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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 ...

Approximate expressions are developed for internally balanced singular values corresponding to the modes of mechanical systems with gryoscopic forces, light damping, and small circulatory forces. The singular values involve input and output coupling, modal frequency, and modal damping, and they serve as a guide for model reduction by modal truncation. The derivation of these singular values is based on perturbation analysis, and the satisfaction of a frequency separation condition is required to insure their validity. Nomenclature A= system matrix B-input matrix C = output matrix A 0 = system matrix for conservative forces A l = system matrix for nonconservative forces T = transformation matrix W c = system controllability grammian WQ = system observability grammian W c^ = controllability grammian for two-mode subsystem € = small parameter associated with nonconservative forces a, = ith balanced singular value d t = ith approximate balanced singular value a>j = ith system frequency C/.. = ith system damping ratio Q = diagonal matrix of system frequencies

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