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 ...
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447-448 A t w IP7Agains, IEEE Truns.Abstract-Various optimal control problems for discrete-time systems with time-lag controls are discussed. Some of the basic features of this type of system are noted. A simple example is given for illustrative
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