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UBC Theses and Dissertations

Learning algorithms for manipulator control 1984

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LEARNING ALGORITHMS FOR MANIPULATOR CONTROL By CHARLES KEVIN HUSCROFT B.A.Sc.(Hons), The U n i v e r s i t y of B r i t i s h C o l u m b i a , 1 9 7 9 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF October © C h a r l e s K e v i n BRITISH COLUMBIA 1984 H u s c r o f t , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department Of Electrical Engineering The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 17, 1984 A b s t r a c t A method of r o b o t m a n i p u l a t o r c o n t r o l i s proposed whereby a l g o r i t h m s are used t o l e a r n sum of p o l y n o m i a l s r e p r e s e n t a t i o n s of m a n i p u l a t o r dynamics and k i n e m a t i c s r e l a t i o n s h i p s . The l e a r n e d r e l a t i o n s h i p s a r e u t i l i z e d t o c o n t r o l t h e m a n i p u l a t o r u s i n g the t e c h n i q u e of R e s o l v e d A c c e l e r a t i o n C o n t r o l . Such l e a r n i n g i s a c h i e v e d w i t h o u t r e c o u r s e t o a n a l y s i s of the m a n i p u l a t o r ; hence the name S e l f - L e a r n e d C o n t r o l . Rates of convergence of s e v e r a l l e a r n i n g a l g o r i t h m s are compared when l e a r n i n g e s t i m a t e s of v a r i o u s n o n - l i n e a r , m u l t i - v a r i a t e f u n c t i o n s . I n t e r f e r e n c e M i n i m i z a t i o n i s found t o be s u p e r i o r t o the G r a d i e n t Method, L e a r n i n g I d e n t i f i c a t i o n and the K l e t t C e r e b e l l a r Model. S i m p l i f i c a t i o n of the i m p l e m e n t a t i o n of I n t e r f e r e n c e M i n i m i z a t i o n i s d e s c r i b e d . A v a r i a n t , P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n , i s i n t r o d u c e d t h a t i s s u i t a b l e f o r c e r t a i n a p p l i c a t i o n s . S e l f - L e a r n e d C o n t r o l w i t h p a t h s p e c i f i c a t i o n i n C a r t e s i a n c o o r d i n a t e s i s demonstrated f o r a s i m u l a t e d two l i n k m a n i p u l a t o r . I t i s shown t h a t sum of p o l y n o m i a l s r e p r e s e n t a t i o n s of the i n v e r s e dynamics, i n v e r s e k i n e m a t i c s and d i r e c t p o s i t i o n k i n e m a t i c s r e l a t i o n s h i p s a r e adequate t o a c h i e v e c o n t r o l comparable t o t h a t a c h i e v e d u s i n g t h e i r a n a l y t i c a l c o u n t e r p a r t s and can be l e a r n e d w i t h o u t a n a l y s i s of t h e m a n i p u l a t o r . F u r t h e r r e s e a r c h i s o u t l i n e d t o a c h i e v e a u t o m a t i c a d a p t a t i o n t o t o o l mass, i m p l e m e n t a t i o n of sum of p o l y n o m i a l s e s t i m a t o r s and enhancement of the K l e t t C e r e b e l l a r Model. T a b l e o f C o n t e n t s A b s t r a c t i i L i s t of F i g u r e s v i i L i s t of T a b l e s x i i Acknowledgements x i v 1. I n t r o d u c t i o n 1 2. I n t e r f e r e n c e M i n i m i z a t i o n and R e l a t e d L e a r n i n g A l g o r i t h m s 6 2.1 L e a r n e d F u n c t i o n a l E s t i m a t i o n u s i n g Sums of P o l y n o m i a l s 6 2.2 An Improved L e a r n i n g A l g o r i t h m - I n t e r f e r e n c e M i n i m i z a t i o n 7 2.3 R e l a t i o n s h i p t o S i m i l a r L e a r n i n g A l g o r i t h m s 9 2.4 C h o i c e of T e s t C o n d i t i o n s f o r C o m p a r i s o n o f A l g o r i t h m s 13 2.5 C o m p a r i s o n o f t h e L e a r n i n g A l g o r i t h m s 18 2.6 I m p l e m e n t a t i o n C o n s i d e r a t i o n s f o r I n t e r f e r e n c e M i n i m i z a t i o n 28 2.7 A p p l i c a t i o n s where P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n i s A p p r o p r i a t e 32 2.8 Summary 33 3. S e l f - L e a r n e d C o n t r o l o f a Two L i n k M a n i p u l a t o r .. 34 3.1 R e s o l v e d A c c e l e r a t i o n C o n t r o l w i t h C a r t e s i a n P a t h S p e c i f i c a t o n 34 3.2 A Two L i n k M a n i p u l a t o r 41 3.3 S i m u l a t i o n o f t h e M a n i p u l a t o r 47 3.4 P a t h S p e c i f i c a t i o n 53 i i i 3.5 M a n i p u l a t o r C o n t r o l u s i n g t h e A n a l y t i c a l I n v e r s e Dynamics and A n a l y t i c a l I n v e r s e K i n e m a t i c s 61 3.5.1 I d e a l Open L o o p C o n t r o l 61 3.5.2 I d e a l C l o s e d Loop C o n t r o l 65 3.5.3 R e a l i z a b l e C l o s e d L o o p C o n t r o l - S t a n d a r d A n a l y t i c a l C o n t r o l 67 3.5.4 C h o i c e of P a r a m e t e r s D e f i n i n g S t a n d a r d A n a l y t i c a l C o n t r o l 72 3.6 Adequacy of a Sum of P o l y n o m i a l s R e p r e s e n t a t i o n of t h e I n v e r s e Dynamics and I n v e r s e K i n e m a t i c s 87 3.6.1 D e r i v a t i o n o f a Sum o f P o l y n o m i a l s R e p r e s e n t a t i o n o f t h e I n v e r s e Dynamics 87 3.6.2 P r e - L e a r n i n g of t h e I n v e r s e Dynamics .. 89 3.6.3 A t t e m p t e d D e r i v a t i o n o f a Sum of P o l y n o m i a l s R e p r e s e n t a t i o n of t h e C a r t e s i a n I n v e r s e D y n a m i c s 93 3.6.4 L i m i t a t i o n o f a Sum o f P o l y n o m i a l s R e p r e s e n t a t i o n o f t h e C a r t e s i a n I n v e r s e Dynamics t o a P o r t i o n o f t h e M a n i p u l a t o r ' s S p a c e 95 3.6.5 P r e - L e a r n i n g of t h e C a r t e s i a n I n v e r s e Dynamics 98 3.6.6 Closed L o o p C o n t r o l u s i n g t h e P r e - L e a r n e d C a r t e s i a n I n v e r s e Dynamics .... 100 3.7 Adequacy of a Sum of P o l y n o m i a l s R e p r e s e n t a t i o n o f t h e D i r e c t P o s i t i o n K i n e m a t i c s 102 3.7.1 D e r i v a t i o n o f a Sum o f P o l y n o m i a l s R e p r e s e n t a t i o n o f t h e D i r e c t P o s i t i o n K i n e m a t i c s 103 3.7.2 P r e - L e a r n i n g o f t h e D i r e c t P o s i t i o n K i n e m a t i c s 107 3.7.3 C l o s e d L o o p C o n t r o l u s i n g t h e P r e - L e a r n e d C a r t e s i a n I n v e r s e Dynamics and P r e - L e a r n e d D i r e c t P o s i t i o n K i n e m a t i c s 112 i v 3.8 S e l f - L e a r n i n g o f t h e C a r t e s i a n I n v e r s e Dynamics 120 3.8.1 A Method f o r S e l f - L e a r n i n g of t h e C a r t e s i a n I n v e r s e Dynamics 120 3.8.2 C l o s e d L o o p C o n t r o l u s i n g t h e S e l f - L e a r n e d C a r t e s i a n I n v e r s e Dynamics .... 133 3.9 S e l f - L e a r n i n g o f t h e D i r e c t P o s i t i o n K i n e m a t i c s 139 3.9.1 A Method f o r S e l f - L e a r n i n g of t h e D i r e c t P o s i t i o n K i n e m a t i c s 139 3.9.2 C l o s e d L o o p C o n t r o l u s i n g t h e S e l f - L e a r n e d C a r t e s i a n I n v e r s e Dynamics and S e l f - L e a r n e d D i r e c t P o s i t i o n K i n e m a t i c s 155 3.10 Summary 165 4. C o n t r i b u t i o n s of t h i s T h e s i s 171 5. S u g g e s t i o n s f o r F u t u r e R e s e a r c h 174 5.1 F u r t h e r I n v e s t i g a t i o n o f S e l f - L e a r n e d C o n t r o l 174 5.1.1 C o n t r o l o f a More Complex M a n i p u l a t o r . 174 5.1.2 A d a p t a t i o n t o T o o l Mass 175 5.2 I m p l e m e n t a t i o n o f Sum of P o l y n o m i a l s E s t i m a t o r s 178 5.2.1 D i g i t a l Computer I m p l e m e n t a t i o n 179 5.2.1 S t o c h a s t i c Computer I m p l e m e n t a t i o n .... 186 5.3 P r o p o s e d M o d i f i c a t i o n s t o t h e K l e t t C e r e b e l l a r M o d e l 190 5.3.1 The K l e t t C e r e b e l l a r M o d e l 190 5.3.2 L e a r n e d O r t h o g o n a l i z a t i o n 193 5.3.3 I n p u t V a r i a b l e S p l i t t i n g 199 v B i b l i o g r a p h y 206 Appendix A: A n a l y t i c a l Kinematics of the Two Link Manipulator 211 Appendix B: A n a l y t i c a l Dynamics of the Two Link .Manipulator 215 Appendix C: Closed Loop C o n t r o l of Li n e 1 u s i n g the S e l f - L e a r n e d C a r t e s i a n Inverse Dynamics a f t e r 200, 400, 600, 800 and 1000 T r a i n i n g Paths 218 Appendix D: View of C i r c l e 1 u s i n g the S e l f - L e a r n e d D i r e c t P o s i t i o n Kinematics a f t e r 200, 400, 600, 800, 1000, 1400, 1800, 2200, 2600, 3000, 3400, 3800, 4200, and 4600 T r a i n i n g Paths 224 v i L i s t of F i g u r e s 2.1 C o n v e r g e n c e r a t e s when t a r g e t f u n c t i o n i s a p o l y n o m i a l w i t h randomly c h o s e n c o e f f i c i e n t s v e r s u s c o n v e r g e n c e r a t e s when t a r g e t f u n c t i o n i s a p o l y n o m i a l w i t h 1's a s c o e f f i c i e n t s , f o r c a s e s=3 and v=3 15 2.2 C o n v e r g e n c e r a t e s f o r Method 5, t h e C e r e b e l l a r M o d e l , as a f u n c t i o n of u f o r t h e c a s e s s=1 and v=3, s=3 and v=3, and s=3 and 17 2. 3 C o n v e r g e n c e r a t e s f o r v a r i o u s methods when s= 1 . . 20 2. 4 C o n v e r g e n c e r a t e s f o r v a r i o u s methods when s= 2 . . 21 2. 5 C o n v e r g e n c e r a t e s f o r v a r i o u s methods when s= 3 . . 22 2. 6 C o n v e r g e n c e r a t e s f o r v a r i o u s methods when s= 4 . . 23 2. 7 C o n v e r g e n c e r a t e s f o r v a r i o u s methods when s= 5 . . 24 2. 8 C o n v e r g e n c e r a t e s f o r v a r i o u s methods when s= 6 . . 25 2. 9 R e d u c t i o n o f e s t i m a t i o n e r r o r , A f , as a f u n c t i o n o f t h e number of t r a i n i n g p o i n t s f o r t h e c a s e s=3 and v=3 u s i n g t h e v a r i o u s methods ( e s t i m a t i o n e r r o r a v e r a g e d o v e r e a c h 100 i t e r a t i o n i n t e r v a l ) 26 2.10 P a t t e r n o f z e r o and n o n - z e r o e l e m e n t s of m a t r i x P f o r t h e c a s e s=3 and v=3 (+ = p o s i t i v e , - = n e g a t i v e , 0 = z e r o ) 29 2.11 T o t a l number of e l e m e n t s , number o f n o n - z e r o e l e m e n t s and o r d e r o f m a t r i x P f o r c a s e s where s=3 31 3.1 A two l i n k m a n i p u l a t o r 42 3.2 B l o c k d i a g r a m o f m a n i p u l a t o r s i m u l a t i o n p r o g r a m . 48 3.3 S t a n d a r d p a t h l i n e 1 i n C a r t e s i a n c o o r d i n a t e s ... 55 3.4 S t a n d a r d p a t h l i n e 1 i n j o i n t c o o r d i n a t e s 56 3.5 S t a n d a r d p a t h c i r c l e 1 i n C a r t e s i a n c o o r d i n a t e s . 58 3.6 S t a n d a r d p a t h c i r c l e 1 i n j o i n t c o o r d i n a t e s 59 3.7 View o f s t a n d a r d p a t h c i r c l e 1 ( t i c k s mark i n t e r v a l s of 0.2 s e c ) 60 v i i 3.8 I d e a l open l o o p c o n t r o l of l i n e 1 62 3.9 T o r q u e p r o f i l e f o r l i n e 1 64 3.10 E r r o r c o r r e c t i n g a c t i o n of i d e a l c l o s e d l o o p c o n t r o l 66 3.11 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1 71' 3.12 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t t c a i c = 0 ' 0 2 s e c 7 3 3.13 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t t c a i c = 0 ' 0 3 s e c 74 3.14 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t t c a i c = 0 - 0 4 s e c 7 5 3.15 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t EXACT=.TRUE 77 3.16 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t PRDICT=.FALSE 78 3.17 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t USEOBS=.TRUE 80 3.18 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, e x c e p t t h a t k 1 =64 and k 2 = l 0 2 4 82 3.19 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t k 1 = 4 and k 2=4 83 3.20 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t k 1 = 8 and k 2 = l 2 8 84 3.21 S t a n d a r d a n a l y t i c a l c o n t r o l o f l i n e 1, e x c e p t t h a t ^ = 3 2 and k 2=32 85 3.22 R e d u c t i o n o f e s t i m a t i o n e r r o r d u r i n g p r e - l e a r n i n g o f t h e i n v e r s e d y n a m i c s ( e s t i m a t i o n e r r o r a v e r a g e d o v e r e a c h 100 i t e r a t i o n i n t e r v a l ) 91 3.23 R e d u c t i o n o f e s t i m a t i o n e r r o r d u r i n g p r e - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s ( e s t i m a t i o n e r r o r a v e r a g e d o v e r e a c h 100 i t e r a t i o n i n t e r v a l ) 99 3.24 C l o s e d l o o p c o n t r o l o f l i n e 1 u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s 101 3.25 View o f c i r c l e 1 u s i n g t h e d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s 106 v i i i 3.26 R e d u c t i o n of e s t i m a t i o n e r r o r d u r i n g p r e - l e a r n i n g of t h e d i r e c t p o s i t i o n k i n e m a t i c s ( e s t i m a t i o n e r r o r a v e r a g e d o v e r e a c h 100 i t e r a t i o n i n t e r v a l ) 109 3.27 View o f c i r c l e 1 u s i n g t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 110 3.28 View o f c i r c l e 1 u s i n g t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g c l o s e d l o o p c o n t r o l of c i r c l e 1 u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 114 3.29 View o f c i r c l e 1 d u r i n g c l o s e d l o o p c o n t r o l o f c i r c l e 1 u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 115 3.30 C l o s e d l o o p c o n t r o l o f l i n e 1 u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 118 3.31 C l o s e d l o o p c o n t r o l o f l i n e 1 u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s w i t h ADJVIS=.TRUE. 119 3.32 C h o i c e s o f a_ and v m f o r f i r s t 250 t r a i n i n g p a t h s u s e d i n s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s 127 3.33 View o f f i r s t 250 t r a i n i n g p a t h s u s e d i n s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s 128 3.34 P r o p o r t i o n o f l e a r n i n g o p p o r t u n i t i e s a t w h i c h l e a r n i n g t o o k p l a c e d u r i n g s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s 129 3.35 P a t h e r r o r s d u r i n g s e l f - l e a r n i n g of t h e C a r t e s i a n i n v e r s e d y n a m i c s 130 3.36 Maximum o u t o f bounds e x c u r s i o n s d u r i n g s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s 131 3.37 A v e r a g e path, e r r o r s d u r i n g c l o s e d l o o p c o n t r o l of t h e s i x s t a n d a r d p a t h s u s i n g t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s a s a f u n c t i o n o f t h e number o f t r a i n i n g p a t h s u s e d 134 3.38 C l o s e d l o o p c o n t r o l o f l i n e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s 135 i x 3.39 C l o s e d l o o p c o n t r o l o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s 136 3.40 A v e r a g e t o r q u e e s t i m a t i o n e r r o r d u r i n g c l o s e d l o o p c o n t r o l o f t h e s i x s t a n d a r d p a t h s u s i n g t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s as a f u n c t i o n o f t h e number of t r a i n i n g p a t h s u s e d ... 138 3.41 P r o p o r t i o n of l e a r n i n g o p p o r t u n i t i e s a t w h i c h l e a r n i n g t o o k p l a c e d u r i n g s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s 144 3.42 P a t h e s t i m a t i o n e r r o r s d u r i n g s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s 145 3.43 View o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 146 3.44 View o f c i r c l e 1 u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 154 3.45 View o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g c l o s e d l o o p c o n t r o l o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 156 3.46 View o f c i r c l e 1 d u r i n g c l o s e d l o o p c o n t r o l o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 157 3.47 View o f c i r c l e 1 u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g c l o s e d l o o p c o n t r o l o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and smoothed f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s ... 160 3.48 View o f c i r c l e 1 d u r i n g c l o s e d l o o p c o n t r o l o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 161 3.49 C l o s e d l o o p c o n t r o l o f l i n e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 163 3.50 C l o s e d l o o p c o n t r o l o f l i n e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s w i t h ADJVIS=.TRUE 164 x 5.1 B l o c k d i a g r a m of d i g i t a l c omputer i m p l e m e n t a t i o n of a sum of p o l y n o m i a l s e s t i m a t o r 184 5.2 S t o c h a s t i c computer c i r c u i t f o r summing t h e p r o d u c t s of two p a i r s of i n p u t v a r i a b l e s 189 5.3 B l o c k d i a g r a m of c e r e b e l l a r s y s t e m 190 5.4 B l o c k d i a g r a m of t h e K l e t t C e r e b e l l a r M odel 191 5.5 S c h e m a t i c o f G r a n u l e c e l l - G o l g i c e l l n etwork a) s howing t h o s e l o c a t i o n s where a m p l i f i c a t i o n c o u l d o c c u r b) showing lumped a m p l i f i c a t i o n s 194 5.6 Example of i n p u t v a r i a b l e s p l i t t i n g 202 5.7 R e p r e s e n t a t i o n s of s i n ( 2 7 r z ) • ••• 202 5.8 E n c o d i n g of j o i n t a n g l e by i n p u t v a r i a b l e s p l i t t i n g 205 5.9 Example o f p r o d u c t s o f s p l i t i n p u t v a r i a b l e s r e p r e s e n t i n g j o i n t a n g l e 205 x i L i s t o f T a b l e s 2.1 Summary of l e a r n i n g a l g o r i t h m s 13 * 2.2 S t e p s i z e f a c t o r s , u, f o r Method 5, t h e C e r e b e l l a r M odel 18 3.1 D a t a d e f i n i n g s t a n d a r d l i n e a r p a t h s .- 54 3.2 D a t a d e f i n i n g s t a n d a r d c i r c u l a r p a t h s 57 3.3 P a t h e r r o r u s i n g i d e a l open l o o p c o n t r o l 63 3.4 P a t h e r r o r u s i n g i d e a l c l o s e d l o o p c o n t r o l 67 3.5 P a t h e r r o r u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l .... 70 3.6 P a t h e r r o r u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l , e x c e p t f o r n o t e d v a r i a t i o n s 86 3.7 C o e f f i c i e n t s f o r t h e d e r i v e d and p r e - l e a r n e d sum o f p o l y n o n i a l s r e p r e s e n t a t i o n s o f t h e i n v e r s e d y n a m i c s f u n c t i o n f o r t o r q u e 92 3.8 C o e f f i c i e n t s f o r t h e d e r i v e d and p r e - l e a r n e d sum o f p o l y n o n i a l s r e p r e s e n t a t i o n s of t h e i n v e r s e d y n a m i c s f u n c t i o n f o r t o r q u e 93 3.9 P a t h e r r o r and t o r q u e e r r o r u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s f o r c l o s e d l o o p c o n t r o l 102 3.10 P a t h e s t i m a t i o n e r r o r u s i n g t h e d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s 105 3.11 P a t h e s t i m a t i o n e r r o r u s i n g t h e model v i s i o n s y s t e m h a v i n g e x a c t p o s i t i o n measurement 107 3.12 C o e f f i c i e n t s f o r t h e d e r i v e d and p r e - l e a r n e d sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s of t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n s f o r p o s i t i o n s x 1 and *2 1 1 1 3.13 P a t h e s t i m a t i o n e r r o r u s i n g t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 112 3.14 P a t h e r r o r u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s f o r c l o s e d l o o p c o n t r o l 113 3.15 P a t h e r r o r and t o r q u e e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s f o r c l o s e d l o o p c o n t r o l 139 x i i 3.16 P a t h e s t i m a t i o n e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 147 3.17 C o e f f i c i e n t s f o r t h e d e r i v e d and f i n a l s e l f - l e a r n e d sum of p o l y n o m i a l s r e p r e s e n t a t i o n s of t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n s f o r p o s i t i o n s x 1 and 148 3.18 P a t h e s t i m a t i o n e r r o r u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 152 3.19 C o e f f i c i e n t s f o r t h e d e r i v e d and smoothed, f i n a l s e l f - l e a r n e d sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s o f t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n s f o r p o s i t i o n s x 1 and x 2 153 3.20 P a t h e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s i n c l o s e d l o o p c o n t r o l 158 3.21 P a t h e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s i n c l o s e d l o o p c o n t r o l 159 3.22 Summary of p a t h e r r o r s u s i n g v a r i o u s c o n t r o l schemes 170 5.1 Number o f t e r m s , m, f o r s y s t e m s o f v a r i o u s o r d e r s , s, and v a r i o u s numbers of i n p u t v a r i a b l e s , v .... 181 5.2 M u l t i p l i c a t i o n s r e q u i r e d f o r e s t i m a t e s o f v a r i o u s o r d e r s o f t h e C a r t e s i a n i n v e r s e d y n a m i c s f o r m a n i p u l a t o r s h a v i n g v a r i o u s d e g r e e s of f r e e d o m .. 182 5.3 M u l t i p l i c a t i o n s r e q u i r e d f o r e s t i m a t e s of v a r i o u s o r d e r s o f t h e d i r e c t p o s i t i o n k i n e m a t i c s f o r m a n i p u l a t o r s h a v i n g v a r i o u s d e g r e e s o f f r e e d o m .. 182 x i i i Acknowledgements I would l i k e t o t h ank my s u p e r v i s o r , D r . P e t e r D. L a w r e n c e , f o r h i s e n t h u s i a s t i c e n c o u r a g e m e n t , a s s i s t a n c e and s u p p o r t . I w o u l d a l s o l i k e t o thank f e l l o w s t u d e n t , James J . C l a r k , f o r t h e many i d e a s he o f f e r e d d u r i n g o u r f r e q u e n t c o n v e r s a t i o n s . My w i f e , D u l c e E s t r e l l a , has been a c o n s t a n t s o u r c e o f e n c o u r a g e m e n t and my d a u g h t e r , A l i c i a E s t r e l l a , has been a c o n s t a n t s o u r c e of i n s p i r a t i o n . I d e d i c a t e t h i s t h e s i s t o my nephew Lawrence Wyatt T e l l i n g , whom I w i l l a l w a y s remember. T h i s r e s e a r c h has been s u p p o r t e d by t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada t h r o u g h a P o s t g r a d u a t e S c h o l a r s h i p t o i t s a u t h o r and G r a n t #A4924, and by t h e U n i v e r s i t y of B r i t i s h C o l u m b i a i n t h e f o r m of a T e a c h i n g A s s i s t a n c e s h i p . x i v 1 INTRODUCTION T h e r e i s c u r r e n t l y much i n t e r e s t i n a d a p t i v e c o n t r o l i n t h e f i e l d o f r o b o t i c s . L a r g e l y , t h i s i s due t o t h e d e s i r e t o a v o i d a n a l y s i s of t h e d y n a m i c s and k i n e m a t i c s of r o b o t m a n i p u l a t o r s . The g o a l o f t h i s t h e s i s i s t o examine a f a m i l y o f l e a r n i n g a l g o r i t h m s and d e m o n s t r a t e t h e i r a b i l i t y t o l e a r n t h e non- l i n e a r , m u l t i - v a r i a t e f u n c t i o n s d e s c r i b i n g t h e d y n a m i c s and k i n e m a t i c s o f a r o b o t m a n i p u l a t o r . F u r t h e r m o r e , i t i s i n t e n d e d t o show t h a t t h i s l e a r n i n g c a n be done w i t h o u t r e c o u r s e t o a n a l y s i s of t h e d y n a m i c s o r k i n e m a t i c s of t h e m a n i p u l a t o r . The s t a r t i n g p o i n t f o r t h i s work was t h e C e r e b e l l a r Model p r o p o s e d by K l e t t [ 2 6 ] . The l e a r n i n g machine p r o p o s e d by K l e t t r e p r e s e n t s an e x t e n s i o n o f p r e v i o u s c e r e b e l l a r m o d e l s s u c h as t h o s e p r o p o s e d by A l b u s [1,2] and Marr [31] and draws upon e a r l i e r work on p e r c e p t r o n s , and o t h e r l e a r n i n g m a c h i n e s [ 4 2 , 4 9 ] . The c e r e b e l l u m i s i n v o l v e d i n m a i n t a i n i n g p o s t u r e and c o o r d i n a t i n g motor a c t i v i t i e s o f t h e body [ 1 , 2 6 , 3 8 ] ; i . e . m a n i p u l a t o r c o n t r o l . Thus i t i s r e a s o n a b l e t o c o n s i d e r a p p l i c a t i o n of t h e K l e t t C e r e b e l l a r Model i n l e a r n e d m a n i p u l a t o r c o n t r o l . C u r r e n t l y , most i n d u s t r i a l r o b o t m a n i p u l a t o r s a r e b a s e d on i n d i v i d u a l j o i n t s e r v o c o n t r o l . P a t h s p e c i f i c a t i o n s i n a t a s k o r i e n t e d c o o r d i n a t e s y s t e m s u c h as C a r t e s i a n c o o r d i n a t e s must be t r a n s f o r m e d i n t o p a t h s p e c i f i c a t i o n s i n t e r m s o f s u c c e s s i v e j o i n t p o s i t i o n s , a t i m e c o n s u m i n g t a s k . I n d i v i d u a l j o i n t s e r v o c o n t r o l t y p i c a l l y n e g l e c t s c o u p l i n g dynamic e f f e c t s and t h u s 1 p e r f o r m a n c e i s l i m i t e d . M a n i p u l a t o r c o n t r o l t e c h n i q u e s have been p r o p o s e d t h a t make use o f t h e c o m p l e t e a n a l y t i c a l d y n a m i c s of a m a n i p u l a t o r ; an example i s R e s o l v e d A c c e l e r a t i o n C o n t r o l as p r o p o s e d by Luh e t a l [ 3 0 ] . The d i s a d v a n t a g e o f s u c h t e c h n i q u e s i s t h a t t h e y r e q u i r e a n a l y s i s o f t h e d y n a m i c s and k i n e m a t i c s of e a c h m a n i p u l a t o r t o be so c o n t r o l l e d . T h i s i s o f t e n a d i f f i c u l t t a s k . F u r t h e r m o r e , t h e p a r a m e t e r s u s e d i n t h e f o r m u l a t i o n o f t h e d y n a m i c s s u c h a s l i n k l e n g t h s , masses, c e n t e r s o f g r a v i t y and moments of i n e r t i a may be d i f f i c u l t t o o b t a i n by anyone o t h e r t h a n t h e m a n u f a c t u r e r o f t h e m a n i p u l a t o r . A d a p t i v e t e c h n i q u e s have been p r o p o s e d t o a v o i d t h e n e c e s s i t y o f m a n i p u l a t o r a n a l y s i s and measurement o f m a n i p u l a t o r p a r a m e t e r s [ 2 9 ] . One s u c h t e c h n i q u e i s t h e A d a p t i v e L i n e a r C o n t r o l l e r p r o p o s e d by K o i v o and Guo [ 2 7 ] . A l o c a l l y v a l i d model of t h e m a n i p u l a t o r d y n a m i c s i s e s t i m a t e d o n - l i n e , b a s e d on r e c e n t o b s e r v a t i o n s of a p p l i e d t o r q u e s and r e s u l t i n g m a n i p u l a t o r m o t i o n . P e r f o r m a n c e i s q u i t e good and t h e method d o e s n o t r e q u i r e a priori a n a l y s i s o f t h e m a n i p u l a t o r o r measurement o f m a n i p u l a t o r p a r a m e t e r s . U n f o r t u n a t e l y , many m a t h e m a t i c a l o p e r a t i o n s a r e n e c e s s a r y f o r t h e o n - l i n e o p e r a t i o n s . I n t u i t i v e l y , t h e A d a p t i v e L i n e a r C o n t r o l l e r would seem t o be i n e f f i c i e n t b e c a u s e a l t h o u g h i t i s c o n t i n u a l l y a d a p t i n g , i . e . l e a r n i n g , i t i s c o n t i n u a l l y f o r g e t t i n g a s w e l l . The l o c a l l y v a l i d model i s c h a n g e d a s t h e m a n i p u l a t o r moves from p l a c e t o p l a c e by i n c o r p o r a t i n g i n f o r m a t i o n f r o m new o b s e r v a t i o n s and f o r g e t t i n g i n f o r m a t i o n from o l d o b s e r v a t i o n s . Thus upon 2 r e p e a t i n g a p a t h , the A d a p t i v e L i n e a r C o n t r o l l e r does not b e n e f i t from the p r e v i o u s e x p e r i e n c e . S e l f - L e a r n e d C o n t r o l as proposed i n t h i s t h e s i s d i f f e r s from p r e v i o u s a d a p t i v e methods i n t h a t a g l o b a l l y v a l i d model of the m a n i p u l a t o r dynamics and k i n e m a t i c s i s l e a r n e d . In t h i s r e s p e c t i t i s somewhat s i m i l a r t o the C e r e b e l l a r Model A r t i c u l a t i o n C o n t r o l l e r proposed by A l b u s [ l , 2 ] . Due t o the use of c o n t i n u o u s f u n c t i o n s as the b a s i s of our l e a r n e d f u n c t i o n a l e s t i m a t e s , however, our method r e q u i r e s f a r fewer w e i g h t i n g c o e f f i c i e n t s than the method proposed by A l b u s and no s p e c i a l e n c o d i n g of i n p u t v a r i a b l e s . Once the dynamics and k i n e m a t i c s have been l e a r n e d t o a degree adequate f o r c o n t r o l p u r p o s e s , no f u r t h e r l e a r n i n g need take p l a c e . T h i s assumes, of c o u r s e , t h a t the m a n i p u l a t o r dynamics and k i n e m a t i c s a r e not c h a n g i n g . I n t e r f e r e n c e M i n i m i z a t i o n , t h e p r i n c i p l e l e a r n i n g a l g o r i t h m used i n t h i s t h e s i s , was d e r i v e d by K l e t t as an i n t e r m e d i a t e s t e p i n t h e d e r i v a t i o n of h i s C e r e b e l l a r Model [ 2 6 ] . Chapter 2 examines t h e convergence r a t e s of I n t e r f e r e n c e M i n i m i z a t i o n and r e l a t e d l e a r n i n g a l g o r i t h m s such as the G r a d i e n t Method [36,57] and L e a r n i n g I d e n t i f i c a t i o n [ 12,39,50], A method of r e d u c i n g the number of c a l c u l a t i o n s r e q u i r e d t o implement I n t e r f e r e n c e M i n i m i z a t i o n i s i n t r o d u c e d . A l s o i n t r o d u c e d a r e P o i n t w i s e c o u n t e r p a r t s of s e v e r a l of t h e s e a l g o r i t h m s t h a t have u t i l i t y i n c e r t a i n a p p l i c a t i o n s . In c h a p t e r 3 the a f o r e m e n t i o n e d l e a r n i n g a l g o r i t h m s a r e a p p l i e d t o a c h i e v e S e l f - L e a r n e d c o n t r o l of a s i m u l a t e d two l i n k m a n i p u l a t o r . S e c t i o n 3.1 d i s c u s s e s R e s o l v e d A c c e l e r a t i o n C o n t r o l 3 u s i n g p a t h s p e c i f i c a t i o n i n C a r t e s i a n c o o r d i n a t e s and s e c t i o n 3.2 p r o v i d e s the a n a l y t i c a l dynamics and a n a l y t i c a l k i n e m a t i c s of the two l i n k m a n i p u l a t o r . S e c t i o n s 3.3 and 3.4 o u t l i n e how the two l i n k m a n i p u l a t o r was s i m u l a t e d and how s t a n d a r d t e s t p a t h s p e c i f i c a t i o n s were g e n e r a t e d . In s e c t i o n 3.5 the a n a l y t i c a l dynamics and a n a l y t i c a l k i n e m a t i c s of the two l i n k m a n i p u l a t o r a r e used i n a s i m u l a t i o n of a r e a l i s t i c i m p l e m e n t a t i o n of R e s o l v e d A c c e l e r a t i o n C o n t r o l t h a t we c a l l S t a n d a r d A n a l y t i c a l C o n t r o l . T h i s e s t a b l i s h e s a performance benchmark f o r subsequent comparison w i t h S e l f - L e a r n e d c o n t r o l . In s e c t i o n s 3.6 and 3.7 i t i s shown t h a t the C a r t e s i a n i n v e r s e dynamics and d i r e c t p o s i t i o n k i n e m a t i c s of the two l i n k m a n i p u l a t o r can be r e p r e s e n t e d a d e q u a t e l y as sums of p o l y n o m i a l s over a workspace c o n s i s t i n g of a s i z a b l e p o r t i o n of the m a n i p u l a t o r ' s r e a c h . F u r t h e r m o r e , i t i s shown t o be p o s s i b l e t o P r e - L e a r n t h e s e sum of p o l y n o m i a l s r e p r e s e n t a t i o n s u s i n g the a n a l y t i c a l C a r t e s i a n i n v e r s e dynamics and a n a l y t i c a l d i r e c t p o s i t i o n k i n e m a t i c s as a g u i d e . F i n a l l y , i n s e c t i o n s 3.8 and 3.9 i t i s shown t h a t t h e s e r e p r e s e n t a t i o n s can be S e l f - L e a r n e d w i t h o u t r e c o u r s e t o the a n a l y t i c a l dynamics or a n a l y t i c a l k i n e m a t i c s . In c o n t r o l u s i n g the S e l f - L e a r n e d C a r t e s i a n i n v e r s e dynamics as shown i n s e c t i o n 3.8, a v i s i o n system i s assumed t o observe the m a n i p u l a t o r and p r o v i d e e r r o r c o r r e c t i n g feedback i n f o r m a t i o n . S e c t i o n 3.9 shows t h a t u t i l i z a t i o n of the S e l f - L e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s p e r m i t s replacement of the v i s i o n system once l e a r n i n g i s c o m p l e t e . The c o n t r i b u t i o n s of t h i s t h e s i s a r e summarized i n c h a p t e r 4 4 and c h a p t e r 5 o f f e r s s e v e r a l d e t a i l e d s u g g e s t i o n s f o r a r e a s o f f u t u r e r e s e a r c h . F i r s t , t h e r e i s t h e need t o a p p l y S e l f - L e a r n i n g t o a r e a l m a n i p u l a t o r h a v i n g more t h a n 2 d e g r e e s o f fr e e d o m and t o e x t e n d S e l f - L e a r n i n g t o a l l o w t h e C a r t e s i a n i n v e r s e d y n a m i c s t o be a d j u s t e d a u t o m a t i c a l l y t o compensate f o r t h e a t t a c h m e n t t o t h e m a n i p u l a t o r of a t o o l o f any mass w i t h i n a g i v e n r a n g e . S e c o n d l y , i n v e s t i g a t i o n i s w a r r a n t e d i n t o t h e i m p l e m e n t a t i o n of t h e r e q u i r e d sum of p o l y n o m i a l s e s t i m a t o r s . F i n a l l y , p r o p o s e d m o d i f i c a t i o n s of t h e K l e t t C e r e b e l l a r M o d e l may i n c r e a s e i t s p l a u s i b i l i t y . 5 2 INTERFERENCE MINIMIZATION AND RELATED LEARNING ALGORITHMS 2.1 LEARNED FUNCTIONAL ESTIMATION USING SUMS OF POLYNOMIALS In a d a p t i v e c o n t r o l , i t i s o f t e n n e c e s s a r y t o l e a r n t o e s t i m a t e unknown f u n c t i o n a l r e l a t i o n s h i p s t h r o u g h a l e a r n i n g t e c h n i q u e . In t h i s t h e s i s we s t u d y t e c h n i q u e s whereby f u n c t i o n a l e s t i m a t i o n i s l e a r n e d b a s e d on a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n . One s u c h t e c h n i q u e i s t h e G r a d i e n t Method u s e d by Widrow e t a l [57] f o r a d a p t i v e f i l t e r i n g . A n o t h e r t e c h n i q u e i s t h e method of L e a r n i n g I d e n t i f i c a t i o n t h a t was f o r m u l a t e d f o r l i n e a r s y s t e m s by Nagumo and Noda [ 3 9 ] , L e a r n i n g I d e n t i f i c a t i o n was e x t e n d e d t o n o n - l i n e a r s y s t e m s by Roy and Sherman [ 5 0 ] . T h e s e and r e l a t e d methods have been i n v e s t i g a t e d by o t h e r a u t h o r s s u c h as K l e t t [ 2 6 ] , Eweda and O d i l e [ 1 4 ] , B i t m e a d and A n d e r s o n [ 8 ] , Chan and Babu [ 1 0 , 1 1 ] , J o h n s o n [25] and B i l l i n g s [ 7 ] . . T h e s e t e c h n i q u e s have been u s e d t o e s t i m a t e a f u n c t i o n o f t i m e s a m p l e s of a c o n t i n u o u s s i g n a l [ 3 9 , 5 0 , 5 7 ] , They c a n a l s o be g e n e r a l i z e d t o e s t i m a t e f u n c t i o n s o f a r b i t r a r y v a r i a b l e s [ 2 6 , 1 0 , 1 1 ] . We a r e i n t e r e s t e d i n t h e l e a r n i n g of n o n - l i n e a r f u n c t i o n s o f s e v e r a l v a r i a b l e s f o r a p p l i c a t i o n i n m a n i p u l a t o r c o n t r o l . T h i s t h e s i s d e s c r i b e s an i m p r o v e d l e a r n i n g a l g o r i t h m of w h i c h t h e p r e v i o u s l y m e n t i o n e d a l g o r i t h m s a r e s i m p l i f i c a t i o n s . They t h u s f o r m a f a m i l y o f l e a r n i n g a l g o r i t h m s . The c o n v e r g e n c e r a t e s of t h e v a r i o u s a l g o r i t h m s a r e compared and f a c t o r s s i m p l i f y i n g t h e i m p l e m e n t a t i o n o f t h e i m p r o v e d l e a r n i n g a l g o r i t h m a r e d i s c u s s e d . 6 2.2 AN IMPROVED LEARNING ALGORITHM - INTERFERENCE MINIMIZATION T h i s t h e s i s i s c o n c e r n e d w i t h a l g o r i t h m s t h a t l e a r n t o e s t i m a t e a f u n c t i o n as a sum o f p o l y n o m i a l s , s p e c i f i c a l l y t h e Ko l m o g o r o v - G a b o r [ 18,50] p o l y n o m i a l s . The e s t i m a t e o f a f u n c t i o n i s formed as a w e i g h t e d sum o f p o l y n o m i a l t e r m s , f = I w / cp / c(z) = w Tp (2.1) k The b a s i s p o l y n o m i a l t e r m s a r e t h e s e t , { p . ( z ) J = { n z . e i } (2.2) * i=0 where, v Z e. = s , e. an i n t e g e r (2.3) i=0 1 1 z Q = 1 (2.4) F o r example, i f s=2 and v=2 t h e n t h e p o l y n o m i a l t e r m s a r e , (p A . ( z ) } = { 1, z ] t z 2 , z ^ f z 2 2 , z ^ 2 } (2.5) The s p a c e s p a n n e d by t h i s s e t i s g i v e n by v, t h e number o f i n p u t v a r i a b l e s , and s, t h e s y s t e m o r d e r . The number o f t e r m s , m, i n s u c h a p o l y n o m i a l i s g i v e n by, ( s + v ) ! m = (2.6) s!v ! An e s t i m a t e f o r m e d i n t h i s manner c a n mimic m u l t i v a r i a t e , non- l i n e a r f u n c t i o n s . L e a r n i n g t a k e s p l a c e by i t e r a t i v e l y a d j u s t i n g t h e w e i g h t v e c t o r a t a s e r i e s o f t r a i n i n g p o i n t s u n t i l t h e e s t i m a t e c o r r e s p o n d s t o t h e t a r g e t f u n c t i o n t h r o u g h o u t t h e s p a c e . An i m p r o v e d l e a r n i n g a l g o r i t h m c an be d e r i v e d as f o l l o w s : A t e a c h t r a i n i n g p o i n t t h e w e i g h t s a r e a d j u s t e d t o e l i m i n a t e t h e 7 e r r o r i n the e s t i m a t e w h i l e m i n i m i z i n g the change i n the e s t i m a t e a t o t h e r p o i n t s i n the space. We c a l l the a l g o r i t h m , I n t e r f e r e n c e M i n i m i z a t i o n . T h i s s t r a t e g y can be e n f o r c e d by m i n i m i z i n g the f u n c t i o n , c = / ( A w T p ) 2 5S + a(Aw Tp - Af) (2.7) S u s i n g the Lagrange m u l t i p l i e r t e c h n i q u e , where S i s the domain of the i n p u t v a r i a b l e s , Aw i s the change i n the weight v e c t o r and Af i s the e r r o r i n the e s t i m a t i o n . The e r r o r i n t h e e s t i m a t e i s g i v e n by, Af = f - f (2.8) * where f i s the t a r g e t f u n c t i o n . S e t t i n g the p a r t i a l d e r i v a t i v e s of c w i t h r e s p e c t t o Aw and a t o z e r o and s o l v i n g y i e l d s the weight adjustment f o r m u l a , A f P ~ 1 p Aw = (2.9) P P P where, P = / p p T 6S (2.10) S The m a t r i x P i s r e a l , symmetric and p o s i t i v e d e f i n i t e and t h u s the i n v e r s e , P 1 e x i s t s . I n t e r f e r e n c e M i n i m i z a t i o n was d e r i v e d by K l e t t [26] as an i n t e r m e d i a t e s t e p i n the d e r i v a t i o n of h i s C e r e b e l l a r Model and can be g e n e r a l i z e d t o a l l o w use of o t h e r b a s i s s e t s . In t h i s work we c o n s i d e r i n depth the s p e c i f i c case where K-G p o l y n o m i a l s form the b a s i s s e t . T h i s e l i c i t s comparison w i t h o t h e r p r e v i o u s l y d e s c r i b e d l e a r n i n g a l g o r i t h m s based on K-G p o l y n o m i a l s and p e r m i t s s i m p l i f i c a t i o n of the i m p l e m e n t a t i o n of 8 I n t e r f e r e n c e M i n i m i z a t i o n as i s shown i n s e c t i o n 2.6. 2.3 RELATIONSHIP TO SIMILAR LEARNING ALGORITHMS S e v e r a l l e a r n i n g a l g o r i t h m s p r e v i o u s l y d e s c r i b e d i n the l i t e r a t u r e can be c o n s i d e r e d t o be s i m p l i f i c a t i o n s of I n t e r f e r e n c e M i n i m i z a t i o n . In (2.9) i t can be seen t h a t the m a t r i x P 1 does not change the magnitude of the weight adjustment v e c t o r Aw, i t o n l y changes the d i r e c t i o n . I f the m a t r i x P 1 i s d e l e t e d from the weight adjustment f o r m u l a , the e s t i m a t i o n e r r o r a t a t r a i n i n g p o i n t i s s t i l l e l i m i n a t e d , however, the e f f e c t of the a d j u stment a t o t h e r p o i n t s i n the space i s not m i n i m i z e d . T h i s s u b - o p t i m a l l e a r n i n g a l g o r i t h m i s L e a r n i n g I d e n t i f i c a t i o n , which was f o r m u l a t e d by Nagumo and Noda [39] u s i n g f i r s t o r d e r p o l y n o m i a l s , and by Roy and Sherman [50] u s i n g K-G p o l y n o m i a l s , M p Aw = -=- (2.11) P P In (2.11) the denominator s e r v e s as a s c a l i n g v a r i a b l e t h a t a d j u s t s the s t e p s i z e . When c h a n g i n g the w e i g h t s , the e r r o r i n the e s t i m a t e i s e x a c t l y e l i m i n a t e d r e g a r d l e s s of the l o c a t i o n of the t r a i n i n g p o i n t . I f the denominator i s r e p l a c e d by a p o s i t i v e c o n s t a n t , then the d i r e c t i o n of t h e weight adjustment v e c t o r i s not changed; one s t i l l moves towards a new weight s e t t h a t e l i m i n a t e s t h e e s t i m a t i o n e r r o r a t the t r a i n i n g p o i n t . The adjustment f o r m u l a becomes, Aw = uAfp (2.12) T h i s a l g o r i t h m i s known as t h e G r a d i e n t Method and has been 9 d i s c u s s e d by a u t h o r s such as Widrow e t a l [57] and Eweda and O d i l e [ 1 4 ] . The g a i n f a c t o r u must be chosen such t h a t a d j u s t m e n t s a r e not so l a r g e as t o cause d i v e r g e n c e a t any t r a i n i n g p o i n t s w i t h i n the space. T h i s can be ensured by c h o o s i n g u such t h a t , u < 2 m i n { l / ( p T p ) } (2.13) S T h i s a l g o r i t h m i s even l e s s o p t i m a l than (2.11) s i n c e many i t e r a t i o n s a r e r e q u i r e d a t most t r a i n i n g p o i n t s i n the space j u s t t o e l i m i n a t e the e s t i m a t i o n e r r o r a t the t r a i n i n g p o i n t . Two v a r i a n t s of I n t e r f e r e n c e M i n i m i z a t i o n f o l l o w i f one s e l e c t s b a s i s f u n c t i o n s such t h a t P 1 i s e q u a l t o the i d e n t i t y m a t r i x . One method i s t o form the e s t i m a t e a s , f = w TQp = w Tq (2.14) where, Q = [/ p p T 6 S ] " 1 / 2 = P ( p ) " l / 2 (2.15) S The m a t r i x Q i s r e a l , symmetric and p o s i t i v e d e f i n i t e . The terms of the v e c t o r q a r e a s e t of o r t h o n o r m a l p o l y n o m i a l s . W i t h such a b a s i s v e c t o r , the m a t r i x P becomes, P(q) = / q q T 6S = / Qpp TQ T SS = P " l / 2 P P _ 1 / 2 = I (2.16) S S The weight adjustment f o r m u l a now becomes, M q Aw = -=- (2.17) q q T h i s method i s e q u i v a l e n t t o I n t e r f e r e n c e M i n i m i z a t i o n as g i v e n p r e v i o u s l y i n ( 2 . 9 ) ; the weight adjustment e l i m i n a t e s the e s t i m a t i o n e r r o r a t the t r a i n i n g p o i n t w h i l e m i n i m i z i n g the 10 change i n the e s t i m a t e a t o t h e r p o i n t s i n the space. I t r e a l l y o n l y r e p r e s e n t s a s i m p l i f i c a t i o n of the weight adjustment f o r m u l a a t the expense of a more complex e s t i m a t i o n f o r m u l a i n v o l v i n g o r t h o n o r m a l p o l y n o m i a l s . We c a l l t h i s method, O r t h o g o n a l I n t e r f e r e n c e M i n i m i z a t i o n . As i n ( 2 . 1 1 ) , the denominator i n (2.17) s e r v e s as a s c a l i n g v a r i a b l e t h a t a d j u s t s the s t e p s i z e , and can be r e p l a c e d w i t h a p o s i t i v e c o n s t a n t ; Aw = uAf q ( 2 . 1 8 ) To a v o i d d i v e r g e n c e a t any t r a i n i n g p o i n t w i t h i n the space, one must choose u such t h a t , u < 2 m i n { l / ( q T q ) } ( 2 . 1 9 ) S D e s p i t e i t s s i m i l a r i t y t o t h e G r a d i e n t Method, t h i s a l g o r i t h m i s s u p e r i o r due t o the use of o r t h o n o r m a l p o l y n o m i a l terms i n q. As w i t h the G r a d i e n t Method, s e v e r a l i t e r a t i o n s a r e r e q u i r e d a t a p o i n t i f one i s t o e l i m i n a t e the e s t i m a t i o n e r r o r , however, each adjustment causes a m i n i m a l change i n the e s t i m a t e a t o t h e r p o i n t s i n the space, t h u s s p e e d i n g c onvergence. We c a l l t h i s a l g o r i t h m the C e r e b e l l a r Model as i t was o r i g i n a l l y proposed by K l e t t [26] as a model of l e a r n i n g i n t h e mammalian c e r e b e l l u m . D i s c r e t e c o u n t e r p a r t s of s e v e r a l of the a l g o r i t h m s can be d e r i v e d as f o l l o w s : At each t r a i n i n g p o i n t the w e i g h t s a r e a d j u s t e d t o e l i m i n a t e e r r o r i n the e s t i m a t e a t t h e p o i n t w h i l e m i n i m i z i n g the change i n the e s t i m a t e a t p r e v i o u s t r a i n i n g p o i n t s , r a t h e r than m i n i m i z i n g t h e change i n the e s t i m a t e a t a l l o t h e r p o i n t s i n the space as was done f o r ( 2 . 7 ) . The Lagrange 11 m u l t i p l i e r t e c h n i q u e c a n be u s e d t o o b t a i n a we i g h t a d j u s t m e n t f o r m u l a i d e n t i c a l t o t h a t o f I n t e r f e r e n c e M i n i m i z a t i o n , e x c e p t t h a t i n p l a c e o f t h e P m a t r i x t h e r e i s a m a t r i x D w h i c h i s u s e d c n a t t h e n t h t r a i n i n g p o i n t ; n - 1 m T D N = 1 / n Z p . p . 1 = [ ( n - 1 ) D N _ 1 + P n P n ]/n ( 2 . 2 0 ) where, DQ = I ( 2 . 2 1 ) We c a l l t h i s a l g o r i t h m , P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n . As w i t h I n t e r f e r e n c e M i n i m i z a t i o n , two v a r i a n t s can be i m p l e m e n t e d i n w h i c h p o l y n o m i a l t e r m s i n t h e b a s i s v e c t o r a r e made o r t h o g o n a l u s i n g t h e m a t r i x , E „ = D ~ 1 / 2 ( 2 . 2 2 ) n n We c a l l t h e s e v a r i a n t s , P o i n t w i s e O r t h o g o n a l I n t e r f e r e n c e M i n i m i z a t i o n and t h e P o i n t w i s e C e r e b e l l a r M o d e l . The v a r i o u s a l g o r i t h m s a r e summarized i n t a b l e 2 . 1 . 1 2 METHOD NAME ESTIMATE WEIGHT ADJUSTMENT NOTES -,T— — — 1 G r a d i e n t Method f=w p Aw=uAfp _T- - A f P 2 . L e a r n i n g f=w p Aw= I d e n t i f i c a t i o n p T p 3 I n t e r f e r e n c e f=w p Aw=-—• = rj— P= / pp dS M i n i m i z a t i o n p T P p S -T- - M ^ - - -1/2 4 O r t h o g o n a l f=w q Aw= q=Qp, Q=P / I n t e r f e r e n c e q T q M i n i m i z a t i o n 5 C e r e b e l l a r M odel f=w q Aw=uAfq q=Qp A f D n _ 1 p n-1 6 P o i n t w i s e f=w Tp Aw=- D = l / n 2 p p T I n t e r f e r e n c e p T D -1p /=0 ' 1 c n r M i n i m i z a t i o n -T- - M ^ - - -1/2 -TZ'-lZ A t . . /-» —1 7 ^ 17 _ r > 1 / <• n r ' n n P o i n t w i s e O r t h o g . f=w q Aw= q=E np, En=D I n t e r f e r e n c e q T q M i n i m i z a t i o n -,T— — _ _ _ 8 P o i n t w i s e f=w q Aw=uAfq ^ = E n P C e r e b e l l a r M odel T a b l e 2.1 Summary of l e a r n i n g a l g o r i t h m s 2.4 CHOICE OF TEST CONDITIONS FOR COMPARISON OF ALGORITHMS I n c o m p a r i n g t h e v a r i o u s l e a r n i n g a l g o r i t h m s , a c o n v e n i e n t f i r s t s t e p i s t o n o r m a l i z e a l l i n p u t v a r i a b l e s . C o m p a r i s o n under s u c h c o n d i t i o n s a l l o w s one t o see c h a r a c t e r i s t i c s t h a t a r e f u n d a m e n t a l p r o p e r t i e s o f t h e a l g o r i t h m s r a t h e r t h a n j u s t a n o m a l i e s o c c u r r i n g due t o p a r t i c u l a r c o m b i n a t i o n s of i n p u t v a r i a b l e r a n g e s . I n p r a c t i c e , i n p u t v a r i a b l e s a r e n o r m a l l y c o n f i n e d t o a f i n i t e r a n g e o f v a l u e s and t h u s t h e r e i s n o t a 13 l o s s of g e n e r a l i t y i n n o r m a l i z i n g a l l i n p u t v a r i a b l e s . In o r d e r t o a l l o w p o s i t i v e and n e g a t i v e q u a n t i t i e s t o be r e p r e s e n t e d , a l l i n p u t s were n o r m a l i z e d s u c h t h a t , -1 < z / < 1 (2.23) C o m p a r i s o n s were made f o r v a r i o u s c o m b i n a t i o n s o f s y s t e m o r d e r , s, and number of i n p u t v a r i a b l e s , v. I n a l l c a s e s t h e f u n c t i o n whose e s t i m a t e was t o be l e a r n e d was c h o s e n t o be t h e K-G p o l y n o m i a l c o r r e s p o n d i n g t o s and v w i t h a l l c o e f f i c i e n t s s e t t o 1. F o r example, i n t h e c a s e s=2 and v=2, t h e f u n c t i o n whose e s t i m a t e was t o be l e a r n e d was, * 2 2 f = 1 + z 1 + z 2 + + z 2 + z 1 z 2 (2.24) We have f o u n d t h a t r e l a t i v e r a t e s o f c o n v e r g e n c e when e s t i m a t i n g t h e s e f u n c t i o n s a r e r e p r e s e n t a t i v e of r e s u l t s when e s t i m a t i n g f u n c t i o n s d e s c r i b e d by K-G p o l y n o m i a l s w i t h c o e f f i c i e n t s r a n d o m l y g e n e r a t e d f r o m a u n i f o r m d i s t r i b u t i o n between -1 and 1. F i g u r e 2.1 compares, f o r e a c h o f t h e l e a r n i n g a l g o r i t h m s , t h e r e s u l t s o f f i v e t r i a l s where t h e c o e f f i c i e n t s of t h e K-G t a r g e t f u n c t i o n were a l l 1's w i t h t h e r e s u l t s o f f i v e t r i a l s where t h e c o e f f i c i e n t s were r a n d o m l y c h o s e n . A l l o f t h e t r i a l s shown a r e f o r s=3 and v=3, but s e v e r a l o t h e r c a s e s showed s i m i l a r r e s u l t s . The s t o p p i n g r u l e f o r t h e v a r i o u s l e a r n i n g s e q u e n c e s was as f o l l o w s : when t h e e r r o r , A f , a t a t r a i n i n g p o i n t was f o u n d t o be l e s s t h a n 0.01 a t 100 s u c c e s s i v e t r a i n i n g p o i n t s t h e n i t was assumed t h a t t h e e r r o r was l e s s t h a n 0.01 t h r o u g h o u t t h e s p a c e and c o n v e r g e n c e was deemed t o have o c c u r r e d a t t h e f i r s t o f t h e 100 s u c h t r a i n i n g p o i n t s . 100 p o i n t s has p r o v e n t o be an a d e q u a t e t e s t f o r c o n v e r g e n c e i n t h e c a s e s c o n s i d e r e d . S e v e r a l 14 8 h CASE s=3 v=3 METHOD 1 METHOD 2 METHOD 5 METHOD 3 I METHOD 6 RANDOM COEF'S • ALL rs Too" Too" Tio" TRIAL0 NUMBER t.oo Figure 2.1 Convergence r a t e s when ta r g e t f u n c t i o n i s polynomial w i t h randomly chosen c o e f f i c i e n t s versus convergenc ra t e s when t a r g e t f u n c t i o n i s a polynomial with 1's a c o e f f i c i e n t s , f o r case s=3 and v=3 15 t r i a l s were d u p l i c a t e d u s i n g a s i m i l a r s t o p p i n g r u l e where a c c e p t a n c e was r e q u i r e d a t more than 100 s u c c e s s i v e t r a i n i n g p o i n t s . Convergence o c c u r r e d a f t e r the same number of p o i n t s as b e f o r e . The t r a i n i n g p o i n t s were s u c c e s s i v e l y chosen from u n i f o r m d i s t r i b u t i o n s of the i n p u t v a r i a b l e s and one weight adjustment was performed a t each p o i n t . For those l e a r n i n g a l g o r i t h m s t h a t r e q u i r e s p e c i f i c a t i o n of the s t e p s i z e f a c t o r u, u was chosen t o be as l a r g e as a l l o w a b l e w i t h o u t c a u s i n g d i v e r g e n c e t o b e p o s s i b l e . For Method 1, the G r a d i e n t Method, the c o n d i t i o n f o r n o n - d i v e r g e n c e g i v e n i n (2.13) was s a t i s f i e d by c h o o s i n g , u = 2/m . (2.25) For Method 5, the C e r e b e l l a r Model, such an e x p l i c i t f o r m u l a * f o r u was not o b t a i n e d ; f o r each c o m b i n a t i o n of s and v, a * c o r r e s p o n d i n g u was o b t a i n e d t h r o u g h s i m u l a t i o n s where, u = 2 m i n { l / ( q T q ) } = 2 m i n { l / ( p T Q T Q p ) } (2.26) S S * In a l l c a s e s u was o b t a i n e d by c o n s i d e r i n g 2000 randomly g e n e r a t e d i n p u t v a r i a b l e v e c t o r s { z ^ . , , 2 }. The u f a c t o r s so o b t a i n e d a r e t a b u l a t e d i n t a b l e 2.2. I t was v e r i f i e d i n s e v e r a l c a s e s t h a t such a c h o i c e of u i s near o p t i m a l f o r Method 5. F i g u r e 2.2 shows the number of s t e p s r e q u i r e d f o r convergence as a f u n c t i o n of u f o r the c a s e s , s=1 and v=3, s=3 and v=3, s=3 and v=1. The c h o i c e of u g i v e n by (2.26) was near o p t i m a l i n a l l c a s e s e x c e p t f o r s=3 and v=3. I t was noted f o r t h i s and o t h e r c a s e s w i t h a l a r g e number of terms, m, t h a t i t was t y p i c a l l y p o s s i b l e t o speed convergence by i n c r e a s i n g u s l i g h t l y from t h a t , 16 8 • i GAIN FACTOR Figure 2.2 Convergence ra t e s f o r Method 5, the C e r e b e l l a r Model, as a f u n c t i o n of u f o r the cases s=1 and v=3, s=3 and v=3, and s=3 and v=1 17 g i v e n by ( 2 . 2 6 ) . T h i s e n t a i l s a degree of r i s k as w i t h such a c h o i c e of u the a l g o r i t h m was d i v e r g e n t a t some p o i n t s i n the space. Thus i f o c c a s i o n a l groups of s e v e r a l s u c c e s s i v e t r a i n i n g p o i n t s were i n d i v e r g e n t r e g i o n s of the space, the t o t a l a l g o r i t h m would d i v e r g e . T h i s was c o n f i r m e d by s e v e r a l t r i a l s i n v o l v i n g 10 or more a d j u s t m e n t s per t r a i n i n g p o i n t i n which use of a u g r e a t e r than t h a t s p e c i f i e d by (2.26) would l e a d t o * d i v e r g e n c e . Thus i t seems r e a s o n a b l e t o c o n s i d e r u as g i v e n by (2.26) t o be the best c h o i c e when u s i n g Method 5, the C e r e b e l l a r Model. s y 6 .082 .022 5 t 5 .112 .036 .015 e m 4 . 161 .064 .034 0 3 .251 .131 .093 .109 .147 — — r d 2 .455 .333 .327 .448 .676 1 . 08 1 .68 3.00 e r 1 1.0 1.18 1 .64 2.69 4.65 8. 31 14.7 27.7 1 2 3 4 5 6 7 8 Number of V a r i a b l e s T a b l e 2.2 S t e p s i z e f a c t o r s , u, f o r Method 5, the C e r e b e l l a r Model 2.5 COMPARISON OF THE LEARNING ALGORITHMS Comparisons were made of a l l of the l e a r n i n g a l g o r i t h m s i n t a b l e 2.1 except methods 4, 7 and 8. Methods 4 and 7 a r e 18 e q u i v a l e n t t o methods 3 and 6, r e s p e c t i v e l y , and converge i n t h e same number of s t e p s . Method 8 was i n c l u d e d i n t a b l e 2.1 f o r c o m p l e t e n e s s o n l y ; i n a c t u a l a p p l i c a t i o n , method 8 appears t o r e q u i r e too much c o m p u t a t i o n t o be p r a c t i c a l . The c h o i c e of u t o ensure n o n - d i v e r g e n c e depends on E n which i s c h a n g i n g as l e a r n i n g t a k e s p l a c e . Thus the o p t i m a l u cannot be c a l c u l a t e d b e f o r e h a n d f o r method 8. F i g u r e s 2.3 t h r o u g h 2.8 show the number of s t e p s r e q u i r e d f o r the v a r i o u s l e a r n i n g a l g o r i t h m s t o c o n v e r g e . The number of i n p u t v a r i a b l e s , v, ranged from 1 t o 8 and the system o r d e r , s, ranged from 1 t o 6 i n the c a s e s i n v e s t i g a t e d . The d a t a r e p r e s e n t the r e s u l t s f o r s i n g l e t r i a l s e xcept f o r the f i r s t o r d e r c a s e s where the r e s u l t s r e p r e s e n t the average of t h r e e t r i a l s . F i g u r e 2.9 shows the r e d u c t i o n of e s t i m a t i o n e r r o r , M , as a f u n c t i o n of the number of i t e r a t i o n s f o r the c a s e s=3 and v=3 u s i n g t h e v a r i o u s l e a r n i n g a l g o r i t h m s . The graph shows th e magnitude of A f , averaged over i n t e r v a l s of 100 t r a i n i n g i t e r a t i o n s . For c a s e s h a v i n g a low system o r d e r t h e r e i s not much d i f f e r e n c e between most of the v a r i o u s a l g o r i t h m s ; t h e d i f f e r e n c e s become pronounced o n l y when h i g h e r o r d e r systems a r e c o n s i d e r e d . Method 1, the G r a d i e n t Method, i s c o n s i s t e n t l y t h e p o o r e s t a l g o r i t h m . Method 3, I n t e r f e r e n c e M i n i m i z a t i o n , i s c o n s i s t e n t l y much b e t t e r than the G r a d i e n t Method; f o r 3 r d , 4 t h , 5 t h and 6 t h o r d e r systems the f a c t o r of improvement i s a p p r o x i m a t e l y 9, 20, 24 and 28, r e s p e c t i v e l y . Method 2, L e a r n i n g I d e n t i f i c a t i o n , i s s i m i l a r i n performance t o the G r a d i e n t Method 19 8 h F I R S T ORDER C A S E S 8 • METHOD 6 K METHOD 5 • METHOD 3 METHOD 2 • METHOD 1 • NUMBER OF 3.00 4.00 8.00 NUMBER OF V A R I A B L E S 7700 8.00 F i g u r e 2.3 Convergence r a t e s f o r v a r i o u s methods when s=1 20 S E C O N D ORDER C A S E S 81 • 1 co8J s I ID Z 8 K • METHOD 6 METHOD 5 METHOO 3 METHOD 2 METHOD 1 NUMBER OF TERMS Too •V.00 Too" 17oo 4.00 8.00 N U M B E R OF V A R I A B L E S 8.00 7.00 F i g u r e 2.4 Convergence r a t e s f o r v a r i o u s methods when s=2 21 T H I R D ORDER C A S E S 8" I 8 8 METHOD 6 METHOD 5 METHOD 3 METHOD 2 METHOD 1 NUMBER OF TERMS 7. oo •f^o i!oo a'.oo Too BTOO i.oo s.oo NUMBER OF VARIABLES 6.00 7.00 gure 2.5 Convergence r a t e s f o r v a r i o u s methods when s=3 FOURTH ORDER CASES X • METHOD 6 METHOD S METHOD 3 METHOD 2 METHOD 1 NUMBER OF TERMS 8.00 4.00 8.00 N U M B E R OF V A R I A B L E S Too" ToT 6.00 F i g u r e 2.6 Convergence r a t e s f o r v a r i o u s methods when s=4 23 F I F T H ORDER C A S E S METHOD METHOD METHOD METHOD METHOD 6 5 3 2 1 NUMBER OF TERMS 7. oo ~r "NUMBER OF VARIABLES BTOO" l.oo 7.00 F i g u r e 2.7 C o n v e r g e n c e . r a t e s f o r v a r i o u s methods when s=5 24 S I X T H ORDER C A S E S X • METHOD 6 METHOD 5 METHOD 3 METHOD 2 METHOD 1 NUMBER OF TERMS Tib" 7.00 iToo iToo BTOO NUMBER OF VARIABLES To7 Figure 2.8 Convergence r a t e s f o r var i o u s methods when s=6 25 (L60 iilo ITio 2~*0 i.oo - 103 TRAINING POINTS o ^ 0 . 0 0 Figure 2.9 Reduction of es t i m a t i o n e r r o r , Af, as a fu n c t i o n of the number of t r a i n i n g p o i n t s f o r the case s=3 and v«=3 using the var i o u s methods (estimation e r r o r averaged over each 100 i t e r a t i o n i n t e r v a l ) 26 i n c a s e s where the system o r d e r , s, i s l a r g e r than the number of i n p u t v a r i a b l e s , v, however, L e a r n i n g I d e n t i f i c a t i o n i s much b e t t e r when v>s, a p p r o a c h i n g Method 3 i n p e r f o r m a n c e . Method 5, the C e r e b e l l a r Model, i s ' i n t e r m e d i a t e i n performance t o the G r a d i e n t Method and I n t e r f e r e n c e M i n i m i z a t i o n , much l i k e L e a r n i n g I d e n t i f i c a t i o n ; e x cept t h a t when s>v t h e C e r e b e l l u m Model i s b e t t e r than L e a r n i n g I d e n t i f i c a t i o n . Method 6, P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n , i s c o n s i s t e n t l y the b e s t method; f o r c a s e s where v=1 t h e P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n i s s i m i l a r t o I n t e r f e r e n c e M i n i m i z a t i o n and f o r c a s e s where v>1 i t i s b e t t e r by a f a c t o r of a p p r o x i m a t e l y 2, even when s=1 . The c h o i c e of which a l g o r i t h m t o use depends on the c o m p l e x i t y of i m p l e m e n t a t i o n as w e l l as t h e a c h i e v a b l e p e rformance. Method 3, I n t e r f e r e n c e M i n i m i z a t i o n , o f f e r s performance second o n l y t o Method 6, P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n . As w i l l be shown i n s e c t i o n 2.6, t h e c o m p l e x i t y of I n t e r f e r e n c e M i n i m i z a t i o n i s not as g r e a t as i t might seem, thus s i m p l i f y i n g i m p l e m e n t a t i o n . P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n , however, e n t a i l s much more c o m p u t a t i o n than the o t h e r a l g o r i t h m s as the m a t r i x D R must be updated and i n v e r t e d y i e l d i n g D 1 a t each t r a i n i n g p o i n t . P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n i s b e t t e r t h a n I n t e r f e r e n c e M i n i m i z a t i o n by o n l y a f a c t o r of 2 and thus t y p i c a l l y does not w a r r a n t c o n s i d e r a t i o n . There a r e , however, c e r t a i n s i t u a t i o n s i n which the use of P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n i s a p p r o p r i a t e . T h i s i s shown i n s e c t i o n 2.7. 27 2.6 IMPLEMENTATION CONSIDERATIONS FOR INTERFERENCE MINIMIZATION I n t e r f e r e n c e M i n i m i z a t i o n a c h i e v e s i t s improved performance over the G r a d i e n t Method and L e a r n i n g I d e n t i f i c a t i o n a t the expense of a more complex a l g o r i t h m . The i n c r e a s e i n c o m p l e x i t y , however, i s not as g r e a t as i t appears a t f i r s t . I f the i n p u t v a r i a b l e s a r e n o r m a l i z e d as i n (2.23), then t h e m a t r i x P 1 i n (2.9) becomes s p a r s e . P 1 has a s p e c i a l form t h a t a l l o w s i t s s p a r s e n e s s t o be e a s i l y u t i l i z e d t o speed up t h e c o m p u t a t i o n s r e q u i r e d i n I n t e r f e r e n c e M i n i m i z a t i o n . C o n s i d e r the elements of t h e m a t r i x P. The element P.. i s i J g i v e n by the i n t e g r a l , P.. = J p . ( z ) p . ( z ) 5S (2.27) J S 1 From the d e f i n i t i o n of P/ C(z) g i v e n i n (2.2), (2.3) and (2.4) we have, 1 1 v , . p.. = n z . * e / * e y * ' 5z,..6z (2.28) l J -1 -1 *=0 * i v I f i t o c c u r s t h a t (e^ jc+ej )̂mod2=1 f o r any v a r i a b l e , z^, then P. .=0. At l e a s t h a l f of the elements P. . a r e z e r o and f o r v>>s the v a s t m a j o r i t y of the elements P̂ . a r e z e r o . An e x a m i n a t i o n of the nonzero elements i n P r e v e a l s a s i g n i f i c a n t p a t t e r n . I f the p o l y n o m i a l s i n the s e t { p ^ ( z ) } a r e o r d e r e d a p p r o p r i a t e l y , the s p e c i a l banded n a t u r e of P w i l l become a p p a r e n t . The m a t r i x has a " k i t e " form i n which a l l t h e nonzero elements appear as a s e r i e s of s m a l l e r s y m m e t r i c a l m a t r i c e s a l o n g the d i a g o n a l . O p e r a t i o n s w i t h P can thus be done as s m a l l e r t a s k s performed on the i n d i v i d u a l s m a l l e r m a t r i c e s . T h i s means t h a t P 1 and Q have 28 the same " k i t e " form as P. As an example, f i g u r e 2.10 shows the -1 format of P and { p , ( z ) } f o r the case s=3 and v=3. {p*(i>} 1 Z 1 Z 1 z 2 z 2 2 3 Z 3 ZT Z 1 Z 1 Z 1 Z 1 2 2 Z 2 Z 1 Z 3 Z 3 Z 2 Z 2 Z 1 Z 1 Z 2 Z 2 Z 2 Z 2 Z 3 Z 3 Z 3 Z 3 Z 1 Z 1 Z 3 Z 2 Z 2 Z 3 Z 3 Z 3 Z 1 Z 3 Z 2 Z 3 z z z .-1 + - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - + - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - + - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - + - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - + - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - + - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - + - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - + - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - + - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - - + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + F i g u r e 2.10 P a t t e r n of z e r o and non-zero elements of m a t r i x P f o r the case s=3 and v=3 (+ = p o s i t i v e , - = n e g a t i v e , 0 = z e r o ) -1 An e m p i r i c a l method of d e s c r i b i n g the form of P 1 has been found. C o n s i d e r a system of o r d e r s, h a v i n g v i n p u t v a r i a b l e s . The number of terms i n {p ^ ( z ) } and hence the s i z e of P i s , (v+s)! m(v,s) = (2.29) v ! s ! I t i s u s e f u l t o d e f i n e the v a l u e of t h i s f u n c t i o n when the arguments a r e z e r o or n e g a t i v e ; m(-v,s) = m(v,-s) = 0 (2.30) m(0,s) = m(s ,0 ) = 1 (2.31) 29 f o r v,s p o s i t i v e . The s i z e and m u l t i p l i c i t y of the s m a l l e r m a t r i c e s i d e n t i t y , f o r m i n g P 1 can then be o b t a i n e d from the terms of the s\2+1 m(v,s) = Z r . d . (2.32) i = 1 1 1 where \ r e p r e s e n t s i n t e g e r d i v i s i o n and where, r. = m(v-s-1+2/,s+2-2/) (2.33) d / = m(v,/-1) (2.34) In the m a t r i x P 1 , the number of d i f f e r e n t s i z e s of s u b m a t r i c e s a l o n g the d i a g o n a l c o r r e s p o n d s t o the number of terms i n the summation of ( 2 . 3 2 ) . There a r e s\2+1 unique s u b m a t r i x s i z e s . The s i z e of each i s g i v e n by the f a c t o r d^ and t h e number of o c c u r r e n c e s of each i s g i v e n by the f a c t o r r^. . The t o t a l number of nonzero elements i n P 1 i s , s\2+1 , n z ( v , s ) = Z r . d . (2.35) i = 1 2 For a system of o r d e r s, n z ( v , s ) i s much s m a l l e r than m(v,s) f o r v>>s, growing i n s i z e as v i s i n c r e a s e d i n manner more l i k e t h a t of m ( v , s ) . T h i s i s shown f o r the case s=3 i n f i g u r e 2.11. The m a t r i x P 1 thus becomes almost d i a g o n a l i n form f o r v>>s. N o r m a l i z a t i o n of the i n p u t v a r i a b l e s as i n (2.23) thus s i g n i f i c a n t l y reduces the number of m u l t i p l i c a t i o n s r e q u i r e d when implementing I n t e r f e r e n c e M i n i m i z a t i o n . 30 F i g u r e 2.11 T o t a l number of e l e m e n t s , number of non-ze elements and o r d e r of m a t r i x P 1 f o r c a s e s where s=3 31 2.7 APPLICATIONS WHERE POINTWISE INTERFERENCE MINIMIZATION IS APPROPRIATE In t h e d e r i v a t i o n o f I n t e r f e r e n c e M i n i m i z a t i o n , t h e e r r o r a t a t r a i n i n g p o i n t was e l i m i n a t e d w h i l e m i n i m i z i n g t h e change i n t h e e s t i m a t e a t o t h e r p o i n t s i n t h e s p a c e . An e q u a l w e i g h t i n g was assumed f o r t h e s e o t h e r p o i n t s . A more g e n e r a l a p p r o a c h i s t o a l l o w a n o n - u n i f o r m w e i g h t i n g [ 2 6 ] , T h i s would be a p p r o p r i a t e when a c c u r a c y i s more c r i t i c a l a t c e r t a i n r e g i o n s o f t h e s p a c e t h a n o t h e r s . W i t h t h i s more g e n e r a l a p p r o a c h e q u a t i o n (2.7) becomes, c = J h ( S ) ( A w T p ) 2 5S + a ( A w T p - A f ) (2.36) S where h ( S ) i s a s t r i c t l y p o s i t i v e w e i g h t i n g f u n c t i o n . The optimum w e i g h t a d j u s t m e n t f o r m u l a r e m a i n s as b e f o r e e x c e p t t h a t t h e d e f i n i t i o n o f t h e m a t r i x P becomes, P = J h ( S ) p p T 8S (2.37) S One a p p l i c a t i o n t h a t demands n o n - u n i f o r m w e i g h t i n g i s when t h e i n p u t v a r i a b l e s have a n o n - u n i f o r m p r o b a b i l i t y d i s t r i b u t i o n and one w i s h e s t o improve t h e e s t i m a t e most r a p i d l y i n t h e r e g i o n s o f t h e s p a c e t h a t a r e most p r o b a b l e . I t i s t h e n a p p r o p r i a t e t o use t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e i n p u t v a r i a b l e s a s h ( S ) . I f h ( S ) i s an even f u n c t i o n of t h e i n p u t v a r i a b l e s f o r m i n g S t h e n P i s s p a r s e a s d e s c r i b e d p r e v i o u s l y . I f t h e d i s t r i b u t i o n h ( S ) i s n o t known b e f o r e h a n d t h e n i t i s a p p r o p r i a t e t o use P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n a s i t e f f e c t i v e l y l e a r n s h ( S ) . I n t h e l i m i t , t h e m a t r i x D n c o n v e r g e s t o be a s c a l e r m u l t i p l e of t h e c o r r e s p o n d i n g m a t r i x P . F o r 32 example, i f the i n p u t v a r i a b l e s a r e n o r m a l i z e d as i n (2.23) t h e n , D = P/2 (2.38) 0 0 P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n combines the l e a r n i n g of P w i t h the l e a r n i n g of the e s t i m a t e . 2.8 SUMMARY A f a m i l y of l e a r n i n g a l g o r i t h m s has been i n t r o d u c e d and t h e i r convergence c h a r a c t e r i s t i c s compared. The method of I n t e r f e r e n c e M i n i m i z a t i o n has been found t o be s u p e r i o r t o e i t h e r the G r a d i e n t Method or L e a r n i n g I d e n t i f i c a t i o n . F u r t h e r , n o r m a l i z a t i o n of the i n p u t v a r i a b l e s s i m p l i f i e s the i m p l e m e n t a t i o n of I n t e r f e r e n c e M i n i m i z a t i o n . P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n has been found t o be the best a l g o r i t h m on the b a s i s of r a p i d c onvergence, however, the improvement over I n t e r f e r e n c e M i n i m i z a t i o n i s s m a l l and would not seem t o j u s t i f y the s i g n i f i c a n t i n c r e a s e i n a l g o r i t h m c o m p l e x i t y . One s i t u a t i o n where i t may be a p p r o p r i a t e t o use P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n i s when one wishes t o have most r a p i d convergence i n t h o s e r e g i o n s of the i n p u t v a r i a b l e space t h a t a r e most p r o b a b l e and the p r o b a b i l i t y d i s t r i b u t i o n , h ( S ) , i s unknown. P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n e f f e c t i v e l y l e a r n s t h i s d i s t r i b u t i o n . 33 3 S E L F - L E A R N E D C O N T R O L O F A T W O L I N K M A N I P U L A T O R One of the p r i n c i p l e c o n t e n t i o n s of t h i s t h e s i s , i s t h a t l e a r n i n g a l g o r i t h m s can be used t o a c h i e v e s e l f - l e a r n e d c o n t r o l of a m a n i p u l a t o r . I t had been h y p o t h e s i z e d t h a t w e i g h t e d sums of p o l y n o m i a l terms c o u l d model t h e m u l t i - v a r i a t e , n o n - l i n e a r f u n c t i o n s t h a t d e s c r i b e the dynamics and k i n e m a t i c s of a m a n i p u l a t o r , and t h a t I n t e r f e r e n c e M i n i m i z a t i o n or r e l a t e d methods c o u l d be used t o l e a r n the w e i g h t i n g c o e f f i c i e n t s , w i t h o u t r e c o u r s e t o a n a l y s i s of the m a n i p u l a t o r , and thus l e a r n the f u n c t i o n s n e c e s s a r y t o a c h i e v e c o n t r o l . 3.1 RESOLVED ACCELERATION CONTROL WITH CARTESIAN PATH SPECIFICATION G i v e n t h a t one has the a b i l i t y t o l e a r n t h e m u l t i - v a r i a t e , n o n - l i n e a r f u n c t i o n s d e s c r i b i n g a m a n i p u l a t o r , i t i s then n e c e s s a r y t o u t i l i z e t h i s c a p a b i l i t y w i t h a s u i t a b l e c o n t r o l t e c h n i q u e . A f a m i l y of s u i t a b l e methods [20,29,30] a r e th o s e c a l l e d v a r i o u s l y , I n v e r s e P l a n t , I n v e r s e System, I n v e r s e Problem , e t c . One example of th e s e methods t h a t has been a p p l i e d t o m a n i p u l a t o r c o n t r o l i s R e s o l v e d A c c e l e r a t i o n C o n t r o l as d e s c r i b e d by Luh e t a l [ 3 0 ] , I t i s w i t h i n t h i s c o n t r o l framework t h a t we propose t o a p p l y l e a r n i n g a l g o r i t h m s t o l e a r n m a n i p u l a t o r c o n t r o l . In g e n e r a l , the dynamics of a m a n i p u l a t o r can be d e s c r i b e d by a group of f u n c t i o n a l r e l a t i o n s h i p s t h a t we c a l l the d i r e c t dynamics; 34 a d i r . d y n ( 7 , a , a ) (3.1) where a / a 6t (3.2) a / a 6t (3.3) a i s a v e c t o r of g e n e r a l i z e d j o i n t a n g l e s and 7 i s a v e c t o r of g e n e r a l i z e d t o r q u e s or f o r c e s , as a p p r o p r i a t e , t h a t a r e a p p l i e d t o the j o i n t s . A p p l i c a t i o n of t o r q u e 7 when the m a n i p u l a t o r i s i n p o s i t i o n a and moving w i t h v e l o c i t y a w i l l r e s u l t i n a c c e l e r a t i o n a. The e x a c t form of the f u n c t i o n s c o m p r i s i n g the d i r e c t dynamics r e l a t i o n s h i p s a r e dependent on the m a n i p u l a t o r c o n f i g u r a t i o n as t o t y p e and number of l i n k s , and the l i n k l e n g t h s , l i n k masses and mass d i s t r i b u t i o n s , j o i n t f r i c t i o n and g r a v i t y . The m a n i p u l a t o r can be c o n t r o l l e d u s i n g i t s i n v e r s e dynamics which a r e d e s c r i b e d by a group of f u n c t i o n a l r e l a t i o n s h i p s t h a t we c a l l the i n v e r s e dynamics; 7 = i n v . d y n ( a , a , a ) (3.4) where, a = 6 a ( t ) (3.5) 8t a = 6 a ( t ) (3.6) 6 t To a c h i e v e a c c e l e r a t i o n a when moving a t v e l o c i t y a and i n p o s i t i o n a, t o r q u e 7 must be a p p l i e d , a and a can be o b t a i n e d be d i f f e r e n t i a t i n g the p a t h s p e c i f i c a t i o n , a ( t ) . In g e n e r a l i t i s not easy t o o b t a i n t h e s e r e l a t i o n s h i p s , e s p e c i a l l y f o r complex m a n i p u l a t o r s h a v i n g s e v e r a l degrees of freedom. I f t h e s e r e l a t i o n s h i p s a r e known e x a c t l y , i t would appear 35 t o be p o s s i b l e t o c o n t r o l a m a n i p u l a t o r by open l o o p c o n t r o l . G i v e n t h a t one s t a r t s w i t h the i n i t i a l l y d e s i r e d p o s i t i o n e q u a l t o the a c t u a l p o s i t i o n , then g i v e n the d e s i r e d p a t h o.^(t) one can c a l c u l a t e t o r q u e 7 ( t ) u s i n g the i n v e r s e dynamics r e l a t i o n s h i p s , a p p l y r ( t ) t o the m a n i p u l a t o r and cause the m a n i p u l a t o r t o move such t h a t a ( t ) e q u a l s a^(t). T h i s i s e x p r e s s e d m a t h e m a t i c a l l y ' as a d e f i n i t i o n of the d i r e c t and i n v e r s e dynamics, namely, a ( t ) = d i r . d y n ( i n v . d y n ( a ^ , a ^ , a ^ ) , a , a ) = a ^ ( t ) (3.7) g i v e n t h a t , a ( t Q ) = a d ( t Q ) (3.8) and, a ( t Q ) = a d ( t Q ) (3.9) In p r a c t i c e such a c o n t r o l t e c h n i q u e i s poor due t o f i n i t e p r e c i s i o n i n c a l c u l a t i o n s , i m p r e c i s e knowledge of parameters i n the i n v e r s e dynamics r e l a t i o n s h i p s such as j o i n t masses, j o i n t l e n g t h s , e t c . and n o i s e - l i k e d i s t u r b a n c e s of the m a n i p u l a t o r by the environment. Even w i t h a v e r y a c c u r a t e a p p r o x i m a t i o n of the i n v e r s e dynamics r e l a t i o n s h i p s and m i n i m a l e n v i r o n m e n t a l d i s t u r b a n c e s , t r a c k i n g e r r o r s w i l l t end t o accumulate and cause f u r t h e r t r a c k i n g e r r o r s when u s i n g open l o o p c o n t r o l . Good c o n t r o l can be a c h i e v e d by u s i n g feedback t o a d j u s t a p p l i e d t o r q u e s t o c o r r e c t f o r p o s i t i o n and v e l o c i t y e r r o r s ; 7 = i n v . d y n ( a ( j + k 1 ( a ^ - a ) + k j ( a ^ - a ) ,a,a) (3.10) A p p l i c a t i o n of such t o r q u e s r e s u l t s i n m a n i p u l a t o r motion as f o l l o w s , a - d i r . d y n ( i n v . d y n ( a d + k 1 ( a d - a ) + k 2 ( a d - a ) , a , a ) , a , a ) ( 3 . 1 1 ) 36 which by the d e f i n i t i o n of the d i r e c t and i n v e r s e dynamics y i e l d s , a = a^+k 1 (a^-a) +k 2(a (j-a) (3.12) 0 = ( a d ~ a ) + k 1 ( a d - a ) + k 2 ( a d - a ) (3.13) D e f i n i n g the p o s i t i o n e r r o r a s , e = a, - a (3.14) a a r e s u l t s i n a second o r d e r , l i n e a r , homogeneous d i f f e r e n t i a l e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t s d e s c r i b i n g the change i n p o s i t i o n e r r o r w i t h t i m e , 0 = e + k,e + k 0 e (3.15) a 1 a 2 a For s t a b i l i t y , or e q u i v a l e n t l y t o have e*a d e c r e a s e w i t h t i m e , one must choose k^O, and f o r a c r i t i c a l l y damped response one 2 must choose k 1 =4k 2. T h i s t e c h n i q u e f o r m a n i p u l a t o r c o n t r o l has a l r e a d y been demonstrated by Luh e t a l [30] f o r a complex m a n i p u l a t o r whose dynamics were known a n a l y t i c a l l y . One of the main g o a l s of t h i s work i s t o dete r m i n e whether t h i s c o n t r o l t e c h n i q u e can be s u c c e s s f u l l y a p p l i e d when the i n v e r s e dynamics r e l a t i o n s h i p s a r e l e a r n e d w i t h o u t r e c o u r s e t o a n a l y s i s of the dynamics of a m a n i p u l a t o r . In r e s o l v e d a c c e l e r a t i o n c o n t r o l as put f o r w a r d by Luh e t a l , t h e p a t h s p e c i f i c a t i o n i s i n j o i n t c o o r d i n a t e s . T y p i c a l l y , i t would be more d e s i r a b l e t o s p e c i f y the m a n i p u l a t o r p a t h i n ano t h e r c o o r d i n a t e system which i s more s u i t a b l e f o r t h e t a s k a t hand. F or example, one may d e s i r e t o s p e c i f y the p a t h of the m a n i p u l a t o r end p o i n t i n C a r t e s i a n c o o r d i n a t e s , r a t h e r than s p e c i f y the p a t h of the m a n i p u l a t o r j o i n t s i n j o i n t c o o r d i n a t e s . 37 M a n i p u l a t o r end p o i n t c o n t r o l i n C a r t e s i a n c o o r d i n a t e s can be a c h i e v e d u s i n g the i n v e r s e k i n e m a t i c s of the m a n i p u l a t o r i n c o n j u n c t i o n w i t h the i n v e r s e dynamics of the m a n i p u l a t o r . The i n v e r s e k i n e m a t i c s a r e d e s c r i b e d by a group of f u n c t i o n a l r e l a t i o n s h i p s as f o l l o w s : a - p o s . i n v . k i n ( x ) (3.16) a = v e l . i n v . k i n ( x , a ) (3.17) a = a c c . i n v . k i n ( x , a , a ) (3.18) G i v e n the v e c t o r s x, x and x d e s c r i b i n g t h e m a n i p u l a t o r end p o i n t p o s i t i o n , v e l o c i t y and a c c e l e r a t i o n i n C a r t e s i a n c o o r d i n a t e s , one can o b t a i n the v e c t o r s a, a and a d e s c r i b i n g the m a n i p u l a t o r j o i n t p o s i t i o n s , v e l o c i t i e s and a c c e l e r a t i o n s . T y p i c a l l y , the c o m b i n a t i o n of j o i n t a n g l e s t h a t w i l l p l a c e the m a n i p u l a t o r end p o i n t a t a g i v e n C a r t e s i a n p o s i t i o n i s not un i q u e . I t i s thus n e c e s s a r y t o a p p l y c o n s t r a i n t s t o p e r m i t t h e i n v e r s e k i n e m a t i c s t o be d e s c r i b e d by f u n c t i o n a l r e l a t i o n s h i p s . For example, a r e v o l u t e j o i n t may be c o n s t r a i n e d t o r e v o l v e t h r o u g h l e s s than it r a d i a n s . Such c o n s t r a i n t s o f t e n p h y s i c a l l y e x i s t because of the manner i n which m a n i p u l a t o r s a r e c o n s t r u c t e d ; r e v o l u t e j o i n t s o f t e n a r e r e s t r i c t e d t o l e s s than a f u l l c i r c l e of r o t a t i o n . For m a n i p u l a t o r s h a v i n g more than t h r e e degrees of freedom, one t y p i c a l l y wants t o c o n t r o l more than j u s t the end p o i n t of the m a n i p u l a t o r . For example, w i t h a s i x degree of freedom m a n i p u l a t o r , one can s p e c i f y the end p o i n t p a t h and the o r i e n t a t i o n of the l a s t l i n k which i s t y p i c a l l y thought of as the hand or t o o l . In t h i s c a s e , t h e c o n s t r a i n t s d e s c r i b i n g the 3 8 o r i e n t a t i o n of the hand, which a r e n e c e s s a r y t o p e r m i t the i n v e r s e k i n e m a t i c s t o be d e s c r i b e d by f u n c t i o n a l r e l a t i o n s h i p s , can be viewed as p a r t of the p a t h s p e c i f i c a t i o n r a t h e r than as c o n s t r a i n t s . The i n v e r s e k i n e m a t i c s and the i n v e r s e dynamics can be combined t o form a group of f u n c t i o n a l r e l a t i o n s h i p s t h a t we c a l l the C a r t e s i a n i n v e r s e dynamics; 7 = c a r t . i n v . d y n ( x , x , x ) (3.19) where, £ = 6 x ( t ) (3.20) 6t x = 6 x ( t ) (3.21) 6t S i m i l a r l y , the d i r e c t k i n e m a t i c s and the d i r e c t dynamics can be combined t o form a group of f u n c t i o n a l r e l a t i o n s h i p s t h a t we c a l l the C a r t e s i a n d i r e c t dynamics; x = c a r t . d i r . d y n ( 7 , x , x ) (3.22) where, x = J x 6t (3.23) x = / x 5t (3.24) The d i r e c t k i n e m a t i c s of the m a n i p u l a t o r a r e d e s c r i b e d by a group of f u n c t i o n a l r e l a t i o n s h i p s as f o l l o w s , x = p o s . d i r . k i n ( a ) (3.25) x = v e l . d i r . k i n ( a , a ) (3.26) x = a c c . d i r . k i n ( a , a , a ) (3.27) G i v e n t h e v e c t o r s a, a and a d e s c r i b i n g t h e m a n i p u l a t o r j o i n t p o s i t i o n s , v e l o c i t i e s and a c c e l e r a t i o n s , one can o b t a i n the v e c t o r s x, x and x d e s c r i b i n g the m a n i p u l a t o r end p o i n t 39 p o s i t i o n , v e l o c i t y and a c c e l e r a t i o n . I n a manner analagous t o t h a t d e s c r i b e d p r e v i o u s l y , one can c o n t r o l t h e m a n i p u l a t o r u s i n g the C a r t e s i a n i n v e r s e dynamics a l o n g w i t h feedback t o c o r r e c t p o s i t i o n and v e l o c i t y e r r o r s ; T = c a r t . i n v . d y n ( x ( j + k 1 ( x ^ - x ) + k 2 (x^-x) ,x,x) (3.28) x= c a r t . d i r . d y n ( c a r t . i n v . d y n ( x ^ + k 1 ( x ^ - x ) + k 2 ( x ^ - x ) , x , x ) , x , x ) (3.29) x = x ( 3 + k 1 ( x d - x ) + k 2 ( x d ~ x ) (3.30) 0 = ( x d - x ) + k 1 ( x d - x ) + k 2 ( x d ~ x ) (3.31) D e f i n i n g the C a r t e s i a n p o s i t i o n e r r o r a s , e x = x d ~ x (3.32) - one can choose k 1 and k 2 as b e f o r e t o a c h i e v e s t a b i l i t y and c r i t i c a l damping of C a r t e s i a n p o s i t i o n e r r o r s . The C a r t e s i a n i n v e r s e dynamics r e l a t i o n s h i p s a r e more complex than the i n v e r s e arm r e l a t i o n s h i p s due t o the i n c l u s i o n of a c o o r d i n a t e t r a n s f o r m a t i o n t h r o u g h use of the i n v e r s e k i n e m a t i c s . Such would be the case i f one wanted t o use t h i s a pproach i n c o n j u n c t i o n w i t h o t h e r c o o r d i n a t e systems, such as c y l i n d r i c a l or s p h e r i c a l , t h a t might be more c o n v e n i e n t g i v e n the t a s k a t hand. The key p o i n t i s t h a t c o n t r o l i s d e s i r e d w i t h r e s p e c t t o a c o o r d i n a t e system t h a t may not be the most n a t u r a l one f o r d e s c r i b i n g m a n i p u l a t o r dynamics. A second major g o a l of t h e work i s thu s t o d e t e r m i n e whether l e a r n e d r e s o l v e d a c c e l e r a t i o n c o n t r o l can be s u c c e s s f u l l y a p p l i e d u s i n g path s p e c i f i c a t i o n i n a t a s k o r i e n t e d c o o r d i n a t e system such as C a r t e s i a n c o o r d i n a t e s . When a p p l y i n g r e s o l v e d a c c e l e r a t i o n c o n t r o l u s i n g 40 C a r t e s i a n c o o r d i n a t e s f o r end p o i n t p a t h s p e c i f i c a t i o n , i t i s n e c e s s a r y t o observe the end p o i n t C a r t e s i a n p o s i t i o n i n o r d e r t o use feedback t o c o r r e c t p o s i t i o n and v e l o c i t y e r r o r s . I d e a l l y , one c o u l d use a v i s i o n system t o measure the end p o i n t p a t h . E q u i v a l e n t l y , one can measure j o i n t p o s i t i o n s , v e l o c i t i e s and a c c e l e r a t i o n s and use the d i r e c t k i n e m a t i c r e l a t i o n s h i p s t o d e t e r m i n e the m a n i p u l a t o r end p o i n t p a t h . The d i r e c t k i n e m a t i c s can be d e t e r m i n e d by a g e n e r a l t e c h n i q u e [44,46] and they a r e much s i m p l e r t o d e r i v e than the i n v e r s e k i n e m a t i c s . T h e i r e v a l u a t i o n r e q u i r e s n o t h i n g more than m u l t i p l i c a t i o n and summation of s i n e and c o s i n e terms. N e v e r t h e l e s s , such a method r e q u i r e s m a n i p u l a t o r a n a l y s i s and time consuming, m a n i p u l a t o r - s p e c i f i c c a l c u l a t i o n s . W i t h I n t e r f e r e n c e M i n i m i z a t i o n i t s h o u l d be p o s s i b l e t o l e a r n the f u n c t i o n a l r e l a t i o n s h i p s t h a t d e s c r i b e the d i r e c t k i n e m a t i c s . T h i s would a l l o w C a r t e s i a n p a t h c o n t r o l of m a n i p u l a t o r w i t h o u t the use of a v i s i o n system and w i t h o u t r e s o r t t o a n a l y s i s and c a l c u l a t i o n of the d i r e c t k i n e m a t i c s . The m a n i p u l a t o r end p o i n t c o u l d then be o b t a i n e d from s i m p l e j o i n t p a t h measurements. 3.2 A TWO LINK MANIPULATOR In o r d e r t o t e s t the f e a s i b i l i t y of l e a r n e d c o n t r o l of a m a n i p u l a t o r , a s i m p l e two l i n k m a n i p u l a t o r was chosen as a t e s t v e h i c l e . The s t r u c t u r e of the two l i n k m a n i p u l a t o r and a s s o c i a t e d c o o r d i n a t e r e f e r e n c e frames a r e shown i n f i g u r e 3.1. 41 F i g u r e 3.1 A two l i n k manipulator Such a simple manipulator was chosen f o r two b a s i c reasons: F i r s t , i t s dynamics and kinematics are known i n a n a l y t i c a l form. T h i s permits the arm to be simulated on a computer, with a l l the advantages t h a t s i m u l a t i o n b r i n g s ; no time consuming hardware development, exact s p e c i f i c a t i o n of parameters, a c c u r a t e o b s e r v a t i o n of a l l v a r i a b l e s , and no p h y s i c a l damage when mistakes are made. A l s o , c o n t r o l performance when using the a n a l y t i c a l dynamics and kinematics can serve as benchmarks a g a i n s t which one can compare c o n t r o l performances when using l e a r n e d dynamics and k i n e m a t i c s . Secondly, as the manipulator was t o be simulated, i t had to be simple t o a v o i d r e q u i r i n g i n o r d i n a t e amounts of computer time f o r ac c u r a t e s i m u l a t i o n of 42 numerous movements. In s p i t e of the s i m p l i c i t y of the two l i n k m a n i p u l a t o r , i t s k i n e m a t i c s and dynamics a r e d e s c r i b e d by h i g h l y n o n - l i n e a r , m u l t i - v a r i a t e f u n c t i o n s as i s t y p i c a l of more complex m a n i p u l a t o r s . We e x p e c t t h a t r e s u l t s u s i n g the two l i n k m a n i p u l a t o r w i l l be r e p r e s e n t a t i v e of p o s s i b l e r e s u l t s u s i n g more complex m a n i p u l a t o r s . The d i r e c t k i n e m a t i c s of the two l i n k m a n i p u l a t o r can be o b t a i n e d by u s i n g b a s i c t r i g o n o m e t r y t o c a l c u l a t e the C a r t e s i a n p o s i t i o n of the end p o i n t as a f u n c t i o n of the j o i n t a n g l e s . These f u n c t i o n s a r e then d i f f e r e n t i a t e d t o y i e l d the C a r t e s i a n v e l o c i t y and a c c e l e r a t i o n of the end p o i n t i n terms of j o i n t a n g l e s , a n g u l a r v e l o c i t i e s and a n g u l a r a c c e l e r a t i o n s . The d i r e c t k i n e m a t i c s a r e as f o l l o w s : x 1 • 1 1 s i n ( a 1 ) + l ^ i n t c ^ + a ^ (3. 33) X 2 • - l ^ o s f a 1 ) - l 2 c o s ( a 1 + a 2 ) (3. 34) x 1 = = l 1 c o s ( a 1 ) d 1 + l 2 c o s ( a 1 + a 2 ) ( d 1 + d l ) (3. 35) x 2 = • 1 1 s i n ( a 1 ) d 1 + l 2 s i n ( a 1 + a 2 ) ( d 1 + d 2 ) (3. 36) x 1 -= l 1 c o s ( a l ) d 1 + l j c o s t c ^ + c ^ ) ( d 1 + 'd2 ) - 1 2 s i n ( a 1 ) d 1 - l 2 s i n ( a 1 + a 2 ) ( d 1 + d 2 ) 2 (3. 37) x 2 = l 1 s i n ( a 1 ) d 1 + l 2 s i n ( a 1 + a 2 ) ( d 1 + a 2 ) + 1 c o s ( a ] ) d 1 + l 2 c o s ( a 1 + a 2 ) ( d 1 + d 2 ) 2 ( 3 . 38) To o b t a i n the i n v e r s e k i n e m a t i c s of the two l i n k m a n i p u l a t o r i t i s n e c e s s a r y t o a p p l y a c o n s t r a i n t t o make f u n c t i o n a l r e l a t i o n s h i p s p o s s i b l e . I n t h e f o l l o w i n g we r e s t r i c t a 2 t o be between 0 and 7r r a d i a n s . The i n v e r s e k i n e m a t i c s can then be w r i t t e n a s , a2 = a r c c o s U x ^ + x 2 2 - 1 ^ - 1 2 2 ) / 2 1 1 1 2 ) (3.39) 43 a 1 = a r c t a n ( x 1 , - x 2 ) _ a r c t a n ( l 2 s i n ( a 2 ) + l 2 c o s ( a 2 ) ) (3.40) d 1 = ( l 2 s i n ( a 1 + a 2 ) x l - l 2 c o s ( a 1 + a 2 ) x 2 ) / l 1 l 2 s i n ( a 2 ) (3.41 ) d 2 = [ ( - l 1 s i n ( a 1 ) x 1 + 1 1 c o s ( a 1 ) x 2 ) / l . l 2 s i n ( a 2 ) ] - d 1 (3.42) 2 a 1 = ( l 2 s i n ( a 1 + a 2 ) x 1 - l 2 c o s ( a 1 + a 2 ) x 2 + 1 1 l 2 c o s ( a 2 ) d 1 + l 2 2 ( d 1 + d 2 ) 2 ) / l 1 l 2 s i n ( a 2 ) (3.43) 2 • 2 d 2 = [ ( - 1 1 s i n (a. )x 1 + l ^ o s f a ^ x ^ - 1^ d 1 - l ^ c o s U ^ ( d 1 + d 2 ) 2 ) / l 1 l 2 s i n ( a 2 ) ] - (3.44) Note that a r c t a n f u n c t i o n has two arguments and i s thus assumed to y i e l d a value between -it and it. Since i t i s the s i n e s and c o s i n e s of a 1 and that are used i n subsequent c a l c u l a t i o n s , i t i s u s e f u l to use the f o l l o w i n g formulas, c o s ( a 2 ) = ( x ^ + x 2 2 - l , 2 - 1 2 2 ) / 2 1 1 1 2 (3.45) s i n ( a 2 ) = (1 " c o s ( a 2 ) 2 ) l / 2 (3.46) 2 2 c o s ( a 1 ) = [ x 1 l l s i n ( a 2 ) - x 2 ( 1 . + l 2 c o s ( a 2 ) ) ] / ( x 1 +x 2 ) (3.47) s i n ( t t l ) = (1 - c o s ( a i ) 2 ) l / 2 (3.48) The d e r i v a t i o n of the d i r e c t and i n v e r s e kinematics i s shown i n appendix A. The i n v e r s e dynamics of the two l i n k manipulator can be obtained u s i n g Lagrangian mechanics [37,46]. The Lagrangian L i s d e f i n e d as the d i f f e r e n c e between the k i n e t i c energy K and the p o t e n t i a l energy P of the system, L = K - P (3.49) The i n v e r s e dynamics equations are obtained as, 44 where a r e t h e c o o r d i n a t e s i n w h i c h t h e k i n e t i c and p o t e n t i a l e n e r g y a r e e x p r e s s e d , d̂ . a r e t h e c o r r e s p o n d i n g v e l o c i t i e s , and T . t h e c o r r e s p o n d i n g f o r c e s o r t o r q u e s . i s e i t h e r a f o r c e o r a t o r q u e , d e p e n d i n g upon whether ai i s a l i n e a r o r a n g u l a r c o o r d i n a t e . T h e s e f o r c e s , t o r q u e s and c o o r d i n a t e s a r e r e f e r r e d t o as g e n e r a l i z e d f o r c e s , t o r q u e s and c o o r d i n a t e s . A p p l i c a t i o n o f L a g r a n g i a n m e c h a n i c s t o a two l i n k m a n i p u l a t o r h a v i n g l i n k l e n g t h s 1 1 and 1 2 , l i n k masses m1 and m 2, and o r i e n t e d as shown i n f i g u r e 3.1 w i t h r e s p e c t t o g, t h e g r a v i t a t i o n a l f i e l d , y i e l d s t h e f o l l o w i n g i n v e r s e d y n a m i c s , T1 = d 1 i a i + d l 2 d 2 + d 1 1 1 a 1 2 + d l 2 2 d 2 2 + d 1 1 2 d 1 d 2 + d l 2 1 d 2 d 1 + d 1 ( 3 * 5 1 ) r 2 = d 2 1 a 1 + d 2 2 d 2 + d 2 n d 1 2 + d 2 2 2 d 2 2 + d 2 1 2 d 1 d 2 + d 2 2 1 d 2 d 1 + d 2 ( 3 * 5 2 ) where, d n = ( m 1 + m 2 ) l l 2 + m 2 l 2 2 + 2 m 2 l 1 l 2 c o s ( a 2 ) (3.53) 2 d l 2 = m 2 l 2 + m 2 l 1 l 2 c o s ( a 2 ) (3.54) d 1 1 1 = 0 (3.55) d l 2 2 = ~ m2^" 1 •'•2 S i n ̂ a2 ^ (3.56) d 1 1 2 = ~ m 2 l 1 1 2 s * n ^ a 2 ^ (3.57) d t 2 1 = - m 2 1 i 1 2 S i n ^ a 2 ^ (3.58) d 1 = ( m 1 + m 2 ) g l 1 s i n ( a 1 ) + m 2 g l 2 s i n ( a 1 + a 2 ) (3.59) d 2 1 = m 2 1 2 2 + m 2 1 1 1 2 c o s ( a 2 ) (3.60) d 2 2 = m 2 1 2 2 ( 3 , S 1 } d 2 1 1 = m 2 l 1 1 2 S ^ n ^ a 2 ^ (3.62) 45 d 2 2 2 = 0 (3.63) d 2 l 2 = 0 (3.64) d 2 2 1 = 0 (3.65) d 2 = m 2 g l 2 s i n ( a 1 + a 2 ) (3.66) The d e r i v a t i o n o f t h e i n v e r s e d y n a m i c s i s shown i n a p p e n d i x B and f o l l o w s t h e a p p r o a c h u s e d by P a u l [ 4 6 ] . N o t e , however, t h a t t h e r e s u l t s d i f f e r s l i g h t l y from t h o s e g i v e n by P a u l as t h e r e i s an a r i t h m e t i c e r r o r i n t h e h i s d e r i v a t i o n . N ote a l s o t h a t t h i s d e r i v a t i o n n e g l e c t s f r i c t i o n a l f o r c e s . T h i s i s n o t n e c e s s a r i l y a s i m p l i f i c a t i o n , t h o u g h , from a c o n t r o l p o i n t o f v i e w . F r i c t i o n w ould t e n d t o damp any o s c i l l a t i o n s t h a t o c c u r whereas w i t h no f r i c t i o n , o s c i l l a t i o n s must be damped o u t by t h e c o n t r o l s y s t e m . The d i r e c t d y n a m i c s c a n be o b t a i n e d t h r o u g h a l g e b r a i c m a n i p u l a t i o n o f t h e i n v e r s e d y n a m i c s and a r e a s f o l l o w s , a 1 = [ d l 2 ( r 2 - g 2 ) + d ^ t g , - Tt)]/AQ (3.67) a 2 = [ d 2 1 ( T 1 - g,) + d n ( g 2 - T 2 ) ] / d 0 (3.68) where, d 0 = d l 2 d 2 1 " ' d 1 1 d 2 2 (3 . 69) 91 = d 1 1 1 d 1 2 + d 1 2 2 d 2 2 + d 1 l 2 d 1 d 2 + d l 2 1 d 2 d 1 + d 1 ( 3 . 70) 9 2 = d 2 1 1 d 1 2 A , . 2 + d 2 2 2 a 2 + d 2 2 1 d 2 d 1 + d 2 ( 3 . 71 ) I t i s t h e d i r e c t d y n a m i c s i n c o n j u n c t i o n w i t h t h e i n t e g r a l e q u a t i o n s , a = / a 6t (3.72) o = / a 8t (3.73) t h a t a l l o w s s i m u l a t i o n o f t h e m a n i p u l a t o r on a computer u t i l i z i n g n u m e r i c a l i n t e g r a t i o n r o u t i n e s . To a c h i e v e l e a r n e d r e s o l v e d a c c e l e r a t i o n c o n t r o l i n a 46 C a r t e s i a n r e f e r e n c e frame i t i s n e c e s s a r y t o l e a r n t h e i n v e r s e d y n a m i c s and t h e i n v e r s e k i n e m a t i c s . To r e p l a c e a v i s i o n s y s t e m o r i t s e q u i v a l e n t f o r f e e d b a c k p u r p o s e s , i t i s n e c e s s a r y t o l e a r n t h e d i r e c t k i n e m a t i c s . I t c a n be se e n t h a t even ' f o r a s i m p l e two l i n k m a n i p u l a t o r t h e f u n c t i o n a l r e l a t i o n s h i p s t o be l e a r n e d a r e m u l t i - v a r i a t e and h i g h l y n o n - l i n e a r . 3.3 SIMULATION OF THE MANIPULATOR T e s t i n g o f v a r i o u s c o n t r o l s y s t e m s was c a r r i e d o u t u s i n g a s i m u l a t i o n o f t h e two l i n k m a n i p u l a t o r . The s i m u l a t i o n was p e r f o r m e d on a - VAX-11/750 computer u s i n g ACSL ( a d v a n c e d c o n t i n u o u s s i m u l a t i o n l a n g u a g e ) . ACSL p e r m i t s t h e s i m u l a t i o n of c o n t i n u o u s t i m e s y s t e m s , s u c h a s t h e m a n i p u l a t o r , as w e l l a s s i m u l a t i o n o f d i s c r e t e t i m e s y s t e m s , s u c h a s t h e d i g i t a l c o m p u t i n g d e v i c e s t h a t w o u l d be u s e d t o implement a m a n i p u l a t o r c o n t r o l s y s t e m [ 5 8 ] . A s c h e m a t i c d i a g r a m o f t h e s i m u l a t i o n i s shown i n f i g u r e 3.2. The v a r i o u s b l o c k s i n f i g u r e 3.2 c a n be t h o u g h t o f as s e p a r a t e p h y s i c a l e n t i t i e s . D e r i v a t i v e b l o c k s r e p r e s e n t c o n t i n u o u s t i m e s y s t e m s ; s y s t e m s whose s t a t e s change c o n t i n u o u s l y , whereas d i s c r e t e b l o c k s r e p r e s e n t d i s c r e t e t i m e s y s t e m s ; s y s t e m s whose s t a t e s c h ange a t d i s c r e t e p o i n t s i n t i m e . 47 useobs CD ^ DISCRETE ri BLOCK OJ SAMPLE dlffer- "le antlatton = *e K a d i f fe r -entiation learning Machine 2 vtalon tyatea) analytical direct dynanlca DERIVATIVE BLOCK ARM analytical dlract kinematic* . taaeh l e a r n i n g Machine 1 taaeh learning Mchlna 2 DISCRETE BLOCK LEARNJ DISCRETE BLOCK LEARN 2 DERIVATIVE BLOCK ARM D e r i v a t i v e b l o c k ARM r e p r e s e n t s the two l i n k m a n i p u l a t o r . G i v e n a statement of the d i r e c t dynamics i n d i f f e r e n t i a l form, a = d i r . d y n ( 7 , a , a ) (3.74) the i n t e g r a t i o n a l g o r i t h m s of ACSL advance the s t a t e s of the system, a and a, a s , a = / a 6t ' (3.75) a = J a 5t (3.76) Of the v a r i o u s i n t e g r a t i o n a l g o r i t h m s a v a i l a b l e i n ACSL, the Adams-Moulton v a r i a b l e s t e p s i z e , v a r i a b l e o r d e r a l g o r i t h m was chosen. The r e l a t i v e e r r o r i n t r o d u c e d a t each i n t e g r a t i o n s t e p was r e s t r i c t e d t o be l e s s than 1.0E-6 ti m e s the maximum v a l u e of the s t a t e up t o t h a t p o i n t i n the s i m u l a t i o n and the a b s o l u t e e r r o r i n t r o d u c e d a t each i n t e g r a t i o n s t e p was r e s t r i c t e d t o be l e s s than 1.0E-6. S i n c e the o r d e r of magnitude of the s t a t e s i s about 1 and FORTRAN s i n g l e p r e c i s i o n r e a l v a r i a b l e s a r e used by ACSL f o r s t o r a g e , t h e s e e r r o r bounds a r e near the l i m i t of what i s a c h i e v a b l e . Other f i x e d s t e p s i z e , f i x e d o r d e r i n t e g r a t i o n a l g o r i t h m s a v a i l a b l e i n ACSL, such as E u l e r , 2nd o r d e r Runge- K u t t a , and 4 t h o r d e r Runge-Kutta, t y p i c a l l y r e q u i r e d more CPU time t o a c h i e v e comparable e r r o r bounds. The Gear's S t i f f v a r i a b l e s t e p s i z e , v a r i a b l e o r d e r a l g o r i t h m was not c o n s i d e r e d as i t i s o n l y a p p r o p r i a t e f o r c e r t a i n d i f f i c u l t i n t e g r a t i o n problems, which i s not the case here [ 5 8 ] . ACSL e n s u r e s t h a t s t a t e s and d e r i v a t i v e s i n d e r i v a t i v e b l o c k s a r e up t o date a t the t i m e s when any o t h e r b l o c k s a r e b e i n g e x e c u t e d . Thus the s i m u l a t e d two l i n k m a n i p u l a t o r a p p e a r s as a c o n t i n u o u s system t o 49 the r e s t of the s i m u l a t i o n . DERIVATIVE BLOCK PTHGEN D e r i v a t i v e b l o c k PTHGEN c o n t a i n s code t o compute the c u r r e n t p a t h s p e c i f i c a t i o n , x^, x^ and x^, g i v e n the c u r r e n t t i m e , t . No i n t e g r a t i o n i s performed by PTHGEN as a dynamic system w i t h s t a t e s i s not b e i n g m o d e l l e d h e r e . A d e r i v a t i v e b l o c k i s used i n t h i s manner s i m p l y t o ensure t h a t the p a t h s p e c i f i c a t i o n i s c u r r e n t whenever another b l o c k i s b e i n g e x e c u t e d w i t h o u t h a v i n g t o d u p l i c a t e the code i n each b l o c k t h a t uses the p a t h s p e c i f i c a t i o n . P a t h s can be e i t h e r l i n e a r or c i r c u l a r as d e s c r i b e d i n s e c t i o n 3 . 4 . DERIVATIVE BLOCK CALC D i s c r e t e b l o c k CALC pe r f o r m s the c o m p u t a t i o n s r e q u i r e d t o c o n t r o l t he m a n i p u l a t o r . I n p u t s a r e the s p e c i f i c a t i o n of the d e s i r e d C a r t e s i a n p a t h , x^, x^ and x^, and the view of the r e s u l t i n g j o i n t p a t h , a, a and a. The out p u t i s the to r q u e 7 t h a t i s a p p l i e d t o the m a n i p u l a t o r . S i n c e CALC i s a d i s c r e t e b l o c k , i t i s ex e c u t e d as d i s c r e t e i n t e r v a l s i n time as f i x e d by t ,_. The output t o r q u e i s thus updated a t i n t e r v a l s of t , and h e l d c o n s t a n t o t h e r w i s e . The time r e q u i r e d t o p e r f o r m the co m p u t a t i o n s of the CALC b l o c k would d e t e r m i n e the lower bound on t , i n an a c t u a l i m p l e m e n t a t i o n of the c o n t r o l system w i t h ca xc a r e a l m a n i p u l a t o r . The many s u b p a r t s of d i s c r e t e b l o c k CALC a l l o w v a r i o u s t y p e s of c o n t r o l t o be per f o r m e d . C o n t r o l can be open l o o p , u s i n g o n l y the d e s i r e d C a r t e s i a n p a t h s p e c i f i c a t i o n or c l o s e d l o o p , u s i n g the view of the r e s u l t i n g j o i n t p a t h f o r e r r o r 50 c o r r e c t i n g feedback. L o g i c a l v a r i a b l e CLOSED d e t e r m i n e s whether c o n t r o l i s open or c l o s e d l o o p . S i n c e p a t h s p e c i f i c a t i o n i s i n C a r t e s i a n c o o r d i n a t e s , feedback i n f o r m a t i o n must a l s o be i n C a r t e s i a n c o o r d i n a t e s . An ap p r o x i m a t e C a r t e s i a n view of the r e s u l t i n g p a t h can be d e t e r m i n e d from the r e s u l t i n g j o i n t p a t h by u s i n g a model of v i s i o n system or by u s i n g a l e a r n e d e q u i v a l e n t of the v i s i o n system based on L e a r n i n g Machine 2. S e l e c t i o n i s made a c c o r d i n g t o l o g i c a l v a r i a b l e VISION. A l t e r n a t i v e l y , the e x a c t r e s u l t i n g C a r t e s i a n p a t h can be o b t a i n e d u s i n g the a n a l y t i c a l d i r e c t k i n e m a t i c s i f l o g i c a l v a r i a b l e EXACT i s t r u e . Due t o c a l c u l a t i o n d e l a y i n a r e a l a p p l i c a t i o n , c o n t r o l u s i n g feedback cannot make use of c u r r e n t o b s e r v a t i o n s of the r e s u l t i n g p a t h but r a t h e r o n l y o b s e r v a t i o n s from a t l e a s t time t c a i c p r e v i o u s . A d e l a y of t ^ i n the o b s e r v a t i o n s of the r e s u l t i n g p a t h i s implemented when l o g i c a l v a r i a b l e DELAY i s t r u e . G i v e n t h a t t h e r e i s a d e l a y i n o b s e r v a t i o n s , one can attempt t o p r e d i c t what the c u r r e n t o b s e r v a t i o n s would be. Such p r e d i c t i o n i s done whenever l o g i c a l v a r i a b l e PRDICT i s t r u e . P o s i t i o n and v e l o c i t y p r e d i c t i o n i s done u s i n g e i t h e r the d e s i r e d v e l o c i t y and a c c e l e r a t i o n or the ob s e r v e d v e l o c i t y and a c c e l e r a t i o n depending on the s t a t e of l o g i c a l v a r i a b l e USEOBS. F i n a l l y , one can c a l c u l a t e the t o r q u e t o a p p l y t o the m a n i p u l a t o r by u s i n g the a n a l y t i c a l C a r t e s i a n i n v e r s e dynamics, c o n s i s t i n g of the a n a l y t i c a l i n v e r s e k i n e m a t i c s and the a n a l y t i c a l i n v e r s e dynamics, o r by u s i n g a l e a r n e d e q u i v a l e n t of the C a r t e s i a n i n v e r s e dynamics, L e a r n i n g Machine 1. The method of c a l c u l a t i o n i s s e l e c t e d by l o g i c a l 5 1 v a r i a b l e INVARM, L e a r n i n g Machine 1 b e i n g s e l e c t e d when INVARM i s f a l s e . DISCRETE BLOCKS LEARN I AND LEARN 2 D i s c r e t e b l o c k s LEARN 1 and LEARN2 c o n t a i n code t h a t t r a i n s L e a r n i n g Machines 1 and 2, r e s p e c t i v e l y , u s i n g o b s e r v a t i o n s of the system. These b l o c k s a r e e x e c u t e d a t i n t e r v a l s of t ^ r n 1 and t l r n 2 when e n a b l e d by l o g i c a l v a r i a b l e s LRNINV and LRNVIS, r e s p e c t i v e l y . DISCRETE BLOCK SAMPLE D i s c r e t e b l o c k SAMPLE c o n t a i n s code t o c o l l e c t s t a t i s t i c a l i n f o r m a t i o n about the performance of t h e c o n t r o l system. I t i s exe c u t e d a t i n t e r v a l s of t „ . t _ was s e t t o be 0.01 sec f o r samp samp a l l s i m u l a t i o n s . I n f o r m a t i o n c o m p i l e d i n c l u d e s r o o t mean square p a t h e r r o r , XERMS = ( m e a n { ( x . - x , . ) 2 } ) 1 / 2 (3.77) VXERMS = (meant ( x . - x , . ) 2 } ) 1 / ^ 2 (3.78) I 0.1 AXERMS = ( m e a n { ( x . - x , . ) 2 } ) 1 ^ 2 (3.79) i ai r o o t mean square p a t h e s t i m a t i o n e r r o r , EXERMS = (mean{(x.-x . ) 2 } ) ^ 2 (3.80) EVERMS = (mean{(x.-x . ) 2 } ) 1 / / 2 (3.81) / ei EAERMS = (mean{(x.-x . ) 2 } ) 1 ^ 2 (3.82) / ei r o o t mean square l e a r n e d p a t h e s t i m a t i o n e r r o r , LXERMS = (mean{(x.-x, - ) 2 } ) 1 / 2 (3.83) / l e i LVERMS = (mean{(x.-x, . ) 2 } ) 1 / 2 (3.84) i Lei LAERMS = (mean{(x.-x, . ) 2 } ) 1 / 2 (3.85) / lei r o o t mean square t o r q u e e s t i m a t i o n e r r o r , TERMS = ( m e a n { ( r d . - r 1 / ) 2 } ) 1 / 2 (3.86) 52 a n d maximum a p p l i e d t o r q u e , TRQMAX = m a x { a b s ( ) } (3.87) A l s o c o m p i l e d a r e s t a t i s t i c s a b o u t how o f t e n and how f a r t h e r e s u l t i n g p a t h m o t i o n i s o u t o f bounds, as bounds on a l l o w a b l e m o t i o n a r e imposed by c e r t a i n o f t h e c o n t r o l t e c h n i q u e s c o n s i d e r e d . More d e t a i l e d d e s c r i p t i o n s o f c e r t a i n a s p e c t s of t h e s i m u l a t i o n a r e i n c l u d e d i n f o l l o w i n g s e c t i o n s when s u c h a s p e c t s o f t h e s i m u l a t i o n a r e c e n t r a l t o t h e p a r t i c u l a r c o n t r o l t e c h n i q u e b e i n g s t u d i e d . 3.4 PATH SPECIFICATION B e f o r e g o i n g on t o d i s c u s s methods o f c o n t r o l l i n g t h e m a n i p u l a t o r i t i s n e c e s s a r y t o d e s c r i b e how p a t h s p e c i f i c a t i o n s a r e g e n e r a t e d . Two t y p e s of p a t h s a r e u s e d t o t e s t c o n t r o l o f t h e m a n i p u l a t o r ; l i n e a r p a t h s a n d c i r c u l a r p a t h s . LINEAR PATHS L i n e a r p a t h s s i m p l y c o n s i s t o f l i n e a r m o t i o n between two p o i n t s g i v e n by t h e c o o r d i n a t e s <x^ n 1 , x ^ n 2 > a n c ^ < x f i i ' x f i 2 > * T n e p a t h c o n s i s t s o f f i v e p h a s e s o f m o t i o n . No m o t i o n o c c u r s u n t i l t i m e t f t , t h e n t h e r e i s a p h a s e o f c o n s t a n t a c c e l e r a t i o n , a m „ . a 0 F max p h a s e o f c o n s t a n t v e l o c i t y , v m a x ' a P n a s e °^ c o n s t a n t d e c e l a r a t i o n , a m a x » - a n c * f i n a l l y a p e r i o d o f no m o t i o n , a g a i n o f d u r a t i o n t n . I f t h e i n i t i a l a nd f i n a l p o i n t s a r e c l o s e t o g e t h e r t h e n v i s n o t r e a c h e d . T h i s o c c u r s when, max K«fir*in1»2'+ < x £ i 2 - x i n 2 » 2 ] , / 2 < W ^ m a x ( 3- 8 e> I n t h i s c a s e t h e r e i s no ph a s e o f c o n s t a n t , n o n - z e r o v e l o c i t y , 53 r a t h e r t h e r e i s s i m p l y a c c e l e r a t i o n u n t i l h a l f w a y a l o n g the p a t h and then d e c e l e r a t i o n u n t i l s t o p p e d . Four s t a n d a r d l i n e a r p a t h s were used f o r t e s t p u r p o s e s . The d a t a d e f i n i n g t h e s e f o u r s t a n d a r d p a t h s i s g i v e n i n t a b l e 3.1. A p l o t of l i n e 1 i n C a r t e s i a n c o o r d i n a t e s i s shown i n f i g u r e 3.3. F i g u r e 3.4 shows l i n e 1 i n j o i n t c o o r d i n a t e s . PATH x i n 1 x i n 2 X f i 1 X f i 2 V max a max fc0 l i n e 1 0.45 -1 .45 -0.45 -0.55 0.80 0.80 0.50 l i n e 2 0.45 -0.55 -0.45 -1 .45 0.80 0.80 0.50 l i n e 3 . -0.45 -1 .00 0.45 -1 .00 0.80 0.80 0.50 l i n e 4 0.00 -1 .45 0.00 -0.55 0.80 0.80 0.50 T a b l e 3.1 Data d e f i n i n g s t a n d a r d l i n e a r p a t h s CIRCULAR PATHS C i r c u l a r p a t h s a r e c e n t e r e d about the o r i g i n of the work space l o c a t e d a t c o o r d i n a t e s < x c i » x C 2 > = < 0 » ~ , > « M o t i o n can be c l o c k w i s e or c o u n t e r c l o c k w i s e . The i n i t i a l p o s i t i o n i s g i v e n by the s p e c i f i e d r a d i u s , p, and the s p e c i f i e d s t a r t i n g a n g l e , 6, as f o l l o w s , x i n 1 = x c 1 + P c o s ( e ) (3.89) x in2 = x c 2 + p s i n ( 0 ) (3.90) 54 s s CO • • (_> LU toe l o X D O 1.60 T SEC g 8 • — * " o LU <"§ o LU — " • Csl X 2 o • X B§ —1 1 '0. S s » , s s CO •i 1 0.80 1.60 T SEC 2.40 3.20 4.00 4.00 F i g u r e 3.3 S t a n d a r d p a t h l i n e 1 i n C a r t e s i a n c o o r d i n a t e s 55 CO C M • • U J Od- or: 0.00 0.80 1.60 T SEC 4.00 CD" CO" CJ o U J U J cn CO £ g <E—•- cxa>- — H CNJ 3 O* • CO' CO 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 • o • CO" gB a - -• g s acM- — * 5 X H-CN • is • CM" 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 i g u r e 3.4 Standard path l i n e 1 i n j o i n t coordinates 56 M o t i o n proceeds a l o n g a c i r c u l a r a r c f o r the f r a c t i o n of a complete r e v o l u t i o n s p e c i f i e d by / 3 . A l o n g the t a n g e n t , the same f i v e phases of motion o c c u r as w i t h l i n e a r p a t h s , except t h a t v. „ and a.„„ r e p l a c e v and a„ , r e s p e c t i v e l y . The maximum tng t n g c max max r J t a n g e n t i a l v e l o c i t i e s and a c c e l e r a t i o n s must be c o n s t r a i n e d , however, such t h a t t he v e c t o r sum of t a n g e n t i a l and r a d i a l a c c e l e r a t i o n never exceeds a„ . T h i s i s a c c o m p l i s h e d by max r J c h o o s i n g , v. = min { v m .(P7a m a v) l / 2} ( 3 . 9 1 ) tng max max a r n o = < a „ , a v 2 ~ V . n r , 4 / P 2 ) l / 2 ( 3 . 9 2 ) tng max t n g where 7 i s a f a c t o r t h a t s p e c i f i e s t he p o r t i o n of a m a x t h a t i s t o be a l l o c a t e d t o c e n t r i p e t a l a c c e l e r a t i o n d u r i n g the c o n s t a n t t a n g e n t i a l v e l o c i t y phase of m o t i o n . Two c i r c u l a r p a t h s were used f o r t e s t p u r p o s e s . The d a t a d e f i n i n g t h e s e two s t a n d a r d p a t h s i s g i v e n i n t a b l e 3 . 2 . A p l o t of c i r c l e 1 i n C a r t e s i a n c o o r d i n a t e s i s shown i n f i g u r e 3 . 5 . F i g u r e 3 . 6 shows c i r c l e 1 i n j o i n t c o o r d i n a t e s and f i g u r e 3 . 7 shows a p l a n view of c i r c l e 1 . P A T H P 0 / 3 7 v m^„ a m a v t n d i r max max u c i r c l e 1 0 . 4 0 0 . 0 0 1 . 0 0 0 . 7 5 0 . 9 0 0 . 9 0 1 . 0 0 cw c i r c l e 2 • 0 . 4 0 3 . 1 4 1 . 0 0 0 . 7 5 0 . 9 0 0 . 9 0 1 . 0 0 ccw Ta b l e 3 . 2 Data d e f i n i n g s t a n d a r d c i r c u l a r p a t h s 5 7 - 5 8 CO" OJ • • o UJ COO cn - O U J to • Q , <r-4 .00 SEC CM in x t~«M CM*" F i g u r e 3.6 S t a n d a r d p a t h c i r c l e 1 i n j o i n t c o o r d i n a t e s 59 O I o C O • • o" • I CSJ X . o CO o in 0 . 5 0 - 0 . 3 0 •0 .10 0 . 1 0 DX(1) M Figure 3.7 View of standard path c i r c l e ( t i c k s mark i n t e r v a l s of 0.2 sec) 60 3.5 MANIPULATOR CONTROL USING THE ANALYTICAL INVERSE DYNAMICS AND ANALYTICAL INVERSE KINEMATICS M a n i p u l a t o r c o n t r o l u s i n g the a n a l y t i c a l i n v e r s e dynamics and a n a l y t i c a l i n v e r s e k i n e m a t i c s , which a r e t o g e t h e r c a l l e d t he a n a l y t i c a l C a r t e s i a n i n v e r s e dynamics, w a r r a n t s i n v e s t i g a t i o n as i t forms a benchmark a g a i n s t which t o compare m a n i p u l a t o r c o n t r o l u s i n g a l e a r n e d sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the C a r t e s i a n i n v e r s e dynamics. 3.5.1 IDEAL OPEN LOOP CONTROL B e f o r e moving on t o more r e a l i s t i c c o n t r o l s c e n a r i o s , i t i s i n s t r u c t i v e t o s i m u l a t e i d e a l open l o o p c o n t r o l of the m a n i p u l a t o r . The meaning of open l o o p c o n t r o l i s c l e a r , however, i d e a l r e q u i r e s e x p l a n a t i o n . By i d e a l , we mean t h a t the a n a l y t i c a l i n v e r s e k i n e m a t i c s and a n a l y t i c a l i n v e r s e dynamics a r e used t o compute the t o r q u e s r e q u i r e d t o a c h i e v e a g i v e n p a t h . I t a l s o means t h a t the t o r q u e i s updated so f r e q u e n t l y t h a t i t i s e f f e c t i v e l y c a l c u l a t e d c o n t i n u o u s l y . To t e s t i d e a l open l o o p c o n t r o l , two s i m u l a t i o n s were c a r r i e d out w i t h CLOSED=.FALSE. , INVARM= .TRUE. and t: ,„ = 0.0001 s e c . S i n c e the C 3 1 C d u r a t i o n of motion f o r the p a t h s was s e v e r a l seconds, such a s m a l l t , a s s u r e s t h a t t o r q u e i s e f f e c t i v e l y c a l c u l a t e d C cL J. C c o n t i n u o u s l y . T a b l e 3.3 shows the p a t h e r r o r t h a t o c c u r r e d f o r the p a t h s , l i n e 1 and c i r c l e 1, and f i g u r e 3.8 shows the r e s u l t i n g p a t h i n C a r t e s i a n c o o r d i n a t e s f o r t h e s i m u l a t i o n of l i n e 1. The e r r o r s t h a t occur a r e m i n i m a l f o r b o t h p a t h s and the r e s u l t i n g p a t h f o r l i n e 1 i s i n d i s t i n g u i s h a b l e from the d e s i r e d p a t h , which i s shown i n f i g u r e 3.3. The e r r o r s can be a t t r i b u t e d 61 s CM C_> 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 8 8 — < / ) § U J 0 cos * — L C\J •— • 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 8 S 8 X o 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 F i g u r e 3.8 I d e a l open loop c o n t r o l of l i n e 1 62 t o f i n i t e p r e c i s i o n i n the c a l c u l a t i o n of the a n a l y t i c a l i n v e r s e k i n e m a t i c s and a n a l y t i c a l i n v e r s e dynamics, i n a c c u r a c y i n the s i m u l a t i o n of the m a n i p u l a t o r , f i n i t e p r e c i s i o n i n the c a l c u l a t i o n of the a n a l y t i c a l d i r e c t k i n e m a t i c s , and the f a c t t h a t t o r q u e i s not updated c o n t i n u o u s l y , a l b e i t t h a t t o r q u e i s updated v e r y f r e q u e n t l y . PATH XERMS VXERMS AXERMS l i n e 1 2.54E-5 3.05E-5 1.05E-4 c i r c l e 1 2.10E-5 3.22E-5 7.78E-5 T a b l e 3.3 P a t h e r r o r u s i n g i d e a l open l o o p c o n t r o l W h i l e such i d e a l open l o o p c o n t r o l i s not a r e a l i s t i c c o n t r o l s t r a t e g y , i t s e r v e s t o t e s t two a s p e c t s of the s i m u l a t i o n program. F i r s t , i t c o n f i r m s t h a t p a t h s p e c i f i c a t i o n i s b e i n g performed c o r r e c t l y . The d e s i r e d a c c e l e r a t i o n and v e l o c i t y a r e inde e d the d e r i v a t i v e s of the d e s i r e d v e l o c i t y and p o s i t i o n , r e s p e c t i v e l y . S e c o n d l y , i t c o n f i r m s t h a t the i n v e r s e k i n e m a t i c s and i n v e r s e dynamics a r e inde e d the i n v e r s e of the d i r e c t dynamics and d i r e c t k i n e m a t i c s used. W h i l e t h i s does not mean t h a t t h e k i n e m a t i c s and dynamics a r e n e c e s s a r i l y c o r r e c t , i t does make i t v e r y p r o b a b l e t h a t , g i v e n t h a t the a n a l y t i c a l k i n e m a t i c s and a n a l y t i c a l dynamics have been d e r i v e d c o r r e c t l y , e r r o r s have not been i n t r o d u c e d i n t h e i m p l e m e n t a t i o n of the s i m u l a t i o n . I f e i t h e r of these a s s e r t i o n s were not t r u e , i t 63 would be h i g h l y u n l i k e l y t h a t t h e r e s u l t i n g p a t h would d u p l i c a t e the d e s i r e d p a t h so c l o s e l y when u s i n g open l o o p c o n t r o l . In a d d i t i o n , i d e a l open l o o p c o n t r o l a l l o w s one t o examine the t o r q u e s t h a t must be a p p l i e d t o the m a n i p u l a t o r t o d r i v e i t t h r o u g h a s p e c i f i e d p a t h . F i g u r e 3 . 9 shows t h e t o r q u e , as f u n c t i o n of t i m e , t h a t i s r e q u i r e d t o d r i v e t h e m a n i p u l a t o r t h r o u g h s t a n d a r d p a t h , l i n e 1 . F i g u r e 3 . 9 Torque p r o f i l e f o r l i n e 1 6 4 3.5.2 IDEAL CLOSED LOOP CONTROL The p o s i t i o n and v e l o c i t y t r a c k i n g e r r o r s t h a t o c c u r when u s i n g i d e a l open l o o p c o n t r o l c an be r e d u c e d by u s i n g e r r o r c o r r e c t i n g f e e d b a c k r e s u l t i n g i n i d e a l c l o s e d l o o p c o n t r o l . I d e a l c l o s e d l o o p c o n t r o l assumes e x a c t measurement of t h e t h e r e s u l t i n g C a r t e s i a n p a t h and c o n t i n u o u s u p d a t i n g of t h e a p p l i e d t o r q u e . To t e s t i d e a l c l o s e d l o o p c o n t r o l , two s i m u l a t i o n s were c a r r i e d o u t w i t h CLOSED=.TRUE., INVARM=.TRUE., EXACT=.TRUE., PRDICT=.FALSE., DELAY=.FALSE., ^ = 16, k 2 = 6 4 a n d t c a i c = 0 - 0 0 0 1 s e c . The r e s u l t i n g p a t h e r r o r s a r e shown i n t a b l e 3.4 and a r e r e d u c e d f r o m t h o s e o c c u r r i n g w i t h i d e a l open l o o p c o n t r o l . I n f a c t , any f u r t h e r r e d u c t i o n o f p a t h e r r o r i s l i m i t e d by t h e p r e c i s i o n o f c a l c u l a t i o n s i n t h e s i m u l a t i o n . The e r r o r c o r r e c t i n g a c t i o n o f c l o s e d l o o p c o n t r o l i s i l l u s t r a t e d by a s i m u l a t i o n i n w h i c h t h e m a n i p u l a t o r i s p e r t u r b e d f r o m t h e d e s i r e d p o s i t i o n o f < x ( j i ' x ( 3 2 > = < U V ~ 1 > a t t i m e t=0.2 s e c and i s r e s t o r e d t o t h e d e s i r e d p o s i t i o n by e r r o r c o r r e c t i n g f e e d b a c k . Such a s i m u l a t i o n i s shown i n f i g u r e 3.10. In t h e o r y i t w o u l d seem t h a t one c o u l d a c h i e v e c r i t i c a l damping of s u c h p o s i t i o n and v e l o c i t y e r r o r s w i t h as s h o r t a t i m e c o n s t a n t a s d e s i r e d s i m p l y by i n c r e a s i n g k 1 and k 2 w h i l e 2 m a i n t a i n i n g t h e c o n d i t i o n k 1 = 4 k 2 . I n p r a c t i c e t h i s c a n n o t be done a s i d e a l c l o s e d l o o p c o n t r o l c a n n o t be a c h i e v e d . 65 o COB =>g (Do -OTP (SI 0.00 0.20 0. T 40 SEC 0.60 0.80 1.00 o tn g X o in o m g CM ><o in 0.00 0.20 0.40 T SEC 0.60 0.80 7.00 F i g u r e 3.10 E r r o r c o r r e c t i n g a c t i o n of i d e a l c l o s e d l o o p c o n t r o l 66 PATH XERMS VXERMS AXERMS l i n e 1 2.37E-6 6.34E-6 1.46E-4 c i r c l e 1 1.56E-6 4.97E-6 1.01E-4 Ta b l e 3.4 Path e r r o r u s i n g i d e a l c l o s e d l o o p c o n t r o l 3.5.3 REALIZABLE CLOSED LOOP CONTROL - STANDARD ANALYTICAL CONTROL In a p r a c t i c a l i m p l e m e n t a t i o n of c l o s e d l o o p c o n t r o l , v a r i o u s c o n s t r a i n t s n e c e s s i t a t e d e v i a t i o n from i d e a l i t y . These c o n t r a i n t s t y p i c a l l y have t h e i r r o o t s i n t e c h n o l o g i c a l l i m i t a t i o n s on the speed a t which c a l c u l a t i o n s can be performed. CALCULATION DELAY A c e r t a i n amount of time p a s s e s between the o b s e r v a t i o n of the m a n i p u l a t o r p o s i t i o n , and the c a l c u l a t i o n of the t o r q u e t o a p p l y t o the m a n i p u l a t o r t o c o r r e c t p o s i t i o n and v e l o c i t y e r r o r s w h i l e t r a c k i n g a p a t h s p e c i f i c a t i o n . There i s thus a d e l a y between o b s e r v a t i o n of m a n i p u l a t o r motion and e r r o r c o r r e c t i n g a c t i o n . Such a d e l a y i s implemented i n the s i m u l a t i o n when l o g i c a l v a r i a b l e DELAY i s t r u e as f o l l o w s , * o ( t ) = 5 s e ( t " t c a l c ) ( 3 ' 9 3 ) U n l e s s one uses p a r a l l e l p r o c e s s i n g or p i p e l i n e d p r o c e s s i n g , the a p p l i e d t o r q u e w i l l be q u a n t i z e d i n time by t h i s same c a l c u l a t i o n i n t e r v a l . To keep computer c o s t t o a minimum, i t i s d e s i r a b l e t o p e r m i t t h i s c a l c u l a t i o n d e l a y t o be as l a r g e as p o s s i b l e w i t h o u t r e s u l t i n g i n poor c o n t r o l . We have chosen t c a ^ c = O . O l sec as a r e a s o n a b l e compromise. INEXACT VISION E x a c t measurement of the m a n i p u l a t o r p a t h f o r use i n e r r o r c o r r e c t i n g feedback i s an u n r e a l i s t i c a s s u m p t i o n . A more r e a l i s t i c a s s u m p t i o n , a l t h o u g h s t i l l r a t h e r o p t i m i s t i c , i s t h a t the v i s i o n system e s t i m a t e s o n l y p o s i t i o n , x*v, so a c c u r a t e l y as t o be c o n s i d e r e d e q u a l t o the e x a c t p o s i t i o n , x. V e l o c i t y and a c c e l e r a t i o n would then be o b t a i n e d by s i m p l e d i f f e r e n t i a t i o n of the o b s e r v e d p o s i t i o n as a f u n c t i o n of t i m e . We thus model the v i s i o n system as f o l l o w s , x e ( t ) = x y ( t ) = x ( t ) (3.94) x e ( t ) - [ x e ( t > - x e ( t - t c a l c ) ] / t c a l c (3.95) J e ( t ) = [ i e ( t > - x e ( t - t c a l c ) ] / t c a l c (3.96) Note t h a t i n a d d i t i o n t o the p r e v i o u s l y mentioned d e l a y of t c a l c ' t n e r e a r e a d d i t i o n a l e f f e c t i v e d e l a y s of *- C B\ c/ 2 ^ o r v e l o c i t y o b s e r v a t i o n s and t c a ^ c f o r a c c e l e r a t i o n o b s e r v a t i o n s . PREDICTION I t i s p o s s i b l e t o reduce the e f f e c t of feedback d e l a y by u s i n g p r e d i c t i o n . S i n c e one knows the d e s i r e d v e l o c i t y and a c c e l e r a t i o n when the o b s e r v a t i o n s a r e made, one can use t h i s i n f o r m a t i o n t o p r e d i c t what t h e o b s e r v e d p o s i t i o n and v e l o c i t y s h o u l d be a t time t ^ l a t e r . These p r e d i c t i o n s can then be used i n t h e c a l c u l a t i o n of t h e t o r q u e t o a p p l y a t time  tcaic l a t e r . To t h i s end, we make use of p o s i t i o n and v e l o c i t y p r e d i c t i o n as f o l l o w s , 5 p ( t + t c a l c ) = * o ( t ) + * d ( t ) t c a l c + * d ( t ) t c a l c 2 / 2 ( 3 ' 9 7 ) x p ( t + t c a l c ) " * o ( t ) + ^ d ( t ) t c a l c ( 3 ' 9 8 ) 68 Note t h a t i t i s a l s o p o s s i b l e t o use the ob s e r v e d v e l o c i t y and a c c e l e r a t i o n f o r p r e d i c t i o n p u r p o s e s , r a t h e r than the d e s i r e d v e l o c i t y and a c c e l e r a t i o n . Such use of o b s e r v a t i o n s , however, was found t o have c e r t a i n d i s a d v a n t a g e s and was not adopted. FEEDBACK GAIN The feedback g a i n s , k 1 and k 2, cannot be made a r b i t r a r i l y l a r g e i n p r a c t i c e . A c h o i c e of k 1=16 and k 2=64 was found t o r e s u l t i n good e r r o r c o r r e c t i o n w i t h a m i n i m a l tendency t o cause i n s t a b i l i t y g i v e n our c h o i c e of pa r a m e t e r s f o r s t a n d a r d a n a l y t i c a l c o n t r o l ; c l o s e d l o o p c o n t r o l u s i n g the a n a l y t i c a l i n v e r s e k i n e m a t i c s , a n a l y t i c a l i n v e r s e dynamics and r e a l i s t i c c o n s t r a i n t s . STANDARD ANALYTICAL CONTROL S t a n d a r d a n a l y t i c a l c o n t r o l i s s i m u l a t e d by s e t t i n g CLOSED=.TRUE., INVARM=.TRUE., EXACT=.FALSE., VISI0N=.TRUE, DELAY=.TRUE., PRDICT=.TRUE., USEOBS=.FALSE., k 1=16, k 2=64 and t c a ^ c = 0 . 0 l s e c . To t e s t s t a n d a r d a n a l y t i c a l c o n t r o l , s i m u l a t i o n s were done of the s i x s t a n d a r d p a t h s . The r e s u l t i n g p a t h e r r o r s a r e shown i n t a b l e 3.5. F i g u r e 3.11 shows t h e r e s u l t i n g p a t h i n C a r t e s i a n c o o r d i n a t e s f o r t he s i m u l a t i o n of l i n e 1. P o s i t i o n and v e l o c i t y t r a c k i n g i s v e r y good but t h e r e i s a s l i g h t o v e r s h o o t i n g of s t e p s i n a c c e l e r a t i o n . T r a c k i n g i s s i g n i f i c a n t l y p o o r e r , though, than t h a t which appeared p o s s i b l e u s i n g i d e a l open l o o p o r i d e a l c l o s e d l o o p c o n t r o l . The performance of s t a n d a r d a n a l y t i c a l c o n t r o l shown here forms a benchmark f o r the e v a l u a t i o n of l e a r n e d c o n t r o l . One s h o u l d hope t o a c h i e v e s i m i l a r r e s u l t s w i t h l e a r n e d c o n t r o l , but s h o u l d not expect t o 6 9 e x c e e d t h e p e r f o r m a n c e o f s t a n d a r d a n a l y t i c a l c o n t r o l . VXERMS AXERMS 2.29E-3 2.44E-3 2.45E-3 2.42E-3 1.50E-3 1.48E-3 9.54E-2 9.58E-2 9.92E-2 9.87E-2 4.73E-2 4.73E-2 PATH XERMS l i n e 1 l i n e 2 l i n e 3 l i n e 4 c i r c l e 1 c i r c l e 2 5.99E-4 7.09E-4 6.75E-4 6.32E-4 6.53E-4 6.18E-4 T a b l e 3.5 P a t h e r r o r u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l 70 o o CM eng 4.00 T SEC F i g u r e 3.11 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1 71 3.5.4 CHOICE OF PARAMETERS DEFINING STANDARD ANALYTICAL CONTROL The p a r a m e t e r s u s e d i n s t a n d a r d a n a l y t i c a l c o n t r o l were c h o s e n so as t o a c h i e v e good c o n t r o l w i t h t h e c o n s t r a i n t t h a t t h e s y s t e m be r e a l i z a b l e . The a p p r o p r i a t e n e s s o f t h e c h o s e n p a r a m e t e r s c a n be d e m o n s t r a t e d by s i m u l a t i o n s u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l e x c e p t t h a t i n d i v i d u a l p a r a m e t e r s a r e v a r i e d . CALCULATION DELAY C a l c u l a t i o n d e l a y i s u n a v o i d a b l e i n a r e a l i m p l e m e n t a t i o n . I t was c h o s e n t o be a s l a r g e as p o s s i b l e w i t h o u t r e s u l t i n g i n p o o r c o n t r o l , as t h i s p e r m i t s i m p l e m e n t a t i o n w i t h l o w e r c o s t c o m p u t i n g h a r d w a r e . F i g u r e s 3.12, 3.13 and 3.14.show s i m u l a t i o n s of l i n e 1 f o r t h e c a s e s t c a i c = 0«02 s e c , t - c a l c ^ * ^ 4 s e c a n d t c a ^ c = 0 . 0 4 s e c , r e s p e c t i v e l y . The p a t h e r r o r f o r t h e s e s i m u l a t i o n s i s shown i n t a b l e 3.6. I t c a n be s e e n t h a t as t c a ^ c i s i n c r e a s e d from 0.01 s e c , t h e s t a n d a r d v a l u e , t h e c o n t r o l becomes p r o g r e s s i v e l y p o o r e r . F o r t h e c a s e t c a l c = ^ " ^ 2 s e c ' p o s i t i o n and v e l o c i t y t r a c k i n g i s q u i t e good, however, t h e r e i s a n o t i c e a b l e o s c i l l a t i o n i n t h e a c c e l e r a t i o n o f t h e m a n i p u l a t o r . T h i s c o u l d c a u s e e x c e s s i v e v i b r a t i o n o f t h e l i n k s o f an a c t u a l m a n i p u l a t o r , an e f f e c t t h a t i s n o t m o d e l l e d h e r e . I t i s assumed t h a t t h e t o r q u e must be u p d a t e d a t a r e a s o n a b l y h i g h r a t e t o a v o i d r e s o n a n c e w i t h t h e v i b r a t i o n a l modes o f m a n i p u l a t o r l i n k s , o t h e r w i s e m a n i p u l a t o r s w o u l d have t o be s p e c i a l l y c o n s t r u c t e d t o damp v i b r a t i o n s . 72 4.00 4.00 4.00 T SEC F i g u r e 3.12 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t t , =0.02 sec c a l c 73 4 . 0 0 4.00 4.00 T SEC F i g u r e 3.13 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t t "i "0.03 sec c a i c 74 4.00 4.00 4.00 T SEC F i g u r e 3.14 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t t , =0.04 sec c a i c INEXACT VISION A c c u r a t e measurement of m a n i p u l a t o r motion i s n e c e s s a r y f o r feedback t o be u s e f u l i n i m p r o v i n g c o n t r o l . E r r o r s i n measurement t r a n s l a t e d i r e c t l y i n t o t r a c k i n g e r r o r s . I t i s thus n e c e s s a r y t o i n v e s t i n a v e r y a c c u r a t e v i s i o n system, or i t s e q u i v a l e n t u s i n g the a n a l y t i c a l d i r e c t k i n e m a t i c s , i f good c o n t r o l i s t o be a c h i e v e d . I t does seem r e a s o n a b l e , though, t h a t only. C a r t e s i a n p o s i t i o n needs t o be measured a c c u r a t e l y . C a r t e s i a n v e l o c i t y and a c c e l e r a t i o n , i f r e q u i r e d , can be a d e q u a t e l y o b t a i n e d by d i f f e r e n t i a t i n g the p o s i t i o n . F i g u r e 3.15 shows a s i m u l a t i o n of l i n e 1 i n which e x a c t v i s i o n i s used f o r feedback purposes by s e t t i n g EXACT=.TRUE. The r e s u l t i n g p a t h e r r o r i s shown i n t a b l e 3.6. I t can be seen t h a t c o n t r o l w i t h e x a c t v i s i o n i s not s i g n i f i c a n t l y b e t t e r than w i t h our model of a v i s i o n system where v e l o c i t y and a c c e l e r a t i o n measurements a r e o b t a i n e d by d i f f e r e n t i a t i n g a c c u r a t e measurements of p o s i t i o n . PREDICTION P r e d i c t i o n has been used i n s t a n d a r d a n a l y t i c a l c o n t r o l . The b e n e f i t s of p r e d i c t i o n a r e shown by a s i m u l a t i o n i n which no p r e d i c t i o n was used. The r e s u l t i n g p a t h i s shown i n f i g u r e 3.16 and t h e p a t h e r r o r i s l i s t e d i n t a b l e 3.6. Lack of p r e d i c t i o n r e s u l t s i n more pronounced o v e r s h o o t i n g of s t e p s i n a c c e l e r a t i o n and a s i g n i f i c a n t i n c r e a s e i n p a t h e r r o r as compared t o r e s u l t s u s i n g p r e d i c t i o n . F o r the two l i n k m a n i p u l a t o r , p r e d i c t i o n r e q u i r e s o n l y 6 m u l t i p l i c a t i o n s and 6 a d d i t i o n s . P r e d i c t i o n was thus adopted f o r use i n s t a n d a r d a n a l y t i c a l c o n t r o l as i t r e q u i r e s a m i n i m a l amount of a d d i t i o n a l c o m putation and 76 CM COo 0.00 0.80 1.60 T SEC o o o 5° —'o 0.00 0.80 1.60 T SEC o m X o in 0.00 0.80 1.60 T SEC 4.00 F i g u r e 3.15 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1 f e x c e p t t h a t EXACT=.TRUE. 77 4.00 4.00 4.00 T SEC F i g u r e 3.16 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t PRDICT=.FALSE. 78 s i g n i f i c a n t l y improves t r a c k i n g . I n s t a n d a r d a n a l y t i c a l c o n t r o l , p r e d i c t i o n i s implemented u s i n g the d e s i r e d v e l o c i t y and a c c e l e r a t i o n . I t i s a l s o p o s s i b l e t o use the o b s e r v e d v e l o c i t y and a c c e l e r a t i o n f o r t h i s p u rpose. F i g u r e 3.17 shows a s i m u l a t i o n of l i n e 1 w i t h USEOBS=.TRUE. The r e s u l t i n g p a t h e r r o r i s shown i n t a b l e 3.6. T r a c k i n g i s no b e t t e r than w i t h p r e d i c t i o n u s i n g the d e s i r e d v e l o c i t y and a c c e l e r a t i o n s p e c i f i c a t i o n . When c o n t r o l i s m a r g i n a l such t h a t t h e r e s u l t i n g v e l o c i t y and a c c e l e r a t i o n d i f f e r somewhat from the d e s i r e d v e l o c i t y and a c c e l e r a t i o n , i t might be more b e n e f i c i a l t o use the o b s e r v e d v e l o c i t y and a c c e l e r a t i o n f o r - p r e d i c t i o n . We see p r e d i c t i o n , however, as a way of i m p r o v i n g c o n t r o l t h a t i s a l r e a d y adequate, not as a way of making in a d e q u a t e c o n t r o l become adequate. A l s o , o b s e r v a t i o n of a c c e l e r a t i o n i s not o t h e r w i s e r e q u i r e d , e x cept when l e a r n i n g , as w i l l be shown l a t e r . O b s e r v a t i o n of a c c e l e r a t i o n i s thus an a d d i t i o n a l c o m p u t a t i o n a l expense w i t h l i t t l e b e n e f i t . Hence we adopted p r e d i c t i o n u s i n g the d e s i r e d v e l o c i t y and a c c e l e r a t i o n f o r use i n s t a n d a r d a n a l y t i c a l c o n t r o l . FEEDBACK GAIN W i t h t h e d e p a r t u r e s from i d e a l i t y p r e s e n t i n r e a l i z a b l e , c l o s e d l o o p c o n t r o l , i t i s not p o s s i b l e t o a r b i t r a r i l y i n c r e a s e t h e feedback g a i n s , k 1 and k 2 , t o a c h i e v e ever b e t t e r c o n t r o l . Because of t h e s e d e p a r t u r e s from i d e a l i t y , the n o n - l i n e a r i t i e s of the m a n i p u l a t o r and v i s i o n system a r e o n l y a p p r o x i m a t e l y c a n c e l l e d by the a n a l y t i c a l i n v e r s e dynamics and a n a l y t i c a l i n v e r s e k i n e m a t i c s . Thus the c h a r a c t e r i z a t i o n of the c o n t r o l 79 F i g u r e 3.17 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t USEOBS=.TRUE. 80 system by a second o r d e r , l i n e a r , homogeneous d i f f e r e n t i a l e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t s as i n (3.31) i s o n l y a p p r o x i m a t e l y t r u e . We found t h a t k 1=16 and k 2=64 gave about the b e s t c o n t r o l i n c o m b i n a t i o n w i t h the o t h e r chosen parameters d e f i n i n g s t a n d a r d a n a l y t i c a l c o n t r o l . F i g u r e 3.18 shows a s i m u l a t i o n of l i n e 1 w i t h k 1=64 and k 2 = l 0 2 4 . The c o n t r o l system i s c l e a r l y becoming o s c i l l a t o r y as the feedback g a i n s a r e i n c r e a s e d . F i g u r e 3.19 shows a s i m u l a t i o n of l i n e 1 where the feedback g a i n s are reduced t o ^=4 and k 2=4. Now the c o n t r o l system i s not o s c i l l a t o r y but the p a t h e r r o r i s i n c r e a s e d from t h a t of s t a n d a r d a n a l y t i c a l c o n t r o l due t o the s l o w e r c o r r e c t i o n of v e l o c i t y and p o s i t i o n e r r o r s . The e f f e c t of d e p a r t u r e from 2 . the c o n s t r a i n t t h a t k 1 =4k 2 i s shown i n f i g u r e s 3.20 and 3.21 where l i n e 1 i s s i m u l a t e d w i t h feedback g a i n s of ^=8, k 2=128 and k 1=32, k 2=32, r e s p e c t i v e l y . The p a t h e r r o r s f o r t h e s e s i m u l a t i o n s w i t h v a r i o u s feedback g a i n s a r e l i s t e d i n t a b l e 3.6. I t can be seen t h a t ^=16 and ^ 2=64 a r e r e a s o n a b l e c h o i c e s f o r use i n s t a n d a r d a n a l y t i c a l c o n t r o l . S t a n d a r d a n a l y t i c a l c o n t r o l has been e s t a b l i s h e d by r e a s o n a b l y c h o o s i n g the parameters n e c e s s a r y t o implement a r e a l i z a b l e , c l o s e d l o o p c o n t r o l system f o r the two l i n k m a n i p u l a t o r . S t a n d a r d a n a l y t i c a l c o n t r o l t h u s s e r v e s as a r e a s o n a b l e benchmark a g a i n s t which t o compare l e a r n e d c o n t r o l . 81 4.00 4.00 4.00 T SEC F i g u r e 3.18 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t k,=64 and k o=!024 82 CM Wo %8 0.00 0.80 1.60 T SEC 4.00 o o —'o 0.00 1.60 T SEC 4.00 tn o o X o in 4.00 F i g u r e 3.19 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t k 1=4 and k 2 = 4 83 o o o o CM COo Csl CJ COo CM is 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 o o CJ COo 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 o in X o tn 0.00 0.80 1.60 T SEC 4.00 F i g u r e 3.20 S t a n d a r d a n a l y t i c a l c o n t r o l of l i n e 1, except t h a t k 1 = 8 and k 2=128 84 Figure 3.21 Standard a n a l y t i c a l c o n t r o l of l i n e 1, except that k =32 and k 0=32 85 ( PATH XERMS VXERMS AXERMS NOTES l i n e 1 5 .99E-4 2. 29E-3 l i n e 1 1 .12E-3 4. 53E-3 l i n e 1 1 .62E-3 9. 70E-3 l i n e 1 2 •87E-3 3. 71E-2 l i n e 1 2 .60E-4 1 . 64E-3 l i n e 1 4 .57E-3 7. 06E-3 l i n e 1 6 .60E-4 2. 05E-3 l i n e 1 1 .55E-4 6. 57E-3 l i n e 1 2 .44E-3 3. 06E-3 l i n e 1 2 .47E-4 2. 64E-3 l i n e 1 1 .36E-3 2. 92E-3 9. 54E- 2 S t a n d a r d 1 . 20E- 1 ' c a l c = ° - 0 2 s e c 2. 06E- 1 ^ a l c = ° - 0 3 s e c 6. 45E- 1 ^ a l c = ° - 0 4 s e c 9. 02E- 2 E x a c t V i s i o n 1 . 12E- 1 No P r e d i c t i o n 9. 66E- 2 Use O b s e r v a t i o n s 4. 38E- 1 k 1 = 64, k 2 = 1024 8. 74E- 2 k 1 = 4, k 2 = 4 9. 35E- 2 k 1 = 8 , k 2 = 128 1 . 09E- 1 k 1 = 32, k 2 = 32 T a b l e 3.6 P a t h e r r o r u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l , e x c e p t f o r n p t e d v a r i a t i o n s 86 3.6 ADEQUACY OF A SUM OF POLYNOMIALS REPRESENTATION OF THE INVERSE DYNAMICS AND INVERSE KINEMATICS A p r i n c i p a l g o a l o f t h i s work i s t o l e a r n a sum of p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e i n v e r s e d y n a m i c s and i n v e r s e k i n e m a t i c s o f t h e two l i n k m a n i p u l a t o r w i t h o u t r e c o u r s e t o a n a l y s i s o f t h e m a n i p u l a t o r . B e f o r e c o n s i d e r i n g how s u c h l e a r n i n g i s t o be p e r f o r m e d , i t i s i n s t r u c t i v e t o c o n s i d e r whether t h e i n v e r s e d y n a m i c s and i n v e r s e k i n e m a t i c s c a n e v e n be a d e q u a t e l y r e p r e s e n t e d by a sum o f p o l y n o m i a l s f o r t h e p u r p o s e s o f c o n t r o l . 3.6.1 DERIVATION OF A SUM OF POLYNOMIALS REPRESENTATION OF THE INVERSE DYNAMICS A l t h o u g h t h e i n v e r s e d y n a m i c s a r e q u i t e c o m p l ex, i t i s s t r a i g h t f o r w a r d t o d e r i v e a sum of p o l y n o m i a l s r e p r e s e n t a t i o n t h a t c l o s e l y a p p r o x i m a t e s t h e a n a l y t i c a l i n v e r s e d y n a m i c s . The a n a l y t i c a l i n v e r s e d y n a m i c s a r e g i v e n by e q u a t i o n s (3.51) t h r o u g h ( 3 . 6 6 ) . To d e r i v e a sum o f p o l y n o m i a l s e q u i v a l e n t i t i s n e c e s s a r y t o c o n s i d e r a two l i n k m a n i p u l a t o r w i t h s p e c i f i c p a r a m e t e r s . To t h i s end we have c h o s e n a two l i n k m a n i p u l a t o r a s f o l l o w s , m1 = 1 Kg (3.99) m 2 = 1 Kg (3.100) 1 1 = 1 m (3.101) 1 2 = 1 m (3.102) The s t a n d a r d g r a v i t a t i o n a l a c c e l e r a t i o n has been assumed; g = 9.81 m / s e c 2 (3.103) F o r t h e s e c h o i c e s o f p a r a m e t e r s , t h e c o e f f i c i e n t s i n t h e 87 a n a l y t i c a l i n v e r s e d y n a m i c s a r e , n e g l e c t i n g u n i t s , a s f o l l o w s , d n = 3 + 2 c o s ( a 2 ) (3.104) d 1 2 = 1 + c o s ( a 2 ) (3.105) d n i = 0 (3.106) d 1 2 2 = - s i n ( a 2 ) (3.107) d 1 1 2 = - s i n ( a 2 ) (3.108) d l 2 1 = - s i n ( a 2 ) (3.109) d 1 = I 9 . 6 2 s i n ( a 1 ) + 9.81 s i n ( a } + a 2 ) (3.110) and, d 2 1 = 1 + c o s ( a 2 ) (3.111) d 2 2 = 1 (3.112) d 2 1 1 = s i n ( a 2 ) (3.113) d 2 2 2 = 0 (3.114) d 2 1 2 = 0 (3.115) d 2 2 1 = 0 (3.116) d 2 = 9 . 8 1 s i n ( a 1 + a 2 ) (3.117) W i t h t h e s e c o e f f i c i e n t s t h e a n a l y t i c a l i n v e r s e d y n a m i c s a r e , T 1 = 3 d 1 + 2 c o s ( a 2 ) d 1 + d 2 + c o s ( a 2 ) d 2 2 - s i n ( a 2 ) d 2 - 2 s i n ( a 2 ) d 1 d 2 + I 9 . 6 2 s i n ( a 1 ) + 9.81 s i n ( a 1 + a 2 ) (3.118) T 2 = d 1 + c o s ( a 2 ) d 1 + d 2 + s i n ( a 2 ) d 1 2 + 9.81s in(a,+a 2) (3.119) The s i n e and c o s i n e t e r m s c a n be a p p r o x i m a t e d o v e r t h e ra n g e -7r < a. < 7r u s i n g t h e i r e q u i v a l e n t s e r i e s r e p r e s e n t a t i o n s , t r u n c a t e d t o 4 t h o r d e r and l o w e r t e r m s , c o s ( a 2 ) « 1 - a 2 2 / 2 + a 2 4 / 2 4 (3.120) s i n ( a 2 ) * a 2 - a 2 3 / 6 (3.121) 8 8 s i n ( a i ) = a. - ( 3 . 122) 3 2 2 3 s i n ( a ^ + a 2 ) — + a 2 - /6 - Q^^2 ~ a l a 2 ^ 2 ~ a2 ^ (3.123) S u b s t i t u t i n g t h e s e a p p r o x i m a t i o n s f o r the s i n e and c o s i n e terms and k e e p i n g the r e s u l t i n g terms of 4*"*1 o r d e r or lower y i e l d s a sum of p o l y n o m i a l s r e p r e s e n t a t i o n f o r the i n v e r s e dynamics, T 1 <* 29.4a 1 - 4 . 9 l a 1 3 + 9.81a 2 - 4 . 9 l a 1 2 a 2 - 4 . 9 l a 1 a 2 2 3 . . . 2 » 2- - 1.64a 2 - 2 a 2 a ^ a 2 - °-2a2 + ^ a1 ~ a 2 a1 + 2 d 2 - 0 . 5 a 2 2 d 2 (3. 124) T 2 * 9 . 8 l a 1 - 1 . 6 4 a 1 3 + 9 . 8 l a 2 - 4 . 9 l a 1 2 a 2 - 4 . 9 l a 1 a 2 2 3 2 2 - 1.64a 2 + 0^2°-] + 2d.j - 0. 5 a 2 d^ + d 2 (3.125) D e m o n s t r a t i o n of the a c c u r a c y a c h i e v a b l e w i t h such a 4 ^ o r d e r sum of p o l y n o m i a l s r e p r e s e n t a t i o n i s done i n the f o l l o w i n g s e c t i o n s . 3.6.2 PRE-LEARNING OF THE INVERSE DYNAMICS Havi n g d e r i v e d a sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the i n v e r s e dynamics, i t seemed n a t u r a l t o det e r m i n e whether t h i s r e p r e s e n t a t i o n c o u l d be l e a r n e d u s i n g the a n a l y t i c a l i n v e r s e dynamics as a g u i d e . We c a l l t h i s p r e - l e a r n i n g as the sum of p o l y n o m i a l s r e p r e s e n t a t i o n i s l e a r n e d o f f - l i n e , b e f o r e b e i n g used i n any c o n t r o l scheme. T h i s was done u s i n g a program a n a l a g o u s t o t h a t used i n c h a p t e r 2 t o i n v e s t i g a t e the convergence r a t e s of t h e v a r i o u s l e a r n i n g a l g o r i t h m s e x c e p t t h a t the t a r g e t f u n c t i o n s used were the two f u n c t i o n s t h a t make up the i n v e r s e dynamics. P r e - l e a r n i n g was c a r r i e d out u s i n g Method 2, L e a r n i n g I d e n t i f i c a t i o n , f o r t h e c a s e s , f = T 1 and f = r 2 . The number of i n p u t v a r i a b l e s was v = 6 and the system o r d e r was 89 s = 4. The i n p u t v a r i a b l e s were, z 1 = a 1 (3.126) z 2 = a 2 (3.127) z 3 = o 1 (3. 128) z 4 = d 2 (3.129) z 5 = a 1 • (3.130) Z g = d 2 (3.131) T r a i n i n g p o i n t s were randomly g e n e r a t e d u n i f o r m l y over the space, -1 < z. < 1, and 20,000 i t e r a t i o n s were performed. Torques T 1 and T 2 were l e a r n e d as sums of p o l y n o m i a l s of the i n p u t v a r i a b l e s , z.. F i g u r e 3.22 shows the r e d u c t i o n of A f 1 and A f 2 as a f u n c t i o n of the number of t r a i n i n g i t e r a t i o n s . The graphs show the magnitudes of A f 1 and A f 2 , averaged over i n t e r v a l s of 100 t r a i n i n g i t e r a t i o n s . I t can be seen t h a t the average e r r o r i n e s t i m a t i n g the t a r g e t f u n c t i o n s ( i n t h i s case the t o r q u e s T 1 and T 2 ) has dropped t o about 0.05 N»m. Note t h a t the e s t i m a t e s of the t o r q u e s do not improve a f t e r about 5000 i t e r a t i o n s . T h i s i m p l i e s t h a t b e t t e r a c c u r a c y cannot be a c h i e v e d w i t h o u t i n c r e a s i n g , s, the system o r d e r . T a b l e 3.7 and 3.8 shows the c o e f f i c i e n t s d e r i v e d p r e v i o u s l y and shown i n e q u a t i o n s (3.124) and (3.125) a l o n g w i t h t h e i r c o u n t e r p a r t s t h a t were p r e - l e a r n e d . C l e a r l y , as l e a r n i n g p r o c e e d s , the c o e f f i c i e n t s a r e c o n v e r g i n g t o v a l u e s c l o s e t o t h o s e o b t a i n e d by d e r i v a t i o n . One s h o u l d not e x p e c t an e x a c t c o r r e s p o n d e n c e as the b e s t 4 t h o r d e r a p p r o x i m a t i o n of the a n a l y t i c a l i n v e r s e dynamics i s not l i k e l y t o be e x a c t l y t h a t 90 0.00 0.40 O^BO L 2 0 -10^ I TERAT IONS 1.60 2.00 o_ Figure 3.22 Reduction of e s t i m a t i o n e r r o r during p r e - l e a r n i n g of the i n v e r s e dynamics (e s t i m a t i o n e r r o r averaged over each 100 i t e r a t i o n i n t e r v a l ) 91 o b t a i n e d by t r u n c a t i n g t h e s e r i e s r e p r e s e n t a t i o n s of t h e s i n e and c o s i n e t e r m s , as done' h e r e . F u r t h e r m o r e , L e a r n i n g I d e n t i f i c a t i o n does not q u i t e c o n v e r g e t o t h e b e s t 4 t n o r d e r a p p r o x i m a t i o n a s i t i s l i m i t e d by t h e c o n s t r a i n t t h a t l e a r n i n g e l i m i n a t e s e s t i m a t i o n e r r o r a t t h e l a s t t r a i n i n g p o i n t . T h e r e s h o u l d be, however, a c l o s e c o r r e s p o n d e n c e between t h e d e r i v e d and p r e - l e a r n e d c o e f f i c i e n t s . POLYNOMIAL DERIVED PRE-LEARNED TERM COEFFICIENT COEFFICIENT z 1 29.4 29.3 z 1 3 -4.91 -4.47 Z 2 9.81 9.67 z z 2  Z 2 Z 1 -4.91 -4.24 z 2 z Z 2 Z 1 -4.91 -4.23 -1 .64 -1 .33 Z 4 Z 3 Z 2 -2.00 -1.81 Z 4 % -1 .00 -0.92 Z 5 5.00 5.00 2 Z 5 Z 2 -1 .00 -0.91 Z 6 2.00 1 .99 z z 2  2 6 Z 2 -0.50 -0.51 o t h e r s 0.00 <0.06 T a b l e 3.7 C o e f f i c i e n t s f o r a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e i n v e r s e d y n a m i c s f u n c t i o n f o r t o r q u e T 1 o b t a i n e d by d e r i v a t i o n and by p r e - l e a r n i n g 92 POLYNOMIAL TERM DERIVED COEFFICIENT PRE-LEARNED COEFFICIENT z1 9.81 9.69 3 z 1 . -1 .64 -1 .36 Z 2 9.81 9.71 2 2 2 Z 1 -4.91 -4.24 2 Z 2 Z 1 -4.91 -4.22 Z 2 3 -1 .64 -1.41 z 2 z Z 3 Z 2 1 .00 0.90 Z 5 2.00 2.00 2 Z 5 Z 2 -0.50 -0.44 Z 6 1 .00 0.99 o t h e r s 0.00 <0.05 T a b l e 3.8 C o e f f i c i e n t s f o r a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e i n v e r s e d y n a m i c s f u n c t i o n f o r t o r q u e r2 o b t a i n e d by d e r i v a t i o n and by p r e - l e a r n i n g 3.6.3 ATTEMPTED DERIVATION OF A SUM OF POLYNOMIALS REPRESENTATION OF THE CARTESIAN INVERSE DYNAMICS To a c h i e v e o u r g o a l o f l e a r n e d c o n t r o l w i t h p a t h s p e c i f i c a t i o n i n C a r t e s i a n c o o r d i n a t e s , i t i s n e c e s s a r y t o l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s . Thus i t i s i n s t r u c t i v e t o a t t e m p t t o d e r i v e a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e i n v e r s e k i n e m a t i c s o f t h e two l i n k m a n i p u l a t o r , w h i c h t o g e t h e r w i t h t h e i n v e r s e d y n a m i c s form t h e C a r t e s i a n i n v e r s e d y n a m i c s o f t h e m a n i p u l a t o r as g i v e n i n ( 3 . 1 9 ) . I t was f o u n d t h a t an 93 adequate sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the C a r t e s i a n i n v e r s e d y n a m i c s . c o u l d o n l y be a c h i e v e d over a p o r t i o n a t a time of the space over which t h e m a n i p u l a t o r i s c a p a b l e . o f moving. The i n v e r s e k i n e m a t i c s of the two l i n k m a n i p u l a t o r a r e g i v e n by e q u a t i o n s (3.39) t h r o u g h ( 3 . 4 4 ) . S i n c e i t i s the s i n e s or c o s i n e s of the a n g u l a r p o s i t i o n s , r a t h e r than the a n g l e s t h e m s e l v e s , t h a t are r e q u i r e d f o r s u b s t i t u t i o n i n t o the i n v e r s e dynamics, i t i s u s e f u l t o c o n s i d e r e q u a t i o n s (3.44) t h r o u g h (3.48) r a t h e r than the i n v e r s e k i n e m a t i c s f u n c t i o n s f o r a n g u l a r p o s i t i o n . -Given t h e l i n k l e n g t h s chosen p r e v i o u s l y , the f o r m u l a s f o r the s i n e s and c o s i n e s of a n g u l a r p o s i t i o n become, c o s ( a 2 ) = x ^ / 2 + x 2 2 / 2 - 1 ( 3 . 1 32) s i n ( a 2 ) = (1 - c o s 2 ( a 2 ) ) l / 2 (3.133) 2 2 c o s ( o ] ) = x 1 s i n ( a 2 ) / ( x 1 +x 2 ) - x 2/2 (3.134) s i n U ^ = (1 - c o s 2 ( a 1 ) ) 1 / 2 (3.135) The f i r s t two f u n c t i o n s g i v e n h e r e , (3.132) and (3.133), can be w e l l r e p r e s e n t e d by a sum of p o l y n o m i a l s over the whole space w i t h i n the m a n i p u l a t o r ' s r e a c h , e x c e p t . v e r y c l o s e t o the o r i g i n . The second two f u n c t i o n s , (3.134) and ( 3 . 1 3 5 ) , appear t o not be 2 2 r e p r e s e n t a b l e by a sum of p o l y n o m i a l s due t o the term ( x 1 +x 2 ) i n t h e denominator of ( 3 . 1 3 4 ) . The i n v e r s e k i n e m a t i c s f u n c t i o n s f o r d 1 , d 2 , d 1 and d 2 a l l have the term s i n ( a 2 ) a p p e a r i n g i n a denominator p o s i t i o n . T h i s means t h a t t h e s e f u n c t i o n s have s i n g u l a r i t i e s a t the o r i g i n and a t t h e l i m i t s of the m a n i p u l a t o r ' s r e a c h where s i n ( a 2 ) i s z e r o . The s i n g u l a r i t i e s a t the o r i g i n and a t the l i m i t s of the 94 m a n i p u l a t o r ' s r e a c h r e f l e c t c o n t r o l d i f f i c u l t i e s t h a t a r e i n h e r e n t w i t h the m a n i p u l a t o r . One s h o u l d not expect t o have good c o n t r o l a t t h e s e p o i n t s u s i n g p a t h s p e c i f i c a t i o n i n C a r t e s i a n c o o r d i n a t e s . One would n o r m a l l y not u t i l i z e the m a n i p u l a t o r a t t h e s e p r o b l e m a t i c p o i n t s i n the space. The d i f f i c u l t y i n r e p r e s e n t i n g c o s ( a 1 ) w i t h a sum of p o l y n o m i a l s i s more s i g n i f i c a n t . Even n e g l e c t i n g p o i n t s near the o r i g i n and near the l i m i t s of t h e m a n i p u l a t o r ' s r e a c h , i t does not appear t o be p o s s i b l e t o r e p r e s e n t f u n c t i o n (3.134) as a sum of p o l y n o m i a l s over the r e s t of the m a n i p u l a t o r ' s r e a c h . I t seems t h a t any sum of p o l y n o m i a l s t h a t i s good i n one quadrant i s bad i n the o p p o s i t e q u a d r a n t . A t t e m p t s t o l e a r n c o s ( a 1 ) over a c i r c u l a r band e x c l u d i n g the o r i g i n and the l i m i t s of the m a n i p u l a t o r ' s r e a c h proved f u t i l e . I t was t h u s n e c e s s a r y t o l i m i t l e a r n i n g of the i n v e r s e k i n e m a t i c s and hence the C a r t e s i a n i n v e r s e dynamics t o a p o r t i o n of t h e space. 3.6.4 LIMITATION OF A SUM OF POLYNOMIALS REPRESENTATION OF THE CARTESIAN INVERSE DYNAMICS TO A PORTION OF THE MANIPULATOR'S SPACE To p e r m i t a sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the C a r t e s i a n i n v e r s e dynamics, i t was n e c e s s a r y t o l i m i t l e a r n i n g t o a p o r t i o n of the m a n i p u l a t o r ' s space. T h i s i s not a g r e a t r e s t r i c t i o n of the t e c h n i q u e as most r e a l m a n i p u l a t o r s have r e v o l u t e j o i n t s t h a t a r e c o n s t r a i n e d t o w i t h i n a f r a c t i o n of a complete c i r c l e of m o t i o n . One n o r m a l l y uses the m a n i p u l a t o r i n a r e s t r i c t e d r e g i o n or workspace, t y p i c a l l y i n f r o n t of or below the m a n i p u l a t o r o r i g i n . F u r t h e r , one can use s e v e r a l d i f f e r e n t 95 sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s o f t h e C a r t e s i a n i n v e r s e d y n a m i c s , e a c h f o r ' a d i f f e r e n t p o r t i o n o f t h e m a n i p u l a t o r ' s s p a c e . We r e s t r i c t e d o u r a t t e n t i o n t o a w o r k s p a c e below t h e o r i g i n o f t h e two l i n k m a n i p u l a t o r , namely t h e r e g i o n s p e c i f i e d by, -0.5 < x 1 < 0.5 m (3.136) -1.5 < x 2 < -0.5 m (3.137) V e l o c i t y and a c c e l e r a t i o n were r e s t r i c t e d t o m a g n i t u d e s o f l e s s 2 t h a n 1 m/sec and 1 m/sec , r e s p e c t i v e l y . The n o r m a l i z e d i n p u t v a r i a b l e s f o r t h e l e a r n i n g a l g o r i t h m s were t h u s , z 1 = 2x 1 (3.138) z 2 = 2 x 2 + 2 (3.139) z 3 = x 1 . (3.140) z 4 = x 2 (3.141 ) z 5 = x 1 (3.142) z 6 = x 2 (3.143) Over t h i s p o r t i o n o f t h e m a n i p u l a t o r ' s r e a c h i t i s p o s s i b l e t o r e p r e s e n t t h e C a r t e s i a n i n v e r s e d y n a m i c s a s sums o f p o l y n o m i a l s . The c o s i n e o f a2 i s a l r e a d y e x a c t l y r e p r e s e n t e d a s a sum o f p o l y n o m i a l s o f C a r t e s i a n c o o r d i n a t e s , cos(a2) = z ^ / 6 + z 2 2 / 6 - z 2 / 2 - 1/2 (3.144) The s i n e o f a2 i s e q u a l t o t h e b i n o m i a l e x p a n s i o n o f t h e r i g h t hand s i d e o f ( 3 . 1 3 3 ) , 7 4 6 sin(a2) = 1 - cos(a2) /2 - cos(a2) /8 - c o s ( a 2 ) /16 ... (3.145) A good a p p r o x i m a t i o n f o r sin(a2) c a n be o b t a i n e d by t r u n c a t i n g t h e s e r i e s (3.145) a t an a p p r o p r i a t e l e n g t h , s u b s t i t u t i n g i n t h e 96 e x p r e s s i o n f o r c o s ( a 2 ) , and then r e t a i n i n g a l l p o l y n o m i a l terms of a g i v e n maximum or d e r or l e s s , say 4 f c^ o r d e r . T h i s i s r a t h e r d i f f i c u l t t o do, however, as a l a r g e number of the terms of (3.145) have s i g n i f i c a n t components of 4*"*1 o r d e r o r l o w e r . Over t h e workspace t h e c o s i n e of a 1 i s , cos(a.,) = 1/2 - z 2/4 + z 1 s i n ( a 2 ) / ( z 1 2 / 2 + z 2 2 / 2 - 2 z 2 + 2 ) (3.146) The t r o u b l e s o m e p o l y n o m i a l i n the denominator of (3.146) can be r e p l a c e d w i t h i t s T a y l o r s e r i e s e q u i v a l e n t t a k e n about the o r i g i n of the workspace, ( z l 2 / 2 + z 2 2 / 2 - 2 z 2 + 2 ) " 1 = \/2 + z 2 / 2 - z ^ / 8 + 3 z 2 2 / 8 - 3 Z ] 2 Z 2 / 1 2 + 3 Z 2 3 / 1 2 ... (3.147) The e x p r e s s i o n f o r c o s ( a ^ ) can then be t r u n c a t e d t o those terms of 4 f ck o r d e r o r lo w e r . G i v e n a sum of p o l y n o m i a l s r e p r e s e n t a t i o n of c o s ( a 1 ) , one can then o b t a i n the s i n e of u s i n g a b i n o m i a l e x p a n s i o n of the r i g h t hand s i d e of (3.135) , as was done f o r s i n (a 1 ), s i n ( a 2 ) = 1 - c o s ( a 2 ) 2 / 2 - c o s ( a 2 ) 4 / 8 - c o s ( a 2 ) 6 / l 6 ... (3.148) The f a c t o r l / s i n ( a 2 ) t h a t appears i n the i n v e r s e k i n e m a t i c s f u n c t i o n s f o r d 1 , d 2 , d 1 and d 2 can be r e p l a c e d w i t h the T a y l o r s e r i e s f o r the r e c i p r o c a l of t h e p o l y n o m i a l a p p r o x i m a t i o n f o r s i n ( a 2 ) , t a k e n about the o r i g i n of the workspace. A c c u r a t e r e p r e s e n t a t i o n of t h e s e s i n e , c o s i n e and co s e c a n t terms can be o b t a i n e d u s i n g t h e p o l y n o m i a l s e r i e s shown h e r e , t r u n c a t e d t o a c e r t a i n maximum o r d e r , 4*"** o r d e r f o r example. S u b s t i t u t i n g t h e s e r e s u l t s i n t o the i n v e r s e k i n e m a t i c s 9 7 r e l a t i o n s h i p s and s u b s e q u e n t l y c o m b i n i n g t h e i n v e r s e k i n e m a t i c s w i t h i n v e r s e d y n a m i c s , one c a n o b t a i n an a c c u r a t e sum of p o l y n o m i a l s e x p r e s s i o n f o r t h e C a r t e s i a n i n v e r s e d y n a m i c s . A g a i n , o n l y t e r m s o f a s c e r t a i n maximum o r d e r o r l e s s need be r e t a i n e d . I t i s c l e a r l y d i f f i c u l t t o p e r f o r m t h e a l g e b r a n e c e s s a r y t o d e r i v e s u c h a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e C a r t e s i a n i n v e r s e d y n a m i c s . The f e a s i b i l i t y of s u c h a r e p r e s e n t a t i o n c a n be d e m o n s t r a t e d , however, by p r e - l e a r n i n g a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e C a r t e s i a n i n v e r s e d y n a m i c s . 3.6.5 PRE-LEARNING OF THE CARTESIAN INVERSE DYNAMICS To d e m o n s t r a t e t h e a d e q u a c y o f sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e C a r t e s i a n i n v e r s e d y n a m i c s , s u c h a r e p r e s e n t a t i o n was p r e - l e a r n e d u s i n g t h e a n a l y t i c a l C a r t e s i a n i n v e r s e d y n a m i c s a s a g u i d e . T h i s was done u s i n g a p r o g r a m a n a l a g o u s t o t h a t u s e d i n c h a p t e r 2 t o i n v e s t i g a t e t h e c o n v e r g e n c e r a t e s o f t h e v a r i o u s l e a r n i n g a l g o r i t h m s e x c e p t t h a t t h e t a r g e t f u n c t i o n s u s e d were t h e two f u n c t i o n s t h a t make up t h e C a r t e s i a n i n v e r s e d y n a m i c s . P r e - l e a r n i n g was c a r r i e d o u t * u s i n g Method 2, L e a r n i n g I d e n t i f i c a t i o n , f o r t h e c a s e s f = r 1 * and f ~ T 2 ' T n e number °f i n p u t v a r i a b l e s was v = 6 and t h e s y s t e m o r d e r was s = 4. The i n p u t v a r i a b l e s were t h o s e g i v e n i n e q u a t i o n s (3.138) t h r o u g h ( 3 . 1 4 3 ) . T r a i n i n g p o i n t s were ra n d o m l y g e n e r a t e d u n i f o r m l y o v e r t h e s p a c e , -1 < z. < 1, and 20,000 i t e r a t i o n s were p e r f o r m e d . F i g u r e 3.23 shows t h e r e d u c t i o n o f A f 1 a n d A f 2 a s a f u n c t i o n o f t h e number o f t r a i n i n g i t e r a t i o n s p e r f o r m e d . The 98 0.00 0.40 0JBC) JL20 «io« ITERATIONS 1.60 2.00 Figure 3.23 Reduction of e s t i m a t i o n e r r o r during p r e - l e a r n i n g of the C a r t e s i a n inverse dynamics (e s t i m a t i o n e r r o r averaged over each 100 i t e r a t i o n i n t e r v a l ) 99 g r a p h s show t h e m a g n i t u d e s o f A f 1 and A f 2 , a v e r a g e d o v e r i n t e r v a l s o f 100 t r a i n i n g i t e r a t i o n s . The a v e r a g e e r r o r i n e s t i m a t i n g t h e t a r g e t f u n c t i o n s ( i n t h i s c a s e t h e t o r q u e s T 1 and r 2 ) has d r o p p e d t o a b o u t 0.6 N.m. I t was f o u n d t h a t t h e C a r t e s i a n i n v e r s e d y n a m i c s , a s p r e - l e a r n e d h e r e , were a d e q u a t e t o a c h i e v e c o n t r o l c o m p a r a b l e t o t h a t u s i n g t h e a n a l y t i c a l C a r t e s i a n i n v e r s e d y n a m i c s . 3.6.6 CLOSED LOOP CONTROL USING THE PRE-LEARNED CARTESIAN INVERSE DYNAMICS To t e s t t h e a d e q u a c y o f t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s , s i m u l a t i o n s were c a r r i e d o ut i n w h i c h t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s were s u b s t i t u t e d f o r t h e a n a l y t i c a l i n v e r s e i n o t h e r w i s e s t a n d a r d a n a l y t i c a l c o n t r o l . S i m u l a t i o n p a r a m e t e r s were s e t as f o l l o w s : CLOSED=.TRUE., INVARM=.FALSE., EXACT=.FALSE., VISION=.TRUE., DELAY=.TRUE., PRDICT=.TRUE., USEOBS=.FALSE., k 1=16, k 2=64 and t c a i c = 0 « u 1 s e c • T n e p r e - l e a r n e d c o e f f i c i e n t s were u s e d i n L e a r n i n g M a c h i n e 1 t o e s t i m a t e t o r q u e s . No l e a r n i n g t o o k p l a c e d u r i n g t h e s i m u l a t i o n s . S i m u l a t i o n s were c a r r i e d o u t f o r a l l s i x s t a n d a r d p a t h s . The r e s u l t i n g p a t h e r r o r s a r e shown i n t a b l e 3.9. Mean p o s i t i o n t r a c k i n g e r r o r i s a b o u t 3 mm compared t o a b o u t 0.5 mm when u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l . V e l o c i t y and a c c e l a r a t i o n t r a c k i n g e r r o r s a r e s i m i l a r l y i n c r e a s e d . A l s o shown i n t a b l e 3.9 i s t h e e r r o r i n e s t i m a t i n g t h e t o r q u e and t h e maximum t o r q u e t h a t i s a p p l i e d d u r i n g e a c h p a t h . The e r r o r i n t h e e s t i m a t i n g t h e t o r q u e i s t y p i c a l l y a b o u t 4 p e r c e n t o f t h e maximum a p p l i e d t o r q u e . F i g u r e 3.24 shows a p l o t o f p a t h 1. I t i s e v i d e n t t h a t 100 F i g u r e 3.24 C l o s e d l o o p c o n t r o l of l i n e 1 u s i n g the p r e - l e a r n e d C a r t e s i a n i n v e r s e dynamics 101 c o n t r o l i s q u i t e s t a b l e as no l a r g e g l i t c h e s a r e o c c u r i n g o t h e r t h a n some o v e r s h o o t i n g o f s t e p s i n a c c e l e r a t i o n . W h i l e t h e p e r f o r m a n c e i s n o t q u i t e as good a s w i t h s t a n d a r d a n a l y t i c a l c o n t r o l , i t i s good enough t o d e m o n s t r a t e t h a t a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e C a r t e s i a n i n v e r s e d y n a m i c s i s a d e q u a t e t o a c h i e v e good c o n t r o l . PATH XERMS VXERMS AXERMS TERMS TRQMAX l i n e 1 3. 82E-3 7. 26E-3 9. 94E-2 3 • 44E-1 8.33 l i n e 2 8. 88E-3 2. 02E-3 1 . 36E-1 4 .08E- 1 9.15 l i n e 3 6. 05E-4 2. 76E-3 1 . 01E-1 1 .65E- 1 14.1 l i n e 4 3. 32E-3 6. 49E-3 1 . 01E-1 4 . 56E-1 6.79 c i r c l e 1 2. 46E-3 6. 19E-3 5. 30E-2 3 .21E- 1 9.62 c i r c l e 2 2. 01E-3 6. 41E-3 5. 56E-2 2 .68E- 1 13.6 T a b l e 3.9 P a t h e r r o r and t o r q u e e r r o r u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s i n c l o s e d l o o p c o n t r o l 3.7 ADEQUACY OF A SUM OF POLYNOMIALS REPRESENTATION OF THE DIRECT POSITION KINEMATICS A n o t h e r p r i n c i p a l g o a l o f t h i s work i s t o l e a r n a sum of p o l y n o m i a l s r e p r e s e n t a t i o n of t h e d i r e c t k i n e m a t i c s of t h e two l i n k m a n i p u l a t o r w i t h o u t r e c o u r s e t o a n a l y s i s of t h e m a n i p u l a t o r . As w i t h t h e i n v e r s e d y n a m i c s and i n v e r s e k i n e m a t i c s , i t i s i n s t r u c t i v e t o f i r s t c o n s i d e r whether t h e 102 d i r e c t p o s i t i o n k i n e m a t i c s can be a d e q u a t e l y r e p r e s e n t e d by a sum of p o l y n o m i a l s f o r the purpose of r e p l a c i n g the v i s i o n system. 3.7.1 DERIVATION OF A SUM OF POLYNOMIALS REPRESENTATION OF THE DIRECT POSITION KINEMATICS A sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the d i r e c t p o s i t i o n k i n e m a t i c s can be d e r i v e d q u i t e e a s i l y . S i n c e the d i r e c t k i n e m a t i c s of any m a n i p u l a t o r can be o b t a i n e d i n a s i m i l a r form by a g e n e r a l t e c h n i q u e , i t would appear t h a t a sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the d i r e c t k i n e m a t i c s i s a c h i e v a b l e f o r a l l m a n i p u l a t o r s . The a n a l y t i c a l d i r e c t p o s i t i o n k i n e m a t i c s a r e g i v e n by e q u a t i o n s (3.32) and ( 3 . 3 3 ) . F o r a two l i n k m a n i p u l a t o r w i t h p a r a m e t e r s as g i v e n i n e q u a t i o n s (3.99) t h r o u g h (3.103), the a n a l y t i c a l d i r e c t k i n e m a t i c s a r e as f o l l o w s , x 1 = s i n ( a 1 ) + s i n ( a 1 + a 2 ) (3.149) x 2 = - c o s l a ^ ) - c o s ( a 1 + a 2 ) (3.150) S i n c e we a r e co n c e r n e d w i t h m o d e l l i n g the d i r e c t p o s i t i o n k i n e m a t i c s over the workspace i n p a r t i c u l a r , t h e i n p u t v a r i a b l e s a sum of p o l y n o m i a l s r e p r e s e n t a t i o n can be chosen as f o l l o w s , z} = (a, + ir/3)/U/2) (3.151) z 2 = ( a 2 - 27r/3 ) / ( 7 r/2) (3.152) < z 1 , z 2 > = <0,0> c o r r e s p o n d s t o t h e c e n t e r of t h e workspace and the workspace i s e n c l o s e d i n the r e g i o n d e f i n e d by -1 < z. < 1. W i t h such a n o r m a l i z a t i o n of the i n p u t v a r i a b l e s the d i r e c t p o s i t i o n k i n e m a t i c s become, 103 x 1 = [sin ( 7 r z ^/2) - i/3cos(irz ^/2) + 5 ^ ( 7 ^ / 2 ) 0 0 5 ( 7 ^ 2 / 2 ) + cos (7T2 ^2)5111(7:22/2) + ^ 3 0 0 5 ( 7 : 2 ^ 2 ) 0 0 5 ( 7 : 2 2 / 2 ) - l / S s i n d r z / 2 ) s i n ( i r z 2 / 2 ) ]/2 (3.153) x 2 = [~cos(7:z 1/2) - i/3sin(7T2^/2) - 0 0 5 ( 7 : 2 ^ 2 ) 0 0 5 ( 7 : 2 2 / 2 ) + s i n ( T : Z 1 / 2 ) s i n (irz2/2) + v /3sin(7:z 1/2)cos(7:z2/2) + /3cos ( 7 ^ / 2 ) s i n (7:z 2/2 ) ]/2 (3.154) The s i n e and c o s i n e terms can be r e p l a c e d w i t h t h e i r s e r i e s e q u i v a l e n t s , t r u n c a t e d t o 4 f c^ o r d e r or lower terms, s i n ( j r z 1 / 2 ) « *zy/2 - vZz,3/4B (3.155) cos(-irz 1/2) <* 1 - T : 2 Z 1 2 / 4 + i r 4 z 1 4 / 3 8 4 ( 3 . 1 56) s i n ( f f 2 2 / 2 ) * KZy/2 - T : 3 Z 1 3/48 (3 . 1 57) c o s ( i r z 2 / 2 ) « 1 - %2z2/k + 7^2^/384 (3.158) S u b s t i t u t i n g t h e s e e x p r e s s i o n s i n t o e q u a t i o n s (3.153) and (3.154), and keep i n g t h o s e terms of 4 t h o r d e r or lower y i e l d s a sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the d i r e c t p o s i t i o n k i n e m a t i c s , x 1 = i r Z j / 2 + T:22/4 - V /3T: 22 12 2/8 - v /3T: 22 2 2/16 - n3z^3/48 - T: 32 2 3/96 - T : 3 Z 1 2 Z 2 / 3 2 - Tp>zyz2/Z2 + v / 3 T : 4 Z 3 2 2 / 1 9 2 + I / 3 T : 4 Z 1 Z 2 3 / 1 9 2 + V/3T : 4 Z 1 2 Z 2 2 / 1 28 + I/3T : 4 Z 2 4/768 (3.159) x 2 = -1 + ]/3nz2/A + T: 22 1 2/8 + %2z2/\S> - y/ZtC'z 2z2/Z2 - V /3T: 32 12 2 2/64 + T: 22 12 2/8 - V /3T: 32 2 3/96 - 7^2^/384 - T: 42 1 32 2/192 - T : 4 Z 1 2 Z 2 2 / 1 2 8 - T : 4 Z 1 Z 2 3 / 1 9 2 - T : 4 Z 2 4 / 7 6 8 (3 . 160) To t e s t t he a c c u r a c y of t h e d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s , s i m u l a t i o n s were c a r r i e d out u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l f o r the s i x s t a n d a r d p a t h s w h i l e u s i n g the 104 d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s t o e s t i m a t e t h e p o s i t i o n a t s u c c e s s i v e p o i n t s i n t i m e . As w i t h t h e m o d e l l e d v i s i o n s y s t e m , v e l o c i t y and a c c e l e r a t i o n e s t i m a t e s a r e o b t a i n e d by s i m p l e d i f f e r e n t i a t i n g o f t h e p o s i t i o n e s t i m a t e s . T a b l e 3.10 shows t h e e r r o r i n p a t h e s t i m a t i o n f o r t h e v a r i o u s s t a n d a r d p a t h s . E s t i m a t e s o f v e l o c i t y and a c c e l e r a t i o n compare w e l l w i t h t h o s e o b t a i n e d u s i n g t h e m o d e l l e d v i s i o n s y s t e m i n w h i c h p o s i t i o n e s t i m a t e s a r e e x a c t . E r r o r s i n p a t h e s t i m a t i o n by t h e v i s i o n s y s t e m a r e shown i n t a b l e 3.11. F i g u r e 3.25 shows t h e r e s u l t i n g v i e w of s t a n d a r d p a t h c i r c l e 1 u s i n g t h e d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s . P A T H LXERMS LVERMS LAERMS l i n e 1 l i n e 2 l i n e 3 l i n e 4 c i r c l e 1 c i r c l e 2 4.89E-3 3.43E-2 8.69E-4 1 .28E-.2 6.60E-3 6.58E-3 7.75E-3 3.64E-2 3.42E-3 1.65E-2 1.53E-3 1.43E-3 5.25E-2 1.09E-1 4.82E-2 6.92E-2 6.08E-2 5.70E-2 T a b l e 3.10 P a t h e s t i m a t i o n e r r o r u s i n g t h e d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s 105 o in o o' • - . •o" I CM X UJO o to o in ).50 •0.30 •0.10 0,10 L E X ( l ) M 0.30 0.50 F i g u r e 3.25 View of c i r c l e 1 u s i n g t h e d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s 106 PATH LXERMS LVERMS LAERMS l i n e 1 0.0 3.01E-3 4.62E-2 l i n e 2 0.0 3.02E-3 4.64E-2 l i n e 3 0.0 3.21E-3 4.80E-2 l i n e 4 0.0 3.20E-3 4.56E-2 c i r c l e 1 0.0 2.90E-3 2.56E-2 c i r c l e 2 0.0 2.89E-3 2.57E-2 T a b l e 3.11 P a t h e s t i m a t i o n e r r o r u s i n g t h e v i s i o n s y s t e m w h i c h has e x a c t p o s i t i o n e s t i m a t i o n 3.7.2 PRE-LEARNING OF THE DIRECT POSITION KINEMATICS H a v i n g d e r i v e d a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s , i t was v e r i f i e d t h a t t h i s r e p r e s e n t a t i o n c o u l d be p r e - l e a r n e d u s i n g t h e a n a l y t i c a l d i r e c t p o s i t i o n k i n e m a t i c s a s a g u i d e . T h i s was done u s i n g a p r o g r a m a n a l a g o u s t o t h a t u s e d i n c h a p t e r 2 t o i n v e s t i g a t e t h e c o n v e r g e n c e r a t e s o f t h e v a r i o u s l e a r n i n g a l g o r i t h m s e x c e p t t h a t t h e t a r g e t f u n c t i o n s u s e d were t h e two f u n c t i o n s t h a t make up t h e d i r e c t p o s i t i o n k i n e m a t i c s . Method 2, L e a r n i n g I d e n t i f i c a t i o n , was u s e d f o r t h e c a s e s , f = x 1 and f = x 2 « The number o f i n p u t v a r i a b l e s was v = 6 and t h e s y s t e m o r d e r was s = 4. The i n p u t v a r i a b l e s were t h o s e g i v e n i n e q u a t i o n s (3.151) and ( 3 . 1 5 2 ) . T r a i n i n g p o i n t s were r a n d o m l y g e n e r a t e d u n i f o r m l y o v e r t h e s p a c e , -1 < z. < 1, and 30,000 i t e r a t i o n s were p e r f o r m e d . 107 F i g u r e 3.26 shows the r e d u c t i o n of the e s t i m a t i o n e r r o r s , Af 1 and A f 2 , as a f u n c t i o n of the number of i t e r a t i o n s . The graphs show the magnitudes of A f 1 and A f 2 , averaged over i n t e r v a l s of 100 t r a i n i n g i t e r a t i o n s . The average e r r o r i n e s t i m a t i n g the t a r g e t f u n c t i o n s ( i n t h i s case t h e p o s i t i o n s x 1 and x 2> d e c r e a s e s t o about 0.03 m. T h i s r e p r e s e n t s the b e s t a c c u r a c y a c h i e v a b l e w i t h a o r d e r system as f u r t h e r i t e r a t i o n s were not r e d u c i n g the e r r o r . T a b l e 3.12 l i s t s t he p r e - l e a r n e d c o e f f i c i e n t s a l o n g s i d e the d e r i v e d c o e f f i c i e n t s from e q u a t i o n s (3.159) and (3.160). I t can be seen t h a t the l e a r n e d c o e f f i c i e n t s a re q u i t e c l o s e t o those d e r i v e d p r e v i o u s l y . G i v e n the s i m i l a r i t y i n c o e f f i c i e n t s , one would expect t h a t p a t h e s t i m a t i o n e r r o r s u s i n g the p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s would be s i m i l a r t o tho s e t h a t o c c u r r e d p r e v i o u s l y w i t h the d e r i v e d d i r e c t p o s i t i o n k i n e m a t i c s . T a b l e 3.13 shows the e r r o r s i n p a t h e s t i m a t i o n f o r the v a r i o u s s t a n d a r d p a t h s when s i m u l a t e d u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l and viewed w i t h the p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F i g u r e 3.27 shows the r e s u l t i n g view of c i r c l e 1. 108 ex -o - Figure 3.26 Reduction of e s t i m a t i o n e r r o r during p r e - l e a r n i n g of the d i r e c t p o s i t i o n kinematics ( e s t i m a t i o n e r r o r averaged over each 100 i t e r a t i o n i n t e r v a l ) 109 110 POLYNOMIAL TERM X(1) COEFFICIENTS DERIVED PRE-LEARNED X( 2 ) COEFFICIENTS DERIVED PRE-LEARNED 1 0.0 0.020 -1 .00 -1 .02 z 1 1 .57 1 .50 0.0 -0.082 2 2 1 0.0 0.044 1 .23 1.18 Z/ -0.646 -0.496 0.0 0. 1 33 - I 4 0.0 0.056 -0.254 -0.226 Z 2 0.785 0.730 1 .36 1 .26 Z 2 2 1 -2.14 -2.04 1 .23 1.14 2 Z 2 Z 1 -0.969 -0.668 -1 .68 -1 .12 Z 2 Z 1 3 0.879 0.613 -0.507 -0.398 2 z 2 -1 .07 -1 .02 0.617 0.580 z 2 z z 2 Z 1 -0.969 -0.667 -0.839 -1.16 1 .32 0.984 -0.761 -0.542 -0.323 -0.221 -0.559 -0.364 z h 0.879 0.632 -0.507 -0.380 z 2 4 0.220 0.146 -0.127 -0.112 T a b l e 3.12 C o e f f i c i e n t s f o r d e r i v e d and p r e - l e a r n e d sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s o f t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n s f o r p o s i t i o n s x. and x 9 111 PATH LXERMS LVERMS LAEPvMS l i n e 1 l i n e 2 l i n e 3 l i n e 4 c i r c l e 1 c i r c l e 2 3.17E-2 4.49E-2 3.74E-2 3.27E-2 3.74E-2 3.73E-2 1.59E-2 3.11E-2 2.52E-2 1.79E-2 2.18E-2 2.40E-2 5.78E-2 7.67E-2 7.00E-2 5.37E-2 4.85E-2 4.98E-2 T a b l e 3.13 P a t h e s t i m a t i o n e r r o r u s i n g t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 3.7.3 CLOSED LOOP CONTROL USING THE PRE-LEARNED CARTESIAN INVERSE DYNAMICS AND PRE-LEARNED DIRECT POSITION KINEMATICS S i m u l a t i o n s v e r i f i e d t h a t c l o s e d l o o p c o n t r o l i s p o s s i b l e u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . S i m u l a t i o n p a r a m e t e r s were s e t a s f o l l o w s : CLOSED=.TRUE., INVARM=.FALSE., EXACT=.FALSE., VI SION=.FALSE., DELAY=.TRUE., PRDICT=.TRUE., USEOBS=.FALSE., k = 1 6 , k~=64 a n d t , =0.01 s e c . A l l s i x s t a n d a r d p a t h s were s i m u l a t e d . The r e s u l t i n g p a t h e r r o r s a r e shown i n t a b l e 3.14. 112 PATH XERMS VXERMS AXERMS l i n e 1 2.77E-2 2.27E-2 1.78E-1 l i n e 2 4.08E-2 6.13E-2 3.10E-1 l i n e 3 3.66E-2 5.87E-2 3.16E-1 l i n e 4 2.82E-2 2.83E-2 1.87E-1 c i r c l e 1 3.52E-2 3.24E-2 1.56E-1 c i r c l e 2 3.37E-2 3.19E-2 1.41E-1 T a b l e 3.14 P a t h e r r o r u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s i n c l o s e d l o o p c o n t r o l T r a c k i n g i s s i g n i f i c a n t l y worse t h a n t h a t a c h i e v e d w i t h t h e v i s i o n s y s t e m , however, t h i s i s t o be e x p e c t e d due t o t h e i n a c c u r a c i e s o f t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . P a t h p o s i t i o n e r r o r s a r e o f t h e same o r d e r o f m a g n i t u d e as p o s i t i o n e s t i m a t i o n e r r o r s , a s shown p r e v i o u s l y i n t a b l e 3.12. The c o n t r o l s y s t e m p e r c e i v e s t h a t i t i s f o r c i n g t h e m a n i p u l a t o r t o c l o s e l y f o l l o w t h e p a t h s p e c i f i c a t i o n . F o r example, f i g u r e 3.28 shows t h e r e s u l t i n g p a t h a s s e e n u s i n g t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . S t a n d a r d p a t h c i r c l e 1 a p p e a r s t o be f o l l o w e d c l o s e l y . Due t o t h e i n a c c u r a c i e s i n t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s , t h e a c t u a l p a t h i s somewhat d i f f e r e n t , a s shown i n f i g u r e 3.29. The m a n i p u l a t o r must move t h r o u g h a d i s t o r t e d t r a j e c t o r y s u c h t h a t t h e p e r c e i v e d t r a j e c t o r y f o l l o w s t h e p a t h s p e c i f i c a t i o n . 113 r 0 . 5 0 -0.30 -0.10 0.10 0.30 0.50 LEX(l) M F i g u r e 3.28 View of c i r c l e 1 u s i n g the p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g c l o s e d l o o p c o n t r o l of c i r c l e 1 using the p r e - l e a r n e d C a r t e s i a n i n v e r s e dynamics and p r e - l e a r n e d d i r e c t p o s i t i o n kinematics 114 o in tn ^O.SO -0.30 -0.10 0.10 0.30 0.50 XU) M Figure 3.29 View of c i r c l e 1 during c l o s e d loop c o n t r o l of c i r c l e 1 using the pre-learned C a r t e s i a n inverse dynamics and pre-learned d i r e c t p o s i t i o n kinematics 115 F i g u r e 3.30 shows t h e r e s u l t i n g p a t h f o r t h e s i m u l a t i o n of l i n e 1. The l a r g e a c c e l e r a t i o n s t h a t i n i t i a l l y o c c u r a r e due t o t h e sudden p e r c e i v e d p o s i t i o n e r r o r t h a t r e s u l t s when one s w i t c h e s f r o m t h e a c c u r a t e v i s i o n s y s t e m t o t h e l e s s a c c u r a t e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . T h i s u n d e s i r a b l e g l i t c h c a n be e l i m i n a t e d by u s i n g I n t e r f e r e n c e M i n i m i z a t i o n o r L e a r n i n g I d e n t i f i c a t i o n t o p e r f o r m one i t e r a t i o n o f a d d i t i o n a l l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s b e f o r e s w i t c h i n g t o use o f t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s f o r f e e d b a c k p u r p o s e s . T h i s i s i n v o k e d by s e t t i n g l o g i c a l v a r i a b l e ADJVIS t o be t r u e . S i n c e I n t e r f e r e n c e M i n i m i z a t i o n and L e a r n i n g I d e n t i f i c a t i o n e l i m i n a t e e s t i m a t i o n e r r o r a t t h e t r a i n i n g p o i n t , t h i s a v o i d s t h e sudden p e r c e i v e d p o s i t i o n e r r o r when s u b s e q u e n t l y s w i t c h i n g t o t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F i g u r e 3.31 shows a s i m u l a t i o n o f l i n e 1 u s i n g t h i s t e c h n i q u e . N o t i c e how t r a c k i n g i s i m p r o v e d n e a r t h e t r a i n i n g p o i n t b ut m i n i m a l l y a f f e c t e d away fr o m t h e t r a i n i n g p o i n t . S i m i l a r g l i t c h e s a t t h e end o f a p a t h where t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s a r e u s e d c a n be a v o i d e d by m e a s u r i n g t h e f i n a l p o s i t i o n e r r o r w i t h t h e v i s i o n s y s t e m and t h e n e x e c u t i n g a s h o r t c o r r e c t i v e p a t h u s i n g t h e v i s i o n s y s t e m f o r f e e d b a c k p u r p o s e s . In f i g u r e s 3.30 and 3.31 t h e r e a r e s e v e r a l o t h e r u n e x p l a i n e d g l i t c h e s i n t h e a c c e l e r a t i o n p r o f i l e s . T h e s e o c c u r b e c a u s e t h e l e a r n e d C a r t e s i a n i n v e r s e t h a t d r i v e s t h e m a n i p u l a t o r i s d o i n g o n l y a n o m i n a l j o b o f c a n c e l l i n g t h e d i r e c t d y n a m i c s and t h e d i r e c t k i n e m a t i c s b a s e d on t h e l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . I t a p p e a r s t h a t a t c e r t a i n p o i n t s i n p a t h 116 s p a c e , t h e m i s m a t c h i n t h e i n v e r s e and d i r e c t components i s c a u s i n g f e e d b a c k t o ° b e c o m e p o s i t i v e , t h u s b r i e f l y r e i n f o r c i n g e r r o r s u n t i l s u c h p r o b l e m a t i c p o i n t s i n p a t h s p a c e a r e l e f t . C l e a r l y , i f t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s were much l e s s a c c u r a t e , c o n t r o l would become u n s t a b l e o v e r l a r g e r e g i o n s o f p a t h s p a c e and p a t h s p e c i f i c a t i o n s c o u l d n o t be f o l l o w e d . N e v e r t h e l e s s , t h e s i m u l a t i o n s show t h a t c o a r s e c o n t r o l i s p o s s i b l e u s i n g t h e p r e - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F i n e r c o n t r o l a p p e a r s t o r e q u i r e o n l y an improvement i n t h e a c c u r a c y of t h e l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . 117 o o —1 - 1 C M • • U J 0 Sg 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 F i g u r e 3.30 C l o s e d l o o p c o n t r o l of l i n e 1 u s i n g the p r e - l e a r n e d C a r t e s i a n i n v e r s e dynamics and p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 118 o o CM O COo , 0.00 0.80 1.60 T SEC 2.40 3 .20 4.00 s s c_> I ± J 0 cos 0.00 0.80 1.60 T SEC 2.40 3.20 4.00 s in 0.00 0.80 1.60 T SEC 2.40 4.00 Figure 3.31 Closed loop c o n t r o l of l i n e 1 using the pre-learned C a r t e s i a n inverse dyanmics and pre-learned d i r e c t p o s i t i o n kinematics w i t h ADJVIS=.TRUE. 119 3.8 SELF-LEARNING OF THE CARTESIAN INVERSE DYNAMICS H a v i n g d e m o n s t r a t e d t h a t a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n of t h e C a r t e s i a n i n v e r s e d y n a m i c s i s a d e q u a t e t o a c h i e v e good c o n t r o l , i t r e m a i n s o n l y t o f i n d a method o f l e a r n i n g t h e C a r t e s i a n i n v e r s e d y n a m i c s w i t h o u t r e c o u r s e t o a n a l y s i s of t h e m a n i p u l a t o r . We have f o u n d t h a t t h i s c a n be done t h r o u g h o b s e r v a t i o n of a p p l i e d t o r q u e s and c o r r e s p o n d i n g m a n i p u l a t o r m o t i o n ; hence t h e name s e l f - l e a r n i n g . 3.8.1 A METHOD FOR SELF-LEARNING OF THE CARTESIAN INVERSE DYNAMICS THE METHOD G i v e n s u f f i c i e n t and v a r i e d o b s e r v a t i o n s o f t h e m a n i p u l a t o r p o s i t i o n , v e l o c i t y , t o r q u e s and r e s u l t i n g a c c e l e r a t i o n , one c a n use I n t e r f e r e n c e M i n i m i z a t i o n o r r e l a t e d methods t o l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s . The c o r r e s p o n d e n c e between p o s i t i o n , v e l o c i t y , a c c e l e r a t i o n and t o r q u e imposed by t h e d i r e c t d y n a m i c s and d i r e c t k i n e m a t i c s , i e . t h e m a n i p u l a t o r and t h e v i s i o n s y s t e m , i s t h e same a s t h a t imposed by t h e C a r t e s i a n i n v e r s e d y n a m i c s ; an n - t u p l e o f o b s e r v a t i o n s v a l i d f o r t h e d i r e c t r e l a t i o n s h i p s i s a l s o a v a l i d n - t u p l e f o r t h e i n v e r s e r e l a t i o n s h i p s . To l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s o v e r a p a t h s p a c e s u c h a s t h a t d e f i n e d by e q u a t i o n s (3.138) t h r o u g h ( 3 . 1 4 3 ) , two c o n d i t i o n s must be met: F i r s t , one must o b t a i n o b s e r v a t i o n s c o r r e s p o n d i n g t o a l l r e g i o n s o f t h e p a t h s p a c e , -1 < z . < 1. H e r e t h e z. a r e t h e n o r m a l i z e d C a r t e s i a n p o s i t i o n s , v e l o c i t i e s a n d a c c e l e r a t i o n s t h a t a c t a s i n p u t v a r i a b l e s i n t h e .sum o f 120 p o l y n o m i a l s r e p r e s e n t a t i o n of t h e C a r t e s i a n i n v e r s e dynamics. O b s e r v a t i o n s c o r r e s p o n d i n g t o the i n f i n i t e number of p o i n t s i n p a t h space a r e f o r t u n a t e l y not n e c e s s a r y ; t h e g e n e r a l i z a t i o n t h a t o c c u r s when u s i n g I n t e r f e r e n c e M i n i m i z a t i o n or r e l a t e d methods w i l l e f f e c t i v e l y i n t e r p o l a t e between t r a i n i n g p o i n t s . There must be s u f f i c i e n t and v a r i e d o b s e r v a t i o n s , however, t o a l l o w a c c u r a t e i n t e r p o l a t i o n over the whole of p a t h space and t o overcome the l e a r n i n g i n t e r f e r e n c e t h a t o c c u r s d u r i n g t r a i n i n g . S e c o n d l y , one must c o n s t r a i n m a n i p u l a t o r motion t o be w i t h i n the bounds -1 < z. < 1 i n o r d e r t h a t l e a r n i n g can t a k e p l a c e . Such c o n s t r a i n t s cannot be imposed u s i n g the as y e t u n l e a r n e d C a r t e s i a n i n v e r s e dynamics and must not r e q u i r e use of the a n a l y t i c a l C a r t e s i a n i n v e r s e dynamics. A s i m p l e method has been found f o r r e s t o r i n g m a n i p u l a t o r motion t o be w i t h i n the p a t h space -1 < z. < 1 whenever i t i s obs e r v e d t h a t m a n i p u l a t o r motion has exceeded the bounds of t h i s p a t h space. The method meets the above mentioned r e q u i r e m e n t s . The c e n t e r of p a t h space i s d e f i n e d by zf. e q u a l t o z e r o . T h i s c o r r e s p o n d s t o z e r o v e l o c i t y and z e r o a c c e l e r a t i o n w h i l e a t C a r t e s i a n p o s i t i o n , <x ,,x 0> = <0,-1> m (3.161) c 1 c2 T h i s c o r r e s p o n d s t o j o i n t p o s i t i o n , < a c 1 ' a c 2 > = < 2»r/3 ,-TT/3> (3.162) By s i m p l e s t a t i c a n a l y s i s of the m a n i p u l a t o r , the c o r r e s p o n d i n g t o r q u e s a r e , T , = -8.50 N-m (3.163) c l T ~ — 8.50 N-m (3.164) c2 121 One can a p p l y uncoupled j o i n t s e r v o c o n t r o l t o m a i n t a i n the m a n i p u l a t o r a t j o i n t p o s i t i o n < a c y a C 2 > o r r e s t o r e the m a n i p u l a t o r t o t h i s p o s i t i o n i f d i s t u r b e d from i t . An example of such uncoupled j o i n t s e r v o c o n t r o l i s the f o l l o w i n g , T 1 = T c 1 - j 1 a. 1 - j 2 ( a 1 ~ a c l ) (3.165) T 2 = r c 2 " j 1 d 2 " j 2 ( a 2 _ a c 2 ) (3.166) S i n c e the s e r v o e d j o i n t p o s i t i o n c o r r e s p o n d s t o t h e c e n t e r of the C a r t e s i a n p a t h space, such s i m p l e j o i n t s e r v o c o n t r o l can be used t o r e t u r n the m a n i p u l a t o r w i t h i n the bounds -1 < z. < 1. The a n g u l a r p o s i t i o n i s e a s i l y measured w i t h p o t e n t i o m e t e r s and the a n g u l a r v e l o c i t y can be o b t a i n e d from t h e s e measurements by s i m p l e d i f f e r e n t i a t i o n . G i v e n . a method f o r c o n s t r a i n i n g t h e m a n i p u l a t o r w i t h i n the p a t h space or more c o r r e c t l y r e t u r n i n g the m a n i p u l a t o r w i t h i n the p a t h space when i t i s obser v e d t o be out of bounds, i t i s o n l y n e c e s s a r y t o f i n d a mechanism t o cause the m a n i p u l a t o r t o move so as t o e x p l o r e a l l the r e g i o n s of p a t h space. We have found t h a t t h i s can be done by l e a r n i n g the sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the C a r t e s i a n i n v e r s e dynamics w h i l e u s i n g the same p a r t i a l l y l e a r n e d C a r t e s i a n i n v e r s e dynamics t o d r i v e the m a n i p u l a t o r over a sequence of randomly g e n e r a t e d p a t h s t h a t encompass the r e g i o n s of p a t h space. I n i t i a l l y , when the s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics a r e v e r y p o o r l y known, the p a t h s p e c i f i c a t i o n s a r e not f o l l o w e d and c o n s t r a i n i n g a c t i o n i s f r e q u e n t l y r e q u i r e d t o r e s t o r e the m a n i p u l a t o r w i t h i n the bounds -1 < Zj < 1. D u r i n g such p o o r l y c o n t r o l l e d t h r a s h i n g , however, l e a r n i n g can t a k e p l a c e whenever the m a n i p u l a t o r i s 122 w i t h i n bounds. I n f a c t t h e m a n i p u l a t o r a c c e l e r a t i o n may i m m e d i a t e l y become o u t o f bounds when d r i v e n by t h e as y e t u n l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s ; l e a r n i n g may i n i t i a l l y o n l y be p o s s i b l e a t t h o s e i n s t a n c e s when t h e c o n s t r a i n i n g a c t i o n has j u s t r e t u r n e d t h e m a n i p u l a t o r w i t h i n b o unds. L a t e r , when t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s become more a c c u r a t e , t h e p a t h s p e c i f i c a t i o n s a r e f o l l o w e d more c l o s e l y . The s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s a r e e v e n t u a l l y , l e a r n e d o v e r t h e whole of p a t h s p a c e i f t h e t r a i n i n g p a t h s c o v e r t h e whole o f p a t h s p a c e . S i m u l a t i o n s were p e r f o r m e d of 1000 t r a i n i n g p a t h s d u r i n g w h i c h s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s was c a r r i e d o u t . S i m u l a t i o n p a r a m e t e r s were s e t as f o l l o w s : CLOSED=.TRUE., INVARM=.FALSE., EXACT=.FALSE., VISION=.TRUE., DELAY=.TRUE., PRDICT=.TRUE., USEOBS=.FALSE., k 1=4, k 2=4, and t c a ^ c = 0 . 0 l s e c . The f e e d b a c k g a i n s were r e d u c e d somewhat from t h o s e u s e d i n s t a n d a r d a n a l y t i c a l c o n t r o l a s t h i s p e r m i t t e d s t a b l e c o n t r o l t o be a c h i e v e d e a r l i e r i n t h e t r a i n i n g p r o c e s s ; h i g h e r f e e d b a c k g a i n s c a u s e b e t t e r p a t h t r a c k i n g , when t h e C a r t e s i a n i n v e r s e d y n a m i c s have been w e l l l e a r n e d but c r e a t e more i n s t a b i l i t y d u r i n g e a r l i e r s t a g e s o f t r a i n i n g when t h e C a r t e s i a n i n v e r s e d y n a m i c s a r e o n l y p a r t i a l l y l e a r n e d . L e a r n i n g p a r a m e t e r s were s e t a s f o l l o w s : LRNINV=.TRUE., t , =0.01 s e c and Af . =0.01 N«m. U s i n g t h e n o m e n c l a t u r e o f l r n 1 min c h a p t e r 2, Method 2, L e a r n i n g I d e n t i f i c a t i o n , was t h e l e a r n i n g a l g o r i t h m u s e d . The number o f i n p u t v a r i a b l e s was v = 6 and t h e s y s t e m o r d e r was s = 4. The i n p u t v a r i a b l e s were t h o s e g i v e n i n 123 e q u a t i o n s (3.138) t h r o u g h ( 3 . 1 4 3 ) . C o n s t r a i n i n g a c t i o n was i n v o k e d whenever t h e m a n i p u l a t o r was o b s e r v e d t o be o u t o f b ounds. The g a i n s u s e d i n t h e u n c o u p l e d j o i n t s e r v o c o n t r o l d e f i n e d i n (3.165) and (3.166) were, j 1 = 4 (3.167) j 2 = 4 (3.168) In a d d i t i o n t h e m a g n i t u d e s o f t h e a p p l i e d t o r q u e s were r e s t r i c t e d a t a l l t i m e s t o be l e s s t h a n TRQLIM = 24 N»m. T h i s m o d e ls t h e f i n i t e d r i v e c a p a b i l i t y p r e s e n t i n r e a l m a n i p u l a t o r s ; i n p r a c t i c e a p p l i e d t o r q u e s a r e l i m i t e d by t h e m o t o r s , h y d r a u l i c s , o r p n e u m a t i c s u s e d . At e a c h i n t e r v a l o f t ^ r n 1 , l e a r n i n g was c o n d i t i o n a l l y p e r f o r m e d . F i r s t , t h e o b s e r v e d m a n i p u l a t o r m o t i o n had t o be w i t h i n t h e bounds -1 < z. < 1 i n o r d e r t h a t l e a r n i n g c o u l d t a k e p l a c e . Then t h e c o e f f i c i e n t s o f t h e p a r t i a l l y l e a r n e d sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e C a r t e s i a n i n v e r s e d y n a m i c s were u s e d o f f - l i n e t o e s t i m a t e t h e t o r q u e s c o r r e s p o n d i n g t o t h e o b s e r v e d m a n i p u l a t o r m o t i o n . I n i t i a l l y t h e s e c o e f f i c i e n t s were a l l z e r o . By o f f - l i n e we mean t h a t t h e s e e s t i m a t e d t o r q u e s a r e n o t a p p l i e d t o t h e m a n i p u l a t o r . N o t e t h a t t h i s means t h a t s e l f - l e a r n i n g d o e s n o t have t o be done i n r e a l t i m e . In s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s , t h e e s t i m a t i o n e r r o r , A f , i s t h e d i f f e r e n c e between t h e e s t i m a t e d t o r q u e and t h e c o r r e s p o n d i n g t o r q u e t h a t was a p p l i e d t o t h e m a n i p u l a t o r a t t h e i n s t a n t t h a t t h e m o t i o n was o b s e r v e d . I f t h i s e s t i m a t i o n e r r o r was g r e a t e r t h a n A f m ^ n f o r e i t h e r o f t h e t o r q u e s t h e n t h e sum of 124 p o l y n o m i a l s r e p r e s e n t a t i o n of t h e C a r t e s i a n i n v e r s e d y n a m i c s f u n c t i o n f o r t h e c o r r e s p o n d i n g t o r q u e was a d j u s t e d u s i n g L e a r n i n g I d e n t i f i c a t i o n . Note t h a t m a n i p u l a t o r m o t i o n was assumed t o be m e a s u r e d e x a c t l y f o r t h e p u r p o s e s of s e l f - l e a r n i n g . T h i s i s n o t s u c h an u n r e a s o n a b l e a s s u m p t i o n as i t m i g h t seem. B e c a u s e s e l f - l e a r n i n g i s done o f f - l i n e w i t h o u t r e a l t i m e c o n t r a i n t s , one c a n d e v o t e more computer t i m e t o o b t a i n i n g a c c u r a t e e s t i m a t e s of v e l o c i t y and a c c e l e r a t i o n b a s e d on c h a n g e s i n o b s e r v e d p o s i t i o n . S i n c e s e l f - l e a r n i n g f r o m a p o i n t of no knowledge o f t h e C a r t e s i a n i n v e r s e d y n a m i c s i s a one t i m e e v e n t , o c c u r r i n g b e f o r e a m a n i p u l a t o r i s p u t t o work, i t i s r e a s o n a b l e t o assume t h a t a d d i t i o n a l c o m p u t i n g c a p a b i l i t y c a n t e m p o r a r i l y be u t i l i z e d . N o t e a l s o t h a t one c a n base a c c e l e r a t i o n and v e l o c i t y e s t i m a t e s on o b s e r v a t i o n s o f p o s i t i o n , a f t e r , a s w e l l a s b e f o r e , t h e p o i n t i n t i m e i n q u e s t i o n . Thus i t i s r e a s o n a b l e t o assume t h a t o b s e r v a t i o n s o f m a n i p u l a t o r m o t i o n f o r s e l f - l e a r n i n g p u r p o s e s c a n be made much more a c c u r a t e l y t h a n o b s e r v a t i o n s f o r use i n c l o s e d l o o p c o n t r o l . L i n e a r p a t h s were u s e d f o r t r a i n i n g p u r p o s e s . The a c c e l e r a t i o n s p e c i f i c a t i o n f o r e a c h p a t h was c h o s e n u n i f o r m l y between t h e l i m i t s , The v e l o c i t y s p e c i f i c a t i o n s were c h o s e n u n i f o r m l y between t h e l i m i t s , 0.1 < a max < 0.9 m/sec (3.169) 0.05 < v, < 0.95 m/sec (3.170) max a max and v max c o u l d n o t be c h o s e n t o o c l o s e t o z e r o as t h e p a t h 125 s p e c i f i c a t i o n would t a k e an i n o r d i n a t e amount of time t o be c a r r i e d o u t . The f i n a l p o s i t i o n of each p a t h was chosen u n i f o r m l y from w i t h i n the r e g i o n d e f i n e d by, 0.375 < a b s ( x - . , - x ,) < 0.45 m (3.171) 111 c l 0.375 < a b s ( x . . ~ - x -) < 0.45 m (3.172) 112 CZ Note t h a t the f i n a l p o s i t i o n of each p a t h i s the i n i t i a l p o s i t i o n of each subsequent p a t h . The upper l i m i t s on a c c e l e r a t i o n , v e l o c i t y , and p o s i t i o n s e r v e t o a v o i d p a t h s p e c i f i c a t i o n s t h a t a r e near the bounds of p a t h space and thus l i k e l y t o i n v o k e c o n s t r a i n i n g a c t i o n even when t r a c k i n g e r r o r s a r e q u i t e s m a l l . F i g u r e 3.32 shows the randomness of the c h o i c e s of a and v f o r t h e f i r s t 250 t r a i n i n g p a t h s . The max max r e s t r i c t i o n t h a t p a t h s b e g i n and end i n a band near the edge of the workspace r e s u l t s i n a f a i r l y u n i f o r m c overage of p o s i t i o n s w i t h i n the workspace by t h e t r a i n i n g p a t h s . F i g u r e 3.33 shows the p o s i t i o n s of the f i r s t 250 t r a i n i n g p a t h s . RESULTS D u r i n g the i n i t i a l s t a g e s of s e l f - l e a r n i n g , the m a n i p u l a t o r i s f r e q u e n t l y out of bounds and t h u s l e a r n i n g cannot t a k e p l a c e a t e v e r y i n t e r v a l of ^ m i * L a t e r , as b e t t e r c o n t r o l i s a c h i e v e d , the p r o p o r t i o n of o p p o r t u n i t i e s a t which l e a r n i n g t a k e s p l a c e i n c r e a s e s . F i n a l l y , as the s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics become q u i t e a c c u r a t e , t h e p r o p o r t i o n of o p p o r t u n i t i e s a t which l e a r n i n g t a k e s p l a c e d e c r e a s e s s i n c e the e s t i m a t i o n e r r o r i s f r e q u e n t l y l e s s than the t h r e s h o l d , A f m i n = 0.01 N«m. The p r o p o r t i o n of o p p o r t u n i t i e s d u r i n g each p a t h a t which l e a r n i n g t a k e s p l a c e i s shown i n f i g u r e 3.34. 126 F i g u r e 3.32 C h o i c e s of a m a x and v m a x f o r f i r s t 250 t r a i n i n g p a t h s used i n s e l f - l e a r n i n g of t h e C a r t e s i a n i n v e r s e dynamics 127 F i g u r e 3.33 View of f i r s t 250 t r a i n i n g p a t h s used i n s e l f - l e a r n i n g of the C a r t e s i a n i n v e r s e dynamics 128 F i g u r e 3.34 P r o p o r t i o n of l e a r n i n g o p p o r t u n i t i e s at which l e a r n i n g took place during s e l f - l e a r n i n g of the C a r t e s i a n inverse dynamics As s e l f - l e a r n i n g proceeds, the path s p e c i f i c a t i o n s are f o l l o w e d more and more c l o s e l y . Figure 3.35 shows the path e r r o r s that occurred during each of the t r a i n i n g paths. Path e r r o r s are c l e a r l y being reduced as t r a i n i n g takes p l a c e . Better c o n t r o l a l s o means tha t the manipulator goes out of bounds l e s s o f t e n . F i g u r e 3.36 shows the maximum out of bounds excursions that occurred during each of the t r a i n i n g paths. The frequency and extent of out of bounds excursions are reduced with t r a i n i n g . 129 0.00 0.20 0.40 0.60 0.80 -io3 TRAINING PATHS F i g u r e 3.35 P a t h e r r o r s d u r i n g s e l f - l e a r n i n g of the C a r t e s i a n i n v e r s e dynamics 130 o CM-'"" CJ UJ CO •-V.O OO X o <LkiL.k J<| . . . . f l . .1. A I i j t | i l l I , I i ,1 L i 1 0.00 0.20 0.40 0.60 0.80 1.00 •IO* TRAINING PATHS 0.20 0.40 0.60 0.80 1.00 •103 TRAINING PATHS o in CD O • J U L J . — I 1 1 — • — i 1 0.00 0.20 0.40 0.60 0.80 1.00 •io3 TRAINING PATHS F i g u r e 3.36 Maximum out of bounds excursions during s e l f - l e a r n i n g of the C a r t e s i a n inverse dynamics 131 I t i s a c c e l e r a t i o n t h a t i s most p o o r l y t r a c k e d and hence most f r e q u e n t l y out of bounds. In f a c t , s l i g h t o v e r s h o o t i n g of s t e p s i n a c c e l e r a t i o n can exceed the bounds on a c c e l e r a t i o n and cause c o n s t r a i n i n g a c t i o n t o be i n v o k e d w h i c h , may then r e s u l t i n v e l o c i t y and p o s i t i o n e x c e e d i n g t h e i r bounds. T h i s o c c u r s because c o n s t r a i n i n g a c t i o n s i m p l y r e t u r n s the m a n i p u l a t o r w i t h i n bounds w i t h o u t c o n c e r n as t o where. When c o n s t r a i n i n g a c t i o n i s c e a s e d , the l a r g e p a t h t r a c k i n g e r r o r t h a t i s l i k e l y t o e x i s t may cause a l a r g e c o r r e c t i v e a c c e l e r a t i o n which a g a i n t a k e s t h e m a n i p u l a t o r out of bounds. Thus a s l i g h t o v e r s h o o t i n g of s t e p s i n a c c e l e r a t i o n can l e a d t o an e p i s o d e of t h r a s h i n g i n which v e l o c i t y and p o s i t i o n bounds a r e a l s o exceeded. Many of the out of bounds e x c u r s i o n s t h a t o c c u r r e d d u r i n g the l a t t e r s t a g e s of t r a i n i n g appeared t o be of t h i s n a t u r e . C o n s t r a i n t s on a c c e l e r a t i o n a r e n e c e s s a r y d u r i n g the i n i t i a l s t a g e s of s e l f - l e a r n i n g . Once the C a r t e s i a n i n v e r s e dynamics a r e w e l l l e a r n e d , however, i t appears t h a t more r e l i a b l e c o n t r o l i s a c h i e v a b l e when no c o n s t r a i n t s on a c c e l e r a t i o n a r e imposed; one then r e l i e s on the l i m i t a t i o n of a p p l i e d t o r q u e s t o l i m i t a c c e l e r a t i o n . A b e t t e r c o n s t r a i n i n g t e c h n i q u e might be t o in v o k e c o n s t r a i n i n g a c t i o n t h a t would i n i t i a l l y damp out m a n i p u l a t o r m o t i o n . A f t e r the m a n i p u l a t o r i s st o p p e d , j o i n t s e r v o i n g c o u l d then be used t o s l o w l y move t h e m a n i p u l a t o r towards the c e n t e r of C a r t e s i a n p a t h space, as b e f o r e . Once the m a n i p u l a t o r i s ob s e r v e d t o a g a i n be w i t h i n C a r t e s i a n p a t h space, the next t r a i n i n g p a t h c o u l d be begun, s t a r t i n g from the c u r r e n t m a n i p u l a t o r p o s i t i o n . T h i s s h o u l d a v o i d the p r o t r a c t e d e p i s o d e s 132 o f t h r a s h i n g t h a t o c c u r r e d u s i n g t h e s i m p l i s t i c c o n s t r a i n i n g t e c h n i q u e d e s c r i b e d p r e v i o u s l y . As m e n t i o n e d p r e v i o u s l y , s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s r e q u i r e d 1000 t r a i n i n g p a t h s . T h i s c o r r e s p o n d e d t o 2948 s e c o n d s (49 m i n u t e s ) of s e l f - l e a r n i n g . I f t ^ r n 1 were i n c r e a s e d t h e n r e q u i r e d s e l f - l e a r n i n g t i m e w o u l d a l s o i n c r e a s e . 3.8.2 CLOSED LOOP CONTROL USING THE SELF-LEARNED CARTESIAN INVERSE DYNAMICS The c o e f f i c i e n t s o f t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s were s a v e d a f t e r 200, 400, 600, 800 and 1000 t r a i n i n g p a t h s . S i m u l a t i o n s o f c l o s e d l o o p c o n t r o l o f a l l o f t h e s i x s t a n d a r d p a t h s were c a r r i e d o u t t o t e s t t h e a d e q u a c y o f t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s a s l e a r n i n g p r o g r e s s e d . S i m u l a t i o n p a r a m e t e r s were s e t as f o l l o w s : CLOSED=.TRUE., INVARM=.FALSE., EXACT=.FALSE., VISION=.TRUE., DELAY=.TRUE., PRDICT=.TRUE., USEOBS=.TRUE., ^ = 1 6 , k 2 = 6 4 a n d t c a l c = ° * ° 1 S S C * The r e s u l t i n g p a t h e r r o r s , a v e r a g e d o v e r t h e s i x s t a n d a r d p a t h s , a r e shown i n f i g u r e 3.37 a s a f u n c t i o n o f t h e number o f t r a i n i n g p a t h s u s e d i n s e l f - l e a r n i n g of t h e C a r t e s i a n i n v e r s e d y n a m i c s . F i g u r e 3.38 shows t h e s i m u l a t i o n o f l i n e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s . P l o t s o f t h e o t h e r s i m u l a t i o n s o f l i n e 1 a r e shown i n a p p e n d i x C. F i g u r e 3.39 shows a p l o t o f t h e s i m u l a t i o n o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s . 133 Figure 3.37 Average path e r r o r s during c l o s e d loop c o n t r o l of the s i x standard paths using the s e l f - l e a r n e d C a r t e s i a n inverse dynamics as a f u n c t i o n of the number of t r a i n i n g paths used. 134 4.00 4.00 4.00 T SEC Figure 3.38 Closed loop c o n t r o l of l i n e 1 using the f i n a l s e l f - learned C a r t e s i a n inverse dynamics 135 F i g u r e 3.39 C l o s e d l o o p c o n t r o l of c i r c l e 1 u s i n g s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics the f i n a l 136 T o r q u e e s t i m a t e s u s i n g t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s were compared t o t o r q u e c a l c u l a t e d w i t h t h e a n a l y t i c a l C a r t e s i a n i n v e r s e d y n a m i c s . T o r q u e e r r o r , a v e r a g e d o v e r t h e s i x s t a n d a r d p a t h s a r e shown i n f i g u r e 3.40 a s a f u n c t i o n of t h e number o f t r a i n i n g p a t h s u s e d . P a t h e r r o r s and t o r q u e e r r o r d e c r e a s e d u r i n g s e l f - l e a r n i n g u n t i l minimum v a l u e s a r e r e a c h e d . A f t e r 1000 t r a i n i n g p a t h s i t a p p e a r s t h a t t h e s e e r r o r s c a n n o t be r e d u c e d f u r t h e r . The c o n t r o l p e r f o r m a n c e a f t e r 1000 t r a i n i n g p a t h s t h u s r e p r e s e n t s t h e l i m i t o f what can be a c h i e v e d w i t h a 4 f c^ o r d e r , s e l f - l e a r n e d sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e C a r t e s i a n i n v e r s e d y n a m i c s . The r e s u l t i n g p a t h e r r o r s f o r t h e s i m u l a t i o n s o f t h e s i x s t a n d a r d p a t h s u s i n g t h e C a r t e s i a n i n v e r s e d y n a m i c s s e l f - l e a r n e d o v e r 1000 t r a i n i n g p a t h s a r e l i s t e d i n t a b l e 3.15. A l s o l i s t e d i n t a b l e 3.15 a r e t h e t o r q u e e r r o r and t h e maximum t o r q u e a p p l i e d d u r i n g e a c h s t a n d a r d p a t h . P e r f o r m a n c e i s o n l y s l i g h t l y p o o r e r t h a n w i t h s t a n d a r d a n a l y t i c a l c o n t r o l . P o s i t i o n t r a c k i n g e r r o r s a r e t y p i c a l l y l e s s t h a n 3 mm. 137 - LU CM ~*0.00 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 .00 -io3 TRAINING PATHS Figure 3.40 Average torque e s t i m a t i o n e r r o r during c l o s e d loop c o n t r o l of the s i x standard paths using the s e l f - l e a r n e d C a r t e s i a n inverse dynamics as a f u n c t i o n of the number of t r a i n i n g paths used. 138 PATH XERMS VXERMS AXERMS TERMS TRQMAX l i n e 1 1 .60E-3 4.02E-3 1.01E-l 1.96E-1 8.27 l i n e 2 2 .15E-3 4.75E-3 1.03E-1 2.47E-1 9.41 l i n e 3 1 .47E-3 3.90E-3 1.08E-1 1.64E-1 14.0 l i n e 4 1 .74E-2 4.09E-2 2.92E-1 3.00E-1 6.64 c i r c l e 1 1 .16E-3 3.85E-3 5.29E-2 1.52E-1 9.59 c i r c l e 2 1 .62E-3 3.45E-3 5.26E-2 2.00E-1 13.5 T a b l e 3.15 P a t h e r r o r and t o r q u e e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s i n c l o s e d l o o p c o n t r o l 3.9 SELF-LEARNING OF THE DIRECT POSITION KINEMATICS A sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s h as been p r e - l e a r n e d a n d f o u n d a d e q u a t e f o r c o a r s e c o n t r o l . I t r e m a i n s o n l y t o f i n d a method o f l e a r n i n g t h e d i r e c t p o s i t i o n k i n e m a t i c s w i t h o u t r e c o u r s e t o a n a l y s i s o f t h e m a n i p u l a t o r . We have f o u n d t h a t t h i s c a n be done t h r o u g h o b s e r v a t i o n of m a n i p u l a t o r j o i n t p o s i t i o n s and t h e c o r r e s p o n d i n g m a n i p u l a t o r end p o i n t p o s i t i o n i n C a r t e s i a n c o o r d i n a t e s a s se e n by t h e v i s i o n s y s t e m . 3.9.1 A METHOD FOR SELF-LEARNING OF THE DIRECT POSITION KINEMATICS THE METHOD G i v e n s u f f i c i e n t and v a r i e d o b s e r v a t i o n s o f t h e m a n i p u l a t o r p o s i t i o n i n j o i n t c o o r d i n a t e s a n d m a n i p u l a t o r p o s i t i o n i n 139 C a r t e s i a n c o o r d i n a t e s , one c a n use I n t e r f e r e n c e M i n i m i z a t i o n o r r e l a t e d methods t o l e a r n t h e d i r e c t p o s i t i o n k i n e m a t i c s . To l e a r n t h e d i r e c t p o s i t i o n k i n e m a t i c s o v e r t h e j o i n t p o s i t i o n s p a c e g i v e n by e q u a t i o n s (3.151) and ( 3 . 1 5 2 ) , t h e same two c o n d i t i o n s must be met a s d u r i n g s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s : F i r s t , one must o b t a i n o b s e r v a t i o n s c o r r e s p o n d i n g t o a l l r e g i o n s o f t h e p o s i t i o n s p a c e , -1 < z. < 1. H e r e t h e z. a r e t h e n o r m a l i z e d j o i n t p o s i t i o n s t h a t a c t as i n p u t v a r i a b l e s i n t h e sum of p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s . S e c o n d l y , one must c o n s t r a i n t h e m a n i p u l a t o r m o t i o n t o be w i t h i n t h e bounds -1 < z. < 1 i n o r d e r t h a t l e a r n i n g c a n t a k e p l a c e . S i n c e we w i s h t o use t h e d i r e c t p o s i t i o n k i n e m a t i c s o n l y o v e r t h e w o r k s p a c e , i t i s o n l y n e c e s s a r y t h a t we l e a r n t h e d i r e c t p o s i t i o n k i n e m a t i c s o v e r t h e work s p a c e , n o t o v e r t h e whole j o i n t p o s i t i o n s p a c e g i v e n by -1 < zi < 1. A l s o , s i n c e t h e w o r k s p a c e i s w i t h i n t h e j o i n t s p a c e g i v e n by e q u a t i o n s (3.151) and ( 3 . 1 5 2 ) , c o n s t r a i n i n g m a n i p u l a t o r m o t i o n w i t h i n t h e w o r k s p a c e a s was done d u r i n g s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s a l s o s e r v e s t o c o n s t r a i n t h e m a n i p u l a t o r w i t h i n t h e j o i n t s p a c e -1 < z. < 1. I t was t h u s p o s s i b l e t o p e r f o r m s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s s i m u l t a n e o u s l y w i t h t h e s e l f - l e a r n i n g of t h e C a r t e s i a n i n v e r s e d y n a m i c s . Note t h a t , u n l i k e t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s , t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s were not u s e d f o r c o n t r o l p u r p o s e s w h i l e s e l f - l e a r n i n g was t a k i n g p l a c e . S i m u l a t i o n s were p e r f o r m e d o f 4200 t r a i n i n g p a t h s d u r i n g 140 w h i c h s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s was c a r r i e d o u t . F o r t h e f i r s t 1000 t r a i n i n g p a t h s s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s was s i m u l t a n e o u s l y c a r r i e d o u t . Thus t h e s i m u l a t i o n p a r a m e t e r s were t h e same as g i v e n p r e v i o u s l y . N o t h i n g was c h a n g e d f o r t h e s i m u l a t i o n of t h e f i n a l 3200 t r a i n i n g p a t h s e x c e p t t h a t s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s was d i s a b l e d by s e t t i n g LRNINV=.FALSE. L e a r n i n g p a r a m e t e r s were s e t as f o l l o w s : LRNVIS=.TRUE., t , ,,=0.01 sec and Af . =0.001 m. Method 3, I n t e r f e r e n c e l r n 2 nun M i n i m i z a t i o n was u s e d . The number of i n p u t v a r i a b l e s was v = 2 and t h e s y s t e m o r d e r was s = 4. The i n p u t v a r i a b l e s were t h o s e g i v e n i n e q u a t i o n s (3.151) and ( 3 . 1 5 2 ) . A t e a c h i n t e r v a l o f t i r n 2 ' l e a r n i n g was c o n d i t i o n a l l y p e r f o r m e d . F i r s t , t h e o b s e r v e d m a n i p u l a t o r j o i n t p o s i t i o n s had t o be w i t h i n t h e bounds -1 < z. < 1 i n o r d e r t h a t l e a r n i n g c o u l d t a k e p l a c e . Then t h e c o e f f i c i e n t s o f t h e p a r t i a l l y l e a r n e d sum of p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s were u s e d o f f - l i n e t o e s t i m a t e t h e C a r t e s i a n p o s i t i o n c o r r e s p o n d i n g t o t h e o b s e r v e d j o i n t p o s i t i o n s . T h e s e c o e f f i c i e n t s were i n i t i a l l y z e r o . I n s e l f - l e a r n i n g of t h e d i r e c t p o s i t i o n k i n e m a t i c s , t h e e s t i m a t i o n e r r o r , A f , i s t h e d i f f e r e n c e between t h e e s t i m a t e d C a r t e s i a n p o s i t i o n and t h e c o r r e s p o n d i n g C a r t e s i a n p o s i t i o n t h a t was o b s e r v e d by t h e v i s i o n s y s t e m a t t h e i n s t a n t t h a t t h e j o i n t p o s i t i o n s were o b s e r v e d . I f t h i s e s t i m a t i o n e r r o r was g r e a t e r t h a t Af . f o r e i t h e r o f t h e 3 m m C a r t e s i a n c o o r d i n a t e s t h e n t h e sum o f p o l y n o m i a l s r e p r e s e n t a t i o n of t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n f o r t h e c o r r e s p o n d i n g 141 C a r t e s i a n c o o r d i n a t e was a d j u s t e d u s i n g I n t e r f e r e n c e M i n i m i z a t i o n . The . t r a i n i n g p a t h s t h a t were u s e d were t h e same a s d e s c r i b e d p r e v i o u s l y f o r s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s . The p r o p o r t i o n o f o p p o r t u n i t i e s d u r i n g e a c h p a t h a t w h i c h l e a r n i n g t a k e s p l a c e i s shown i n f i g u r e 3.41. U n l i k e d u r i n g s e l f - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s , t h i s p r o p o r t i o n r e m a i n s r e l a t i v e l y c o n s t a n t d u r i n g s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s . T h i s r e f l e c t s t h e f a c t t h a t t h e m a n i p u l a t o r e x c e e d s t h e bounds on i n p u t v a r i a b l e s r e p r e s e n t i n g n o r m a l i z e d j o i n t p o s i t i o n s f o r t h e d i r e c t p o s i t i o n k i n e m a t i c s , l e s s o f t e n t h a n i t e x c e e d s t h e more e x t e n s i v e bounds on i n p u t v a r i a b l e s r e p r e s e n t i n g n o r m a l i z e d C a r t e s i a n end p o i n t p o s i t i o n s f o r t h e C a r t e s i a n i n v e r s e d y n a m i c s . A l s o , t y p i c a l l e a r n e d p o s i t i o n e s t i m a t i o n e r r o r o f l e s s t h a n 0.001 m t h r o u g h o u t t h e w o r k s p a c e c o u l d n o t be a c h i e v e d w i t h a 4 f c^ o r d e r s y s t e m and t h u s p o s i t i o n e s t i m a t i o n e r r o r was i n f r e q u e n t l y l e s s t h a n t h e t h r e s h o l d , Af . = 0.001 m. I t i s p r o b a b l e t h a t Af . c o u l d have been mm c mm i n c r e a s e d , t o say 0.01 m, w i t h o u t i n c r e a s i n g t h e number of t r a i n i n g p a t h s n e c e s s a r y t o s e l f - l e a r n t h e d i r e c t p o s i t i o n k i n e m a t i c s w h i l e r e d u c i n g t h e number o f l e a r n i n g i t e r a t i o n s 0.01 m d i d l i t t l e t o s p e e d c o n v e r g e n c e . The c o e f f i c i e n t s of t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s were s a v e d a t v a r i o u s p o i n t s d u r i n g t h e l e a r n i n g RESULTS r e q u i r e d . T r a i n i n g a t p o i n t s where Af min was between 0.001 m and 142 p r o c e s s . S i m u l a t i o n s o f c i r c l e 1 were c a r r i e d o u t u s i n g s t a n d a r d a n a l y t i c a l c o n t r o l t o e n s u r e a c c u r a t e t r a c k i n g o f t h e p a t h s p e c i f i c a t i o n . The s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s were u s e d t o o b t a i n a view o'f c i r c l e 1 f o r e a c h o f t h e s i m u l a t i o n s . F i g u r e 3.42 shows t h e improvement i n t h e a c c u r a c y o f p a t h e s t i m a t i o n as a f u n c t i o n o f t h e number o f t r a i n i n g p a t h s u s e d i n s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s . F i g u r e 3.43 shows t h e view o f c i r c l e 1 u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c ^ . . P l o t s of t h e v i e w u s i n g t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s f o r t h e o t h e r s i m u l a t i o n s o f c i r c l e 1 a r e shown i n a p p e n d i x D. I t c a n be se e n t h a t t h e s e l f - l e a r n e d p a t h e s t i m a t e s i n i t i a l l y become more a c c u r a t e w i t h t r a i n i n g b u t a r e n o t i m p r o v i n g a f t e r 4200 t r a i n i n g p a t h s . The p e r f o r m a n c e a f t e r 4200 t r a i n i n g p a t h s i s t h u s r e p r e s e n t a t i v e o f t h e b e s t a c c u r a c y t h a t c a n be a c h i e v e d u s i n g a s e l f - l e a r n e d , 4 f c^ o r d e r sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s . T a b l e 3.16 l i s t s t h e p a t h e s t i m a t i o n e r r o r s u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . 143 1.00 (i\20 ( M O CL60 euro 1.00 -io3 TRAINING PATHS Figure 3.41 P r o p o r t i o n of l e a r n i n g o p p o r t u n i t i e s at which l e a r n i n g took place during s e l f - l e a r n i n g of the d i r e c t p o s i t i o n kinematics 144 F i g u r e 3.42 P a t h e s t i m a t i o n e r r o r s d u r i n g s e l f - l e a r n i n g of the d i r e c t p o s i t i o n k i n e m a t i c s 145 o w o • o O' i F i g u r e 3.43 View of c i r c l e 1 u s i n g the f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 146 PATH LXERMS LVEPvMS LAERMS l i n e 1 l i n e 2 l i n e 3 1 i n e 4 c i r c l e 1 c i r c l e 2 1.37E-1 9.96E-2 2.75E-2 7.56E-2 4.97E-2 5.28E-2 1.66E-1 1.13E-1 4.65E-.2 7.86E-2 9.61E-2 9.01E-2 4/17E-1 2.90E-1 1.64E-1 2.52E-1 3.1OE-1 3.02E-1 T a b l e 3.16 P a t h e s t i m a t i o n e r r o r u s i n g the f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s T a b l e 3.17 l i s t s t he c o e f f i c i e n t s f o r the f i n a l s e l f - l e a r n e d sum of p o l y n o m i a l s r e p r e s e n t a t i o n of the d i r e c t p o s i t i o n k i n e m a t i c s a l o n g s i d e the d e r i v e d c o e f f i c i e n t s . The corr e s p o n d e n c e i s q u i t e good e x c e p t f o r the h i g h e r o r d e r terms. The i n n a c c u r a c y i n the h i g h e r o r d e r terms i s not s u r p r i s i n g as these terms a r e most s i g n i f i c a n t near the bounds on ẑ . f o r the s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s ; t h e s e bounds a r e o u t s i d e t h e workspace and thus l i t t l e s e l f - l e a r a i n g took p l a c e near them. S e l f - l e a r n i n g of the d i r e c t p o s i t i o n k i n e m a t i c s r e q u i r e d 4200 t r a i n i n g p a t h s . T h i s c o r r e s p o n d e d t o 12250 seconds (3 hours 24 minutes) of s e l f - l e a r n i n g . 147 POLYNOMIAL X ( 1 ) COEFFICIENTS X ( 2 ) COEFFICIENTS TERM DERIVED SELF-LEARNED DERIVED SELF-LEARNED 1 0.0 -0.009 -1 .00 -1 .04 z 1 1 .57 1 .67 0.0 -0.006 2 Z 1 0.0 -0.040 1 .23 2.20 « , 3 -0.646 -1 .77 0.0 0. 155 - I * 0.0 0.209 -0.254 -0.731 Z 2 0.785 0.787 1 .36 1 .58 Z 2 Z 1 -2.14 -2.60 1 .23 1 .62 z z 2  Z 2 Z 1 -0.969 -0.276 -1 .68 -0.750 z z 3  Z 2 Z 1 0.879 1 .49 -0.507 -1.41 Z 2 " -1 .07 -0.879 0.617 0.277 z 2 z z 2 Z 1 -0.969 -0.051 -0.839 -0.699 2 2 z 2 Z 1 1 .32 -0.492 -0.761 -2.05 Z 2 3 -0.323 -0.653 -0.559 -1 .54 z 3 z z 2 Z 1 0.879 1 .45 -0.507 -0.633 « 2 * 0.220 0.982 -0.127 1.13 T a b l e 3.17 C o e f f i c i e n t s f o r t h e d e r i v e d and f i n a l s e l f - l e a r n e d sum of p o l y n o m i a l s r e p r e s e n t a t i o n s of t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n s f o r p o s i t i o n s x. and x« 148 From * t h e d e r i v a t i o n and p r e - l e a r n i n g o f t h e C a r t e s i a n i n v e r s e d y n a m i c s and d i r e c t p o s i t i o n k i n e m a t i c s , one would e x p e c t t h e d i r e c t p o s i t i o n k i n e m a t i c s t o be e a s i e r t o l e a r n . S e v e r a l f a c t o r s combine, however, t o make i t more d i f f i c u l t t o s e l f - l e a r n a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n of t h e d i r e c t p o s i t i o n k i n e m a t i c s t h a t a r e a d e q u a t e f o r c o n t r o l p u r p o s e s t h a n t o do l i k e w i s e f o r t h e t h e C a r t e s i a n i n v e r s e d y n a m i c s . F i r s t , t h e C a r t e s i a n i n v e r s e d y n a m i c s do not need t o be known v e r y a c c u r a t e l y . t o a c h i e v e good c o n t r o l . F o r f e e d b a c k e r r o r c o r r e c t i o n t o be r e a s o n a b l y e f f e c t i v e i t a p p e a r s t o o n l y be n e c e s s a r y t h a t s i g n o f t h e d i f f e r e n t i a l , S T ^ . / S C L , be known a c c u r a t e l y t h r o u g h o u t p a t h s p a c e ; t h e m a g n i t u d e needs o n l y t o be r o u g h l y known. S e c o n d l y , t h e d i r e c t p o s i t i o n k i n e m a t i c s must be v e r y a c c u r a t e l y known. E r r o r s i n t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s t r a n s l a t e d i r e c t l y i n t o t r a c k i n g e r r o r s and, i f s i g n i f i c a n t enough, o b v i a t e t h e c a n c e l l a t i o n o f s y s t e m non- l i n e a r i t i e s r e q u i r e d f o r s t a b l e r e s o l v e d a c c e l e r a t i o n c o n t r o l of t h e m a n i p u l a t o r . F i n a l l y , t h e t r a i n i n g p a t h s a r e l e s s e f f e c t i v e i n p r o v i d i n g s u i t a b l e t r a i n i n g p o i n t s f o r t h e s e l f - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s . When s e l f - l e a r n i n g t h e C a r t e s i a n i n v e r s e d y n a m i c s , t h e t r a i n i n g p a t h s r e s u l t i n c h a n g e s t o a l l o f t h e s i x i n p u t v a r i a b l e s b e i n g u s e d . F u r t h e r m o r e , s t e p s i n a c c e l e r a t i o n o c c u r . T h i s means t h a t t r a i n i n g p o i n t s c l o s e t o g e t h e r i n t i m e a r e o f t e n f a r a p a r t i n t e r m s o f t h e s p a c e r e p r e s e n t e d by t h e s i x i n p u t v a r i a b l e s . When s e l f - l e a r n i n g t h e d i r e c t p o s i t i o n k i n e m a t i c s t h e r e a r e o n l y two i n p u t v a r i a b l e s and t h e s e v a r i a b l e s a r e n o t c h a n g i n g q u i c k l y . T r a i n i n g p o i n t s 149 c l o s e t o g e t h e r i n t i m e a r e t h u s c l o s e t o g e t h e r i n t e r m s o f t h e s p a c e r e p r e s e n t e d by t h e two i n p u t v a r i a b l e s . S i n c e I n t e r f e r e n c e M i n i m i z a t i o n e l i m i n a t e s e s t i m a t i o n e r r o r e x a c t l y a t a t r a i n i n g p o i n t and t h e sum o f p o l y n o m i a l s e s t i m a t e i s c o n t i n u o u s , t h e r e w i l l be l i t t l e e s t i m a t i o n e r r o r a t n e a r b y s u b s e q u e n t t r a i n i n g p o i n t s . S e l f - l e a r n i n g o n l y t a k e s p l a c e when e s t i m a t i o n e r r o r s o c c u r and hence i t r e q u i r e s more t i m e t o s e l f - l e a r n t h e d i r e c t p o s i t i o n k i n e m a t i c s . I t a l s o seems s u r p r i s i n g t h a t t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s a r e n o t as a c c u r a t e a s t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . A g a i n , t h i s i s due t o d i f f e r e n c e i n t h e way t r a i n i n g p o i n t s a r e c h o s e n . In s e l f - l e a r n i n g , t h e f i n a l t r a i n i n g p o i n t s a l l o c c u r on t h e f i n a l t r a i n i n g p a t h w h i c h r e p r e s e n t s o n l y one r e g i o n o f t h e s p a c e . Thus t h e s e l f - l e a r n e d k i n e m a t i c s a r e q u i t e a c c u r a t e i n t h e r e g i o n o f t h e l a s t t r a i n i n g p a t h a t t h e e x p e n s e o f i n c r e a s e d i n a c c u r a c y a t o t h e r p o i n t s i n t h e w o r k s p a c e . The random c h o o s i n g of t r a i n i n g p o i n t s d u r i n g p r e - l e a r n i n g o f t h e d i r e c t p o s i t i o n k i n e m a t i c s does n o t f a v o r one p a r t i c u l a r r e g i o n o f t h e w o r k s p a c e and t h u s r e s u l t s i n a mean a c c u r a c y t h a t i s b e t t e r . I t i s p o s s i b l e t o m o d i f y s e l f - l e a r n i n g so t h a t t r a i n i n g p o i n t s a r e l e s s f r e q u e n t and t h e r e f o r e more d i s t r i b u t e d o v e r t h e w o r k s p a c e d u r i n g t h e f i n a l s t a g e s o f s e l f - l e a r n i n g . One c a n a l s o a c h i e v e a s i m i l a r a v e r a g i n g o r s m o o t h i n g e f f e c t d u r i n g t h e f i n a l s t a g e s o f s e l f - l e a r n i n g by i n t r o d u c i n g a g a i n f a c t o r o f l e s s t h a n 1 i n t h e w e i g h t a d j u s t m e n t f o r m u l a f o r I n t e r f e r e n c e M i n i m i z a t i o n s u c h t h a t e s t i m a t i o n e r r o r s a r e o n l y r e d u c e d a t t r a i n i n g p o i n t s , not 150 e l i m i n a t e d . T h e s e methods s h o u l d o n l y be a p p l i e d d u r i n g t h e f i n a l s t a g e s o f s e l f - l e a r n i n g a s t h e y would r e d u c e t h e r a t e of c o n v e r g e n c e d u r i n g t h e e a r l y s t a g e s o f s e l f - l e a r n i n g . To t e s t t h e s e methods f o r a v e r a g i n g o r s m o o t h i n g , an a d d i t i o n a l 400 t r a i n i n g p a t h s were s i m u l a t e d d u r i n g w h i c h s e l f - l e a r n i n g o f . t h e d i r e c t p o s i t i o n k i n e m a t i c s was c a r r i e d o u t . S i m u l a t i o n and l e a r n i n g p a r a m e t e r s were t h e same as f o r t r a i n i n g p a t h s 1001 t o 4200, e x c e p t t h a t t h e l e a r n i n g i n t e r v a l was c h a n g e d t o t i r n 2 = ^ 1 ' 1 s e c • I n a d d i t i o n t h e l e a r n i n g a l g o r i t h m a d j u s t m e n t f o r m u l a was m o d i f i e d from t h a t o f n o r m a l I n t e r f e r e n c e M i n i m i z a t i o n by i n c l u s i o n of a g a i n f a c t o r of 1/10 a s f o l l o w s , A f P ~ 1 p Aw = 1/10 -= =- (3.173) P P P A f t e r c o m p l e t i o n of t h i s a d d i t i o n a l s m o o t h i n g , s i m u l a t i o n s were c a r r i e d out t o d e t e r m i n e t h e e x t e n t of improvement. S t a n d a r d a n a l y t i c a l c o n t r o l was u s e d t o s i m u l a t e t h e s i x s t a n d a r d p a t h s and e n s u r e a c c u r a t e t r a c k i n g o f t h e p a t h s p e c i f i - c a t i o n s . The smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s were u s e d t o o b t a i n a v i e w of e a c h s t a n d a r d p a t h . T a b l e 3.18 l i s t s t h e p a t h e s t i m a t i o n e r r o r s f o r e a c h p a t h u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F i g u r e 3.44 shows t h e c o r r e s p o n d i n g v i e w o f c i r c l e 1. T a b l e 3.19 l i s t s t h e c o e f f i c i e n t s f o r t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s a f t e r a d d i t i o n a l s m o o t h i n g . As i n t a b l e 3.17, t h e c o r r e s p o n d e n c e i s q u i t e good e x c e p t f o r h i g h e r o r d e r t e r m s . C l e a r l y , t h e a d d i t i o n a l s e l f - l e a r n i n g w i t h s m o o t h i n g has 151 i m p r o v e d t h e a c c u r a c y o f t h e sum o f p o l y n o m i a l s r e p r e s e n t a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s . I t i s d o u b t f u l , t h o u g h , t h a t a d d i t i o n a l s u c h m o d i f i c a t i o n s of s e l f - l e a r n i n g c o u l d y i e l d f u r t h e r i mprovements. P r o b a b l y a b e t t e r way t o a c h i e v e improvements i s t o use a h i g h e r o r d e r sum o f p o l y n o m i a l s , s a y 6 o r d e r , t h a t c a n more a c c u r a t e l y r e p r e s e n t t h e d i r e c t p o s i t i o n k i n e m a t i c s , w i t h o u t r e q u i r i n g s p e c i a l v a r i a t i o n s of s e l f - l e a r n i n g . PATH LXERMS LVERMS LAERMS l i n e 1 5.32E-2 7.43E-2 2.73E-1 l i n e 2 3.08E-2 4.42E-2 1.77E-1 l i n e 3 1.16E-2 2.37E-2 9.48E-2 l i n e 4 2.33E-2 3.37E-2 1.41E-1 c i r c l e 1 1.04E-2 3.48E-2 1.47E-1 c i r c l e 2 1.17E-2 3.42E-2 1.45E-1 T a b l e 3.18 P a t h e s t i m a t i o n e r r o r u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n , k i n e m a t i c s 152 POLYNOMIAL TERM X( 1) COEFFICIENTS DERIVED SELF-LEARNED & SMOOTHED X(2 ) COEFFICIENTS DERIVED SELF-LEARNED & SMOOTHED 1 0.0 0.009 -1 .00 -0.993 z 1 1 .57 1 .64 0.0 -0.015 2 Z 1 0.0 -0.186 1 .23 1-11 3 z 1 -0.646 -1.61 0.0 0.234 4 Z 1 0.0 0.345 -0.254 -0.437 Z 2 0.785 0.809 1 .36 1 .39 Z 2 Z 1 -2.14 -2.34 1 .23 1 .28 2 Z 2 Z 1 -0.969 -0.453 -1 .68 -0.799 z z 3  Z 2 Z 1 0.879 1 .36 -0.507 -1.13 2 Z 2 -1 .07 -1 .26 0.617 0.407 z 2 z z 2 Z 1 -0.969 -0.076 -0.839 -0.794 2 2 z 2 Z 1 1 .32 -0.312 -0.761 -1 .84 Z 2 3 -0.323 -0.568 -0.559 -1.21 z 3 z Z 2 Z 1 0.879 1 .21 -0.507 -0.383 4 z 2 0.220 1 .38 -0.127 1 .00 T a b l e 3.19 C o e f f i c i e n t s f o r t h e d e r i v e d and smoothed, f i n a l s e l f - l e a r n e d sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s o f t h e d i r e c t p o s i t i o n k i n e m a t i c s f u n c t i o n s f o r p o s i t i o n s x 1 and x~ 153 F i g u r e 3.44 View of c i r c l e 1 u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s 154 3.9.2 CLOSED LOOP CONTROL USING THE SELF-LEARNED CARTESIAN INVERSE DYNAMICS AND THE SELF-LEARNED DIRECT POSITION KINEMATICS S i m u l a t i o n s v e r i f i e d t h a t c l o s e d l o o p c o n t r o l i s p o s s i b l e u s i n g t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . We c a l l t h i s b l i n d c o n t r o l . S i m u l a t i o n p a r a m e t e r s were s e t as f o l l o w s : CLOSED=.TRUE., INVARM=.FALSE., EXACT=.FALSE., VI SION=.FALSE., DELAY=.TRUE., PRDICT=.TRUE., USEOBS=.FALSE., k =8, k 0=16 and t , =0.01 s e c . 1 Z C 3 X C The s t a n d a r d f e e d b a c k g a i n s , k 1=16 and k 2=64, were deemed t o o h i g h a s t h e y r e s u l t e d i n i n s t a b i l i t y . The b e s t r e s u l t s were o b t a i n e d by r e d u c i n g t h e f e e d b a c k g a i n s t o t h e v a l u e s g i v e n a b o v e . A l l s i x s t a n d a r d p a t h s were s i m u l a t e d . The r e s u l t i n g p a t h e r r o r s a r e shown i n t a b l e 3.20. T r a c k i n g i s worse t h a n t h a t a c h i e v e d w i t h t h e p r e - l e a r n e d d i r e c t k i n e m a t i c s , however, t h i s i s t o be e x p e c t e d as t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s a r e l e s s a c c u r a t e t h a n t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . P a t h e r r o r s a r e o f t h e same o r d e r o f m a g n i t u d e as p o s i t i o n e s t i m a t i o n e r r o r s , a s shown p r e v i o u s l y i n t a b l e 3.16. F i g u r e 3.45 shows t h e r e s u l t i n g p a t h as s e e n u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g b l i n d c o n t r o l o f c i r c l e 1. C o n t r o l i s c l e a r l y m a r g i n a l a s t h e c o n t r o l s y s t e m i s u n a b l e t o make t h e p e r c e i v e d p a t h f o l l o w t h e p a t h s p e c i f i c a t i o n t h a t w e l l . The a c t u a l p a t h i s shown i n f i g u r e 3.46 and i s somewhat worse t h a n t h e p e r c e i v e d p a t h , a s e x p e c t e d . 155 I o rs. o o O) O' I CNJ—' © in ^0.50 -0.30 -0.10 0.10 0.30 0.50 LEX(1) M F i g u r e 3.45 View of c i r c l e 1 using the f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics during c l o s e d loop c o n t r o l of c i r c l e 1 using the f i n a l s e l f - l e a r n e d C a r t e s i a n inverse dynamics and f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics 156 Figure 3.46 View of c i r c l e 1 during c l o s e d loop c o n t r o l of c i r c l e 1 using the f i n a l s e l f - l e a r n e d C a r t e s i a n inverse dynamics a.nd f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics 157 PATH XERMS VXERMS AXERMS l i n e 1 l i n e 2 l i n e 3 l i n e 4 c i r c l e 1 c i r c l e 2 1.37E-1 9.96E-2 2.75E-2 7.56E-2 4.97E-2 5.28E-2 1.66E-1 1.13E-1 4.65E-2 7.86E-2 9.61E-2 9.01E-2 4.17E-1 2.90E-1 1.64E-1 2.52E-1 3.10E-1 3.02E-1 T a b l e 3.20 P a t h e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s i n c l o s e d l o o p c o n t r o l A d d i t i o n a l s i m u l a t i o n s were c a r r i e d o u t u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n s k i n e m a t i c s i n c l o s e d l o o p c o n t r o l . S i m u l a t i o n p a r a m e t e r s were n o t a l t e r e d . A l l s i x s t a n d a r d p a t h s were s i m u l a t e d . The r e s u l t i n g p a t h e r r o r s a r e l i s t e d i n t a b l e 3.21. T r a c k i n g i s now a s good a s t h a t a c h i e v e d w i t h t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . T h i s i s t o be e x p e c t e d a s t h e a d d i t i o n a l a v e r a g i n g has r e n d e r e d t h e smoothed, f i n a l d i r e c t p o s i t i o n k i n e m a t i c s t o be a s a c c u r a t e a s t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . P a t h e r r o r s a r e o f t h e same o r d e r o f m a g n i t u d e a s p o s i t i o n e s t i m a t i o n e r r o r s , a s g i v e n p r e v i o u s l y i n t a b l e 3.18. F i g u r e 3.47 shows t h e r e s u l t i n g p a t h a s seen u s i n g t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g b l i n d c o n t r o l o f c i r c l e 1. C o n t r o l i s much b e t t e r now 158 t h a n when u s i n g t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s w i t h o u t s m o o t h i n g , s i n c e t h e c o n t r o l s y s t e m i s a b l e t o make t h e p e r c e i v e d p a t h f o l l o w t h e p a t h s p e c i f i c a t i o n . The a c t u a l p a t h i s shown i n f i g u r e 3.48 and i s somewhat worse t h a n t h e p e r c e i v e d p a t h , as e x p e c t e d . PATH XERMS VXERMS AXERMS l i n e 1 3.83E-2 9.27E-2 4.04E-1 l i n e 2 8.41E-2 1.18E-2 2.34E+0 l i n e 3 1.15E-2 2.71E-2 1.41E-1 l i n e 4 1.85E-2 3.53E-2 1.65E-1 c i r c l e 1 1.11E-2 3.86E-2 1.71E-1 c i r c l e 2 1.40E-2 4.13E-2 1.79E-1 T a b l e 3.21 P a t h e r r o r u s i n g t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s i n c l o s e d l o o p c o n t r o l 159 o in o o C O (\J— . * x7 LU o CO o in 0 . 5 0 - 0 . 3 0 - 0 . 1 0 L E X ( i ) M 0 . 1 0 0 . 3 0 0 . 5 0 F i g u r e 3.47 View of c i r c l e 1 u s i n g the smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i nematics d u r i n g c l o s e d loop c o n t r o l of c i r c l e 1 u s i n g the f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics 160 o in & on o f x O ' I F i g u r e 3.48 View of c i r c l e 1 d u r i n g c l o s e d l oop c o n t r o l of c i r c l e 1 u s i n g the f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics 161 F i g u r e 3.49 shows t h e r e s u l t i n g p a t h d u r i n g b l i n d c o n t r o l o f l i n e 1. The l a r g e a c c e l e r a t i o n s t h a t i n i t i a l l y o c c u r a r e due t o t h e sudden p e r c e i v e d p o s i t i o n e r r o r t h a t r e s u l t s when one s w i t c h e s f r o m t h e a c c u r a t e v i s i o n s y s t e m t o t h e l e s s a c c u r a t e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . As was done b e f o r e when u s i n g t h e p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s , t h i s g l i t c h c a n be e l i m i n a t e d by u t i l i z i n g an i n i t i a l i t e r a t i o n o f L e a r n i n g I d e n t i f i c a t i o n or I n t e r f e r e n c e M i n i m i z a t i o n b e f o r e s w i t c h i n g f r o m t h e v i s i o n s y s t e m t o t h e smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F i g u r e 3.50 shows t h e r e s u l t i n g p a t h d u r i n g b l i n d c o n t r o l o f l i n e 1 where t h i s t e c h n i q u e was e n a b l e d by s e t t i n g ADJVIS=.TRUE. 162 Figure 3.49 Closed loop c o n t r o l of l i n e 1 using the f i n a l s e l f - l e a r n e d C a r t e s i a n inverse dynamics and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics 163 4.00 ' 0 . 0 0 0 . 8 0 1 . 6 0 2 . 4 0 3 . 2 0 4 . 0 0 T SEC in in 4 . 0 0 T SEC Figure 3.50 Closed loop c o n t r o l of l i n e 1 us i n g the f i n a l s e l f - l e a r n e d C a r t e s i a n inverse dynamics and smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics with ADJVIS=.TRUE. 164 These s i m u l a t i o n s show t h a t c o a r s e c o n t r o l i s p o s s i b l e u s i n g the s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics and s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F i n e r c o n t r o l appears t o r e q u i r e o n l y an improvement i n the a c c u r a c y of the s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . By u s i n g a v e r a g i n g or smoothing t e c h n i q u e s d u r i n g the f i n a l s t a g e s of s e l f - l e a r n i n g , i t i s p o s s i b l e t o improve the a c c u r a c y of the s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s t o the p o i n t t h a t they a r e e q u a l t o or b e t t e r than the p r e - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . Good c o n t r o l i s p o s s i b l e u s i n g the smoothed, f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . F u r t h e r improvements a r e l i k e l y i f h i g h e r o r d e r sum of p o l y n o m i a l s r e p r e s e n t a t i o n s a r e used t h a t can more a c c u r a t e l y r e p r e s e n t the d i r e c t p o s i t i o n k i n e m a t i c s over the workspace. S i n c e t h e r e a r e o n l y two i n p u t v a r i a b l e s when e s t i m a t i n g the d i r e c t p o s i t i o n k i n e m a t i c s , one can use h i g h o r d e r , sum of p o l y n o m i a l s r e p r e s e n t a t i o n s w i t h o u t i n c u r r i n g the c o m p u t a t i o n a l c o s t s of h a v i n g a l a r g e number of terms i n the sums. 3.10 SUMMARY The c o n c e p t of s e l f - l e a r n e d c o n t r o l has been i n t r o d u c e d and i t s performance demonstrated u s i n g a s i m u l a t e d two l i n k m a n i p u l a t o r . T h i s performance has been compared t o the c o n t r o l performance a c h i e v e d u s i n g t h e a n a l y t i c a l dynamics and k i n e m a t i c s of the two l i n k m a n i p u l a t o r . C o n t r o l u s i n g the a n a l y t i c a l dynamics and k i n e m a t i c s has been e x t e n s i v e l y t e s t e d t o e n s u r e t h a t s t a n d a r d a n a l y t i c a l c o n t r o l forms a r e a l i s t i c 165 benchmark a g a i n s t which to compare s e l f - l e a r n e d c o n t r o l . S e l f - l e a r n i n g i s based on the f a c t that i t i s p o s s i b l e to repre s e n t the n o n - l i n e a r , m u l t i - v a r i a t e dynamics and kinematics r e l a t i o n s h i p s of the two l i n k manipualator as weighted sums of po l y n o m i a l s . Using I n t e r f e r e n c e M i n i m i z a t i o n or r e l a t e d l e a r n i n g a l g o r i t h m s , i t i s p o s s i b l e to l e a r n the weighting c o e f f i c i e n t s of the sums of polynomials and thus l e a r n the f u n c t i o n a l r e l a t i o n s h i p s necessary to achieve c o n t r o l . Furthermore, i t i s p o s s i b l e to l e a r n these f u n c t i o n a l r e l a t i o n s h i p s without recourse to a n a l y s i s of the manipulator. The i n v e r s e ^ dynamics of the two manipulator can be w e l l r e presented as a sum of polynomials over the whole of the manipulator's reach. T h i s would seem to be t r u e i n general f o r manipulators as t h e i r i n v e r s e dynamics appear i n a s i m i l a r format [28,46]. The sum of polynomials r e p r e s e n t a t i o n can be thought of as merely r e p l a c i n g the v a r i o u s s i n e and c o s i n e terms by t h e i r s e r i e s e q u i v a l e n t s and combining a l i k e terms. T h i s was demonstrated by d e r i v a t i o n of a sum of polynomials r e p r e s e n t a t i o n by j u s t such a technique. The sum of polynomials r e p r e s e n t a t i o n thus y i e l d s an e f f i c i e n t computational format f o r e v a l u a t i o n of the f u n c t i o n a l r e l a t i o n s h i p s as w e l l as p e r m i t t i n g the l e a r n i n g of these r e l a t i o n s h i p s with I n t e r f e r e n c e M i n i m i z a t i o n . The i n v e r s e kinematics of the two l i n k manipulator can only be represented by sums of polynomials over a p o r t i o n of the manipulator's reach at a time. The i n v e r s e kinematics of manipulators i n general are d i f f i c u l t to o b t a i n a n a l y t i c a l l y and 166 a r e l i k e l y t o c o n t a i n r e l a t i o n s h i p s t h a t cannot be r e p r e s e n t e d by sums of p o l y n o m i a l s over the whole of a m a n i p u l a t o r ' s r e a c h . N e v e r t h e l e s s we found t h a t the i n v e r s e k i n e m a t i c s of the two l i n k m a n i p u l a t o r can be w e l l r e p r e s e n t e d as a sum of p o l y n o m i a l s over a workspace c o n s i s t i n g of a s i z a b l e p o r t i o n of t h e m a n i p u l a t o r ' s r e a c h . Hence the C a r t e s i a n i n v e r s e dynamics can be s e l f - l e a r n e d over t h i s workspace. The workspace was not s p e c i a l l y chosen; o t h e r p o r t i o n s of the m a n i p u l a t o r ' s r e a c h c o u l d have been used. I t would be p o s s i b l e t o s u b d i v i d e the whole m a n i p u l a t o r space i n t o p o r t i o n s and s e l f - l e a r n unique sum of p o l y n o m i a l s r e p r e s e n t a t i o n s f o r each p o r t i o n . One s p l i t t i n g t e c h n i q u e t h a t may be v i a b l e i s d i s c u s s e d i n c h a p t e r 5. Another s o l u t i o n t h a t may be p o s s i b l e i s t o use a more g e n e r a l type of l e a r n e d f u n c t i o n a l e s t i m a t i o n , perhaps based on r e p r e s e n t a t i o n c o n s i s t i n g of a r a t i o of two sums of p o l y n o m i a l s . I t would appear t h a t s e l f - l e a r n i n g can be a c h i e v e d f o r m a n i p u l a t o r s i n g e n e r a l over a p o r t i o n of t h e space w i t h i n t h e i r r e a c h and over the whole of t h e i r r e a c h i f a s p l i t t i n g t e c h n i q u e or o t h e r more g e n e r a l type of l e a r n e d f u n c t i o n a l e s t i m a t i o n can be used. The c o n t r o l method w i t h i n which t h e s e s e l f - l e a r n e d r e l a t i o n s h i p s a r e a p p l i e d i s r e s o l v e d a c c e l e r a t i o n c o n t r o l . The s e l f - l e a r n e d i n v e r s e dynamics a r e used t o c a n c e l t h e non- l i n e a r i t i e s of the m a n i p u l a t o r dynamics y i e l d i n g a combined system which i s a p p r o x i m a t e l y l i n e a r and can be c o n t r o l l e d u s i n g s i m p l e e r r o r c o r r e c t i n g feedback. S e l f - l e a r n e d c o n t r o l u s i n g p a t h s p e c i f i c a t i o n i n j o i n t c o o r d i n a t e s would appear t o be easy t o a p p l y i n g e n e r a l s i n c e 1 6 7 the i n v e r s e dynamics appear i n g e n e r a l t o be w e l l r e p r e s e n t e d by a sum of p o l y n o m i a l s over the whole of a m a n i p u l a t o r ' s r e a c h . I n t h i s work we have implemented s e l f - l e a r n e d c o n t r o l u s i n g p a t h s p e c i f i c a t i o n i n C a r t e s i a n c o o r d i n a t e s , which i s more c o n v e n i e n t t o use but more d i f f i c u l t t o implement as i t r e q u i r e s s e l f - l e a r n i n g of the C a r t e s i a n i n v e r s e dynamics which c o n t a i n the d i f f i c u l t t o l e a r n , i n v e r s e k i n e m a t i c s . P a t h s p e c i f i c a t i o n i n C a r t e s i a n c o o r d i n a t e s a l s o n e c e s s i t a t e s the use of a v i s i o n system, o r i t s e q u i v a l e n t u s i n g the a n a l y t i c a l d i r e c t p o s i t i o n k i n e m a t i c s , t o observe the m a n i p u l a t o r p o s i t i o n i n C a r t e s i a n c o o r d i n a t e s f o r use i n e r r o r c o r r e c t i n g feedback. The v i s i o n system or i t s e q u i v a l e n t must be v e r y a c c u r a t e as i n a c c u r a c i e s t r a n s l a t e d i r e c t l y i n t o p a t h t r a c k i n g e r r o r s . The d i r e c t p o s i t i o n k i n e m a t i c s of the two l i n k m a n i p u l a t o r can be w e l l r e p r e s e n t e d u s i n g sums of p o l y n o m i a l s over the whole of the m a n i p u l a t o r ' s r e a c h . T h i s appears t o be p o s s i b l e f o r m a n i p u l a t o r s i n g e n e r a l s i n c e t h e i r a n a l y t i c a l d i r e c t p o s i t i o n k i n e m a t i c s c o n s i s t of sums and p r o d u c t s of s i n e and c o s i n e terms. The d i r e c t p o s i t i o n k i n e m a t i c s of the two l i n k m a n i p u l a t o r were s e l f - l e a r n e d and used i n p l a c e of the v i s i o n system i n c l o s e d l o o p c o n t r o l . T a b l e 3.22 summarizes t h e p a t h e r r o r , averaged over the s i x s t a n d a r d p a t h s , t h a t r e s u l t s when u s i n g the v a r i o u s c o n t r o l schemes. I t can be seen t h a t performance u s i n g the s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics i s q u i t e comparable t o t h a t a c h i e v e d u s i n g the a n a l y t i c a l C a r t e s i a n i n v e r s e dynamics. C l o s e d l o o p c o n t r o l u s i n g the s e l f - l e a r n e d C a r t e s i a n i n v e r s e dynamics i n 168 c o n j u n c t i o n w i t h an a c c u r a t e v i s i o n s y s t e m , o r i t s e q u i v a l e n t , i s a v e r y v i a b l e c o n t r o l t e c h n i q u e . When t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s a r e u s e d i n s t e a d of t h e v i s i o n s y s t e m , p e r f o r m a n c e i s d e g r a d e d , r e f l e c t i n g t h e i n a c c u r a c i e s i n t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s . N e v e r t h e l e s s i t s e r v e s t o d e m o n s t r a t e t h e t e c h n i q u e . We f o u n d t h a t an a v e r a g i n g o r s m o o t h i n g t e c h n i q u e a p p l i e d i n t h e f i n a l s t a g e s o f s e l f - l e a r n i n g s i g n i f i c a n t l y i m p r o v e d t h e a c c u r a c y o f t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s and t h e r e s u l t i n g c o n t r o l p e r f o r m a n c e . To a c h i e v e b e t t e r p e r f o r m a n c e o n l y r e q u i r e s t h a t t h e s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s be more a c c u r a t e , a s c o u l d be a c h i e v e d u s i n g h i g h e r o r d e r , sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s . 169 XERMS VXERMS AXERMS CONTROL SCHEME 6.48E-4 2.10E-3 8.06E-2 3.52E-3 5.22E-3 9.10E-2 4.23E-3 1.01E-2 1.18E-1 3.37E-2 3.92E-2 2.15E-1 7.37E-2 8.60E-2 2.89E-1 2.96E-2 4.11E-2 5.67E-1 S t a n d a r d A n a l y t i c a l C o n t r o l P r e - L e a r n e d C a r t e s i a n I n v e r s e D y n a m i c s S e l f - L e a r n e d C a r t e s i a n I n v e r s e D y namics P r e - L e a r n e d C a r t e s i a n I n v e r s e D y n a m i c s & P r e - L e a r n e d D i r e c t P o s i t i o n K i n e m a t i c s S e l f - L e a r n e d C a r t e s i a n I n v e r s e D y n a m i c s & S e l f - L e a r n e d D i r e c t P o s i t i o n K i n e m a t i c s S e l f - L e a r n e d C a r t e s i a n I n v e r s e D y n a m i c s & Smoothed S e l f - L e a r n e d D i r e c t P o s i t i o n K i n e m a t i c s T a b l e 3.22 Summary o f p a t h e r r o r s u s i n g v a r i o u s c o n t r o l shemes 170 4 CONTRIBUTIONS OF THIS THESIS T h i s t h e s i s has shown t h a t I n t e r f e r e n c e M i n i m i z a t i o n and r e l a t e d l e a r n i n g a l g o r i t h m s can be used t o l e a r n t he n o n - l i n e a r , m u l t i - v a r i a t e f u n c t i o n s d e s c r i b i n g t he dynamics and k i n e m a t i c s of a ro b o t m a n i p u l a t o r . I t has been shown t h a t t h i s l e a r n i n g can be done w i t h o u t r e c o u r s e t o a n a l y s i s of the dynamics and k i n e m a t i c s of the m a n i p u l a t o r . T h i s i s p o s s i b l e t h rough l e a r n i n g based on o b s e r v a t i o n of a p p l i e d t o r q u e s and c o r r e s p o n d i n g m a n i p u l a t o r m o t i o n ; hence the name s e l f - l e a r n i n g . A f a m i l y of l e a r n i n g a l g o r i t h m s has been s t u d i e d . I t has been shown t h a t the G r a d i e n t Method and L e a r n i n g I d e n t i f i c a t i o n a re s i m p l i f i c a t i o n s of I n t e r f e r e n c e M i n i m i z a t i o n , t he p r i n c i p a l l e a r n i n g a l g o r i t h m used i n t h i s t h e s i s . The convergence r a t e s of I n t e r f e r e n c e M i n i m i z a t i o n and t h e s e r e l a t e d l e a r n i n g a l g o r i t h m s have been compared f o r sum of p o l y n o m i a l s e s t i m a t e s of v a r i o u s o r d e r s , h a v i n g v a r i o u s numbers of i n p u t v a r i a b l e s . A method has been documented f o r r e d u c i n g t he number of c a l c u l a t i o n s r e q u i r e d when imp l e m e n t i n g I n t e r f e r e n c e M i n i m i z a t i o n . A new v a r i a n t of I n t e r f e r e n c e M i n i m i z a t i o n c a l l e d P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n has been i n t r o d u c e d and compared w i t h t he o t h e r l e a r n i n g a l g o r i t h m s . A p p l i c a t i o n s where P o i n t w i s e I n t e r f e r e n c e M i n i m i z a t i o n i s a p p r o p r i a t e have been d e s c r i b e d . The concept of s e l f - l e a r n i n g has been i n t r o d u c e d and i t s v i a b i l i t y d emonstrated u s i n g a s i m u l a t e d two l i n k m a n i p u l a t o r . I t has been shown t h a t t h e f u n c t i o n a l r e l a t i o n s h i p s f o r m i n g the i n v e r s e dynamics, the i n v e r s e k i n e m a t i c s (which t o g e t h e r form 171 t h e C a r t e s i a n i n v e r s e d y n a m i c s ) and t h e d i r e c t p o s i t i o n k i n e m a t i c s of t h e two l i n k m a n i p u l a t o r c a n be a d e q u a t e l y t h r e p r e s e n t e d by 4 o r d e r sums o f p o l y n o m i a l s . T h i s c an be done o v e r a l l o f t h e two l i n k m a n i p u l a t o r ' s r e a c h or a p o r t i o n t h e r e o f s u f f i c i e n t t o be c o n s i d e r e d a u s e f u l w o r k s p a c e . S e l f - l e a r n i n g of t h e C a r t e s i a n i n v e r s e d y n a m i c s p e r m i t s r e s o l v e d a c c e l e r a t i o n c o n t r o l t o be a p p l i e d u s i n g p a t h s p e c i f i c a t i o n s c o n v e n i e n t l y g i v e n i n C a r t e s i a n c o o r d i n a t e s . P e r f o r m a n c e u s i n g t h e s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s w i t h o n l y a 4 f c^ o r d e r r e p r e s e n t a t i o n has been f o u n d t o be a l m o s t as good as p e r f o r m a n c e u s i n g t h e a n a l y t i c a l C a r t e s i a n i n v e r s e d y n a m i c s . The C a r t e s i a n i n v e r s e d y n a m i c s can be l e a r n e d by u t i l i z i n g t r a i n i n g p o i n t s c o n s i s t i n g of n - t u p l e s o f a p p l i e d t o r q u e s and r e s u l t i n g end p o i n t C a r t e s i a n p o s i t i o n s , v e l o c i t i e s and a c c e l e r a t i o n s a s imposed by t h e d i r e c t d y n a m i c s of t h e m a n i p u l a t o r and o b s e r v e d u s i n g a v i s i o n s y s t e m . The d i r e c t p o s i t i o n k i n e m a t i c s c a n be l e a r n e d by u t i l i z i n g t r a i n i n g p o i n t s c o n s i s t i n g o f n - t u p l e s o f measured j o i n t p o s i t i o n s and c o r r e s p o n d i n g end p o i n t C a r t e s i a n p o s i t i o n s o b s e r v e d u s i n g a v i s i o n s y s t e m . A s i m p l e t e c h n i q u e has been d e m o n s t r a t e d t h a t p e r m i t s t h e m a n i p u l a t o r end p o i n t m o t i o n t o be c o n s t r a i n e d w i t h i n a d e s i r e d r a n g e o f C a r t e s i a n p o s i t i o n s , v e l o c i t i e s , and a c c e l e r a t i o n s d u r i n g t h e e a r l y s t a g e s o f s e l f - l e a r n i n g when t h e C a r t e s i a n i n v e r s e d y n a m i c s a r e n o t known w e l l enough t o c o n t r o l t h e m a n i p u l a t o r . I t has been shown t h a t t h e v i s i o n s y s t e m u s e d d u r i n g s e l f - l e a r n i n g c a n be r e p l a c e d w i t h t h e f i n a l s e l f - l e a r n e d d i r e c t p o s i t i o n k i n e m a t i c s d u r i n g s u b s e q u e n t use o f t h e 1 7 2 m a n i p u l a t o r . W h i l e s e l f - l e a r n e d c o n t r o l h as o n l y been d e m o n s t r a t e d f o r a s i m u l a t e d two l i n k m a n i p u l a t o r , e v i d e n c e has been p r e s e n t e d t h a t w o u l d l e a d one t o b e l i e v e t h a t s e l f - l e a r n e d c o n t r o l i s a g e n e r a l a p p r o a c h t h a t c a n be a p p l i e d t o m a n i p u l a t o r s h a v i n g more d e g r e e s o f f r e e d o m u s i n g p a t h s p e c i f i c a t i o n i n a c o o r d i n a t e s y s t e m t h a t i s c o n v e n i e n t f o r t h e t a s k s t o be c a r r i e d o u t . D e t a i l e d s u g g e s t i o n s f o r f u t u r e r e s e a r c h a r e p r e s e n t e d ( i n t h e f o l l o w i n g c h a p t e r , c h a p t e r 5) c o v e r i n g s u c h t o p i c s a s f u r t h e r i n v e s t i g a t i o n o f s e l f - l e a r n e d c o n t r o l ( e s p e c i a l l y a u t o m a t i c a d a p t a t i o n f o r an unknown - t o o l m a s s ) , h a r d w a r e i m p l e m e n t a t i o n of t h e r e q u i r e d sum o f p o l y n o m i a l s e s t i m a t o r s , and m o d i f i c a t i o n o f t h e K l e t t C e r e b e l l a r M odel ( t h e m a j o r i n s p i r a t i o n f o r t h i s t h e s i s ) t o i n c r e a s e i t s p l a u s i b i l i t y . 173 5 SUGGESTIONS FOR FUTURE R E S E A R C H The c h a r a c t e r i z a t i o n o f a f a m i l y of l e a r n i n g a l g o r i t h m s and f o r m u l a t i o n and d e m o n s t r a t i o n o f s e l f - l e a r n e d c o n t r o l of t h e two l i n k m a n i p u l a t o r u t i l i z i n g t h e s e l e a r n i n g a l g o r i t h m s i s t h e p r i n c i p a l c o n t r i b u t i o n o f t h i s t h e s i s , however, p e r h a p s j u s t a s i m p o r t a n t a r e c e r t a i n p r o p o s a l s r e l a t e d t o t h i s work t h a t may form t h e b a s i s f o r f u t u r e r e s e a r c h . 5.1 FURTHER INVESTIGATION OF SELF-LEARNED CONTROL 5.1.1 CONTROL OF A MORE COMPLEX MANIPULATOR From t h e o u t s e t i t was hoped t h a t s e l f - l e a r n e d c o n t r o l would p r o v e a v i a b l e c o n t r o l scheme f o r complex i n d u s t r i a l m a n i p u l a t o r s h a v i n g up t o 6 d e g r e e s o f f r e e d o m . The two l i n k m a n i p u l a t o r was m e r e l y a p r e l i m i n a r y t e s t v e h i c l e . We b e l i e v e t h a t no s i m p l i f y i n g a s s u m p t i o n s have been made t h a t would p r o h i b i t t h e e x t e n s i o n o f s e l f - l e a r n e d c o n t r o l t o a r e a l m a n i p u l a t o r h a v i n g up t o 6 d e g r e e s of f r e e d o m . The l o g i c a l n e x t s t e p w o u l d be t o a t t e m p t t o a p p l y s e l f - l e a r n e d c o n t r o l t o a r e a l m a n i p u l a t o r w i t h more t h a n 2 d e g r e e s o f f r e e d o m . Many o f t h e a s p e c t s o f s e l f - l e a r n i n g n eed t o be s t u d i e d f u r t h e r . I t was a p p a r e n t t h a t many p a r a m e t e r s had an e f f e c t on t h e r a t e o f s e l f - l e a r n i n g . One m i g h t e x p e r i m e n t by c h a n g i n g f e e d b a c k g a i n s d u r i n g s e l f - l e a r n i n g a s t h e optimum f e e d b a c k g a i n i n t e r m s o f m i n i m i z i n g o u t o f bounds e x c u r s i o n s , i e . i n s t a b i l i t y , w h i l e m i n i m i z i n g p a t h t r a c k i n g e r r o r , c h a n g e s as s e l f - l e a r n i n g p r o g r e s s e s . I t i s a l s o a p p a r e n t t h a t t h e c h o i c e of 174 t r a i n i n g p a t h s , f r e q u e n c y of l e a r n i n g i t e r a t i o n s , and f o r c e f u l n e s s of c o n s t r a i n i n g a c t i o n have an e f f e c t on t h e l e a r n i n g r a t e . As d i s c u s s e d i n s e c t i o n 3.9.1, t h e c o n s t r a i n i n g t e c h n i q u e u s e d i n t h e s e s i m u l a t i o n s was v e r y s i m p l i s t i c and i m p r o v e d t e c h n i q u e s were p r o p o s e d . I n v e s t i g a t i o n i s w a r r a n t e d i n t o c h a n g i n g p a r a m e t e r s o f t h e c o n s t r a i n i n g t e c h n i q u e u s e d h e r e o r d e v e l o p m e n t o f b e t t e r t e c h n i q u e s i n o r d e r t o m i n i m i z e s u c h q u a n t i t i e s a s maximum ou t o f bounds e x c u r s i o n s , r i s k of o u t of bounds e x c u r s i o n s t h a t m i g h t damage a r e a l m a n i p u l a t o r , e t c . F o r example t h e c o n s t r a i n i n g t e c h n i q u e u s e d h e r e was f o u n d t o l i m i t o u t o f bounds e x c u r s i o n s q u i t e w e l l b u t t h e r e may be s i t u a t i o n s where l a r g e u n a c c e p t a b l e o u t o f bounds e x c u r s i o n s o c c u r d u r i n g s e l f - l e a r n i n g u s i n g t h i s c o n s t r a i n i n g t e c h n i q u e ; i t has n o t been p r o v e n o t h e r w i s e . I t i s a l s o a p p a r e n t t h a t a v e r a g i n g t e c h n i q u e s a s d i s c u s s e d i n s e c t i o n 3.9.1 c a n i m prove t h e mean e s t i m a t i o n a c c u r a c y a c h i e v a b l e w i t h a g i v e n o r d e r sum o f p o l y n o m i a l s r e p r e s e n t a t i o n . I t may a l s o be p o s s i b l e t o use l e a r n i n g a l g o r i t h m s b a s e d on e s t i m a t i o n u s i n g f u n c t i o n s o t h e r t h a n p o l y n o m i a l s . The c o s t s t o a p p l y when e v a l u a t i n g t h e numerous p o s s i b i l i t i e s i n t e r m s o f r e q u i r e d c o m p u t i n g c a p a b i l i t y , s e l f - l e a r n i n g t i m e , c o n t r o l p e r f o r m a n c e , an d a d e q u a c y o f c o n s t r a i n i n g a c t i o n , e t c . w i l l become more a p p a r e n t upon a p p l i c a t i o n t o a r e a l m a n i p u l a t o r . 5.1.2 ADAPTATION TO TOOL MASS The i n i t i a l i n t e n t o f t h i s t h e s i s was t o d e m o n s t r a t e s e l f - l e a r n i n g o f t h e i n v e r s e d y n a m i c s o r t h e more d i f f i c u l t C a r t e s i a n i n v e r s e d y n a m i c s . A v i s i o n s y s t e m was t o be u s e d f o r f e e d b a c k 175 p u r p o s e s . I t soon became a p p a r e n t t h a t s e l f - l e a r n i n g c o u l d be a p p l i e d i n o t h e r a r e a s , f o r example t o l e a r n t o e s t i m a t e t h e d i r e c t p o s i t i o n k i n e m a t i c s and t h u s r e p l a c e t h e v i s i o n s y s t e m , o r f r e e i t f o r use i n s p e c i f i c a r e a s o f t h e w o r k s p a c e where h i g h r e s o l u t i o n i s r e q u i r e d t o a c h i e v e f i n e p a t h t r a c k i n g . A n d t h e r a r e a t h a t i n v i t e s t h e a p p l i c a t i o n o f s e l f - l e a r n i n g i s i n a d a p t a t i o n of c o n t r o l a c c o r d i n g t o t h e t o o l mass. C o n s i d e r a g a i n t h e two l i n k m a n i p u l a t o r . One c a n i m a g i n e t h a t t h e s e c o n d l i n k mass c o n s i s t s of t h e mass of t h e f i n a l l i n k a n d t h e mass of .the t o o l t h a t t h e m a n i p u l a t o r i s h o l d i n g , m 2 = m l i n k + m t o o l ( 5 ' 1 ) The l i n k masses a l o n g w i t h o t h e r m a n i p u l a t o r p a r a m e t e r s s u c h as l i n k l e n g t h s , e t c . were c o n s i d e r e d t o be c o n s t a n t s i n f u n c t i o n s f o r m i n g t h e C a r t e s i a n i n v e r s e d y n a m i c s . One c a n a l s o t h i n k o f m t o o l a s a n o t h e r i n p u t v a r i a b l e i n t h e C a r t e s i a n i n v e r s e d y n a m i c s , r = c a r t . i n v . d y n ( x , x , x , m t o o ^ ) (5.2) G i v e n t h a t m t o o ^ i s s p e c i f i e d , one s h o u l d be a b l e t o s e l f - l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s f o r a r a n g e o f t o o l m a sses. A g o o d t e c h n i q u e m i g h t be t o i n i t i a l l y f i x t h e i n p u t v a r i a b l e , m t o o l ' a n d a t t a c h a t o o l o f t h e c o r r e s p o n d i n g mass t o t h e m a n i p u l a t o r w h i l e s e l f - l e a r n i n g t h e C a r t e s i a n i n v e r s e d y n a m i c s . Once good c o n t r o l i s a c h i e v e d , one c o u l d t h e n b e g i n t o v a r y t h e mass o f t h e a t t a c h e d t o o l and c o r r e s p o n d i n g l y t h e v a r i a b l e m t o o i a n d t h u s s e l f - l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s f o r a range o f t o o l m a s s e s . To a d j u s t t h e C a r t e s i a n i n v e r s e d y n a m i c s t o be a p p r o p r i a t e f o r t h e mass o f t h e t o o l i n u s e , one need o n l y know 176 m t o o l a n d a P P ^ v ^ a s a n i n p u t . v a r i a b l e o f t h e f i n a l s e l f - l e a r n e d C a r t e s i a n i n v e r s e d y n a m i c s . A l o g i c a l n e x t s t e p i s t o a s k whether t h e t o o l mass c a n be d e t e r m i n e d s i m p l y by o b s e r v i n g m a n i p u l a t o r m o t i o n and a p p l i e d t o r q u e s . I f one c o u l d d i s c e r n m t o o i b a s e d on o b s e r v a t i o n s o f m a n i p u l a t o r m o t i o n and a p p l i e d t o r q u e s t h e n c o n t r o l of t h e m a n i p u l a t o r c o u l d be a d j u s t e d a p p r o p r i a t e l y when a t o o l of unknown mass i s a t t a c h e d , a v e r y d e s i r a b l e c a p a b i l i t y . C o n s i d e r t h e i n v e r s e d y n a m i c s o f t h e two l i n k m a n i p u l a t o r as g i v e n i n e q u a t i o n s (3.51) t h r o u g h ( 3 . 6 6 ) . One c o u l d s u b s t i t u t e e q u a t i o n (5.1) i n t o t h e i n v e r s e d y n a m i c s r e l a t i o n s h i p s and s o l v e f o r m t o o l y i e l d i n g a f u n c t i o n a l r e l a t i o n s h i p t h a t we w i l l c a l l t h e t o o l d y n a m i c s , m t o o l = t o o l . d y n ( 7 , a , a , a ) (5.3) A p r e l i m i n a r y l o o k s u g g e s t s t h a t t h i s f u n c t i o n w i l l p r o b a b l y not be r e p r e s e n t a b l e o v e r t h e whole o f t h e m a n i p u l a t o r ' s r e a c h u s i n g a sum o f p o l y n o m i a l s . I t i s l i k e l y , however, t h a t i t c a n be so r e p r e s e n t e d o v e r a p o r t i o n o f t h e m a n i p u l a t o r ' s r e a c h , s u c h a s t h e w o r k s p a c e t h a t was c h o s e n so t h a t t h e C a r t e s i a n i n v e r s e d y n a m i c s were r e p r e s e n t a b l e . I f s o , t h e n t h e t o o l d y n a m i c s c o u l d be s e l f - l e a r n e d . The p r o c e s s o f g e n e r a l i z a t i o n o f s e l f - l e a r n i n g need not s t o p h e r e . C l e a r l y , s e l f - l e a r n i n g o f t h e t o o l d y n a m i c s i s q u i t e a n a l o g o u s t o f o r c e f e e d b a c k . One m e r e l y s u b s t i t u t e s an unknown a p p l i e d f o r c e f o r t h e unknown a p p l i e d t o o l mass. W i t h an a p p l i e d f o r c e one n e e d s t o e s t i m a t e i t s d i r e c t i o n a s w e l l a s m agnitude and h e n c e one must l e a r n s e v e r a l f u n c t i o n a l r e l a t i o n s h i p s . 177 C o n c e p t u a l l y , t h o u g h , t h e t e c h n i q u e would be s i m i l a r . A w e a l t h of p o s s i b l e a p p l i c a t i o n s f o r s e l f - l e a r n i n g a r e a p p a r e n t . T h e s e examples a r e from t h e f i e l d of m a n i p u l a t o r c o n t r o l ; a p p l i c a t i o n s may w e l l e x i s t i n o t h e r f i e l d s , anywhere t h a t l e a r n i n g of n o n - l i n e a r , m u l t i - v a r i a t e f u n c t i o n a l r e l a t i o n s h i p s i s r e q u i r e d . The v i a b i l i t y o f t h e s e l f - l e a r n i n g i n t h e s e a p p l i c a t i o n s , however, r e m a i n s t o be d e m o n s t r a t e d . In some c a s e s s e l f - l e a r n i n g may t h e o r e t i c a l l y be a p p l i c a b l e b u t d i f f i c u l t t o implement b e c a u s e o f t h e number o f c a l c u l a t i o n s r e q u i r e d when d e a l i n g w i t h many i n p u t v a r i a b l e s . . In o t h e r c a s e s t h e f u n c t i o n s t o be e s t i m a t e d may o n l y be r e p r e s e n t a b l e a s sums of p o l y n o m i a l s o v e r n a r r o w p o r t i o n s o f t h e i r domains making s e l f - l e a r n i n g d i f f i c u l t t o a p p l y w i t h . t h e l e a r n i n g a l g o r i t h m s g i v e n h e r e . 5.2 IMPLEMENTATION OF SUM OF POLYNOMIALS ESTIMATORS A p r i n c i p a l r e a s o n f o r i n v e s t i g a t i n g s e l f - l e a r n e d c o n t r o l u s i n g sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s i s t h e r e g u l a r i t y o f t h e r e s u l t i n g c o m p u t a t i o n a l s t r u c t u r e . The t h r e e l e a r n i n g a l g o r i t h m s , I n t e r f e r e n c e M i n i m i z a t i o n , L e a r n i n g I d e n t i f i c a t i o n and t h e G r a d i e n t Method a r e a l l b a s e d on t h e same sum o f p o l y n o m i a l s e s t i m a t o r , f = w Tp (5.4) In a c o n t r o l a p p l i c a t i o n i t i s t h e e v a l u a t i o n of t h e e s t i m a t e t h a t i s t h e t i m e c r i t i c a l t a s k . L e a r n i n g of t h e w e i g h t i n g c o e f f i c i e n t s c a n t a k e p l a c e o f f - l i n e u t i l i z i n g a method a p p r o p r i a t e f o r t h e c o m p u t i n g c a p a b i l i t y a v a i l a b l e and t h e r a t e 178 of l e a r n i n g d e s i r e d ; f o r example, I n t e r f e r e n c e M i n i m i z a t i o n c o n v e r g e s much f a s t e r than the G r a d i e n t Method but uses a more complex, and thus time consuming weight a d j u s t m e n t f o r m u l a . Due t o t h e r e g u l a r i t y of t h e sum of p o l y n o m i a l s e s t i m a t o r and i t s a p p l i c a b i l i t y t o problems of v a r i o u s s i z e s , i n v e s t i g a t i o n i s w a r r a n t e d i n t o the i m p l e m e n t a t i o n of sum of p o l y n o m i a l s e s t i m a t o r s u s i n g s p e c i a l l y d e v e l o p e d hardware. P r e l i m i n a r y work has been done f o r i m p l e m e n t a t i o n u s i n g d i g i t a l computing hardware and s t o c h a s t i c computing hardware. More work i s r e q u i r e d t o demonstrate the v i a b i l i t y of t h e s e approaches. 5.2.1 DIGITAL COMPUTER IMPLEMENTATION In a d i g i t a l computer i m p l e m e n t a t i o n , t h e r a t e l i m i t i n g f a c t o r i s the number of m u l t i p l i c a t i o n s r e q u i r e d t o e v a l u a t e the sum of p o l y n o m i a l s . F o r t u n a t e l y , e v a l u a t i o n of the sum of p o l y n o m i a l s does not r e q u i r e as many m u l t i p l i c a t i o n s as one might assume. R e c a l l t h a t , f o r a system of o r d e r s h a v i n g v i n p u t v a r i a b l e s , the p o l y n o m i a l terms a r e t h e s e t , v { p , ( z ) } = { n z . e i } (5.5) * i=0 where, v Z e. *= s . e, an i n t e g e r (5.6) £ = , . i - 0 1 z n = 1 (5.7) The i n p u t v a r i a b l e s and t h e c o n s t a n t 1 form t h e f i r s t v+1 terms. Each of the r e m a i n i n g h i g h e r o r d e r terms can be formed as the p r o d u c t of two lower o r d e r terms. The t o t a l number of terms i s , 179 (s+v)! m = (5.8) v! s! T a b l e 5.1 shows the number of terms f o r systems of v a r i o u s o r d e r s , s, and h a v i n g v a r i o u s numbers of i n p u t v a r i a b l e s , v. The number of m u t i p l i c a t i o n s r e q u i r e d t o form the p o l y n o m i a l terms i s m - (v+1). In a m a n i p u l a t o r c o n t r o l a p p l i c a t i o n the same p o l y n o m i a l terms w i l l t y p i c a l l y be used i n t h e e s t i m a t i o n of s e v e r a l f u n c t i o n s . I n m o d e l l i n g the C a r t e s i a n i n v e r s e dynamics of a m a n i p u l a t o r w i t h r degrees of freedom, the number of i n p u t v a r i a b l e s w i l l be v = 3 r , c o r r e s p o n d i n g t o an a c c e l e r a t i o n , v e l o c i t y and p o s i t i o n f o r each degree of freedom. There w i l l be r f u n c t i o n s f o r m i n g the C a r t e s i a n i n v e r s e dynamics. For such a m a n i p u l a t o r , one pass r e q u i r i n g m-v-1 m u l t i p l i c a t i o n s i s n e c e s s a r y t o form the p o l y n o m i a l terms and r passes r e q u i r i n g m m u l t i p l i c a t i o n s and a d d i t i o n s each a r e n e c e s s a r y t o accumulate the w e i g h t e d sum of t h e s e terms t o complete the e s t i m a t e s of the r f u n c t i o n s . For e s t i m a t e s of o r d e r s f o r each of the f u n c t i o n s f o r m i n g the C a r t e s i a n i n v e r s e dynamics, the number of m u l t i p l i c a t i o n s r e q u i r e d i s t h u s , ( r + 1 ) ( s + 3 r ) ! c i d ( r , s ) = - (3r+1) (5.9) s ! ( 3 r ) ! In m o d e l l i n g the d i r e c t p o s i t i o n k i n e m a t i c s of a m a n i p u l a t o r h a v i n g r degrees of freedom, t h e number of i n p u t v a r i a b l e s w i l l be v=r, c o r r e s p o n d i n g t o the p o s i t i o n f o r each degree of freedom. There w i l l be r f u n c t i o n s f o r m i n g the d i r e c t p o s i t i o n k i n e m a t i c s . For e s t i m a t e s of o r d e r s f o r each of the f u n c t i o n s f o r m i n g the 180 d i r e c t p o s i t i o n k i n e m a t i c s , t h e number of m u l t i p l i c a t i o n s i s t h u s , ( r + 1 ) ( s + r ) ! d p k ( r , s ) = - (r+1) (5.10) s! r ! System O r d e r 1 2 3 4 . 5 6 N u 1 2 3 4 5 6 7 m 2 3 6 10 15 21 28 b 3 4 10 20 35 56 84 e 4 5 15 35 70 126 210 r 5 6 21 56 126 252 462 6 7 28 84 210 462 924 0 7 8 36 120 330 792 1716 f 8 9 45 165 495 1287 3003 9 10 55 220 715 2002 5005 V 10 1 1 66 286 1001 3003 8008 a 1 1 12 78 364 1365 4368 12376 r 1 2 1 3 91 455 1820 6188 18564 i 13 14 105 560 2380 8568 27132 a 1 4 1 5 120 680 3060 1 1628 38760 b 1 5 16 1 36 816 3876 15504 54264 1 16 17 153 969 4845 20349 74613 e 1 7 18 171 1 140 5985 26334 100947 s 18 19 190 1330 7315 33649 134596 T a b l e 5.1 Number o f t e r m s , m, f o r sytems of v a r i o u s o r d e r s , s, and v a r i o u s numbers o f i n p u t v a r i a b l e s , v T a b l e 5.2 summarizes t h e number o f m u l t i p l i c a t i o n s r e q u i r e d t o e v a l u a t e t h e e s t i m a t e s of t h e C a r t e s i a n i n v e r s e d y n a m i c s f o r e s t i m a t o r s o f v a r i o u s o r d e r s , s, and h a v i n g i n p u t v a r i a b l e s c o r r e s p o n d i n g t o v a r i o u s numbers o f d e g r e e s o f fr e e d o m , r . L i k e w i s e , t a b l e 5.3 summarizes t h e number o f m u l t i p l i c a t i o n s r e q u i r e d t o e v a l u a t e t h e e s t i m a t e s o f t h e d i r e c t p o s i t i o n 181 k i n e m a t i c s . Sytem O r d e r D F 1 2 3 4 5 6 e r 1 4 16 36 66 108 1 64 g o e 2 14 77 245 623 1379 2765 r f e 3 30 210 870 2850 7998 20010 e d 4 52 442 2262 9087 30927 92807 e o 5 80 800 4880 23240 93008 325568 s m 6 1 1 4 1311 9291 51 186 235524 942153 T a b l e 5.2 M u l t i p l i c a t i o n s r e q u i r e d f o r e s t i m a t e s o f v a r i o u s o r d e r s of t h e C a r t e s i a n i n v e r s e d y n a m i c s f o r m a n i p u l a t o r s h a v i n g v a r i o u s d e g r e e s of f r e e d o m S y s t e m O r d e r D F 1 2 3 4 5 6 e r 1 4 6 8 10 12 1 4 g o e 2 9 18 30 45 63 84 r f e 3 16 40 80 140 224 336 e d 4 25 75 175 350 630 1050 e o 5 36 126 336 756 1512 2772 s m 6 49 196 588 1470 3234 6468 F i g u r e 5.3 M u l t i p l i c a t i o n s r e q u i r e d f o r e s t i m a t e s o f v a r i o u s o r d e r s o f t h e d i r e c t p o s i t i o n k i n e m a t i c s f o r m a n i p u l a t o r s h a v i n g v a r i o u s d e g r e e s o f f r e e d o m F o r a 4 o r d e r model o f a 2 d e g r e e o f f r e e d o m m a n i p u l a t o r s u c h a s t h e two l i n k m a n i p u l a t o r s t u d i e d h e r e , 623 m u l t i p l i c a t i o n s a r e r e q u i r e d . A s s u m i n g t h a t t c a i c ^ * 0 1 s e c » t h i s c o r r e s p o n d s t o 16 usee p e r m u l t i p l i c a t i o n . I f s e l f - l e a r n e d 182 c o n t r o l c a n be e x t e n d e d t o a 6 d e g r e e of f r e e d o m m a n i p u l a t o r , 51186 m u l t i p l i c a t i o n s w o u l d be r e q u i r e d . F o r t ^ i ^ O ' ^ 1 s e c , t h i s c o r r e s p o n d s t o 195 n s e c p e r m u l t i p l i c a t i o n , q u i t e a h i g h r a t e , b u t n e v e r t h e l e s s a c h i e v a b l e w i t h c u r r e n t t e c h n o l o g y , e s p e c i a l l y i f o n l y f i x e d p o i n t o p e r a t i o n s a r e n e c e s s a r y . E v a l u a t i o n o f t h e d i r e c t p o s i t i o n k i n e m a t i c s e s t i m a t e s r e q u i r e s f a r f e wer m u l t i p l i c a t i o n s and t h u s i s e a s i e r t o a c h i e v e . The b a s i c components o f a d i g i t a l computer i m p l e m e n t a t i o n of a g e n e r a l p u r p o s e sum o f p o l y n o m i a l s e s t i m a t o r a r e a m u l t i p l i e r , an a c c u m u l a t o r , a s e q u e n c e c o n t r o l l e r t o p e r f o r m f e t c h i n g o f o p e r a n d s and s t o r a g e o f r e s u l t s , and memory. A l l p o r t i o n s o f t h i s s y s t e m w o u l d be d e s i g n e d f o r v e r y h i g h s p e e d o p e r a t i o n ; i f o n l y low s p e e d o p e r a t i o n were r e q u i r e d , a s t a n d a r d g e n e r a l p u r p o s e computer c o u l d be u s e d . T h i s d e d i c a t e d s y s t e m would be s u p e r v i s e d by a g e n e r a l p u r p o s e c o m p u t e r . L e a r n i n g , w h i c h i s a l e s s t i m e c r i t i c a l , b u t more a l g o r i t h m i c a l l y complex t a s k , i s assumed t o be p e r f o r m e d u s i n g one's c h o s e n l e a r n i n g a l g o r i t h m u t i l i z i n g t h e s u p e r v i s i n g c o m p u t e r . F i g u r e 5.1 shows a b l o c k d i a g r a m o f t h e p r o p o s e d d i g i t a l c o m p u t i n g h a r d w a r e . The d i g i t a l sum o f p o l y n o m i a l s e s t i m a t o r would have two modes of o p e r a t i o n . I n t h e f i r s t mode, t h e s e t o f K-G p o l y n o m i a l t e r m s i s g e n e r a t e d . To b e g i n t h i s o p e r a t i o n , t h e s u p e r v i s i n g computer p l a c e s t h e i n i t i a l t e r m s , c o n s i s t i n g o f t h e c o n s t a n t 1 and t h e v i n p u t v a r i a b l e s , i n t o t h e two o p e r a n d m emories. I t i s assumed t h a t t h e s u p e r v i s i n g computer h a s a c c e s s t o t h e o p e r a n d memories and p o i n t e r memories a s w e l l r e g i s t e r s w i t h i n t h e s e q u e n c e c o n t r o l l e r a n d a c c u m u l a t o r . The h i g h e r o r d e r 183 SEQUENCE CONTROLLER ACCUMULATOR / \ r e s u l t a d d r e s s 2 MULTIPLIER a d d r e s s 1 / \ / \ \ / POINTER MEMORY 1 \ / \ / \. / POINTER MEMORY 2 \ / \ / \ / \ / p P W W 0 0 E E L L I I Y Y G G N \ N H H 0 r e s u l t 0 T T M / M I I S S A A E E L L T T S S 1 r E E T T o p e r a n d 1 o p e r a n d 2 OPERAND MEMORY 1 OPERAND MEMORY 2 F i g u r e 5.1 B l o c k d i a g r a m o f d i g i t a l c omputer i m p l e m e n t a t i o n of a sum o f p o l y n o m i a l s e s t i m a t o r 184 t e r m s i n t h e s e t of p o l y n o m i a l s a r e formed i n t u r n as p r o d u c t s o f l o w e r o r d e r t e r m s and appended t o t h e s e t of p o l y n o m i a l s i n t h e two o p e r a n d memories. S e l e c t i o n of a p p r o p r i a t e p a i r s of l o w e r o r d e r p o l y n o m i a l t e r m s i s p e r f o r m e d by t h e p o i n t e r m e mories. The s p e c i a l p a t t e r n of a d d r e s s e s s t o r e d i n t h e p o i n t e r memories i s f i x e d f o r a s y s t e m of g i v e n o r d e r , s, and h a v i n g a s p e c i f i c number of i n p u t v a r i a b l e s , v . T h e s e a d d r e s s e s c a n be g e n e r a t e d once by t h e s u p e r v i s i n g computer and p l a c e d i n t h e p o i n t e r memories. D u r i n g o p e r a t i o n , t h e s e q u e n c e c o n t r o l l e r i n d e x e s t h r o u g h t h e p o i n t e r memories w i t h e a s i l y g e n e r a t e d s e q u e n t i a l a d d r e s s e s . The s e t o f p o l y n o m i a l t e r m s i s d u p l i c a t e d i n e a c h o f t h e o p e r a n d memories so t h a t b o t h o f t h e r e q u i r e d l o w e r o r d e r p o l y n o m i a l t e r m o p e r a n d s c an be f e t c h e d a t t h e same t i m e . The h i g h e r o r d e r r e s u l t i s assumed t o be w r i t t e n back i n t o b o t h o p e r a n d memories s i m u l t a n e o u s l y . I n t h e s e c o n d mode o f o p e r a t i o n , e a c h e s t i m a t e i s formed a s a w e i g h t e d sum o f p o l y n o m i a l t e r m s . The two o p e r a n d memories a r e now a d d r e s s e d s e q u e n t i a l l y , d i r e c t l y by t h e s e q u e n c e c o n t r o l l e r . O p e r a n d s r e a d f r o m o p e r a n d memory 1 c o n s i s t o f t h e p o l y n o m i a l t e r m s w h i l e t h o s e r e a d f r o m o p e r a n d memory 2 c o n s i s t o f t h e c o r r e s p o n d i n g w e i g h t i n g c o e f f i c i e n t s f o r t h e f u n c t i o n t o be e s t i m a t e d . T h e i r p r o d u c t s a r e summed i n t h e a c c u m u l a t o r and r e a d f r o m t h e a c c u m u l a t o r by t h e s u p e r v i s i n g computer a t t h e c o m p l e t i o n o f e a c h p a s s . S e v e r a l s u c h p a s s e s a r e made u s i n g a d i f f e r e n t w e i g h t s e t e a c h t i m e u n t i l e s t i m a t e s have been e v a l u a t e d f o r a l l r f u n c t i o n s . N ote t h a t t h e number o f words o f memory r e q u i r e d u s i n g t h i s a p p r o a c h i s , 185 ( r + 4 ) ( s + 2 r ) ! m e m(r,s)= (5.11) s ! r ! A key f a c t o r i n d e t e r m i n i n g the v i a b i l i t y of t h i s approach f o r v e r y h i g h speed o p e r a t i o n i s the p r e c i s i o n r e q u i r e d i n the c o m p u t a t i o n s . In our s i m u l a t i o n s of s e l f - l e a r n e d c o n t r o l , a p p r o x i m a t e l y 7 d e c i m a l d i g i t s of p r e c i s i o n were used. Our s e l f - l e a r n e d e s t i m a t e s of the C a r t e s i a n i n v e r s e dynamics, however, were o n l y a c c u r a t e r e p r e s e n t a t i o n s of the a n a l y t i c a l f u n c t i o n s t o about 2 or 3 d e c i m a l d i g i t s and y e t were adequate f o r c o n t r o l p u r p o s e s . Input v a r i a b l e s a r e n o r m a l i z e d between -1 and 1, and the w e i g h t i n g c o e f f i c i e n t s c o u l d a l s o be n o r m a l i z e d . I t would t h u s appear t h a t f i x e d p o i n t b i n a r y o p e r a t i o n s w i t h a p r e c i s i o n of perhaps 8 t o 10 b i n a r y d i g i t s would be s u f f i c i e n t f o r c o n t r o l p u r p o s e s . Such f i x e d p o i n t o p e r a t i o n s s h o u l d be a c h i e v a b l e a t t h e r a t e r e q u i r e d u s i n g c u r r e n t t e c h n o l o g y . I f h i g h e r p r e c i s i o n o r f l o a t i n g p o i n t r e p r e s e n t a t i o n were found t o be n e c e s s a r y , i t would be much more d i f f i c u l t t o implement the d i g i t a l sum of p o l y n o m i a l s e s t i m a t o r . 5.2.2 STOCHASTIC COMPUTER IMPLEMENTATION I t may a l s o be p o s s i b l e t o implement a sum of p o l y n o m i a l s e s t i m a t o r u s i n g s t o c h a s t i c computing t e c h n i q u e s . The s t o c h a s t i c computer [19,48] i s d i s t i n c t form e i t h e r the a n a l o g or d i g i t a l computer as numbers a r e r e p r e s e n t e d by the p r o b a b i l i t y of o c c u r r e n c e of a 1 i n the b i t sequences on the d a t a l i n e s p e r m i t t i n g m u l t i p l i c a t i o n t o be performed by a s i n g l e g a t e . For example, c o n s i d e r a s i t u a t i o n i n which the numbers A and B, which a r e between 0 a n d M , are r e p r e s e n t e d as the 186 p r o b a b i l i t i e s o f t h e o c c u r e n c e o f a 1 on t h e i r r e s p e c t i v e d a t a l i n e s , Prob(A=1) = A/M (5.12) Prob(B=1) = B/M (5.13) I f t h e s e s i g n a l s a r e f e d i n t o an AND g a t e t h e n i t s o u t p u t , C, w i l l be a 1 whenever b o t h i n p u t s a r e 1's. The p r o b a b i l i t y of t h e o u t p u t b e i n g a 1 i s , Prob(C=1) = Prob(A=1 and B=1) (5.14) I f t h e p r o b a b i l i t i e s on t h e i n p u t l i n e s A and B a r e i n d e p e n d e n t t h e n , Prob(C=1) = P r o b ( A = 1 ) - P r o b ( B = 1 ) (5.15) and t h e number r e p r e s e n t e d on t h e o u t p u t l i n e i s t h e n o r m a l i z e d p r o d u c t of A and B, C = AB/M 2 (5.16) T h i s i s a u n i p o l a r f o r m o f r e p r e s e n t a t i o n . A b i p o l a r f o r m o f r e p r e s e n t a t i o n i s a l s o p o s s i b l e w h i c h p e r m i t s s i g n e d m u l t i p l i c a t i o n u s i n g a s i n g l e EXCLUSIVE OR g a t e [ 1 9 ] . N o r m a l i z e d summation c a n a l s o be p e r f o r m e d w i t h s i m i l a r l y s i m p l e c i r c u i t r y . A d e q u a t e l y random i n p u t s i g n a l s f o r a s t o c h a s t i c computer can be f o r m e d u s i n g a pseudorandom s e q u e n c e g e n e r a t o r and s u i t a b l e l o g i c c i r c u i t r y [ 1 9 ] . T h i s i s done by r e p e a t e d l y c o m p a r i n g an N b i t b i n a r y r e p r e s e n t a t i o n o f t h e i n p u t s i g n a l w i t h an N b i t b i n a r y random number o b t a i n e d f r o m a pseudorandom s e q u e n c e g e n e r a t o r . The s u c c e s s i v e r e s u l t s o f c o m p a r i s o n form a B e r n o u l l i s e q u e n c e whose p r o b a b i l i t y o f a 1 c o r r e s p o n d s t o t h e i n p u t s i g n a l . We have shown t h a t a d e q u a t e l y random i n p u t s i g n a l s c a n a l s o 187 be o b t a i n e d by r a n d o m i z a t i o n of p u l s e - r a t e - e n c o d e d s i g n a l s u s i n g j i t t e r s t a g e s [ 2 3 ] , S i m p l e hardware and p a r a l l e l o p e r a t i o n make s t o c h a s t i c m u l t i p l i c a t i o n and summation a p o t e n t i a l p a r a l l e l p r o c e s s i n g t e c h n i q u e . I t s s i m p l i c i t y makes i s a t t r a c t i v e f o r VLSI i m p l e m e n t a t i o n . A p r i n c i p a l drawback, however, i s t h a t m u l t i p l i c a t i o n s and summations may be v e r y s l o w . C o n s i d e r t h e example g i v e n i n f i g u r e 5.2. P r o d u c t s a r e formed o f two p a i r s o f i n p u t s and t h e s e p r o d u c t s a r e t h e n summed (more c o r r e c t l y a v e r a g e d ) . A s s u m i n g t h a t t h e p r o b a b i l i t i e s o f a 1 a t a l l i n p u t s a r e i n d e p e n d e n t , t h e p r o b a b i l i t y o f a 1 a t t h e o u t p u t G i s , P r o b ( G = l ) = [ P r o b ( A = 1 ) - P r o b ( B = 1 ) + P r o b ( D = 1 ) - P r o b ( E = 1 ) ] / 2 (5.17) T h i s s t r u c t u r e c o u l d be e x t e n d e d t o implement a sum o f p o l y n o m i a l s e s t i m a t o r . To make use o f t h e r e s u l t a t t h e o u t p u t i t i s n e c e s s a r y t o e s t i m a t e t h i s p r o b a b i l i t y by e s t i m a t i n g t h e mean p r o p o r t i o n of 1's t h a t o c c u r a t t h e o u t p u t . F o r example, i f N s a m p l e s a r e t a k e n o f t h e o u t p u t and 0 o f t h e s e s a m p l e s a r e 1's, an e s t i m a t e of t h e p r o b a b i l i t y o f t h e o u t p u t b e i n g a 1 i s , MEAN = O/N (5.18) S i n c e t h e o u t p u t i s a B e r n o u l l i s e q u e n c e , t h e s t a n d a r d d e v i a t i o n i n t h i s e s t i m a t e o f t h e p r o b a b i l i t y i s , SDEV = [ P r o b ( G = l ) ( l - P r o b ( G = D ) / N ] 1 / / 2 (5.19) To h a l v e t h e s t a n d a r d d e v i a t i o n i n one' s e s t i m a t e o f t h e o u t p u t p r o b a b i l i t y one must q u a d r u p l e t h e number o f o b s e r v a t i o n s o f t h e o u t p u t . T h u s w h i l e s t o c h a s t i c c o m p u t i n g may be c o m p a r a t i v e l y 188 f a s t f o r low a c c u r a c y c o m p u t a t i o n s i n v o l v i n g many i n p u t v a r i a b l e s , i t becomes v e r y slow i f h i g h a c c u r a c y c o m p u t a t i o n s a r e r e q u i r e d . As d i s c u s s e d i n s e c t i o n 5.2.1, i t appears t h a t the a c c u r a c y of e s t i m a t e s r e q u i r e d t o a c h i e v e adequate c o n t r o l i s not v e r y g r e a t . Thus s t o c h a s t i c computer i m p l e m e n t a t i o n of a sum of p o l y n o m i a l s e s t i m a t o r may be f e a s i b l e . Note t h a t , as when u s i n g f i x e d p o i n t a r i t h m e t i c i n a d i g i t a l computer, narrow range r e s t r i c t i o n s a r e imposed on a l l v a r i a b l e s a t a l l p o i n t s w i t h i n a s t o c h a s t i c computer. F u r t h e r i n v e s t i g a t i o n i s r e q u i r e d t o dete r m i n e the f e a s i b i l i t y of such an approach. B > INPUTS ARE INDEPENDENT BERNOULLI SEQUENCES ) -CL y PSEUDORANDOM F i g u r e 5.2 S t o c h a s t i c computer c i r c u i t f o r summing the product! of two p a i r s of i n p u t v a r i a b l e s 189 5.3 MODIFICATION OF THE KLETT CEREBELLAR MODEL The s t a r t i n g p o i n t f o r t h i s work was the K l e t t C e r e b e l l a r M odel. D u r i n g the c o u r s e of t h i s r e s e a r c h , s e v e r a l i n s i g h t s have o c c u r r e d about ways of m o d i f y i n g the C e r e b e l l a r Model t o make i t more p l a u s i b l e as a model of the mammalian c e r e b e l l u m . 5.3.1 THE KLETT CEREBELLAR MODEL K l e t t [26] models the mammalian c e r e b e l l u m as a sum of p o l y n o m i a l s e s t i m a t o r u s i n g o r t h o g o n a l p o l y n o m i a l s . F i g u r e 5.3 shows a b l o c k diagram of the c e r e b e l l a r system. The C e r e b e l l a r Model i n c l u d e s o n l y t h o s e n e u r a l pathways t h a t a r e h i g h l i g h t e d . These a r e the p r i n c i p a l pathways. PARALLEL FIBERS GRANULE CELLS 75+ GOLGI CELLS BASKET t STELLATE CELLS \ KEY MOSS; FIBERS -J- EXCITATORY SYNAPSE — INHIBITORY SYNAPSE -V SUBCORTICAL NUCLEAR CELLS CEREBELLAR CORTEX TO MOTOR CENTERS VIA VARIOUS ROUTES CLIMBING FIBERS F i g u r e 5.3 B l o c k diagram of c e r e b e l l a r system 190 A b l o c k diagram of the C e r e b e l l a r Model i s shown i n f i g u r e 5.4. P u r k i n j e c e l l s a r e e x c i t e d by t h e p a r a l l e l f i b e r s of the g r a n u l e c e l l s and i n h i b i t e d by the b a s k e t and s t e l l a t e c e l l s w h ich a r e a l s o e x c i t e d by p a r a l l e l f i b e r s . The net e f f e c t would be e q u i v a l e n t i f P u r k i n j e c e l l s c o u l d be e x c i t e d or i n h i b i t e d by the p a r a l l e l f i b e r s t h u s e l i m i n a t i n g the need f o r b a s k e t and s t e l l a t e c e l l s . P u r k i n j e c e l l s and t h e i r a s s o c i a t e d b a s k e t and s t e l l a t e c e l l s a r e thus m o d e l l e d as w e i g h t e d summation p o i n t s h a v i n g e i t h e r p o s i t i v e or n e g a t i v e w e i g h t i n g c o e f f i c i e n t s , f = I w^q^ = w Tq (5.20) k I t i s t h r o u g h m o d i f i c a t i o n of t h e s e w e i g h t i n g c o e f f i c i e n t s by c o r r e c t i v e c l i m b i n g f i b e r a c t i v i t y t h a t l e a r n i n g i s assumed t o t a k e p l a c e , Aw = uAfq (5.21) Note t h a t such l e a r n i n g i s p e r f o r m e d based on o n l y l o c a l l y a v a i l a b l e i n f o r m a t i o n i n each P u r k i n j e c e l l . CLIMBING F i g u r e 5.4 B l o c k diagram of t h e K l e t t ' C e r e b e l l a r Model 191 G r a n u l e c e l l s a r e e x c i t e d by t h e Mossy f i b e r s . The G r a n u l e c e l l s a r e m o d e l l e d as f o r m i n g t h e p r o d u c t o f t h e v a r i a b l e s r e p r e s e n t e d by Mossy f i b e r a c t i v i t y . T h i s p r o d u c t i s summed w i t h t h e n e g a t i v e , i n h i b i t o r y G o l g i C e l l a c t i v i t y and t h e r e s u l t o u t p u t as P a r a l l e l f i b e r a c t i v i t y . The G o l g i c e l l s a r e t h e m s e l v e s e x c i t e d by p a r a l l e l f i b e r a c t i v i t y . The G r a n u l e c e l l - G o l g i c e l l network i s m o d e l l e d as a n e g t i v e f e e d b a c k n e twork, q = p - Gq (5.22) S t e a d y s t a t e P a r a l l e l f i b e r a c t i v i t y i s t h u s , q = ( I + G ) " 1 p = Qp (5.23) I f t h e P a r a l l e l f i b e r - G o l g i c e l l s y n a p t i c w e i g h t s r e p r e s e n t e d by t h e m a t r i x G a r e c h o s e n a p p r o p r i a t e l y t h e n t h e t h e m a t r i x Q becomes an o r t h o g o n a l i z a t i o n m a t r i x . T h i s r e s u l t s i n i m p r o v e d l e a r n i n g p e r f o r m a n c e as shown p r e v i o u s l y i n c h a p t e r 2. Such o r t h o g o n a l i z a t i o n r e s u l t s i f , Q = [ J p p T 6 S ] ~ 1 / 2 (5.24) S T h i s r e q u i r e s t h a t G be c h o s e n a s , G = [/ p p T 6 S ] 1 / 2 - I (5.25) S In t h e K l e t t C e r e b e l l a r M o d e l , t h e s e s p e c i a l P a r a l l e l f i b e r G o l g i c e l l s y n a p t i c w e i g h t s a r e assumed, n o t l e a r n e d . T h e r e i s e v i d e n c e t h a t m u l t i p l i c a t i o n and summation r e q u i r e d h e r e c a n be p e r f o r m e d by n e u r o n s and n e u r a l n e t w o r k s [26 ] . I t i s c e r t a i n l y p o s s i b l e t o p e r f o r m m u l t i p l i c a t i o n o r summation o f p u l s e - r a t e - e n c o d e d v a r i a b l e s w i t h s i m p l e d i g i t a l l o g i c g a t e s t h a t a r e a n a l o g o u s t o n e u r o n s u s i n g s t o c h a s t i c c o m p u t i n g t e c h n i q u e s [ 1 9 , 4 8 ] . 192 5.3.2 LEARNED ORTHOGONALIZATION A p r i n c i p a l c r i t i c i s m o f t h e K l e t t C e r e b e l l a r M odel i s t h a t i t assumes t h e s p e c i a l s y n a p t i c w e i g h t s i n t h e G r a n u l e c e l l G o l g i c e l l n e t w o r k . T h i s makes t h e C e r e b e l l a r M odel r a t h e r i m p l a u s i b l e as i t w o u l d r e q u i r e an enormous amount of i n f o r m a t i o n t o g e n e t i c a l l y s p e c i f y t h e s e w e i g h t s . To c o u n t e r t h i s argument we have f o u n d t h a t o r t h o g o n a l i z a t i o n i s n o t n e c c e s a r y t o p e r m i t l e a r n i n g t o t a k e p l a c e . W i t h o u t o r t h o g o n a l i z a t i o n , t h e C e r e b e l l a r M odel becomes e q u i v a l e n t t o t h e G r a d i e n t Method o u t l i n e d i n c h a p t e r 2. The G r a d i e n t Method i s a l e s s o p t i m a l l e a r n i n g a l g o r i t h m i n t e r m s o f c o n v e r g e n c e r a t e s . N e v e r t h e l e s s , i t i s c a p a b l e o f l e a r n i n g a c c u r a t e sum o f p o l y n o m i a l s e s t i m a t e s , g i v e n enough t r a i n i n g . F u r t h e r m o r e , we have f o u n d t h a t i t i s p o s s i b l e t o l e a r n t h e o r t h o g o n a l i z a t i o n m a t r i x Q i n a manner t h a t c o u l d c o n c e i v a b l y be p e r f o r m e d i n t h e c e r e b e l l u m . The l e a r n i n g c a n be done u s i n g i n f o r m a t i o n t h a t i s l o c a l l y a v a i l a b l e i n t h e G r a n u l e c e l l s and G o l g i c e l l s . F i g u r e 5.5a shows a s c h e m a t i c of t h e G r a n u l e c e l l - G o l g i c e l l n e t w o r k showing t h o s e l o c a t i o n s where a m p l i f i c a t i o n c o u l d o c c u r . F i g u r e 5.5b shows t h e same s c h e m a t i c but w i t h a m p l i f i c a t i o n s lumped as much a s p o s s i b l e . W i t h t h i s model t h e P a r a l l e l f i b e r a c t i v i t y i s g i v e n by, q = a (b.p. - L g..q.) (5.26) j * i J 1 Note t h a t an i n d i v i d u a l G r a n u l e c e l l i s assumed t o n o t s y n a p s e w i t h t h o s e G o l g i c e l l s t h a t i t i s b e i n g i n h i b i t e d by. 193 Figure 5.5 Schematic of Granule c e l l - G o l g i c e l l network a) showing those l o c a t i o n s where a m p l i f i c a t i o n could occur b) showing lumped a m p l i f i c a t i o n s 194 E x p r e s s i n g t h i s i n m a t r i x form y i e l d s , q = A ( B p - G q ) : (5.27) A and B a r e d i a g o n a l m a t r i c e s and G i s a n t i - d i a g o n a l ( i e . t h e d i a g o n a l e l e m e n t s a r e z e r o ) . S o l v i n g f o r q i n t e r m s o f p y i e l d s , q = ( l + A G ) _ 1 A B p (5.28) To e n s u r e t h a t q i s an o r t h o n o r m a l b a s i s s e t , t h e P o i n t w i s e C e r e b e l l a r M o d e l d i s c u s s e d i n c h a p t e r 2 i s u s e d . We t h u s d e s i r e t h a t , q = E 5 (5.29) where, -1/2 n-1 E = D , / z (5.30) n n D = l / n L p . p . T (5.31) n /=0 ' ' T h e r e f o r e we must e n s u r e t h a t , (I+A G ) _ 1 A B = E o = D " 1 / 2 (5.32) n n n n n n or e q u i v a l e n t l y , B ~ 1 A " 1 ( I + A G ) = D n , / 2 (5.33) n n n n n An a d j u s t m e n t scheme has been f o u n d t h a t meets t h e s e r e q u i r e m e n t s w h i l e u s i n g o n l y l o c a l l y a v a i l a b l e i n f o r m a t i o n . A f t e r t h e n t h a d j u s t m e n t , t h e e l e m e n t s of t h e m a t r i c e s G R , A n and B n a r e a s f o l l o w s ; % i j '  U \ l * k i * k j ( 5 ' 3 4 ) a .. - 0 (5.35) mj b . = 0 (5.36) nij f o r i±j , and, g . = 0 (5.37) 195 a n/y = [ w \ l 0 * k i * k j l ~ } ( 5 - 3 8 ) b n / . - 1 (5.39) f o r i=j . Note t h a t we a r e a s s u m i n g t h a t t h e g a i n s ^ n / y a r e f i x e d , n o t l e a r n e d . In m a t r i x n o t a t i o n t h e G o l g i f e e d b a c k m a t r i x i s t h u s a s f o l l o w s , G n = l / n X^* T " A n " 1 ( 5 ' 4 0 ) A:=0 I t c a n be c o n f i r m e d t h a t s u c h a c h o i c e o f g a i n s r e s u l t s i n t h e d e s i r e d m a t r i c e s by showing t h a t e q u a t i o n (5.33) i s s a t i s f i e d . F i r s t we c a n remove t h e m a t r i x B n 1 from e q u a t i o n (5.33) as i t i s e q u a l t o t h e i d e n t i t y m a t r i x . B ~ 1 A ~ 1 ( I + A G ) = A ~ 1 ( I + A G ) (5.41) n n n n n n n Then t h i s r e s u l t can be s i m p l i f i e d . A n " 1 ( I + A n 6 n ) = A n " ' + [ l / n / A ^ T ] " ^ ( 5 ' 4 2 ) A n " 1 ( I + A n G n ) = ^\L[h^ (5'43) k = 0 S u b s t i t u t i n g i n e q u a t i o n (5.28) y i e l d s , A n " 1 ( I + A n 6 n > = 1 / n ^ P ( I + A n G n } " ' A n ] T ( 5 * 4 4 ) L e t us assume t h a t t h e m a t r i c e s A n and G n c o n v e r g e s u c h t h a t t h e y a p p e a r a s c o n s t a n t s ( t o t h e d e g r e e o f p r e c i s i o n t h a t we a r e c o n c e r n e d w i t h ) a f t e r a f i n i t e number o f a d j u s t m e n t s . I f we c o n s i d e r e q u a t i o n (5.44) a s n-°° r we c a n assume t h a t t h e m a t r i c e s A and G have f i x e d v a l u e s f o r a l l terms o f t h e summation n n w i t h o u t i n t r o d u c i n g s i g n i f i c a n t e r r o r i n t o t h e sum. The m a t r i c e s 196 i n t h e summation o f (5.44) c a n t h u s be f a c t o r e d o u t . A o . ~ 1 ( l + A c . G J = t l / n ^ P * P * T ] [ ( I + A = O G C O > " 1 A O O ] T ( 5 - 4 5 > R e a r r a n g e m e n t t h e n y i e l d s , A 0 0 " 1 ( l + A o o G a ) [ A c a - 1 ( I + A a G o o ) ] T = (5.46) The m a t r i x A^ i s s y m m e t r i c a l . I t i s n o t o b v i o u s t h a t t h e m a t r i x (I+A G ) i s a l s o s y m m e t r i c a l . L e t us assume t h a t (I+A G ) i s s y m m e t r i c a l . Then t h e t r a n s p o s e i n t h e l e f t hand s i d e o f (5.46) can be removed y i e l d i n g , [ A / 1 (I + A ^ G J ] 2 = }/nnLpkpkT (5.47) k=0 Thus i n t h e l i m i t we have, [A " 1 ( I + A G ) ] 2 = (5.48) OO 00 OO OO A _ 1 ( I + A G ) = D 1 / 2 = E _ 1 / 2 (5.49) OO 00 CO CO oo R e c a l l i n g f r o m c h a p t e r 2 t h a t E ^ e q u a l s a s c a l e r m u l t i p l e , _ o f Q , i t i s a p p a r e n t t h a t t h e r e s u l t i s as d e s i r e d . We have n o t a c t u a l l y been a b l e t o p r o v e t h a t t h i s a d j u s t m e n t scheme w i l l work as we had t o assume c o n v e r g e n c e o f t h e m a t r i c e s A and G , and we had t o assume t h a t (I+A G ) was n n oo oo s y m m e t r i c a l . I f t h e s e a s s u m p t i o n s a r e t r u e , t h e n i t i s p o s s i b l e t o l e a r n o r t h o g o n a l i z a t i o n . In t h e c e r e b e l l u m i t d o e s n o t a p p e a r t o be p o s s i b l e t o p e r f o r m t h e a d j u s t m e n t s of t h e g a i n s a s shown i n e q u a t i o n s (5.34) t h r o u g h ( 5 . 3 9 ) . A more p l a u s i b l e h y p o t h e s i s i s t h a t t h e g a i n s a r e a d j u s t e d by e x p o n e n t i a l a v e r a g i n g where, q.Ak) = ( 1-e)g. . (Jfc-1 ) + ep.(Jfc)q.U) (5.50) 3 z j 1J 1 J 197 f o r i * j , and, a f . U ) = ( l - e ) a / . ) + ep. (it )q ( k) (5.51) f o r /=/, and t h e o t h e r a d j u s t m e n t r e l a t i o n s h i p s a r e as b e f o r e , e i s a s m a l l p o s i t i v e g a i n f a c t o r . T h e s e a d j u s t m e n t s might o c c u r c o n t i n u a l l y but w i t h s u c h a s m a l l g a i n f a c t o r , e, t h a t a d a p t a t i o n i s s l o w , r e s u l t i n g i n b e h a v i o r a n a l o g o u s t o a low p a s s f i l t e r . T h e s e a d j u s t m e n t s m i g h t a l s o be t r i g g e r e d t o o c c u r when l e a r n i n g i s t a k i n g p l a c e a t t h e P u r k i n j e c e l l s a s i n d i c a t e d by C l i m b i n g f i b e r a c t i v i t y . T h r o u g h s i m u l a t i o n we have f o u n d t h a t t h i s p r o p o s e d method f o r l e a r n e d o r t h o g o n a l i z a t i o n w i l l work w i t h e i t h e r t r u e a v e r a g i n g o r e x p o n e n t i a l a v e r a g i n g . S u c c e s i v e v a l u e s of t h e i n p u t v a r i a b l e s zf. u s e d t o g e n e r a t e t h e p o l y n o m i a l t e r m s i n p were c h o s e n r a n d o m l y . I t has a l s o been f o u n d t o work r e g a r d l e s s of w hether t h e p o l y n o m i a l t e r m s i n p a r e s t r i c t l y p o s i t i v e o r n o t . I t t h u s may p r o v e a u s e f u l method o f o r t h o g o n a l i z i n g s i g n a l s , q u i t e a p a r t f r o m i t s v a l i d i t y as a model o f p a r t of t h e c e r e b e l l u m . S e v e r a l p r o b l e m s r e m a i n i n d e f e n d i n g t h i s scheme as a c e r e b e l l a r m o d e l . F i r s t , u s i n g s u c h l e a r n e d o r t h o g o n a l i z a t i o n , t h e r e s u l t i n g v e c t o r , q, i s n o t s t r i c t l y p o s i t i v e . S i g n a l s c a r r i e d on t h e p a r a l l e l f i b e r s o f t h e c e r e b e l l u m a r e u n i p o l a r i n n a t u r e , a l b e i t t h a t t h e y may e f f e c t i v e l y r e p r e s e n t p o s i t i v e and n e g a t i v e v a l u e s o v e r a f i n i t e r a n g e t h r o u g h o f f s e t by a p o s i t i v e b i a s . A more p l a u s i b l e o r t h o g o n a l i z a t i o n scheme would be one t h a t e n s u r e s t h a t q - ( 1 , . . . , 1 ) i s o r t h o g o n a l . S e c o n d l y , i t i s n o t c l e a r t h a t a l l G o l g i c e l l s s y n a p s e w i t h mossy f i b e r s s u c h 198 t h a t e a c h G o l g i c e l l has a c c e s s t o t h e s i g n a l p̂ . upon w h i c h a d j u s t m e n t of G o l g i f e e d b a c k g a i n s a r e b a s e d i n t h i s m o d e l . A more p l a u s i b l e model i n t e r m s of h a v i n g g a i n a d j u s t m e n t s b e i n g b a s e d on l o c a l l y a c c e s s i b l e s i g n a l s i s one where t h e g a i n s * a r e a d j u s t e d a s f o l l o w s , gn/y = 1/nI?0q*; ( 5 ' 5 2 ) a .. = 0 (5.53) TUJ b p . . - 0 (5.54) f o r /*j , and, g n / y = 0 (5.55) n-1 _, a . = [1/n I q ] (5.56) n i J k=0 J n-1 . b . . = [1/n Z p. ] (5.57) n i J k=0 K l Now t h e g a i n s a r e a s s u r e d l y a d j u s t e d on t h e b a s i s of l o c a l l y a v a i l a b l e i n f o r m a t i o n . T h i s method c a n n o t be p r o v e n t o work, however, p r e c i s e l y b e c a u s e t h e q̂ . a r e not s t r i c t l y p o s i t i v e . I t has been f o u n d n o t t o work i n s i m u l a t i o n s , f o r t h e same r e a s o n . I f a method c a n be f o u n d f o r o f f s e t t i n g q t o e n s u r e t h a t t h e q̂ . a r e s t r i c t l y p o s i t i v e , a v a r i a n t o f t h e s e methods may t h e n be a f u n c t i o n a l and p l a u s i b l e model f o r l e a r n i n g o r t h o g o n a l i z a t i o n i n t h e c e r e b e l l u m . 5.3.3 INPUT VARIABLE SPLITTING In o r d e r t o s e l f - l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s of t h e two l i n k m a n i p u l a t o r , we had t o r e s t r i c t o u r s e l v e s t o a p o r t i o n of t h e m a n i p u l a t o r ' s r e a c h . T h i s was n e c e s s a r y b e c a u s e 199 c e r t a i n f u n c t i o n a l r e l a t i o n s h i p s i n t h e i n v e r s e k i n e m a t i c s o f th e two l i n k m a n i p u l a t o r c o u l d n o t be r e p r e s e n t e d a s a sum of p o l y n o m i a l s o v e r t h e whole o f t h e m a n i p u l a t o r ' s r e a c h . T h i s r e p r e s e n t s a l i m i t a t i o n imposed by t h e use o f K-G p o l y n o m i a l s a s a b a s i s s e t . As m e n t i o n e d i n c h a p t e r 3, one c o u l d use s e v e r a l u n i q u e sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s f o r e a c h p o r t i o n of t h e m a n i p u l a t o r ' s r e a c h . One would t h u s be m o d e l l i n g t h e f u n c t i o n a l r e l a t i o n s h i p s o v e r t h e whole s p a c e as p i e c e w i s e l i n e a r , q u a d r a t i c , o r q u a r t i c , e t c . , a c c o r d i n g t o t h e o r d e r of t h e sum of p o l y n o m i a l s r e p r e s e n t a t i o n s u s e d i n e a c h r e g i o n . C l e a r l y a more g e n e r a l c l a s s o f f u n c t i o n s c a n be m o d e l l e d i n t h i s way. I t may be p o s s i b l e t h a t one c o u l d u s e a method t h a t we c a l l i n p u t v a r i a b l e s p l i t t i n g t o a c h i e v e t h e same end. By i n p u t v a r i a b l e s p l i t t i n g , we mean t h a t a s i n g l e i n p u t v a r i a b l e f r o m a c o n t r o l p o i n t o f v i e w , i s e n c o d e d on s e v e r a l i n p u t v a r i a b l e s f r o m t h e p o i n t of v i e w o f t h e sum o f p o l y n o m i a l s e s t i m a t o r . I t i s b e s t i l l u s t r a t e d by an example. C o n s i d e r a sum o f p o l y n o m i a l s r e p r e s e n t a t i o n f o r t h e f u n c t i o n , f = sin ( 2 7 r z ) (5.58) o v e r t h e r a n g e , -1 < z < 1 (5.59) C l e a r l y a f i f t h o r d e r sum o f p o l y n o m i a l s r e p r e s e n t a t i o n i s * r e q u i r e d t o a c h i e v e a m i n i m a l l i k e n e s s o f f . An example of s u c h a r e p r e s e n t a t i o n i s , s i n ( 2 ; r z ) * 5.7z - 2 8 . 4 z 3 + 2 2 . 7 z 5 (5.60) 200 w h i c h was o b t a i n e d by n o t i n g t h a t even power t e r m s were not n e e d e d and t h e n c h o o s i n g t h e c o e f f i c i e n t s o f t h e odd power terms s u c h t h a t a match i s a c h i e v e d a t z e q u a l t o 1/4, 1/2, and 1. Now assume t h a t t h e i n f o r m a t i o n o f v a r i a b l e z i s s p l i t among 4 i n p u t v a r i a b l e s a s f o l l o w s , z. = max{-1, m i n { l , 4z + 5 - 2i}} (5.61) N o t e t h a t o v e r most o f t h e r a n g e o f z, t h e v a r i a b l e s ẑ . a r e s a t u r a t e d a t e i t h e r -1 o r 1. I t i s o n l y o v e r s p e c i f i c p o r t i o n s o f t h e s p a c e t h a t e a c h z. v a r i e s . I t i s p o s s i b l e t o r e p r e s e n t s i n ( 2 7 r z ) a s a sum o f p o l y n o m i a l s i n z. , namely, 2 2 2 2 s i n ( 2 7 r z ) =* - z 1 + z 2 ~ z 3 + z 4 (5.62) F i g u r e 5.6 shows t h e v a r i a b l e s , ẑ . , as f u n c t i o n s of- z. F i g u r e 5.7 shows a p l o t o f s i n ( 2 7 r z ) a l o n g w i t h t h e two sum o f p o l y n o m i a l s r e p r e s e n t a t i o n s . I t c a n be s e e n t h a t t h e r e p r e s e n t a t i o n u s i n g i n p u t v a r i a b l e s p l i t t i n g i s much b e t t e r . I t r e m a i n s t o be d e t e r m i n e d whether I n t e r f e r e n c e M i n i m i z a t i o n or r e l a t e d l e a r n i n g a l g o r i t h m s c a n be made t o work when i n p u t v a r i a b l e s a r e s p l i t i n t h i s manner. E f f o r t s t o d e v e l o p s u c h methods o f l e a r n i n g p i e c e w i s e n o n - l i n e a r e s t i m a t e s o f f u n c t i o n s a r e w a r r a n t e d a s i t c o u l d overcome t h e l i m i t a t i o n s we have e n c o u n t e r e d u s i n g e s t i m a t e s b a s e d on sums of K-G p o l y n o m i a l s . I t m ight make i t p o s s i b l e t o l e a r n t h e C a r t e s i a n i n v e r s e d y n a m i c s o v e r t h e whole of t h e two l i n k m a n i p u l a t o r ' s r e a c h , and l i k e w i s e i n o t h e r s i t u a t i o n s where f u n c t i o n s a r e not w e l l r e p r e s e n t e d as sums of p o l y n o i a l s o v e r t h e r e g i o n of i n t e r e s t . 201 F i g u r e 5.6 Example of i n p u t v a r i a b l e s p l i t t i n g S M . 0 0 - 0 . 6 0 - 0 . 2 0 0 . 2 0 0 . 6 0 J . 0 0 F i g u r e 5.7 R e p r e s e n t a t i o n s of s i n ( 2 j r z ) 202 W i t h r e g a r d s t o t h e C e r e b e l l a r M o d e l , i t i s u n l i k e l y t h a t i n p u t v a r i a b l e s a r e s p l i t i n t h e p r e v i o u s l y d e s c r i b e d f a s h i o n . A more p l a u s i b l e s c e n a r i o i s o u t l i n e d i n t h e f o l l o w i n g . C o n s i d e r an i n p u t v a r i a b l e t o t h e c e r e b e l l u m f r o m a c o n t r o l p o i n t of v i e w . One s u c h v a r i a b l e c o u l d be a j o i n t a n g l e , say a t t h e e lbow. The p o s i t i o n of t h e j o i n t c a n be e f f e c t i v e l y e n c o d e d by s i g n a l s r e p r e s e n t i n g t h e t e n s i o n o r e x t e n s i o n i n t h e v a r i o u s m u s c l e s a c t i n g upon t h e j o i n t . S i n c e t h e r e a r e numerous m u s c l e s and t h e y a r e g e n e r a l l y p a i r e d i n t o o p p o n e n t s , i t i s r e a s o n a b l e t o assume t h a t some s i g n a l s would d e c r e a s e and o t h e r s would i n c r e a s e as t h e j o i n t r o t a t e d f r o m i t s minimum t o maximum a n g u l a r d i s p l a c e m e n t . A l s o i t i s r e a s o n a b l e t o assume t h a t some s i g n a l s w o u l d change from low l e v e l s t o s a t u r a t e d h i g h l e v e l s o v e r narrow r a n g e s o f j o i n t movement w h i l e o t h e r s would change s i m i l a r l y o v e r l a r g e r a n g e s o f j o i n t movement. F i g u r e 5.8 shows a s i m p l e example where t h e j o i n t a n g l e , i s e n c o d e d on s e v e r a l d i f f e r e n t i n p u t v a r i a b l e s . I f t h e s e v a r i a b l e s a r e t a k e n two a t a t i m e and m u l t i p l i e d t o g e t h e r , t h e p r o d u c t s form f u n c t i o n s s u c h as t h o s e shown i n f i g u r e 5.9. Note how t h e s e p r o d u c t s r e s e m b l e s p l i n e f u n c t i o n s . Some a r e narrow and some a r e r a t h e r w i d e . C o n s i d e r now a more r e a l i s t i c p h y s i o l o g i c a l s c e n a r i o i n w h i c h a p a r a m e t e r s u c h a s a j o i n t a n g l e i s e n c o d e d on t h e s i g n a l s coming from p e r h a p s t h o u s a n d s of m u s c l e f i b e r s e n s o r s . I f t h e s e s i g n a l s a r e t a k e n r a n d o m l y , f o u r o r f i v e a t a t i m e as t h e y a r e i n t h e Mossy F i b e r - G r a n u l e C e l l n etwork i n t h e c e r e b e l l u m , and m u l t i p l i e d t o g e t h e r , t h e s e t of b a s i s f u n c t i o n s t h a t r e s u l t p r o b a b l y c o n t a i n s many s p l i n e - l i k e 203 f u n c t i o n s . T h i s h y p o t h e s i s r e m a i n s t o be t e s t e d , however. As p o i n t e d o u t by K l e t t [ 2 6 ] , s p l i n e f u n c t i o n s have d e s i r a b l e p r o p e r t i e s i n terms of m i n i m i z i n g l e a r n i n g i n t e r f e r e n c e . S p l i n e f u n c t i o n s c an be u s e d t o f o r m e s t i m a t e s o f a more g e n e r a l c l a s s o f f u n c t i o n s t h a n K-G p o l y n o m i a l s . Many s u c h s p l i n e f u n c t i o n s may be r e q u i r e d i n o r d e r t o e f f e c t i v e l y s p an t h e domain o f t h e f u n c t i o n s t o be e s t i m a t e d , however, many s u c h s p l i n e s c o u l d be g e n e r a t e d i n t h e c e r e b e l l u m . I n c o r p o r a t i o n o f s p l i n e f u n c t i o n s i n t o a m o d i f i e d K l e t t C e r e b e l l a r Model would r e n d e r t h e model more s i m i l a r t o t h e p r e v i o u s model p r o p o s e d by A l b u s [ l ] . The m o d i f i e d K l e t t M o d e l w o u l d r e t a i n t h e a d v a n t a g e o f r e q u i r i n g many fewer w e i g h t i n g c o e f f i c i e n t s t h a n t h e A l b u s M o d e l , b e c a u s e of i t s use o f c o n t i n u o u s v a r i a b l e s v e r s u s e s s e n t i a l l y b i n a r y v a r i a b l e s i n t h e A l b u s M o d e l . Use o f s p l i n e f u n c t i o n s w o u l d f r e e t h e K l e t t M o d e l f r o m i t s p r e v i o u s d i s a d v a n t a g e ^ o f o n l y b e i n g a p p l i c a b l e f o r e s t i m a t i n g f u n c t i o n s t h a t a r e w e l l r e p r e s e n t e d by p o l y n o m i a l s ; i t wo u l d g a i n t h e g e n e r a l i t y . o f t h e A l b u s M o d e l i n terms o f f u n c t i o n s t h a t c a n be l e a r n e d . F i n a l l y , t h e A l b u s M o d e l i s b a s e d on t h e i m p l a u s a b l e a s s u m p t i o n t h a t e a c h d i s c e r n a b l e p o i n t i n i t s i n p u t s p a c e i s e n c o d e d by a s e t o f m a x i m a l l y a c t i v e i n p u t v a r i a b l e s . The m o d i f i e d K l e t t M o d e l , i n c o n t r a s t , would be b a s e d on a p a t t e r n o f s e n s o r y d a t a e n c o d i n g t h a t i s more c o n s i s t e n t w i t h p h y s i o l o g i c a l e v i d e n c e . 204 Figure 5.8 Encoding of j o i n t angle by input v a r i a b l e s p l i t t i n g Z (3) *Z (4) Z (5) *Z (6) 0 . 2 0 0 . 4 0 0 . 6 0 0 . 6 0 NORMALIZED JOINT ANGLE 1 . 0 0 Figure 5.9 Example of products of s p l i t input v a r i a b l e s representing j o i n t angle 205 BIBLIOGRAPHY [ I ] J . S . A l b u s , T h e o r e t i c a l and E x p e r i m e n t a l A s p e c t s of a C e r e b e l l a r M o d e l , Ph.D. T h e s i s , U n i v e r s i t y of M a r y l a n d , December 1972. [2] J . S . A l b u s , B r a i n s , B e h a v i o r & R o b o t i c s , BYTE P u b l i c a t i o n s , P e t e r b o r o u g h , New H a m p s h i r e , 1981. [3] J . H . 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Man C y b e r . , SMC-3, p.455, September 1973. 209 [57] B. Widrow, J.M. M c C o o l , M.G. L a r i m o r e & C.R. J o h n s o n J r . , " S t a t i o n a r y and N o n s t a t i o n a r y L e a r n i n g C h a r a c t e r i s t i c s of t h e LMS A d a p t i v e F i l t e r " , P r o c . I E E E , V o l . 6 4 , p.1151, A u g u s t 1976. [58] ACSL U s e r G u i d e / R e f e r e n c e M a n u a l , M i t c h e l l and G a u t h i e r , A s s o c . , I n c . , C o n c o r d , Mass. 210 APPENDIX A A N A L Y T I C A L K I N E M A T I C S O F T H E T W O L I N K M A N I P U L A T O R T H E T W O L I N K M A N I P U L A T O R T H E D I R E C T K I N E M A T I C S From the f i g u r e the d i r e c t p o s i t i o n kinematics can be immediately o b t a i n e d , X j = l j S i n ( a j ) .+ l 2 s i n ( a 1 + o 2 ) x 2 «= - l j c o s t t t j ) - l j c o s t a ^ o j ) D i f f e r e n t i a t i o n of these r e l a t i o n s h i p s y i e l d s the d i r e c t v e l o c i t y k i n e m a t i c s , l j c o s f a j ) • l ^ o s ^ + a j ) , l j C O s C a ^ O j ) l j S i n U j ) + l 2 s i n ( a 1 + o 2 ) , l j S i n U j + O j ) which can be rearranged as, I J C O S U J ) , l2cos(at+a2) l j S i n f a j ) , ^ s i n U j + a j ) a 1 + a 2 211 and r e p r e s e n t e d a s , - - — — " M1 _*2_ _ d 1 + d 2_ The d i r e c t v e l o c i t y k i n e m a t i c s can be d i f f e r e n t i a t e d t o y i e l d the d i r e c t a c c e l e r a t i o n k i n e m a t i c s , x 1 l 1 c o s ( a 1 ) l 1 s i n ( a l ) , l 2 c o s ( a 1 + a 2 ) , l 2 s i n ( a 1 + a 2 ) 1 a 1 + a 2 - l 1 s i n ( a 1 ) , - l j S i n C a ^ a j ) l l c o s ( a 1 ) l j c o s f t ^ + a , , ) • 2 a 1 ( d 1 + d 2 V which can be r e p r e s e n t e d a s , - - - M1 -I _ a i + a 2 . + M, "1 ( d 1 + d 2 ) 2 THE INVERSE KINEMATICS R e f e r r i n g a g a i n t o the f i g u r e , the c o s i n e r u l e can be a p p l i e d t o y i e l d , c o s U - a 2 ) = ( X 2 - l l 2 - l 2 2 ) / ( - l 1 l 2 ) and t h u s , c o s ( a 2 ) = ( X 2 - l 1 2 - l 2 2 ) / l 1 l 2 By the r u l e of P y t h a g o r a s we have t h a t , 2 2 2 X Z = X ^ + X 2 and t h u s , c o s ( a 2 ) = ( x 1 2 + x 2 2 - l 1 2 - l 2 2 ) / l 1 l 2 or e q u i v a l e n t l y , 2 2 2 2 a 2 = a r c c o s ( ( x 1 +x 2 -1 1 -1 2 J / ^ l ^ w h ich i s t h e i n v e r s e p o s i t i o n k i n e m a t i c s f u n c t i o n f o r a 2 . The i n v e r s e p o s i t i o n k i n e m a t i c s f u n c t i o n f o r a 1 can be 212 o b t a i n e d by n o t i n g t h a t , tanU) = ( l 2 s i n ( a 2 ) ) / ( l 1 + l 2 c o s ( a 2 ) ) and, tan(tf) = x^/(-x2) S i n c e , o 1 = ^ - <t> t h i s means t h a t , a 1 = a r c t a n ( x 1 , - x 2 ) - a r c t a n ( l 2 s i n ( a 2 ) , 1 1 + l 2 c o s ( a 2 ) ) Note t h a t the a r c t a n f u n c t i o n used here has two arguments and i s thus assumed t o y i e l d a r e s u l t i n the range -TT t o TT. An a l t e r n a t i v e r e p r e s e n t a t i o n of the i n v e r s e p o s i t i o n k i n e m a t i c s f o r a 1 can be o b t a i n e d by n o t i n g t h a t , c o s ( a 1 ) = cos(\/>-0) = cos(\//)cos(0) + s i n (\p) s i n (4>) By i n s p e c t i o n of t h e f i g u r e we have, c o s ( ^ ) = - x 2 / X and, s i n ( ^ ) = X j / X A p p l i c a t i o n of t h e c o s i n e r u l e y i e l d s , cosU) = ( X 2 + f 1 2 - l 2 2 ) / 2 l 1 X A p p l i c a t i o n of t h e s i n e r u l e y i e l d s , s i n ( # ) / l 2 = s i n ( i r - a 2 ) / X = s i n ( a 2 ) / X and hence, s i n ( 0 ) = l 2 s i n ( a 2 ) / X S u b s t i t u t i n g t h e s e e x p r e s s i o n s i n t o our p r e v i o u s e x p r e s s i o n f o r c o s t a ^ y i e l d s , c o s U , ) = - x 2 ( X 2 + l 1 2 - l 2 2 ) / 2 l 1 X 2 + x 1 l 2 s i n ( a 2 ) / X 2 2 S u b s i t i t u t i n g i n t h e p r e v i o u s e x p r e s s i o n f o r X and combining 213 terms t h a t a r e e q u a l t o c o s ( o 2 ) y i e l d s , 2 2 c o s f c ^ ) = [ x 1 l 1 s i n ( a 2 > - x 2 ( l 1 + l 2 c o s ( a 2 ) ) ] / ( x 1 +x 2 ) a n o t h e r form of the i n v e r s e k i n e m a t i c s f u n c t i o n of a 1 . From the d i r e c t v e l o c i t y k i n e m a t i c s we have, x 1 X . = M 1 which can be r e a r r a n g e d t o y i e l d the i n v e r s e v e l o c i t y k i n e m a t i c s a s , "1 d 1 + d 2 = M -1 1 or e q u i v a l e n t l y , d 1 a 1 + a 2 l 1 l 2 s i n ( a 2 ) l 2 s i n ( a 1 + a 2 ) , - ^ c o s ( + a 2 ) - l l s i n ( a 1 ) l 1 c o s ( a 1 ) From t h e d i r e c t a c c e l e r a t i o n k i n e m a t i c s we have, 2 a1 + M, x = M 1 d 1 + o 2 (a,+a2)2 which can be r e a r r a n g e d t o y i e l d the i n v e r s e a c c e l e r a t i o n k i n e m a t i c s a s , ""1 d 1 + d 2 = M -1 M 1 1 > 42 • 2 (d}+a2)2 or e q u i v a l e n t l y , 0, '1 1 2 l 1 l 2 s i n ( a 2 ) l 1 l 2 s i n ( a 2 ) l 2 s i n ( a 1 + a 2 ) , - l j c o s f t ^ + a . , ) - l 1 s i n ( o 1 ) l 1 c o s ( a 1 ) • L X 2 J - l ^ j c o s f a j ) , - 1 ^ , ^ l j C O S f d j ) . 2 ( d ^ d j ) 214 APPENDIX B ANALYTICAL DYNAMICS OF THE TWO LINK MANIPULATOR INVERSE DYNAMICS The Lagrangian [46,37] i s defined as the d i f f e r e n c e between the k i n e t i c energy of a system, K, and the p o t e n t i a l energy of a system, P, L = K - P The in v e r s e dynamics equations are obtained as, r. = 6_ [SL 1 - 3L 6t j a a j da. The d e r i v a t i o n of the inverse dynamics equations begins by noting t h a t the k i n e t i c energy of mass m1 i s , K 1 <= 1/2 m}vy2 = 1/2 n ^ l ^ d , 2 which i s equal to 0 when not moving. The p o t e n t i a l energy of mass m2 i s , Pj » -n^gh « - m 1 g l 1 c o s ( a 1 ) which i s equal to 0 when x 2«0, i e . at the height of the o r i g i n . The k i n e t i c energy of mass m2 i s , 2 K 2 « 1/2 n»2v2 The square of the v e l o c i t y of mass m2 can be obtained from the d i r e c t v e l o c i t y kinematics equations given i n appendix A by noti n g t h a t , v 2 2 - i* + x 2 2 and hence, v 2 2 « l , 2 * , 2 " * l 2 2 ( d 1 2 + 2 d 1 a 2 + d 2 2 ) + 2 l l l 2 o 1 ( a 1 + d 2 ) c o s ( a 2 ) Thus the k i n e t i c energy of mass m2 i s , 215 K 2 = 1/2 n ^ l ^ d , 2 + 1/2 m 2 l 2 2 ( d 1 2 + 2 d 1 d 2 + d 2 2 ) 2 + n ^ ^ ^ c o s f a ^ ( d , +a^a2) The p o t e n t i a l energy of mass m2 i s , P 2 = m2gh = - m 2 g l ^ o s t a ^ ) - m 2 g l 2 c o s ( a 1 + a 2 ) Combining terms y i e l d s the Lagrangian of the two l i n k manipulator, L = 1/2 ( m 1 + m 2 ) l 1 2 d ] 2 + 1/2 m ^ 2 ( a , 2 + 2 d , o 2 + d 2 2 ) 2 + m 2 l 1 l 2 c o s ( a 2 ) ( d 1 +a^a2) + (m 1+m 2)gl 1cos(o l) + m2gl2cos(a^+a2) Various d e r i v a t i v e s of the Lagrangian need to be obtained and combined t o y i e l d the inverse dynamics f u n c t i o n f o r torque Ty. The d e r i v a t i v e s are, 2 2 2 3L/3d 1 «= (m 1+m 2)l 1 d 1 + m 2 l 2 d 1 + m 2 1 2 d 2 + 2 m 2 l 1 l 2 c o s ( a 2 ) d 1 + m ^ l j C O s t a ^ d j 6/6t(bL/dd}) = [(m}+m2)l,2 + m 2 l 2 2 + 2 m 2 l 1 l 2 c o s ( o 2 ) 3 d , + [ m 2 l 2 + n i j l ^ j c o s t a j ) ] d 2 2 - 2 m 2 l 1 l 2 s i n ( a 2 ) d 1 d 2 - n» 2l 1l 2sin(o 2)d 2 3L/3a 1 = - ( m l + m 2 ) g l l s i n ( a 1 ) - m 2 9 1 2 s i n ^ a i + a 2 ^ Combining these d e r i v a t i v e s y i e l d s the inverse dynamics equation f o r torque T 1 , T1 = d 1 1 £ 1 + d l 2 a 2 + d 1 1 1 A 1 2 + d l 2 2 A 2 2 + d 1 l 2 d 1 A 2 + d 1 2 1 A 2 d 1 + d 1 where, d n = ( n ^ + m ^ l j 2 + m 2 l 2 2 + 2 m 2 l 1 l 2 c o s ( o 2 ) d 1 2 = m 2 1 2 + m 2 1 i ^ 2 C O S ^ a 2 ^ d l 2 2 " ~ m 2 l 1 l 2 s i n ^ a 2 ^ 216 d 1 1 2 = - m 2 l 1 l 2 s i n ( a 2 ) d121 = - « > 2 1 l 1 2 s i n ( a 2 ) d 1 = ( m 1 + m 2 ) g l 1 s i n ( a l ) + m 2 g l 2 s i n ( a , + a 2 ) S i m i l a r l y , v a r i o u s d e r i v a t i v e s of the L a g r a n g i a n need t o be o b t a i n e d and combined t o y i e l d the i n v e r s e dynamics f u n c t i o n f o r t o r q u e T 2 . The d e r i v a t i v e s a r e , 2 2 9 L / 9 d 2 = n» 2 12 d1 + m 2 1 2 d 2 + m 2 1 1 1 2 C 0 S ^ a 2 ^ d1 2 5 / 6 t ( 3 L / 9 d 2 ) = [ m 2 l 2 + m 2 l 1 1 2 c o s ^ a 2 ^ ^ 1 2 + m 2 l 2 &2 ~ m 2 ^ i ^ 2 S ^ n ^ a 2 ^ d 1 d 2 2 9 L / 9 a 2 = - m 2 l 1 l 2 S i n ( a 2 ) d 1 - n ^ l 1 l 2 s i n ( o 2 ) d 1 d 2 - m 2 g l 2 s i n ( a 1 + a 2 ) Combining t h e s e d e r i v a t i v e s y i e l d s the i n v e r s e dynamics e q u a t i o n f o r t o r q u e T 2 , r 2 = d 2 l d 1 + d 2 2 o 2 + d 2 n d 1 2 + d 2 2 2 d 2 2 + d 2 ! 2 d 1 d 2 + d 2 2 1 d 2 d 1 + d 2 where, d 2 1 = m 2 l 2 2 ^ l ^ c o s U . , ) d 2 2 = m 2 l 2 2 d 2 l 1 = m 2 l 1 l 2 s i n ( o 2 ) d 2 2 2 * 0 d 2 l 2 * 0 d221 = 0 d 2 = m 2 g l 2 s i n ( a 1 + a 2 ) T h i s d e r i v a t i o n of t h e i n v e r s e dynamics f o l l o w s the approach used by P a u l [ 4 6 ] . Note, however, t h a t t h e r e s u l t s d i f f e r s l i g h t l y from t h o s e g i v e n by P a u l as t h e r e i s an a r i t h m e t i c e r r o r i n h i s d e r i v a t i o n . 217 APPENDIX C C L O S E D L O O P C O N T R O L O F L I N E 1 U S I N G T H E S E L F - L E A R N E D C A R T E S I A N I N V E R S E D Y N A M I C S A F T E R 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 A N D 1 0 0 0 T R A I N I N G P A T H S 218 0.00 S S o COB UJo COB ^ d - CNJ =>§ =>§ 0.00 0.60 0.80 1.60 T SEC 1.60 T SEC 2.40 3.20 4.00 4.00 in 8 o" 1.60 T SEC 4.00 Closed loop c o n t r o l of l i n e 1 using the s e l f - l e a r n e d Cartesian inverse dynamics a f t e r 200 t r a i n i n g paths 219 8 8 CM C M COo CM 0.00 M — 1 r — i 0.80 1.60 T SEC 2.40 3.20 4.00 8 8 4.00 Closed loop c o n t r o l of l i n e 1 using the s e l f - l e a r n e d C a r t e s i a n inverse dynamics a f t e r 400 t r a i n i n g paths 220 s s CM • • CJ L U 0 COS CM CJ COo CM 6.00 0.80 J'SEC 2 , 4 0 3 . 2 0 4 . 0 0 8 8 CJ 0.00 0.80 1.60 T SEC 2.40 4.00 8 8 X o in 0.00 0.80 1.60 _ T.W T SEC 3.20 4.00 Closed loop c o n t r o l of l i n e 1 using the s e l f - l e a r n e d Cartesian inverse dynamics a f t e r 600 t r a i n i n g paths 221 S 8 OJ • • CJ COo 2 R 0.00 8 8 CJ 3 ° 0.80 1.60 2.40 T SEC 3.20 4.00 4.00 o m 8 o'" X o OT 0.00 0.80 1.60 2.40 T SEC 3.20 4.00 Closed loop c o n t r o l of l i n e 1 using the s e l f - l e a r n e d C artesian inverse dynamics a f t e r 800 t r a i n i n g paths 222 8 8 C M UJo 0.00 8 8 CJ UJo CO 5 >8 0.80 T SEC 3.20 4.00 4.00 in 8 X o in Closed loop c o n t r o l of l i n e 1 using the s e l f - l e a r n e d C a r t e s i a n inverse dynamics a f t e r 1000 t r a i n i n g paths 223 APPENDIX D V I E W O F C I R C L E 1 U S I N G T H E S E L F - L E A R N E D D I R E C T ' P O S I T I O N K I N E M A T I C S A F T E R 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 , 1 0 0 0 , 1 4 0 0 , 1 8 0 0 , 2 2 0 0 , 2 6 0 0 , 3 0 0 0 , 3 4 0 0 , 3 8 0 0 , 4 2 0 0 , A N D 4 6 0 0 T R A I N I N G P A T H S 224 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 200 t r a i n i n g paths 225 8 CM in 2Eo* • X UJCO s -0.80 -0.48 •0.16 0.16 LEX CI) M 0.48 0.80 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 400 t r a i n i n g paths 226 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 600 t r a i n i n g paths 227 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t pos kinematics a f t e r 800 t r a i n i n g paths 228 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 1000 t r a i n i n g paths 2 2 9 o (SI O" I CO O" I ( S I - X • LU CO t ).80 -0.48 -0.16 LEX(l) M 0.16 0.48 0.80 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 1400 t r a i n i n g paths 230 o CM O" I CM tn oo CM— 9 S 1 1 J T T .80 -0.48 -0.16 0.16 L E X ( I ) M 0.48 0.80 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 1800 t r a i n i n g paths 231 LEX(l ) M View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 2200 t r a i n i n g paths 232 8 on CM in I OD o' I CM— LU K80 : ( M 8 ^OAB OAS (M8 0.80 L E X ( l ) M View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 2600 t r a i n i n g paths 233 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 3000 t r a i n i n g paths 234 8 • View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 3400 t r a i n i n g paths 235 t O" o x T LU s 0.80 -0.48 -0.16 ^ LEX(1) M 0.16 0.48 — i 0.80 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 3800 t r a i n i n g paths 236 8 L E X ( l ) M View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 4200 t r a i n i n g paths 237 View of c i r c l e 1 using the s e l f - l e a r n e d d i r e c t p o s i t i o n kinematics a f t e r 4600 t r a i n i n g paths ( f i n a l 400 with smoothing) 238

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