Adaptive nonlinear control of one-link flexible arm

When a flexible arm is r o t a t e d b y a motor a b o u t an axis th rough the arm's fixed end , t ,ransverse vibrat ion may occur. The motor t o r q u e should be controlled in such a way that t h e motor rotates b y a specified angle, while simultaneously stabil izing v ibra t ion of t h e flexible arm so that it is arrested at t h e end of rotat ion. In this paper, we propose determinis t ic and adapt ive control Iaws for one-link flexible arm. Using the fact that m a t r i x dD(6)ldt -2C(B, 8 , 6 , 6 ) is skew symmetric, controllers which have a simplified s t r u c t u r e w i t h less computa t iona l bu rden are proposed by us ing Lyapunov stabiIity theory. I . INTRODUCTION Today the ratio of payload to weight of industrial robot amounts from 1 : l O to 1:30 and less. Compared with man it is very lowone person is able to have a r,atio of 1:3 or more [l]. One of the goals for the next generation of robots will be a ration in this dimension. This might be possible only by developing lightweight robots. When a lightweight robot arm carries heavy end-effector, elastic deformation and transverse vibration may occur. Nevertheless, the end-effwtor of the robot h a s to reach each point in the working space without overshooting. This requires new advanced control algorithm: which can be developed only on the basis of appropriate models for the dynamic behavior. During recent decades, many control algorithms were developed to control flexible arm. But almost approaches are based on PD or PID control even without the analysis of stability. The reason is that the dynamics of flexible arm is too complicated to analyze the stability. Cannon and Schmitz [2] controlled a flexible arm by using an optical sensor to detect end-effector position and optimal control theory biBed on the quadratic performance index approach. Yoshiyuki et al.131 designed a finite-dimensional dyna.mic compensator using sensor outputs to construct a feedback controller on the basis of control theory of distributed-parameter system. But iibove problems of controlling flexible robots has been faced by using linear feedback techniques. Recently t,here has been a few nonlinear control laws [4] [5] by using the techniques similar to the computed torque method which has been used for the rigid arm. On the other hand, there has been many papers concerning adaptive control for rigid arm (6,11,7,8,9] (10,131 In this paper, at first we derive dynamic equation for one link flexible arm by using finite-element and Lagrangian approach. See [5] for further details. Then we propose new generalized deterministic and adaptive control laws for flexible robot arm by using a Lyapunov-like function which is similar to that one which was suggested by Spong e l a1.[12] who used this control scheme for the control of rigid robot manipulator. In proposed contr0.l schemes, we use the fact the matrix d D ( 6 ) l d t ZC(0, e , 6,6) is skew symmetric to simplify the structure of the proposed controllers and the stability of over all system is discussed. The paper is organized as follows. Section 2 presents the dynamic equations of a one-link flexible arm modeled by using the assumed mode technique. In section 3, deterministic control law for joint angular displacement is designed. In section 4, adaptive control law for joint angular displacement is design and its stability is discussed. Conclusion are drawn in the final section. 1ARM In this section, we shall derive dynamic model for one-link flexible arm by using the assumed mode technique. The one-link flexible arm is shown in Fig.1. It has a total length equal to L, a hub at one end of inertia J h , mass per unit length p , Young's modulus E, the link cross sectional area A , and constant moment of inertia I . Assuming Bernoulli-Euler beam theory, and small elastic displacements, the link is modeled as where E = z / L is the normalized position along the link. Assuming separability in time and space, i.e. modal analysis gives the following general solution P ~ E I 6 ( t ) = e z p ( j w t ) , w z = __ pAL4 (3) 6(€) = ClsinPE + Czcosp( f Cssinhpt + CrcoshPE (4) The boundary conditions to the problem specify an infinite set of admissible values for the parameter p, each of which determines an associated eigenfrequency w of the beam. From the boundary condition of one-link flexible arm