examine a flexible manipulator with a tip load whose centre of mass is different from the point of attachment to the beam. A reference frame system is selected to highlight the role of the various components of the offset. Finally, we revisit the problem of geometric stiffening. This is a motion induced effect that captures the role of centrifugal forces. Models that ignore geometric stiffening erroneously predict instability and allow the beam to rotate at its critical speed.