P oc r e e d i ng s of t h e Fi st r As i a I n t e na r t i o na l S y mp os i u m o n Mc c ha t r o mc s ( AI SM 2 0 0 4 ) S e pt e mb e r 2 7 -3 0, 20 0 4, Xi ' a n, Ch i na Equi l i br i um Conf i g ur a t i o ns o f a Pl a na r 3- DOE Pa r a l l e l Ro bo t wi t h Fl exural Pi vot s T ua n - J i e Li Xi a o - Yo n g Zho u S c h o o l o f El e c t r o me c h a n i c a l En g i n e e r i n g , Xi d i a n Un i v e r s i t y , Xi ' a n 71 0 0 71 , Ch i n a e - ma i l : l i 9 6 8 6 _c n @s i na. c o m r o bo t s r e l a t e s t o a na 妙z i ng a nd o p t i mi z i n g t he ki n e ma t i c h a v e be e n s t u d i e d. I n t he l a t e 8 0' s , a n e w f i e l d o f bo t h r e s e a r c h a nd a p p l i c a t i o n s ha s be e n o p e n e d by g e o me t y r wi t ho u t r e g a r d t o t he e q ui l i b ium r c o nf i gu at r i o ns Cl a v e l ' t wh o d e v e l o p e d t h e f a mo u s De l t a of t he p a r ll a e l obo r t wi t h f l e x ur a l p i v o t s o n c e t h e e x t e r n a l r o o t s o f t he s ys t e m o f t r an s c e nd e n t a l e q u a t i on s wi t h ma n i p ul a t o r wh i c h ha s t h e a bi l i t y t o mo ve i n h i g h s pe ed. Re c e n t l y , t h e ma c h i n e - t o o l i nd u s t y r d i s c o v e r e d t h e p o t e n t i a l a d v a n t a g e s o f p a r a l l e l r o bo t s a nd mo s t ma j o r ma c h i n e - t o o l c omp a n i e s a r e i n t h e p r o es c s o f e x t e n s i v e t e s t s a n d e v a l ua t i o n o f t h e i r p a r a l l e l ma c h i n e s f o r pa r a l l e l k i n e ma t i c ma c h i n e s t " t . I n t h e ie f l d o f r o b o t i c s , t h e t a s k r e q ui r e d o f t h e h i g h s p e e d a n d a c c u r a c y c a n b e p e r f o r me d b y v i tu r e o f p a r a l l e l r o b o t s e mp l o y i n g p a r a l l e l me c h a n i s m. A t r i g o n o me t ic r f u nc t i o ns wi t h o u t i ni t i a l va l u e s e l e c t i o n i n p a r a l l e l r o b o t t y p i c a l l y c o n s i s t s o f mo v i n g p l a t f o r m t he s pec i f i ed i nt er val s t o de t e c t t he e nt i r e s e t of t he i ni t i al t ha t i s c o n n e c t e d t o a f i x e d ba s e b y s e v e r a l l i mbs o r l e g s . T y p i c a l l y , t h e l i mb t h a t c o mp o s e s t h e pa r a l l e l me c ha n i s m s u c h a s t y pe o f t he De l t a a n d St e wa t r ma n i p ul a t o r h a s t h e s e r i a l s t uc r t u r e . Ev e n i f a l o t o f pa r a l l e l me c ha ni s ms ha ve b e e n p r o po s e 氏a l mo s t t h e 6 - do f pa r a l l e l me c ha n i s ms h a v e b e e n s t u d i e d . Ho we ve r , i n s e v e al r a p pl i c a t i o n s , a 6 - do f me c h a n i s m i s n o t r e q u i r e d a n d r o bo t s whi c h ha v e l e s s t h a n 6 d o f Abs t r a c t : Al l p r e v i o us r e s e a r c h o n de s i g n of pa r a le l l o a d i s a s s i g ne d. Thi s a r t i c l e a d d r e s s e s t he e q ui l i b r i u m c o nf i gu r a t i o n s of a pl a n a r 3 - d of pa r al l e l ob r ot wi t h f l e x ur a l p i vo t s . A s ys t e m o f f o u r c o u p l e d t r a ns c e n de n t a l e q ua t i o ns wi t h t r i g o n o me t r i c f u nc t i o ns i n f o u r u n kn o wns i s de r i ve d f om r t h e i n v e r s e s t a t i c a n a l ys i s . Th e i n i t i al c o nf i g u at r i o ns of t h e p a r a le l ob r o t a e r d i s c u s s e d f i r s t l y , t he n a n ume r i c l a me t h od i s p r e s e n t e d b a s e d o n t he Ne wt on ' s i t e r a t i ve me t ho d, whi c h c a n f nd i a l t he ea r l c o nf i gu r a t i o ns a n d t he e q u i ib l ium r of t he p a r a le l ob r ot wi t h f l e x u r e hi n ge s .Fi na l l y , a nu me r i c l a e x a mpl e i s p ov r i d e d. Ke ywor ds : Pl a n ar Pa r al l el Robot , P a r a l l e lMechani s m, Pr i v e t , I n ve r s e S t at i c Ana lys i s ,E qui ibr l ium Conf i gur a t i on, Numer i ca l Sol ut i o n c a n b e u s e d [ " 1 I nt r oduct i on F l e x u r e h i n g e 1 6 1 i s a s i mp l e , d e p e n d a b l e a n d Co mpa r e d wi t h s e r i a l r o b o t s , pa r a l l e l r o b o t s h a v e i nge ni o us mec ha ni c a l s t uc r t ur e whi c h e mul a t es t he po t e n t i a l l y h i gh e r p r e c i s i o n , g r e a t e r s t r u c t ur a l f u nc t i on o f a o r d i na y r h i n g e b y c on c e n t r a t i n g le f x i on t o j u s t a f e w r e g i o n s . Be c a u s e o f i t s o u t s t a n d i n g pe r f o r ma nc e s : n o g a p, n o me c ha ni c a l f r i c t i on , no ig r i d i t y , h i g h e r s p e e d a n d a c el c e r a t i o n , a n d l a r g e r c a pa c i t y . Th e r e f o r e , t h e y ha v e r e c e i ve d i n c r e a s e d i nt er es t f r om bot h r es ear cher s a nd i ndus t ies. r Fr om t h e f i r s t p a r a l l e l r o b o t p r o p o s e d b y Go u g h l t t a n d S t e wa tt r ' l , a l o t o f p a r a l l e l r o b o t s o r d e s i g n me t h o d s c r a wl and no n ee d of l ubr i ca t i on whi l e i n mo t i o n, h i gh r e s o l ut i o n c a n be o b t a i n e d. 一 722- C o mpl i a n t r o bo t s a r e an o v e l c l a s s o f r o bo t s . Th e us e o f le f x i bl e me mbe r s t o g a i n mo bi l i t y ha s br o u g h t n o t a b l e a d va n c e s i n t h e r o bo t d e s i gn Co mp l e x mo t i o n a n d i n p u t / o u t p u t c h a r a c t e is r t i c s ma y be o b t a i n e d f r o m c o mp l i a n t r o bo t s wi t h f e we r pa r t s t h a n t h e i r r i g i d - b o d y c o u n t e r p a ts r . Al l pr e vi o us r e s e a r c h o n d e s i g n o f pa r a l l e l r o b o t s r e l a t e s t o a n ly a z i n g a n d o p t i mi z i n g t h e k i n e ma t i c ge o me t y r wi t h o u t r e g a r d t o t h e e q ui l i br iu m c o n f i gu r a t i o ns o f t h e pa r a l l e l r o bo t wi t h le f xu r l a p i vo t s o n c e t h e e x t e na r l l o a d i s a s s i gn e d. Th e d e t er mi na t i o n o f e n t i e r e q ui l i b ium r c o n f i g ur a t i o n s i s a mo e r d i ic f ul t t h e or e t i c lp a r o bl e m b e c a u s e i t de a l s wi t h in f di n g ll a t h e ea r lr o o t s o f t h e s ys t e m o f t r a n s c e nd e n t le a q ua t i o n s d e iv r e d f r o m t h e i n v e r s e s t a t i c a n a l ys i s o f c o mpl i a n t r o bo t s . ext er na l l oad F B , 1 - \, 、 一 、 乍 G l n p , } } F ' X} n I . , 0} Fi gur e 1. The me cha ni s m ar c hi t e ct ur e 2 Pl anar Paral l el Robot s wi t h Fl exural Pi vot s 2. 1 Mec hani s m archi t e ct ur e Th e me c h a n i s m a r c h i t e c t u r e o f t h e p a r a l l e l r o b o t a n ly a z e d i n t h e pr es e n t wor k i s i l l us t r a t e d i n Fi g ur e 1 . Th e mo vi n g p l a t f o r m, wh i c h h a s a r e gu l a r t ia r n gu l a r s ha pe , c o r es r p on d s t o t h e e n d - e fe c t o r . Th i s pl a n a r r o b o t i s c a t e g or iz e d a s a RPR t y pe , b e c a u s e o n e l i mb c o n s i s t s o f t h e p is r ma t i c j o i n t a n d t wo n o nc o n s e c u t i v e r e vo l u t ej o i n t s . A, B a n d C a r e le f xi b l e j o i n t s , wh i c h a r e mo d e l e d a s p i n j o i n t s r e s t r a i n e d b y t o r s i o n l a s pr in g s . Th e c o ns i d e ed r me c h a ni s m i s a 3 - d o f pa r a l l e l r o b o t a c c o r d i n g t o F怕ur e 2 . The c o nf ig u r a t i o n whe n t he l i mb l e ng t h i s n e g a t i v e B t , 0 2,B 3 - Th e a n g u l a r p o s i t i o n s o f t h e l i mb s , a n d a r e po s i t i v e i n c o u nt e r - c l oc k wi s e di r ec ti on B t u,0 2 0,0 3 0— T h e i n i t i a l a n g u l a r po s i t i o n s o f t h e l i mbs wh e n t h e s p r i n gs a r e n o t Li l ' I . Th e e x t e na r ll o a d i s a p p l i e d o n t h e en c t r o i d o f l oaded t h e mo vi n g pl a t f o r m. Th e me c ha n i s m ma y r e ma i n i n e q ui l i b ium r a s s o o n a s t h e t o r q u e s e x e te r d by t he l i mbs u po n t h e s pr in gs a r e e q ua lt o t h e mo me n t s o f f o r c e s t h a t t h e mo vi n g p l a t f o r m t r a n s mi t s t o t he k , , k 2, k 3 - Th e s p r i n g c o n s t a n t s l a yi n g i n p i vo t s A,B a n d C r e s p e ti c v el y d2一 一一Th e s i d e l e n g t h o f t h e mo v i n g l i mbs. Fo r t h e s a k e o f c o n ve n i e nc e, t h e f o l l o wi n g va ia r bl e s a r e e x p l i c i t l y d e f i n e d: F - Th e e x t e na r l l o a d , a p p l i e d o n t he c e n t r o i d o f t h e mo v i n g p l a t f o r m. Fi , F 2,凡— Th e c o mp o n e n t s t h a t t h e mo v i n g pl a t f o r m t r a n s mi t s t o t h e l i mbs Fx,凡— x a n d y c o mp o n e n t s o f t h e pl a t f or m ( d, , 0 ) - Th e p o s i t i o n o f p i v o t B ( a, b - ) Th e p o s i t i o n o f p i v o t C 1 ,,1 2,1 3 - Th e l i mb l e n g t h s , i . e . 1 ,=' A D,1 2 =1 B E, 1 3 =1 C C. Re f e r r i n g t o F i g u r e l , t h e l i mb l e ng t hs a r e r e ga r d e d a s pos i t i v e , a nd a s n e ga t i v e s h o wn i n Fi gu r e 2. 2 . 2 I nv e r s e s t a t i c a na l ys i s Th e me c h a n i s m be i n g a n a l y z e d h e r e h a s t wo i nd e pe nde n t l oops . The f our s c al a r l o op- cl os ur e 一 723一 1 , C O S 0 1 +d2 C O S 么一1 2 c o s 凡一d,=0 1 1 s i n瑞+d2 s i n氏一1 2 s i n仇=0 1 , s i n B , +d2 s i n ( B 4 +; r / 3 ) 一1 3 s i n 0 3一b=0 (l) (2) equa t i ons of t h e me c ha ni s m a r e a s f ol l ows 。 + k , ( B 1 1 , - 0 1 o ) s i n B ・ k 2 ( B 2 - 8 2 已s i n 1 , C O S 0 1 +d 2 C o s ( B 4+刁3 ) 一1 3 C O S 0 3一a=0 ( 3) 凡 F2= F3= k3 ( B:一03 0)C O S 热=0( 1 4 ) 1 2 1 3 k 1 ( B 1 - 0 1 0 ) s i n ( 2 a i 3 一 8 , + 8 4 ’ 十 (5) k 2 ( B 2一0 2 0 ) s i n ( B 2一04一刁3 ) + 1 , (6) 1 , k 3 ( B 3一0 3 0 ) C O S 0 1一 些鱼竺业C o q 1 , k 2 ( B 2一0 2 0 ) 1 , k , ( B ,一0 1 0 ) 1 1 T h e e q ui l i b iu r m o f t h e mo me nt s a bo u t t h e g r o un d e d pi vo t o f e a c h l i mb y i e l d s : F,= s i n仇 =0 l 2 1 , s i n B , +d2 s i n ( B 4 +) r / 3 ) 一1 3 s i n 0 3一b二0 ( 4) k 1 ( B 1一0 1 0 ) k 3 ( B 3一0 3 0 ) k , ( B 3一01 0 ) s i n ( 氏一么)=0 1 , (7) 1 3 wh e r e 1 , ,1 2 a n d 1 3 d e s c r i b e d b y Eq u a t i o n s ( 1 1 ) , Th e e q u i l i br i um e q ua t i o n s o f t he mo v i n g pl a t f o r m a r e: ( 1 2) a nd ( 1 3 ) Th e Eq ua t i on s ( 1 4 ) d e s c r i b e t h e s t a t i c s o f t he 凡 =F, c o s B ,+凡c o s 0 2 +F3 c o s t 9 3 ( 9) f o u r u n k n o wn s〔巩,仇,仇,氏 ) W hen the n s t t u c t i v e p a r a me t e r s ( 0 1 0, 0 2 0,0 3 0,k , ,k 2, ( 1 0 ) 棍 F , s i n ( 2 , r / 3 一 0 1 + 0 4 ) +F 2 s i n ( 0 2 一 co Fx =-F, s i n B ,一F, s i n 02一F3 s i n 0 3 ( 8) me c h a ni s m a n d r e p r e s e n t a c o u pl e d s y s t e m o f f o u r n o n - l i ne a r eq ua t i o n s wi t h t r i go n o me t r i c f un c t i o ns i n 氏一刁3 ) 十凡s i n ( 氏一热)=。 S o l vi n g Eq ua t i o ns ( 1 ) a nd ( 2) s i mu l t a n e o us l y, we have d, ,d2, a b ) a n d t h e e x t e na r l l o a d ( Fx , F) a r e gi v e n, t h e s o l u t i o n s o f Eq ua t i o n s ( 1 4) c a n c o mp l e t e l y d e t e r mi n e t h e e q u i l i b r i u m c o n f i g u r a t i o n s o f t he me c ha ni s m. 一 - - - 2 了之 d, s i n B ,一d2 s i n ( B ,一0 4 ) s i n ( 热一伏) (11) 3 I ni t i a l Co nf i gu r a t i ons (12) Subs t i t ut i ng Equa t i on ( 1 1 ) i nt o Eq ua t i on ( 3) , we ge t d, s i n 0 2一d2 s i n ( B 2一04 ). c o s 鱼+ s i n ( B 2一0 1 ) c o s 热 ( 1 3) d2 c o s ( 氏+刁3 ) 一a l ,= Wh e n F二0,巩=马。,热=仇。a n d仇= t h e me c ha n i s m l i e s i n i t s unl o a d c o n f i gu r a t i on t hi s c a s e 氏。 ma y be d e t e r mi n e d from 际Forthe l 子. d , s i n 0 2一d 2 s i n ( B 2一0 4 ) s m( 热一风) f ol l o wi n g e q ua t i o n d e r i v e d f r o m t h e l o o pequat ions : C, s i n断 +C c o s 翰 +C =0 (15) wher e c os氏 S u bs t i t u t i n g Eq u a t i o ns ( 5 ) , ( 6 ) , ( 7) i n t o Eq u a t i o n s ( 8 ) , ( 9 ) a n d ( 1 0 ) , t og e t h e r wi t h Eq ua t i o n ( 4 ) c o n s t i t u t e t h e f o l l o wi ng f o u r c o u pl e d e q u a t i o n s : 一 724- C , 一 d 2 暗 ・ 零t a n 0 3 0 ・ c o s O 2 0 ( s i n B j o - c o s B i o t a n 0 3] sui t 口2 0一伪 of 石- 2 C2= ‘ 合 t a n 0 3 , - F( x )=( t i ( x ) , 几( x ) , 一了 =0 s i n 0 2 0 ( s i n 0 , 0 - c o s B t 0 t a n 0 3 , ) 1 s i n ( 姚。一佑。) i n 0 2 0 ( s i n B , o - c o s B , 0 W0 3 0 ) + C3 =d,s s i n ( 0 2 。一0 1 0) a t a n已。一b ( 1 6) wh e r e x=( 00 2 , 二了 .T h e s o l v i n g p r o c e s s p r e s e n t e d i s o u t l i n e d a s f ol l o ws : ( 1 ) S p e c 吻 t h e n u me r i c a l p r e c i s i o n ‘,t h e a l l o wa bl e i t e r a t i o n c yc l e s N; Th e n t he l i mb l e n g t h s c a n b e o bt a i n e d t h r o u gh ( 2 ) Di v i d e t h e v a r i a b l e s x i n t o s ma l l s e g me n t s i n s u b s t i t u t i n g B ,二0 1 0, 02 = 0 2 0, 0 3 = 0 3 0 t h e s p e c i f i e d i n t e r v a l s , we g e t di s c r e t e poi n t s o f x 0,=04 , i n t o Eq ua t i o ns ( 1 1 ) , ( 1 2) a nd ( 1 3 ) . a s t h e i n i t i a l v a l u e s o f i t e r a t i o n x 1 0 1 ; ( 3 ) E v a l u a t e F( xt k i ), ( k=0 , 1 , 2 , 二. ).I f 4 So l ut i on of Tr a ns c e nd e nt a l Equa t i ons Eq u a t i o n s ( 1 4 ) a n d ( 1 5 ) a r e t h e t r a ns c e nd e n t a l e q u a t i o n s wi t h t r i g on o me t r i c f u nc t i o n s . I n g e n e r a l , i t er a t i ve met hods ar e us ed t o r es ol ve t r ans cende nt al eq ua t i ons . Ho we ve r , t he i t er a t i ve me t hod n e ed s t he i ni t i al val ues and t he s el ect i on of t hos e i s r a t her di f f i c ul t . Th e s ol ut i on of t he t r a ns c e nde n t a l e q u a t i o n s wi t h t r i go n o me t r i c f u n c t i o ns i s r a t h e r t r o u bl e s ome a l t h ou g h t h e d e gr e e o f e q ua t i o ns i s l o we r . S ome p r o c e d u r e s o f e q u a t i o n ma n i pu l a t i o n, s uc h as t he i de nt i c al t r a ns f o r m a nd el i mi na t i o ns , ma y b r i n g a b o u t u n d e s i r a bl e c o n s e q u e n c e s , s i n c e i t ma y l e a d t o f i n a l e q ua t i on s t h a t d o no t r e c o g ni z e s o me c o n f i g u r a t i o ns a s s o l u t i o n s o r r e ga r d s ome o t h e r s a s s ol u t i o n s wh i l e t h e y a r e n o t . So t h e pr o c e s s o f s ol u t i o n t e s t i s n e c e s s a y, r a n d t h e s ol ut i o n pr oc edur e i s r a t h er t i r es ome. I n t hi s ar ti cl e a mor e conveni ent met hod i s pr e s e n t ed bas ed on t he Ne wt on' s i t e at r i ve me t hod, whi c h c a n wor k o u t a l l t h e r e a l r o o t s of t h e s ys t e m o f t r a ns c e n d e n t a l e q u a t i o n s wi t h t r i g o no me t ic r f unct i ons wi t hout i ni t i al val ue s el ect i on i n t he s p e c i f i e d i n t e va r l s t o d e t e c t t h e e n t i r e s e t o f c o n f i g u at r i o n s o f t h e r o bo t , a n d a v o i d pr o d uc i n g I F ( x ( k ' ) N < 二 , d e t e c t t h e s a m e x ( k ) e x i s t s i n t h e s e t o f s o l u t i o n s o r n o t . I f n o t , t h e n x( " } i s a n e w s ol u t i o n a n d o u t p ut t e d t o t h e s e t o f s ol u t i o n s , t ur n t o s t e p ( 2 ) ; ( 4 ) I f t h e i t e r a t i o n c yc l e s n? N , t u r n t o s t e p ( 2 ) ; ( 5 ) I f F' ( x( k ) ) i s n o n s i n g u l a r , e v a l u a t e x ( k . t ) 二x ( k ) 一F( x ( k ) ) 一 , F( x ( k ) ).I f x( k " ) l i e s i n s i d e t h e i n t e va r l s , t h e n x ( k ) =x ( k . u, t u r n t o s t e p ( 3) ; ( 6 ) Re p e a t s t e p s ( 1 ) t h r o u g h ( 5 ) u n t i l a l l t h e d i s c r e t e po i n t s o f x h a ve b e e n us e d. Th e r e s ul t a n t s e t o f s o l u t i o n s i s r e g a r d e d a s a l l t h e r e a l r o o t s o f t h e s ys t e m o f t r a ns c e n d e n t a l e q u a t i o n s wi t h t r i go no me t r i c f u n c t i o n s i n t h e s p e c i f i e d i nt e va r l s . Th e a b o v e me t h o d i s s u p e r l i n e a r c o nv e r ge nc e , a n d c a n e a s i l y b e i mpl e me n t e d wi t h c o mpu t e r s . Thi s numer i cal met hod c an wor k out al l t he r eal r oot s of t h e s y s t e m o f t r a n s c e n d e n t a l e q u a t i o ns wi t h t r i go n ome t r i c f u nc t i o n s wi t ho u t i n i t i a l va l u e s e l e c t i o n i n t h e s p e c i f i e d i n t e va r l s . e xt r a neous or l os s s ol ut i ons . Si n c e t h e s o l u t i o ns o f t h e p r o bl e m d e s c r i be d b y n ec es s a y r t o e s t a bl i s h t he i n t e va r l s i ns i de whi c h t he s ol ut i on s ear c h mus t be under taken. Consi der t he p r o bl e m o f s e a r c h i n g a l l z e r o s o f t h e t r a n s c e n d e nt a l e q u a t i o n s wi t h t r i go n o me t r i c f u nc t i o n s a s f ol l o ws As a n e xampl e, a s s ume f or t h e pa r a met er s t h e val ue s = 0. 5 a=0 . 5, n=1,丸二1 d, =1 ,棍 =l , 内 氏 Equa t i ons ( 1 4) a nd (1 5) a r e a co un t a bl e i nf ni i t e, i t i s 5 Numer i cal Exampl e k3=1 =二 a 27 r 7汇 , ana for a a g o一丁 ,U 3 0= — t h e e x t e na r l f o r c e s F =1 ,凡 二 一 725- ( r a d ) 1 1 ] 巩 (rad) 热 (rad) 马 (rad) By } I . 0 4 5 7 8 4 6} 12 . 2 3 1 9 9 2 2} 12 . 3 1 4 0 6 7 6 2 ] 1.652975512 . 6 6 9 4 7 7 7} 13 . 4 0 5 6 0 5 1} 11 . 9 6 5 7 8 6 4 1_ ,。, , 。 , ;} . 4 0 9 1 6 9 0} 10 . 9 9 4 2 7 6 5}1 . 9 7 8 4 0 4 0一 口 4.4112156 10 I 1 . 1 8 4 2 7 7 3} 1一 1 . 6 8 8 9 3 0 5} . 7 6 9 9 8 5 4} 10 . 7 1 6 0 3 8 9}1 . 4 9 3 1 4 4 5一 口 1.240044512 I- 0 . 1 1 5 5 3 5 6 1 . 0 7 1 8 9 6 1} I 一 , t 2 5 9 0 4 0 8 . 7 8 7 7 3 8 4} 11 . 6 5 9 2 3 4 7}1 口 4.2663029 14 . 6 3 1 4 0 0 0一 I- 1 . 9 8 7 0 2 9 8}- 2 . 2 9 8 1 8 7 3 1.0198843 . 9 9 9 6 0 9 2} 口 1.3824119 14 12 . 0 5 9 6 3 8 8}1 . 7 7 7 0 7 9 1一 I2 . 0 0 6 2 4 2 7}- 2 . 5 6 5 2 3 2 5}1 . 2 7 6 2 3 3 7 5 . 1 8 8 1 1 1 7} 口 2.3103089 I I1 . 8 5 0 8 0 8 6}1 . 5 2 1 5 3 6 1{ 1 1 - 2 . 4 4 8 2 6 2 5 1 . 4 7 2 9 2 6 7} 1 - 2 . 6 4 0 5 9 6 8 门 4.4735981 1 . 7 8 8 2 9 2 5} 11 . 8 3 7 4 6 1 2}2 . 0 6 7 4 7 1 7} 1- 2 . 5 2 8 7 7 5 5} 12 . 9 6 6 4 9 4 7} 11 . 5 2 4 3 5 9 5 国 5.4744796 5 . 0 0 2 4 9 3 4}2 . 1 6 7 5 8 6 1} 14 . 1 3 2 8 2 8 0 3 . 0 1 8 0 8 9 4} I - 4 . 4 3 2 9 1 6 5 . 2 6 0 4 2 3 012 同 2.7956667 1 . 6 1 1 3 8 1 8}, 0 3 6 8 5 5 2- 0 . 8 2 8 9 4 7 2}0 . 1 5 3 2 2 4 6} 1 - 0 . 8 4 6 1 9 0 8 . 3 4 4 9 8 2 811 . 0 0 0 9 1 9 40 . 9 4 5 8 2 1 4}0 国 2.1066861!6 . 5 7 3 0 5 3 9一 1一 。 . 1 4 4 6 2 0 1}- 0 . 5 2 6 8 8 1 9} 1 - 0 . 7 7 0 6 2 2 6 同 24161775!5 . 8 3 9 6 0 9 34 . 0 5 0 1 7 6 1}0 . 9 1 6 2 0 5 9一 }- 0 . 2 1 4 7 8 7 7}- 0 . 5 9 2 1 8 3 4 0.8629393 同 6.2491286 !1 . 1 2 2 5 5 2 6 1 . 9 9 2 6 0 8 9} 12 . 5 0 3 9 1 7 9一 I1 . 5 2 0 8 6 3 9}0 . 2 7 2 8 3 1 9} 1 一 , . 3 7 1 0 4 5 8 国 5.5210033' 6 . 0 1 6 2 8 1 7}1 . 1 3 1 4 2 8 4}0 . 9 0 2 7 3 7 4一 I0 . 4 1 3 5 6 8 2} } - 0 . 4 0 5 4 7 8 7} I - 0 . 9 0 7 2 4 7 2 国 15. . 2 1 5 2 9 8 1}1 . 3 8 0 3 9 2 1} {0 . 8 3 2 1 4 4 6I 0 . 6 1 2 0 2 2 6} 10 . 1 8 6 8 5 0 6} 1 - 0 . 9 2 3 7 5 4 1 6 0 6 0 9 7 713 .一‘, ,尹-11了J } . Ta bl e 2 . So l ut i on s o f t he e qui l i br i um c o nf i g ur a t i o ns 5. 1 I ni t i a l c o nf i gur a t i o ns o f t he r o bot We o b t a i n t wo r e a l r o o t s b y s o l v i n g Eq u a t i o n ( 1 5 ) i n t h e i n t e r v a l s [ 0 , 2 , r ] f o r 0 4 ,,t h e n t h e c o r r e s po n d i n g l i mb l e n乡h s c a n b e go te n f r o m Eq u a t i o n s ( 1 1 ) , ( 1 2 ) a nd ( 1 3 ) , a s l i s t e d i n Ta bl e 1 . Th e t wo i ni t i a l c o n f i gu r a t i o n s o f t h e r ob o t a r e i l l u s t r a t e d i n Fi g u r e 3. 0 , (rod) Sol ut i on 1 Sol ut i on 2 3. 6652 1 1 1 2 1. 41 42 0. 8660 c or r e s po n di n g l i mb l e n gt h s . Re s ul t s a r e l i s t e d i n Ta bl e 2. Th e p os i t i o ns a n d o r i e n t a t i o n s o f t h e mo vi ng pl a t f o r m i n t h e pl a n e ( x, y) a r e s h o wn i n Fi gu r e 4 by t hos e of t he s i de DE. D, l , s, G, 0. 7071 G, 岛 5. 9378 0. 5624 0. 3953 2637 0. 及 Ta bl e l . So l ut i o ns of t he i ni t i a l c o nf i gu r a t i o ns ' V一 一一, - A 0. 7 5. 2 Equi l i br i um c o nf i g ur a t i ons of t he r obot Th e s ol ut i o n s o f Eq ua t i o n s ( 1 4 ) c a n c o mpl e t e l y d e t e r mi n e t h e e q ui l i br i u m c o n f i g u ra t i o ns ofthe r obot under t he ext er nal l oa ds . Sol ut i ons gh t son r n e dBZ 1 0 , 川 f o rB, a ng t o [ 0 , 4 z / 3 ] f o r B 3 , a n d[ 0 , v c ] f o r 0 4Accordi f or i n t he i nt er val s t he n umer i ca l me t hod me nt i one d a bo ve , we ob t ai n a l l t h e 1 4 r e a l r o o t s o f Eq ua t i o n s ( 1 4) a nd t h e - 一 -一, 一一一入三一一 二 口A "j .一 Fi gur e 3. The t wo i ni t i a l c o nf i g ur at i o ns of t he r obo t 6 Concl us i ons Co mpl i a n t pa r a l l el r o bo t s a r e a n o v e l c l a s s o f r o b o t s . Th e u s e o f le f x i bl e me mb e r s t o ga i n mo bi l i t y h a s b r o u g ht no t a bl e a d va n c e s i n t h e r o b o t d e s i gn . Th e r ob o t s wi t h le f x ur al pi vo t s r e l y o n t he d e le f c t i o n o f 一 726- fe l xi bl e me mb e r s , e n e r g y i s s t o r e d i n t h e f o r m o f s t r a i n e n e r g y wi t hi n t h e c o mpl i a n t me mb e r s . Thi s s t o r e d e n e r gy i s s i mi l a r t o t h e s t r a i n e n e r g y i n a d e le f c t e d s p r i n g, a nd t h e e fe c t s o f s p r i n g s a r e i n t e g r a t e d i n t o ac omp l i a n t r o bo t ' s d e s i gn. I n t hi s ma n n e r , e n e r gy c a n e a s i l y b e s t o r e d o r t r a ns f o r me d, Ac kno wl e dg me n t s Thi s r e s e a r c h i s s po n s o r e d by t h e S ha a n xi Pr o vi n c i a l t o be r e l e as e d i n a di fe r e n t ma nne r . Thi s a ti r c l e ha s Re f e r ence s a d d r e s s e d t h e e q ui l i br i um c o n f i g u r a t i o ns o f t h e pl a n a r pa r a l l e l r o bo t wi t h le f x ur a l pi v o t s o n c e t h e e x t e na r l l o a d i s ss a i g n e d. Th e r e s ul t s i n di c a t e t h e e q ui l i b r i u m c o n igu f r a t i o n s o f c o mpl i a nt r o b o t s a r e l . Gou g h V. E,Co n ri t b ut i o n t o di s c us s i o n of pa pe r s o n mul t i - s t a bl e. Na t u ral Sc i e nc e Founda t i on of Chi na ( Gr a nt No 2 0 0 3 E3 4 ) a n d s u p p o te r d b y t h e Yo u t h S c i e n t i f i c Res e ar c h Wor ks t a t i on of Xi di a n Uni ve r s i t y. r e s e a r c h i n a u t o mo t i v e s t ab i l i t y , c o n rol t a n d t y r e p e r f or ma n c e , I n: Pr oc . Aut o Di v . I n s t Me c h a n i c al En gi ne e r s , 39 2 - 3 95 , 1 9 5 6 2. S t e wa r t D. , A p l a t f o r m wi t h 6 de g ee r s of f r e e do m, I n: Pr oc . o f t h e I n s t i t u t e o f Me c ha n i c a l En gi ne e r i n g, Lo n d o n , UK, 1 8 0 ( p a r t 1 , 1 5 ) : 371 - 38 6, 1 9 6 5 y 4 、亡、见 I n: Pr oc . o f 1 8 t h I n t . S ymp. o n I n du s t r i a l Ro b o t , La us a n ne , | 乌 D \乌 3. Cl a v e l R. , DEL T A, a f a s t r o bo t wi t h p a r a l l e l g e ome t r l 91 - 1 00, 1 988- 4. Ry u S. J . a n d Ki n J . W. e t a l , Ec l i p s e : a n o v e r a c t u a t e d p a al r l e l me c ha n i s m f or ap r i d ma c hi n i ng , AS ME I n t . 681 - 689,1 998 5 . Ka n g B. a nd Ch u J . e t a l , De s i g n of h i g h s p e e d p l a na r p a r a l l e l ma n i pu l a t o r a n d mul t i pl e s i mu l t a ne o u s ! 氏 马 、、巧 - 1 Me c h a ni c a l En gi n e e r i n g Con gr e s s an d Ex po s i t i o n, s pe c i ic f a t i o n c o n t ol r , I n: Pr oc . o f t h e 20 01 I EEE I n t . Con . - 2 o n Ro bo t i c s &Au t o mat i o n, Ko r e a , 2 7 23- 2 7 28 , 2 0 01 \ DB 6 . Hi n kl e y D. , A mu l t i f unc t i o na l le f xu r e b i n g e f o r d e pl o yi n g o mn i di r e c t i o n a l s o l a r a r ays r , I n: 4 2 n d AI AA 1 AS ME/ ASCEI AHS I ASC St r u c t u r e s , S t r u c t u r a l Dyn ami c s 0 1 2 3 4 a n d Ma t e r i al s Co n f e r e n c e a nd Ex hi b i t , Se a t t l e , WA, 1 书, 2001 F i gur e 4 . Th e e qui l i br i um c o nf i gur at i o ns of t he 7. LI Tu a n- pe , An a l ys i s o n t he s t r u c t ur a l t o pol o gi c a l mov i ng pl a t f o r m c h a r a c t e r i s t i c s a n d t he mo bi l i yo t f c o mpl i a n t me c h a n i s ms , Me c h a ni c a l Sc i e n c e a nd T e c h no l o g y, 22 ( 1 ) : 1 0 7- 1 0 9 , 2 00 3 一 727一
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