Feedback control of vibrations in a moving flexible robot arm with rotary and prismatic joints

Abstract

A r o b o t w i t h a l o n g e x t e n d i b l e f l e x i b l e arm which can a l s o u n d e r g o b o t h v e r t i c a l t r a n s l a t i o n and r o t a r y m o t i o n i s c o n s i d e r e d . F i r s t , a d i s t r ibuted-parameter model for the robot arm dynamics i s developed. I t i s f o u n d t h a t t h e e x t e n d i n g m o t i o n could enhance the arm v i b r a t i o n s . Then, a Galerkin type approximation based on a n a p p r o p r i a t e timed e p e n d e n t b a s i s f o r t h e s o l u t i o n s p a c e is used t o obtain an approximate finite-dimensional model f o r s i m u l a t i o n s t u d i e s . A f e e d b a c k c o n t r o l f o r damping t h e m o t i o n -i n d u c e d v i b r a t i o n s i s d e r i v e d by cons i d e r i n g t h e time r a t e -o f -c h a n g e o f t h e t o t a l v ib r a t i o n a l e n e r g y of t h e f l e x i b l e arm. The paper concludes with some s i m u l a t i o n r e s u l t s f o r a speci a l c a s e w i t h t h e p r o p o s e d c o n t r o l law. -1 . INTRODUCTION Recent development i n space r o b o t s w i t h l i g h tw e i g h t f l e x i b l e members and high-speed robots req u i r e s t h e c o n s i d e r a t i o n of t h e e f f e c t s o f m o t i o ni n d u c e d v i b r a t i o n s on m a n i p u l a t i o n [ l ] -[ S I . A b a s i c problem i s t o f i n d means f o r damping o u t t h e v i b r at i o n s as q u i c k l y a s p o s s i b l e so t h a t t h e e n d -e f f e ct o r c a n p e r f o r m i t s tasks without delay. In almost a l l t h e s t u d i e s so f a r , t h e f l e x i b l e members o f t h e r o b o t are assumed t o h a v e f i x e d l e n g t h s a n d t h e i r m o t i o n s a r e d e s c r i b e d by beam e q u a t i o n s d e f i n e d on f i x e d s p a t i a l domains. I n a r e c e n t s t u d y [ 6 ] i n v o l vi n g a r o b o t w i t h a p r i s m a t i c j o i n t a n d a long moving f l e x i b l e arm, i t was found t h a t t h e e x t e n d i n g and c o n t r a c t i n g m o t i o n s h a v e d e s t a b i l i z i n g a n d s t a b i l i zi n g e f f e c t s on t h e v i b r a t o r y m o t i o n s r e s p e c t i v e l y . Here, w e c o n s i d e r a similar r o b o t w i t h a long ext e n d i b l e f l e x i b l e arm which can also undergo both v e r t i c a l t r a n s l a t i o n and rotary motions (see Fig.1). The paper begins with the development .of a mathem a t i c a l model f o r t h e f l e x i b l e r o b o t arm. Then, t h e q u a l i t a t i v e b e h a v i o r o f t h e m o t i o n -i n d u c e d v i b r at i o n s i s examined. A Galerkin-type approximation based on an appropriate time-dependent basis i s used t o o b t a i n an approximate finite-dimensional model f o r computer simulation studies. A f e e d b a c k c o n t r o l f o r damping t h e m o t i o n -i n d u c e d v i b r a t i o n s is d e r i v e d by considering the time-rate-of-change of t h e t o t a l v i b r a t i o n a l e n e r g y of the f l e x i b l e arm.-2. MATHEMATICAL MODEL A r o b o t w i t h ...

Dynamic modeling and simulation of two cooperating structurally-flexible robotic manipulators

“…This simplifies the dynamic model considerably as it precludes the need to include axial shortening terms to properly account for the contact forces arising at the grasp point. 18 Substituing equations (11) and (12) into equations (5)- (9) and by applying the Galerkin criterion with the mode shapes as the weighting functions, the equations of motion can be obtained as R" I + m + 3 ) x ( ' I + m + 3 ) is the stiffness matrix, n e R"+ m+3 is the vector containing nonlinear terms, E e U (n + m+3)x3 is the control distribution matrix, u = [u x , u 2 , u 3 ] T e R 3 is the joint torque vector, / e R 3x <' I+ ' n+3 ) j s the Jacobian matrix, and f = [f x ,f y , t] T eU 3 is the nonconservative reaction force vector. Equation (13) can be used to study the dynamics of the right manipulator.…”

Dynamics and control analysis of a single flexible link robot with translational joints

“…The Galerkin method is used to model the robot. The authors also proposed a feedback control law in 2 . Kwon and Book 3 present a single link robot which is described and modeled by using assumed modes method (AMM).…”

Dynamic Modeling and Control of a Flexible Link Manipulators with Translational and Rotational Joints

“…Dynamic model analysis is based on a Galerkin approximation with time dependent basis functions. They also proposed a feedback control _______  law in [2]. Kwon and Book [3] present a single link robot which is described and modeled by using assumed modes method (AMM).…”