【正文】 function: w a s mi n i mi s e d , w h e r e ( u , v, w ) a r e t h e c o o r d i n a t e s o f t h e c e n t r e o f t h e b a l l t o h e d e t e r mi n e d , ( x / , y ~ , z ~ ) a r e t h e c o o r d i na te s o f t he i t h p o i n t o n t he s ur fa c e o f t he b a l l a nd r i s t h e b a l l d i a me t e r . I n t h e t e s t s p e r f o r me d , i t w a s f o u n d s u ffi c i e n t to me a s ur e o nl y f o ur p o i n ts ( i = 4 ) o n t he s ur fa c e to determine the ball centre. 。 a r e b o t h a l o n g t he a xi s zo a nd a l s o c a nno t b e i d e n ti fi e d i nd e p e nd e n tl y. I t r e q ui r e s , t he r e fo r e , o nl y e i g h t i n d e p e n d e n t ki n e ma t i c p a r a me t e r s to g o fr o m t h e w o r l d fr a me to t he fi r s t fr a me o f t he P U M A us i n g t hi s p a t h. P a th 2 : A s a n a l te r na ti v e , a tr a ns fo r m ma y b e d e fi ne d d i r e c tl y fr o m the w or ld fra me to the ba se fra me X b , Yb , Zb . S i nce thi s i s a fr a me to fr a me tr a ns fo r m i t r e q ui r e s s i x p a r a me te r s , s uc h as the Euler form: T h e f o l l o w i n g D H t r a n s f o r m f r o m x b , Y b , z b t O X l , Y l , z l w o ul d i nv o l v e fo ur p a r a me te r s , b ut A 0 ~ ma y b e r e s o l v e d i n to 4 ~ , , 0 b , ~ , a nd A d ~ r e s ol v e d i nto P xb , P yb , P zb , r e d uc i n g t he p a r a me te r c o u nt to tw o . It i s s e e n t ha t t hi s p a th a l s o r e q ui r e s e i g h t p a r a m e t e r s a s i n p a t h i , b u t a d i f f e r e n t s e t . E i t h e r o f t h e a b o v e m e t h o d s m a y b e u s e d t o m o v e f r o m t h e w o r l d f r a me t o t h e s e c o n d fr a me o f t h e P U M A . I n t h i s w o r k , t h e s e c o n d p a t h i s c h o s e n . T h e t o o l t r a n s f o r m i s a n E u l e r t r a n s f o r m w h i c h r e q u i r e s t h e s p e c i f i c a t i o n o f s i x parameters: T he t o ta l n u mb e r o f p a r a me te r s us e d i n t h e k i ne ma ti c mo d e l b e c o me s 3 0 , a nd t he i r no mi na l v a l ue s a r e d e fi ne d i n Ta b l e 1 . Identification Methodology T he ki ne ma ti c p a ra me te r i de nti fi c a ti o n w ill b e p er fo r me d a s a mul ti d i me ns i o na l mi ni mi s a ti o n p r o c e s s , s i nc e thi s a v o i d s the c a l c ul a ti o n o f t he s ys te m J a c o b i a n. T he p r o c e s s i s as fo l l o w s : 1 . B e g i n w i t h a g u e s s s e t o f ki n e ma t i c p a r a me t e r s , s u c h a s the nominal set. 2. Select an arbitrary set of joint angles for the PUMA. 3. Calculate the pose of the PUMA endeffector. 4 . M e a s u r e t h e a c t u a l p o s e o f t h e P U M A e n d e f f e c t o r f o r t h e s a me s e t o f j o i n t a n g l e s . I n g e n e r a l , t h e me a s u r e d a n d predicted pose will be different. 5 . M o d i f y t h e ki n e ma ti c p a r a me t e r s i n a n o r d e r l y ma n n e r i n o r d e r t o b e s t f i t ( i n a l e a s t s q u a r e s s e n s e ) t h e m e a s u r e d pose to the predicted pose. T he pro cess is a pplied not to a si n gle se t o f joi nt a ngles b ut to a n u mb e r o f j o i n t a n gl e s . T he to ta l n u mb e r o f j o i n t a n gl e s e t s r e q u i r e d , w h i c h a l s o e q u a l s t h e n u m b e r o f p h y s i c a l measurement made, must satisfy K p i s t h e n u mb e r o f k i n e ma t i c p a r a me t e r s t o b e i d e n t i f i e d N i s t h e n u m b e r o f m e a s u r e m e n t s ( p o s e s ) t a k e n D r r e p r e s e n t s t h e n u mb e r o f d e g r e e s o f f r e e d o m p r e s e n t i n each measurement. In the s ys te m d esc rib ed i n thi s pa per, the numb e r o f d e gr ee s of freedom is given by since full pose is meas ure d. In prac tice, ma ny more meas ure me nts s h o u l d b e t a k e n t o o f f s e t t h e e f f e c t o f n o i s e i n t h e e xp e r i me n t a l me a s u r e me n t s . T h e o p ti mi s a ti o n p r o c e d ur e u s e d i s kno w n a s ZX S S O , a nd i s a s ta nd a r d l i b r ar y f u nc ti o n i n t he IMSL package . Pose Measurement It i s ap p are nt fr o m the a b ov e tha t a me a ns to de ter mi ne the f u l l p o s e o f t h e P U M A i s r e q u i r e d i n o r d e r t o p e r f o r m t h e c a l i b r a ti o n. T hi s me t ho d w i l l no w b e d e s c r i b e d i n d e ta i l . T he e nd e ffe c to r c o nsi s ts o f a n a rr a nge me nt o f fi v e p re ci si o nto o li ng b a l l s a s s h o w n i