Asian Journal of Control, Vol. 10, No. 4, pp. 456 461, July 2008 Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.045 –Brief Paper– INFINITE DIMENSIONAL OBSERVER FOR A FLEXIBLE ROBOT ARM WITH A TIP LOAD Tu Duc Nguyen and Olav Egeland ABSTRACT This note addresses observer design for a flexible robot arm with a tip load. The robot arm is modeled as an Euler–Bernoulli beam. The beam is clamped to a motor at one end and attached to a force actuator at the other. Based on boundary measurements, an exponentially stable observer is proposed. The existence, uniqueness, and stability of solutions of the observer are proven using semigroup theory. The results are illustrated by simulation. Key Words: Distributed parameter system, beam equation, infinite dimensional observer, semigroup. I. INTRODUCTION In recent years, the demand for high speed and low energy consuming systems has motivated the in- troduction of flexible parts in many mechanical system, e.g. robot arms with flexible links, spacecraft structures, etc. The dynamics of this class of systems are often de- scribed by a combination of ordinary differential equa- tions (ODEs), partial differential equations (PDEs), and a set of static boundary conditions. Due to the dynamic coupling between the rigid and flexible subsystems, the design of high-performance controllers for such systems is complicated. The idea was first applied by Chen [1] to the systems described by wave equation (e.g. strings), and later extended to the Euler–Bernoulli beam equation and the Timoshenko beam equation by numerous authors, among others [2–8]. To implement some of the pro- posed feedback control laws (e.g. [5]), full information Manuscript received August 31, 2006; revised March 22, 2007; accepted August 7, 2007. Both authors are with Faculty of Information Tech- nology, Mathematics and Electrical Engineering, Norwe- gian University of Science and Technology (NTNU), O. Bragstads plass 2D, N-7491, Trondheim, Norway (e-mail: Tu.Duc.Nguyen@itk.ntnu.no). This work was funded by the Research Council of Norway. The authors thank the anonymous reviewers for useful com- ments on the first version. or part of the states is needed. This is not possible in practice. Typically, the physical observations are finite- dimensional, while the states of the system are infinite- dimensional. Thus, the feedback control laws cannot be implemented directly. One way to overcome this barrier is using an observer and the separation principles (see e.g. [9]). Observer design based on Lyapunov theory is well known and widely used for both linear systems and nonlinear systems. In [7, 10, 11] observer design for flexible-link robot described by ODEs is stud- ied. Balas [12] considered observer design for linear flexible structures described by finite element models (FEM). Demetriou[13] presented a method for con- struction of observer for linear second order lumped and distributed parameter systems using parameter- dependent Lyapunov functions. Kristiansen [14] ap- plied contraction theory [15] in observer design for a class of linear distributed parameter systems. Xu et al. [16] considered infinite dimensional observers for vibrating systems, and Kalman type observers were proposed. Here, as opposed to the work of [7, 10–12] ob- server design for the flexible-link robot is based on an infinite-dimensional model. This note extends previous results [17] on observer design for the one-dimensional beam equation. The note is organized as follows. First, a model of the flexible robot arm is presented. Then, observer design is studied. After that, simulation results are presented. Finally, concluding remarks are given. 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society