A mathematical model for control of flexible robot arms

Abstract

Structural flexibility in robotic systems has been emerging as an issue of increasing concern, for it is only realistic to include the vibration of such a system in the design of control to secure a certain degree of accuracy. The demands for high speed and low cost are driving the research for control of lightweight flexible robots. In this paper, we first formulate a mathematical model for flexible robot arms. This model describes a one-dimensional vibrating robot arm with a moving base. In general, a Cartesian robot consists of components which are flexible robot arms. There have been many investigations of the subject. Amongst them we list a few, such as works of Cannon and Schmitz [1] in 1984 and more recent work of Z.H. Luo, etc. [6,7,8]. Many of these works approach the subject from the design points of view. They have specific “goal items” to be controlled and designed controls accordingly. Our approach is more theoretical and general. First, we take a fourth order partial differential equation, the beam equation, to model the dynamics of the Cartesian flexible robot arm with several boundary conditions. Then, we consider a corresponding state-space control system in which the parameter matrix has its entries differential operators. In this setting we are able to determine the spectrum of the parameter matrix (see Section 2), and subsequently show that the system is both controllable and observable (see Section 3). In this infinite dimensional control analysis, one needs a heavy dose of functional analysis and operator theory in order to investigate the controllability and observability. This work has laid down a foundation for the design of a real-time closed loop feedback control for a flexible Cartesian robot. It is becoming more urgent that the traditional design of robot arms dependent on only the kinematics needs a makeover to include the dynamics of the system into the control. Our work fits nicely in this thrust of research which is becoming the focus of the research of dynamical robotics. The results of this article are taken from [4]. Further work is presently being pursued.