Control Of Robots With Redundant Degrees Of Freedom

T h i s p a p e r a d d r e s s e s t h e problem of p a t h p lanning f o r r o b o t s w i t h redundant d e g r e e s of freedom. I t is assumed t h a t t h e motors f o r each j o i n t are c a p a b l e of a c h i e v i n g t h e commanded v e l o c i t y w i t h i n l i m i t s . 'Thus t h e dynamic model i s s i m p l i f i e d and t h e main c o m p l e x i t y i s t h a t of t h e k i n e m a t i c r e l a t i o n s h i p s . p r imary i n t e r e s t i s t h e problem of moving t h e end e f f e c t o r f rom p o i n t A t o p o i n t B i n minimum t i m e . A sub-opt imal s o l u t i o n i s proposed and d i s c u s s e d . Of An i m p o r t a n t problem i n r o b o t i c s i s t h a t of t r a j e c t o r y p lanning . One approach i s f o r t h e u s e r t o u s e a t e a c h i n g pendant and manual ly d i r e c t t h e r o b o t t h r o u g h a p a t h , s t o r i n g s e v e r a l p o i n t s a l o n g t h e way. While t h e t r a j e c t o r y a c h i e v e d may a l l o w t h e r o b o t t o per form t h e g i v e n t a s k , t h e r e i s no g u a r a n t e e a t a l l r e g a r d i n g t h e e f f i c i e n c y of t h e o p e r a t i o n . Some r e s e a r c h e s have a d d r e s s e d t h e q u e s t i o n of o p t i m a l i t y . S h i n and McKay (1) and Huang and McClamrock ( 2 ) a d d r e s s e d t h e s i t u a t i o n where t h e g e o m e t r i c p a t h i s s p e c i f i e d but t h e speed a l o n g t h e p a t h remains t o be de te rmined . They use v a r i o u s methods of o p t i m i z a t i o n t o s o l v e t h i s problem. T h e i r work d o e s n o t a d d r e s s t h e problem of d e t e r m i n i n g t h e g e o m e t r i c p a t h t o begin w i t h . T h i s i s a n impor tan t p a r t of t h e problem. Geer ing e t a1 ( 3 ) used P o n t r y a g i n ' s maximum p r i n c i p l e t o s o l v e f o r t h e minimum-time t r a j e c t o r y of a two l i n k mechanism. T h e i r r e s u l t s a r e v e r y good and g i v e i n t e r e s t i n g i n s i g h t i n t o t h e problem. U n f o r t u n a t e l y t h i s method i s e x t r e m e l y complex f o r r o b o t s w i t h a l a r g e r number of l i n k s . S o l u t i o n s would r e q u i r e i t e r a t i v e s o l u t i o n s of a two-point boundaryv a l u e problem. What i s proposed i s a n approach u s i n g a s i m p l e dynamic model i .e . one in which comanded j o i n t v e l o c i t i e s can be a c h i e v e d r a p i d l y by t h e i r r e s p e c t i v e motors . The o b j e c t i v e t h e n becomes one of u t i l i z i n g minimum j o i n t motion t o a c h i e v e t h e s p e c i f i e d C a r t e s i a n mot ion . 11. MODEL CONSIDERED D e f i n e t h e f o l l o w i n g v e c t o r s : " ; : I X 2 = 1: ; i i s t h e v e c t o r of C a r t e s i a n c o o r d i n a t e s i n d i c a t i n g t h e p o s i t i o n o f t h e end e f f i c i e n t , and