Arbitrary large-angle satellite attitude maneuvers using an optimal reaction wheel power criterion

An i n t e r e s t i n g and i n s t r u c t i v e s p e c i a l case o f t h e more genera l t h r e e a x i s maneuver i s t h e s i n g l e a x i s o r s l e w i n g maneuver. wheels a r e taken t o be f i x e d and m o t i o n occurs about a t h i r d p r i n c i p a l a x i s which c o i n c i d e s i n d i r e c t i o n w i t h t h e system a n g u l a r momentuin v e c t o r . Two o f t h e t h r e e Consider t h e r i g i d body c o n f i g u r a t i o n o f F i g u r e 1 , where ? . denotes t h e p r i n c i p a l a x i s about which t h e maneuvFr i s t o o c c u r , I i s t h e I n t r o d u c t i o n s p a c e c r a f t mass moinent o f i n e r t i a about a, J i s t h e r e a c t i o n wheel a x i a l moment o f i n e r t i a about The c a p a b i l i t y t o c o n t r o l and/or change t h e spacecra f t a t t i t u d e and angu lar v e l o c i t y s t a t e s i s r e q u i r e d i n a wide v a r i e t y of m iss ions . Accord ing ly , t h e genera l s u b j e c t has r e c e i v e d t h e a t t e n t i o n o f numerous i n v e s t i g a t o r s . I f , however, we r e s t r i c t o u r s e l v e s t o a r h i t r a r y , l a r g e a n g l e maneuvers per for ined i n some opt ima l sense by a r e a c t i o n wheel system, we f i n d few l i t e r a r y e f f o r t s . Vada l i and J u n k i n s l have r e c e n t l y t r e a t e d t h i s maneuver c l a s s w i t h t h e c o n d i t i o n o f o p t i i n a l i t y b e i n g t h e m i n i m i z a t i o n o f t i l ne i n t e g r a l s i n v o l v i n g t h e motor t o r q u e s and t h e t i m e d e r i v a t i v e s t h e r e o f . A l t e r n a t i v e l y , t h e present a u t h o r s have proposed a performance index which i s t h e i n t e g r a l o f t h e motor power squared over t h e maneuver t ime. (A l though t h e i n t e g r a l o f power o v e r t i m e would seem t o be t h e i d e a l index , i t does n o t y i e l d a un ique t o r q u e h i s t o r y . ) Reference 7 , an a n a l y t i c a l s o l u t i o n t o t h e s i n g l e a x i s , minimuin power maneuver i s presented. d isadvantage coinmon t o t h e work o f b o t h References 1 and 2 i s t h e f a c t t h a t t h e f o r m u l a t i o n s do n o t i n c l u d e t h e a b i l i t y t o s p e c i f y a (maximum t o r q u e c o n s t r a i n t ; t h i s i n a b i l i t y i s e s p e c i a l l y t roublesome i n t h e minimum Dower f o r m u l a t i o n . I n A a , 8 and @ a r e wheel and s p a c e c r a f t i n e r t i a l a n g u l a r d isp lacements , and u i s t h e motor t o r q u e e x e r t e d by t h e m o t o r on t h e wheel. be cons idered i s t h e d e t e r m i n a t i o n o f u ( t ) such The problem t o t h a t t h e per formance i n d e x 0 i s m in imized w h i l e boundary c o n d i t i o n s on t h e spacecra f t and wheel s t a t e s a t t h e i n i t i a l and f i n a l t i i nes a r e s a t i s f i e d . React ion Wheel J /*Spacecraf t I s i n c e h i g h to rques a r e r e q u i r e d d u r i n g p e r i o d s o f l o w r e l a t i v e wheel speeds. I n t h e present paper, t h e s i n g l e a x i s maneuver i s re fo r inu la ted ( v i a t h e P o n t r y a g i n Maximum P r i n c i p l e ) so as t o i n c o r p o r a t e t i m e dependent motor t o r q u e c o n s t r a i n t s . These c o n s t r a i n t s (nay he des igned i n such a way t h a t t h e t o r q u e vs. t i m e curve beg ins and ends w i t h a zero v a l u e and s l o p e , t h u s t e n d i n g t o m i n i m i z e t h e e x c i t a t i o n o f spacecra f t e l a s t i c degrees o f F i g . 1 The S i n g l e A x i s Spacecra f t Model * I n s t r u c t o r , Departinent o f Eng ineer ing Science and Mechanics, Assoc ia te Member A I A A . tAssoc ia te Pro fessor , Department o f Eng ineer ing Science and Mechanics, Member AIAA.