A practical approach to feedback control for a mob.pdf

1A pr a c t i c a l a pp r o a c h t o f e e db a c k c o nt r o l f o r a mo b i l e r o bo t h t r a i l e La m i r a ux a La F r a s t r s e a r m e t t o c ol a m e r ng a t r a i l e r . m s of t r a j e c t t r a c a nd s t e e r i ng t o a gi c i t i on a r e a s s e d. e c i s s ue i s s by a n i t e r a t i ve t r a j e c t t r a c P e r t i o ns a r e t a i c c a l t he m o i m e l r e s s on t he m e r i l a r e i l l r a t e t i y of a n t r c t i o n c r ol f or c s e m s r i s e t o a l o t o f f or t he t8 r s . Br t s c i o n m a de s t a i z a t i on a gi c i t i on a c l e a s k f or s s e m s , t i t c be f e d b y a s i m p l e c i t a t ef e e k. Al t e r i ve s a s t i m e - va r f e e k or c e e k n t d. S e e f or a s i n m e r m on c r On t he t r a c a t r a j e c t f or a c s ys t e m m e e t Br t s c i o n a i t i s a n e a s i e r t a s k. A l o t of a l s o a s s e d t m f or t he t i c rc a s e o f m e r Al l t e c r ol l a ws t he s a m e a s s t he e o n of t he s e m i se t l y a nd no t i o n m a t he s e m t e fr om i t s t r a j e c t w l i ng wi t h m e r c ol t a ke i a c c t i i n t he a t i c se i o d a m e t t o s t a i z e a c a r a a c i t i r t oc ol t or f i e l d s t i o a nd e d o n i t e r a t i ve t r a j e c t t r a c s e of a c l e m a t he t a s k of r e a c a c t i o n e ve n m f i c a nd r e e a h t a s k e ng a ny m n t we a r t s c e e d on i t e r a t i ve t r a j e c t t r a c t o l e a d a r ng a t r a i l e r t o a c t i o n. t r a j e c t e s a r e c d b y am o n c r i i n a nd t a a c l e s t a r e gi i n i I n t o wi t a ny l nt a t we r e f e r t o t e f e r e f or a i l s . we a s s t t he e o n of a t r a j e c t i ss t t e d t o t i m we c f or t e t i o ns i s y s i m p l ea nd y a l . I t s e s c on wi t h i s e d a s f o S e c t i c r i e i m e l s e l a r e a nd i t s t r a i l e r : t wo s ys t e m s wi l l be c i d e r e d ( F i ). S e c t i l s wi t h t he c ol s c e a nd t he a na l s o f s t a i t y a nd r s . I n S e c t i o n 4,we s e nt e i m e l r e s s e s c r i p t i on of t he s e l a r e i s a t wo l m e r A t r a i l e r i s c o n t r i t wo f e r e nt s e m s on t he e : o n s e m A, t r a i l e r i s c a t he l a xi s of t he r F i , t e a s o n s e i t i s hi t c t a F i l , . A i s t he t i c r c a s e o f B, f h 0 . s e m i s s i r fr om a c r ol of vi e w a e e s m c x c t i F or t r e a s we l s e a t e l y wi t h s e m s . m e na t o c ol t he l i r a nd a r t i e s （v r ， o f t he r e t i e s a r e m m e a s d by t r i c s e e a s t he a we e n t he r nd t he t r a i l e r i s gi by a n c a l e nc t i on a nd e t i o n （ x r , y r , of t he r r e c d by i t i he e r t i e s . Wi t h t e t i o t he c ol s ys t e m i s :c o ss i ns i n ( ) c o s ( )r r rr r rr rr r r rt tx vy vv ll = （ 1 ）3F i : Hi l a r e w i t h i t s t r a i l e l G c r ol s c he c i ng r e a l s ys t e m s , t o t a ke i a c c t i o ns i o n e o e m a y ha ve m a ny a s i m f e c t i on of t he m s l i p of t he l s , i t i a e f f e c t s . e t i o ns c a n be m e d b ya a t e r m i n t he c ol s e m ( l ) ,le a t o a s e m o f t he f , )x f x u = +e m a y be e i t e r m i t i c or a r a i a In t he f i r s t c a s e , t t i o n i s o nl y t o a e o f t he s e m e o n, e a s i n t e c c a s e , i t c s fr om a r a r of t he s ys t e m . We wi l l s e e l a t e r t s e c m i s a t e r f i t f or o ur e i m e l s e m s t e e r a r r om a s t a r t c t i o n t o a m a ny c i d e r t ha tt he t i on i s t he i i a l t a we e n t he r nd t he t t o n of t he s e m i s f e c t l y To s t he m , t i gn a ni s a i on of t he s t a t e a nd t i m e t m a t he a n a s c a l l y s t a i b r i um o f t he c l d l s e m . i f we i t he y i e r m i n t c l d l s ys t e m , we t wi l l ha We c a n c t t i f t he t i o n i s s m a l l a nd e r m i t i c , t he e i b r i um i ft e i s s t i l l wi l l be c l t o t he a nd i f t he t i on i s a r a i a 4t he e l i b r i um wi l l a n e i b r i um s t we t a he t i on o f t e ne w e i b r i um or s t t i m e va r m e t a r e c nt l i ng wi t h a c l e s c a n be d i n t he ne i t he go a l a nd t t l y i t o e c i s i r e e t r a j e c t e s o f t he c l d l s e m . L e c e t c o s t a t e f e e k c a be a e d i n t he c a s e o f r e a l r a c y i n t he t y l e a ds t o i e a c c e l e r a t i o m e t we t o r e a c h a c t i o n t n t he s e e s i s t he o wi We f i r s t l d a c i s i o n fr e e h we e n t he c e t i o n a nd t he go a l us i ng a c i s i r e e m o t i o n c r i b e d i n , t t he t r a j e c t wi t h a s i m t r a c c r ol l a w. At t he e nd o f t he m o n,t he r r e a c h e t l y t he a of t he i t i t I f t he r e a c c t i o n i s t oo f a r f r om t he a t r a j e c t t we e a s we ha ve or t he f e r We wi l l c r i be t r a j e c t t r a c c ol l a w a nd t gi s i s s a gl o i t e r a t i ve s c e Th e t r c t t r i n g c r ol l n t s e c t i we l wi t h s e m A. t i o ns a r e e a s i e r f or s e s e e S e c t i F i : T r a c c ol l a w f or a s i r l o t o f t r a c c ol l a ws n d f or l e d m e r t t r a i l e r . of t a l t y s i m p l e , gi ve e xc e l l e nt r e s s .I f( ), ,x y a r e t he c e s of t he r e f e r e r i n t he f r a m e of t he r e a l r F i ), a nd i f 0 0,r a r e t he i of t he r e f e r e t r a j e c t t c ol l a w he o wi ng e s s i o n:0103 2c i nr rr rv k xk k = + = + + （ 2 ）i of c r ol l a w i s t he f o wi t he r r d,t he t r a i l e r ne e d be s t a i z e d ( s e e . S o we a 2) t o t he r i r d, we i ne a vi r t r , , )r r rx y ( F i ) h i ss e t r i c a l t o t he r e a l t h r e s t t o t he l a of t he t r a i l e r :52 c s i r t tr r t tr t rx x ly y = = = + t he r e a l r r d, t he vi r t r r d a nd t t s ys t e m ( , , , )r r rx y i s a t i c a l l y e e nt t o t he r e a l t he t r a c c ol l a w ( 2) t o t he vi r t r i : Vi r t r t i o n a r i s e s i s t he t r a i l e r r e a l l y a l wa ys s t a t he r r d ? o wi ng s e c t i o n wi l l a r t t i o t l i t y s of t he t r e c i he r e t he c a s e of a r d m o n ( 0) , t he r d m o ng e e nt by t he vi r t r t r a a t i o n. Le t us 0 0 0( , , , , )r r r r rx y a r e f e r e t r a j e c t a nd , , , , , )r r r r t he r e a l m o n o ft he s e m . We a s s t t he r f o ws e t l y i t s r e f e r e t r a j e c t 0 0 0 0( , , , , , ) ( , , , , )r r r r rv x y r r r r = a nd we f o ur a t t e on on t he t r a i l e t i = . e o n of t t i on i s e a s i l y d f r om s e m ( 1)wi t h 0 ( S e m A) :00 ( s i n s i n )2 c ) s i n( )2 2 = += i s t r e a s i ng i f 2 2 2 + 2 （ 3 ）s e m i s m c r a i by t he i i t i e 2 （ 4 ）6s o t 2 + a 3) i s e e nt t 0 00 220 22 5 ）F i s t he i n on h i s r e a s i ng f or a gi va l ue c a n s e e t t i n c i ns a l l t i o f t he t r a i l e r i b y t he 4). t he c t i m i t e a s i l y t o s t i s a na s c a l l y s t a ue f or t he i a i f t he r e a l or vi r t r o ws i t s r e f e r e f r d t r a j e c t t r a i l e r i s s t a a nd wi l l c ge t r d i t s r e f e r e t r a j e c t g u r e 4: St a bi l i t y d om a i n f o r i r t u al r f or s e m t he t r a i l e r i s c t he r t he f e r c r i o n i s e s i m we c a n r e c e t he vi r t r b y t he t r a i l e r . I n t c a s e i d, t t i e s of t he r , )r a nd of t he t r a i l e r ( , )t a r e c t e d b y t m a c i t i on of t he t r i s t b y t o wi ng s e m :c os c )s i n s i n( )r r r r r rr r r r t rx x l ly y l lr = += += + +a nd t he s t a i t y a na l s c a n be a e d a s we l l , b y c i ng t he m o t he hi t c f o wi ng s e c t i o n a s s e s t he r s of t e r a t i ve s c e s t ne s s of t he i t e r ve s c he a r e t o s t he r s of t he i t e r a t i ve s c he m e we ha c r i b e d a F or t hi s ,we ne e d t o ha ve a m of t he t i o ns a r i s i ng he r m t he t i o ns b y a e of c a of t e m , l e a t o e r m i t i c va r i a t i o n t he ve c t or fi e l I n o ur e i m e nt r r a t i o ns f or i a t o s y i n t he c s e m e t i o ns a r e y f i c t o m F or t r e a s on,we m a ke t i m p l e e s a t 00( ( ) , ( ) )( ( ) , ( ) )c s sc sd q s q sd q s q e s i s t he c i r a i s s a a l t he h, q a q a r er e s t i y t he r e a l a nd r e f e r e c t i o c sd i s a t a t t i o n s e of t he s e m a , a r e t i ve c a fi r s ti i t y m e a ns t t he t a we e n t he r e a l a nd t he r e f e r e c t i o ns i i t o t he t a c e d on t he h. s e c i i t y i se d by t he t r a j e c t t r a c c r ol l a w t t he s ys t e m t o go t oo f a ra wa y f r om i t s r e f e r e t r a j e c t Le t us t t e e s a r e yr e a l i s t i c a nd f i t a l o t of t i o n m s d t o he l e of t he hs a t e d a t e a c h i t e r a t i o n. t e e r i ng m e t we us e t o c t e hs ve r i f i e s a t a l t ya c c ng s m a l l - t i m e c a i t y. m e a ns t i f t he i s s i c i e yc l t o t he s t a r t i ng c i t i o n, t he c d t r a j e c t r e m a i ns i n a ne i t he s t a r t i ng c i t i I n we gi ve a n e s t i m a t e i n t e r m s of t a i f 1q a qa r e t wo s uf f i c i e y c l c t i o t he l e of t he h we e n t i f i e 1 2( ( , ) ) ( , )c q q d q qe i s a t i ve c t a i f i s t he s e o f c i t i o ns r e a c a f t e r i m o we ha ve t o wi ng i i t i e s :11,( , )( ) ( , )c s g o a lc s i g o a l c s i g o a ld q qd q q d q e i i t i e s e t t , )i g o a lq q i s by a s e 2 . . .( )i of t i ve e r s i =8a nd c t r d 4 3 a f t e r e t e r a t i o we do i n a s c a l s t a i t y of t he go a l c i t i t e s e s t he e t e of a s t a i n a r c i t i r e s s s e a l l y c s f r om t he ve r y a l m of t i o ns we ha ve c n. Le r e t t i s a t i on m i n a t i m e va r c r ol l a m a ke i t l i t s a s c a l s t a i t y. e i m e l r e s s he f o wi ng s e c t i o n s t t he c i n o f c r c e i s ve r y s m a l l r i me n t al r e s ul t s e nt e i m e l r e s s i wi t h o ur r i l a r e t ng at r a i l e r , s ys t e m s A a . F i s 5 a s e xa m p l e s o f f i r s t d by t he m on we e n t he i i a lF i : S e m A: t he i i a l a nd c t i nd t he f i r s t h t o be t r a c i : S e m B: t he i i a l a nd c t i t he f i r s t h t o be t r a c a nd t he f i m a t i o i n c k) a nd t he c i t i i n y), i t he l a s tc d m a i n t he s e c